binding dict | chosen string | hard_negative bool | metrics dict | negative_category string | negative_mode string | pair_id string | pair_index int64 | prompt string | rejected string | rejection unknown | safe_lean_text_only bool | schema_version int64 | source dict | split string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
{
"chosen_proof_sha256": "08f7647cfd9ab05023d94bc2fcca4631870d5f71d7f24de348f6c58aa5102a1d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "49e29b7526fb817a02fc6d72904a3c4272... | by
rw [isPosSemidef_def, Matrix.posSemidef_iff_dotProduct_mulVec]
apply and_congr (B.isSymm_iff_isHermitian_toMatrix b)
rw [isNonneg_def]
refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
· rw [star_dotProduct_toMatrix₂_mulVec]
exact h _
· rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]
exact h _ | false | {
"chosen_tokens": 58,
"rejected_tokens": 63,
"token_jaccard": 0.864865,
"token_length_ratio": 1.086207
} | markdown_preamble_trailing | preamble | 006a588191b537987ce94f10d10cf124332130be53a18cec7aeaffc2d310f19f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Matrix.PosDef
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as ... | Here is the proof:
by
rw [isPosSemidef_def, Matrix.posSemidef_iff_dotProduct_mulVec]
apply and_congr (B.isSymm_iff_isHermitian_toMatrix b)
rw [isNonneg_def]
refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
· rw [star_dotProduct_toMatrix₂_mulVec]
exact h _
· rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]
exact ... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/SesquilinearForm",
"family_id": "linearmap",
"file_id": "mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Star.lean",
"sample_id": "9b67a823dcea17b7fc9288545c47193c34fe1b90d1bd9cf3ef9be1964ef22f5d"
} | train |
{
"chosen_proof_sha256": "343a166fea2e90fed4c3043781c3d8f443aa011a6cdb85e131c437406e24ed78",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1f5f28afd315b861373a4cc9f83e458915... | by
have : C a • D f = D (C a * f) := by simp
rw [this, C_mul', D.map_smul] | false | {
"chosen_tokens": 29,
"rejected_tokens": 36,
"token_jaccard": 0.916667,
"token_length_ratio": 1.241379
} | markdown_preamble_trailing | fence | 008161ec5aa1fb2766571828af325a0c8509ce89105a1ad79eadcdffbbbdc6f5 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Derivative
public import Mathlib.Algebra.Polynomial.Module.AEval
public import Mathlib.R... | ```lean
by
have : C a • D f = D (C a * f) := by simp
rw [this, C_mul', D.map_smul]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Polynomial",
"family_id": "c_smul_derivation_apply",
"file_id": "mathlib/Mathlib/Algebra/Polynomial/Derivation.lean",
"sample_id": "f98433b4daad566e797079ce130b71338bc97d14ae011a049aadb8ae222adbed"
} | train |
{
"chosen_proof_sha256": "667083b2ccb953eab7042f8125e1495f0c41208219db3ae4328356bea107af18",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e486eab1ccf683613dfe0b4c7f4076b42f... | by
rw [div_def, ← @Rat.round_cast ℝ _ _, ← @Rat.round_cast ℝ _ _]
simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, add_mul] | false | {
"chosen_tokens": 36,
"rejected_tokens": 40,
"token_jaccard": 0.857143,
"token_length_ratio": 1.111111
} | missing_wrong_import | import | 00b4d46a98db70243ec6d7266484f8f7eb851885d9aaa9d16dbcf198f8f67cd7 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Archimedean.Real.Basic
public import Mathlib.Data.Complex.Basic
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.NumberTheory.Zsq... | import Mathlib.WrongImport
by
rw [div_def, ← @Rat.round_cast ℝ _ _, ← @Rat.round_cast ℝ _ _]
simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, add_mul] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/Zsqrtd",
"family_id": "tocomplex_im_div",
"file_id": "mathlib/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean",
"sample_id": "76ce3bf080d160d098641f9edded107effa5e468a4e154be6bb90faf62e605f0"
} | train |
{
"chosen_proof_sha256": "8f3a85b33714251a1a29f92568064949152f5e84c72c953dd09287cb0b6f1455",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2a102d88cbe43376559a5015393d625cba... | by
rw [NormNum.IsNat.to_eq h rfl]
exact Nat.cast_nonneg' n | false | {
"chosen_tokens": 16,
"rejected_tokens": 21,
"token_jaccard": 0.736842,
"token_length_ratio": 1.3125
} | markdown_preamble_trailing | preamble | 018ece668935a62be4abb3cf4d8d6f329c266da880680cdb88ac456c40f56ad6 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Invertible
public import Mathlib.Algebra.Order.Ring.Cast
public import Mathlib.Tactic.HaveI
public import Mathlib.Tactic.NormNum.Core
Namespace:... | Here is the proof:
by
rw [NormNum.IsNat.to_eq h rfl]
exact Nat.cast_nonneg' n | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Tactic/Positivity",
"family_id": "nonneg_of_isnat",
"file_id": "mathlib/Mathlib/Tactic/Positivity/Core.lean",
"sample_id": "309747378fad33151e6310726ff0174aa0d05877efaa42bd524d8d5cf49ca2f9"
} | train |
{
"chosen_proof_sha256": "3d1a7e2cbfbf665394764afad32089ad753a926b64cde1fb10b4f271be3fc87d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ac00d8bac021edc80086045826c5231fb9... | by
obtain (hs | hs) := hs
· exact hs.2
· exact hs.1.isCompact | false | {
"chosen_tokens": 21,
"rejected_tokens": 25,
"token_jaccard": 0.8125,
"token_length_ratio": 1.190476
} | missing_wrong_import | import | 01e20b3186c068c1f21ac0d461eb91dad6eacf8557baadfe969cd4871e4b9f7b | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.Spectral.Basic
public import Mathlib.Topology.WithTopology
public import Mathlib.Topology.JacobsonSpace
public import Mathlib.Data.Set.Card
Namespace... | import Mathlib.WrongImport
by
obtain (hs | hs) := hs
· exact hs.2
· exact hs.1.isCompact | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/Spectral",
"family_id": "iscompact_of_mem_constructibletopologysubbasis",
"file_id": "mathlib/Mathlib/Topology/Spectral/ConstructibleTopology.lean",
"sample_id": "0f5b96fc655dafea499598a2e613044abf5991cec77163c042656873851c8051"
} | train |
{
"chosen_proof_sha256": "255334aceee88256ccde5808751019138f069b260876b485bc11ea482a1aed8d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e0497951fa64a3a8bf8993417b48930a17... | by
intro b hb
have hg' := hg b hb
obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hg'⟩ := hg'
refine ⟨c₁, hc₁_mem, c₂, hc₂_mem, ?_⟩
filter_upwards [hg', (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg, hfg]
with x hx₁ hx₂ hx₃
intro u hu
rw [hx₂ u hu.1, hx₃]
exact hx₁ u hu | true | {
"chosen_tokens": 95,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.031579
} | circular_dependency | circular_dependency | 01f45cf449314a1f396201cad53d979f4eb7eb8729d86df2e208459d7c12a39e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecialFunctions.Pow.Real
public import Mathlib.Algebra.Order.ToIntervalMod
public import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Al... | by
exact congr_of_eventuallyEq | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "5e5c1b87bdcc6b043dfed46c9a7c541bf3d9a6766666dc2a8e16b19fdb4ccfca",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Computability/AkraBazzi",
"family_id": "congr_of_eventuallyeq",
"file_id": "mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean",
"sample_id": "04f65d5b50a300cc6227733efdcd3f39e044ddd8767d10f8047fdbe408fc2df7"
} | train |
{
"chosen_proof_sha256": "61dd4a4b69745b7e0fa68c4103cf026764280161c8ce4459f05b5e5caa88e47f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ff92774dd6570cff0844a95accc32cba20... | by
simp_rw [univLE_iff_cardinal_le]; apply le_total | false | {
"chosen_tokens": 8,
"rejected_tokens": 13,
"token_jaccard": 0.615385,
"token_length_ratio": 1.625
} | markdown_preamble_trailing | trailing | 0208076cd7af211c6929eeda1e2a5d13d9881f1f09cbf03935e09d4a89f55e1b | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Logic.UnivLE
public import Mathlib.SetTheory.Ordinal.Univ
Namespace:
(root)
Local context:
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released und... | by
simp_rw [univLE_iff_cardinal_le]; apply le_total
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "SetTheory/Cardinal",
"family_id": "univle_total",
"file_id": "mathlib/Mathlib/SetTheory/Cardinal/UnivLE.lean",
"sample_id": "7de96c1ef7802d716a78d8f34f82e51e820ae7e56b3d2af666ad2f6428d02fa0"
} | train |
{
"chosen_proof_sha256": "1c6409609dd76697d622025294c8bc5145bcfe414ffbcf9cd053bd8d9c3b4785",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
let w'' : V₂.op ⋙ eB.op.functor ≅ eT.op.functor ⋙ V₃.op := NatIso.op w'
have : (w ≫ₕ w'.hom).op = (w.op ≫ᵥ w''.hom) := by ext; simp [w'']
rw [← guitartExact_op_iff, ← guitartExact_op_iff w,
← vComp_iff_of_equivalences _ _ _ w'', this]
rfl | false | {
"chosen_tokens": 80,
"rejected_tokens": 2,
"token_jaccard": 0.025,
"token_length_ratio": 0.025
} | sorry_admit_sorryAx_proof_hole | sorry | 0258a2a508961b1612849636bcad3885eb05b24688e91001df56fe74d3e18880 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.GuitartExact.Opposite
Namespace:
CategoryTheory.TwoSquare.GuitartExact
Local context:
/-
Copyright (c) 2026 Joël Riou. All rights reserved.
Re... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/GuitartExact",
"family_id": "hcomp_iff_of_equivalences",
"file_id": "mathlib/Mathlib/CategoryTheory/GuitartExact/HorizontalComposition.lean",
"sample_id": "de181494cf6df2f46264b7108f8b5fe8ade01b1f3c5c162ddf30528ffc055308"
} | train |
{
"chosen_proof_sha256": "2933327708396b2202fe5681ecb1091bcb221fad36d86c22e4ed4f768e81a87c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
ext <;> simp | true | {
"chosen_tokens": 6,
"rejected_tokens": 8,
"token_jaccard": 0.083333,
"token_length_ratio": 1.333333
} | unsolved_goals | unsolved_goals | 02667132161f29e3e1dba5dd662419fad71580543d17984d1f9c0b446c0abaec | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Bicategory.LocallyDiscrete
public import Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo
Namespace:
CategoryTheory.Pseudofunctor... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "3a85eef59907097f2969ba69d7196a93b4304624baf94c3312cb153590fd8dc3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Bicategory",
"family_id": "map_id_map",
"file_id": "mathlib/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean",
"sample_id": "77f925e6d7fb9f855b12d8cba8f5cb36cbebbc68e7684132b1a6af65dd6fe312"
} | train |
{
"chosen_proof_sha256": "306e987910ff023d43690d85687bf5ae9a7b00372cbbe2300a8443541f4fbbe4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3b8de9776eeb7f86a13700a0710fcf6249... | by
intro Y f hf
simp only [Presheaf.χ_app, Opposite.op_unop] at hf ⊢
choose a ha using fun Z (g : Z ⟶ Y) (hg : (Sieve.pullback f ((χ m).app X x)).arrows g) => hg
refine ⟨(hF _ hf).amalgamate a ?_, ?_⟩
· introv Y₁ h
apply (mono_iff_injective (m.app (.op Z))).mp inferInstance
simp_rw [NatTrans.naturalit... | false | {
"chosen_tokens": 165,
"rejected_tokens": 169,
"token_jaccard": 0.958333,
"token_length_ratio": 1.024242
} | missing_wrong_import | import | 02b4094cf6a650f96ac5a6e3e983121316efa3c3c4fd9d7c36f69d7a3987b7e3 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Sites.Closed
public import Mathlib.CategoryTheory.Sites.Equivalence
public import Mathlib.CategoryTheory.Subobject.Classifier.Defs
public import... | import Mathlib.WrongImport
by
intro Y f hf
simp only [Presheaf.χ_app, Opposite.op_unop] at hf ⊢
choose a ha using fun Z (g : Z ⟶ Y) (hg : (Sieve.pullback f ((χ m).app X x)).arrows g) => hg
refine ⟨(hF _ hf).amalgamate a ?_, ?_⟩
· introv Y₁ h
apply (mono_iff_injective (m.app (.op Z))).mp inferInstance
... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Topos",
"family_id": "grothendiecktopology",
"file_id": "mathlib/Mathlib/CategoryTheory/Topos/Sheaf.lean",
"sample_id": "20b48f91a7f4ad693944c040d924608c454aa3ef8bafb869719f264b357fd194"
} | train |
{
"chosen_proof_sha256": "665f9b24ed35095c9258c237fa3353b33a4af46379e1ca75f0d890346defc187",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f402ded8ec1b3fd49c322fa61f9e9685ec... | by
apply EqualizerCondition.mk
intro Z B π _ _
refine ⟨fun a b h ↦ ?_, fun ⟨a, ha⟩ ↦ ?_⟩
· simp only [yonedaPresheaf, comp, Quiver.Hom.unop_op, TypeCat.Fun.coe_mk,
Set.coe_setOf, mapToEqualizer, Set.mem_setOf_eq, ConcreteCategory.hom_ofHom, Subtype.mk.injEq,
mk.injEq] at h
simp only [yonedaPresh... | true | {
"chosen_tokens": 325,
"rejected_tokens": 2,
"token_jaccard": 0.009615,
"token_length_ratio": 0.006154
} | invalid_tactic | invalid_tactic | 031569d80dcce1eeb2a0ab012dc89afd319979620e1bf1a8d9b4bcce58098c02 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Preserves.Opposites
public import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
public import Mathlib.Condensed.Basic
public impo... | by
__oprover_invalid_tactic_df00ef8b543a | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "7e4c508849f8e6007a94799fa3f1a0eb791e8921b5c410b8fd282a7b2b82ab3e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Condensed",
"family_id": "equalizercondition_yonedapresheaf",
"file_id": "mathlib/Mathlib/Condensed/TopComparison.lean",
"sample_id": "df00ef8b543a3838a8dcfd15337dcf5374c33e276a9693b751546ead28ce4bfb"
} | train |
{
"chosen_proof_sha256": "a7995287f4a56e7a4f65ad1427f3f2d4f2887e143a6beb10ba5ae796bad00236",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [← subsingleton_iff_zero_eq_one, OreLocalization.one_def,
OreLocalization.zero_def, oreDiv_eq_iff]
simp | false | {
"chosen_tokens": 17,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.176471
} | by_exact_placeholder | placeholder | 0339380785d74e10a0a891ffeb27765d4a0522dac924f45152effc53d4080fc6 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Submonoid.DistribMulAction
public import Mathlib.GroupTheory.OreLocalization.Basic
public import Mathlib.Algebra.GroupWithZero.Defs
Namespace:
O... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/OreLocalization",
"family_id": "subsingleton_iff",
"file_id": "mathlib/Mathlib/RingTheory/OreLocalization/Basic.lean",
"sample_id": "cc3728dc6e72e50fd255371dff3e5f651d3ce2d9c638da326516235746fae11d"
} | train |
{
"chosen_proof_sha256": "4525ed279745c1a825a2d966ade6667dde0e9859d5d810c453aa56b8f6012282",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4ae32efd86a43fc03720d3484c6c7ad3df... | by
simpa only using! congr_fun (f.isCompat' φ) x | true | {
"chosen_tokens": 13,
"rejected_tokens": 5,
"token_jaccard": 0.125,
"token_length_ratio": 0.384615
} | circular_dependency | circular_dependency | 03461c44c88cb9a1380cb0dfcdd1fdd3b1fa120bf5139ca5cd6765379579bf3e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorProduct.RightExactness
public import Mathlib.RingTheory.Congruence.Hom
public import Mathlib.RingTheory.FiniteType
public import Mathlib.Ri... | by
exact PolynomialLaw.isCompat_apply | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "09b14f7b49ae29a5a60e049189e78be13343a428b834c70791d0b5537b9de052",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/PolynomialLaw",
"family_id": "polynomiallaw",
"file_id": "mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean",
"sample_id": "06f153f627e2ad13220f5061c7800c30d1bdd6c1da9c7ac7f9963af2c8d4a748"
} | train |
{
"chosen_proof_sha256": "faf07d4b9f81d896a4dedf3d8bef12a774f393144e0c5f30d0f0b0458195b2b5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "79e65930fe73a626b39d9d7e987131fdfa... | by
introv W' hf hg hh hi w
simpa only [IsPushout.flip_iff, IsPullback.flip_iff, and_comm] using
H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip | false | {
"chosen_tokens": 42,
"rejected_tokens": 46,
"token_jaccard": 0.909091,
"token_length_ratio": 1.095238
} | missing_wrong_import | import | 03669718c3d3faaf071366f6cd0ec55acbcead621fe0c78cd84310a1c4f84c3c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Extensive
public import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
public import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
Names... | import Mathlib.WrongImport
by
introv W' hf hg hh hi w
simpa only [IsPushout.flip_iff, IsPullback.flip_iff, and_comm] using
H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Adhesive",
"family_id": "ispushout",
"file_id": "mathlib/Mathlib/CategoryTheory/Adhesive/Basic.lean",
"sample_id": "332d57c10825bff92a5f9c100471ccaa87a97a30c29a674c9e1339e0c80d29bd"
} | train |
{
"chosen_proof_sha256": "8f9fde22f4f5e3c6a342c774b51f2e9a3888b3631f985c3a4bfd65b729742429",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9f5d681bdf15caa79e692f23201793dd5e... | by
ext : 1
simp [derivative'] | true | {
"chosen_tokens": 8,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.375
} | unknown_identifier | unknown_identifier | 03988e09076bf188ebdad8f32cae975a6b519a883bea2ca78ba354b3ef45f06b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Derivative
public import Mathlib.Algebra.Polynomial.Module.AEval
public import Mathlib.R... | by
exact __oprover_missing_bbc6b49baeed | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1bd067824a37314782da7518e377e1fd33ad11304939633f5b47140ea3e90fd6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Polynomial",
"family_id": "mkderivation_one_eq_derivative",
"file_id": "mathlib/Mathlib/Algebra/Polynomial/Derivation.lean",
"sample_id": "bbc6b49baeedf3e55b05cb3f8d67c144875e1981fa10f0ed9325586b2eeede01"
} | train |
{
"chosen_proof_sha256": "5416e2b8428ca9ff8cc9626edea3896864ca77c250a86dc31c56a1ba22b8daf5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a2c1aaa8d0f8e5dae27717b6de1757cb5f... | by
unfold Measure.mconv
rw [integral_map (by fun_prop) hf.1, integral_prod]
exact (integrable_map_measure hf.1 (by fun_prop)).mp hf | true | {
"chosen_tokens": 32,
"rejected_tokens": 2,
"token_jaccard": 0.05,
"token_length_ratio": 0.0625
} | invalid_tactic | invalid_tactic | 03a2139af4db8c546781b6d035205f2757bd776fd1019d5707999d001d400ac5 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Group.Convolution
public import Mathlib.MeasureTheory.Integral.Prod
Namespace:
MeasureTheory
Local context:
/-
Copyright (c) 2025 Rémy Degenne.... | by
__oprover_invalid_tactic_91fa9e01efb7 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "a7719d524d4f92f6ff0164abd392f193e5fa01840c627a209ed9ff60798ea72a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Group",
"family_id": "integral_mconv",
"file_id": "mathlib/Mathlib/MeasureTheory/Group/IntegralConvolution.lean",
"sample_id": "91fa9e01efb72d4bb46c66a3c3321fad75748aa4aba9170a103de81feb5b067d"
} | train |
{
"chosen_proof_sha256": "91dab1cd2e2e53fc5f8d17a5c93b18cccb419f8155621f1a414c77fee50249a1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
have : x < 0 := lt_of_tendsto_atBot 0 hx
intro n
simpa [abs_of_neg this, lt_neg] using! lt_of_tendsto_atBot (-n) hx | true | {
"chosen_tokens": 27,
"rejected_tokens": 8,
"token_jaccard": 0.153846,
"token_length_ratio": 0.296296
} | unsolved_goals | unsolved_goals | 03b4b63db8e58f06409315b64b5d42c967e16e1402ac2260952d2a9b06f48669 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Ring.StandardPart
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Order.Filter.FilterProduct
Namespace:
Hyperreal
Loc... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "39839f0bf7bb15e2c15464e4b0aebb811f80512edef7ba8a27da7a2779825d38",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Real",
"family_id": "archimedeanclassmk_neg_of_tendsto_atbot",
"file_id": "mathlib/Mathlib/Analysis/Real/Hyperreal.lean",
"sample_id": "9f0aba5e8c958a1721ba7a16d006099909f296bcaa86d09dd7b80102ebf2e9d4"
} | train |
{
"chosen_proof_sha256": "2f6287791c3459636988da88fa63c514ca770a07a4d1537e256b0a039a8246d2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rcases l.exists_antitone_basis with ⟨u, hu⟩
rcases (Filter.hasBasis_self.mpr hl'.exists_measurable_subset).exists_antitone_subbasis with
⟨v, v_meas, hv⟩
simp only [hu.tendsto_iff hv.toHasBasis, true_imp_iff, true_and, setOf_forall, setOf_exists]
exact .iInter fun n ↦ .iUnion fun _ ↦ .biInter (to_countabl... | false | {
"chosen_tokens": 83,
"rejected_tokens": 2,
"token_jaccard": 0.021739,
"token_length_ratio": 0.024096
} | by_exact_placeholder | by? | 03c4c28dae193b43ef87179c737549426d04fa775fcbfd7ace25e33dfdabf23e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.MeasurableSpace.Constructions
public import Mathlib.Order.Filter.AtTopBot.CompleteLattice
public import Mathlib.Order.Filter.AtTopBot.CountablyGe... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "MeasureTheory/MeasurableSpace",
"family_id": "measurableset_tendsto",
"file_id": "mathlib/Mathlib/MeasureTheory/MeasurableSpace/MeasurablyGenerated.lean",
"sample_id": "f6e07b9557819a58269647c6b5189639d71a69d808a07410892e1fb416291719"
} | train |
{
"chosen_proof_sha256": "318c2cb9db795a2dd46d3f5a27d1f76a3d76756692909ad6bddc78fa03561671",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7ad57aa84d793cc3401f5f4a8dc65aac7c... | by
constructor
rw [← trans_dualMap_symm_flip e]
exact LinearEquiv.bijective _ | true | {
"chosen_tokens": 13,
"rejected_tokens": 3,
"token_jaccard": 0.142857,
"token_length_ratio": 0.230769
} | circular_dependency | circular_dependency | 03e83ffab3b24511547a5c1a3e2a00b485a60092b86eeaa712e710a6b54d8fe7 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Dual.Lemmas
Namespace:
LinearEquiv
Local context:
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as ... | by
exact isReflexive_of_equiv_dual_of_isReflexive | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "cdb778f32decca19a855e9a5065b72a08874ee34b39c0ee27b4c189e9421d036",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/PerfectPairing",
"family_id": "isreflexive_of_equiv_dual_of_isreflexive",
"file_id": "mathlib/Mathlib/LinearAlgebra/PerfectPairing/Basic.lean",
"sample_id": "7da1d9db371103784bbc6daf0c80df8acd3ad5e9a823b6012d8ef842a94b24f4"
} | train |
{
"chosen_proof_sha256": "7361ae6d5ff40c196b32b028010a5c56393ce19a055fbd12b76df3365e97ad61",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6729f70f1e32fd9d52ce0ff9275d6acf57... | by
refine ⟨fun h ↦ eq_univ_iff_forall.mpr h.mem_periodicPts, fun h ↦ ?_⟩
rw [Finite.injective_iff_surjective, ← range_eq_univ, ← univ_subset_iff, ← h]
apply periodicPts_subset_range | true | {
"chosen_tokens": 36,
"rejected_tokens": 3,
"token_jaccard": 0.038462,
"token_length_ratio": 0.083333
} | unknown_identifier | unknown_identifier | 049191310a0dddf7afcd055460d88623cc382f2ac12ebb46054d2e2d91633a24 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GCDMonoid.Finset
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.Data.PNat.Basic
public import Mathlib.Dynamics.PeriodicPts.Defs
publi... | by
exact __oprover_missing_108e499b9132 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "e9369201a8ae13c9e08dbb556769b0fb5ac5f4aa659c3b39e30c09ae5f57502a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Dynamics/PeriodicPts",
"family_id": "injective_iff_periodicpts_eq_univ",
"file_id": "mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean",
"sample_id": "108e499b9132e898676c8d66a09fbbcb3ff6a762c243e0f235bc6414a3079513"
} | train |
{
"chosen_proof_sha256": "c66eb3df5899011ef7909b5b28a8dbc7ead7f1cd7984dec7b597a92d174e6203",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e4be22c9b9981e432ef9024a9996296ed8... | by
rw [SModEq.sub_mem] at hxy ⊢
rw [← smul_sub]
exact smul_mem_smul hc hxy | true | {
"chosen_tokens": 19,
"rejected_tokens": 3,
"token_jaccard": 0.125,
"token_length_ratio": 0.157895
} | unknown_identifier | unknown_identifier | 0497dfba73125893b1e63c80e2a829a70b784bd03d43948f3bedeea907ff7248 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Operations
public import Mathlib.LinearAlgebra.SModEq.Basic
Namespace:
SModEq
Local context:
/-
Copyright (c) 2025 Jiedong Jiang. All rights ... | by
exact __oprover_missing_bf0246d237d6 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "998eeff469aca5126fc003e01a64a753056863204f1986fc278ccae4f1bdc251",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/SModEq",
"family_id": "smul",
"file_id": "mathlib/Mathlib/LinearAlgebra/SModEq/Pointwise.lean",
"sample_id": "bf0246d237d6d9933a85d317c9082d92e62a7779e9ca287e29ec9e8f9ef938b0"
} | train |
{
"chosen_proof_sha256": "a0b20474523f493304d2e5c3c4796a60a88d5f5165ad276749d342179c8ed04e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0f8900d3e899ee934779d7ba7219c40f26... | by
classical
rw [separatingLeft_iff_ker_eq_bot, ker_eq_bot]
let e := LinearEquiv.prodComm R _ _ ≪≫ₗ Module.dualProdDualEquivDual R (Module.Dual R M) M
let h_d := e.symm.toLinearMap.comp (dualProd R M)
refine (Function.Injective.of_comp_iff e.symm.injective
(dualProd R M)).symm.trans ?_
rw [← LinearEquiv... | true | {
"chosen_tokens": 174,
"rejected_tokens": 5,
"token_jaccard": 0.046875,
"token_length_ratio": 0.028736
} | wrong_namespace_qualification | wrong_namespace_qualification | 04a1f03b51301ea5b5c4fc1791ce61e223b3c20ca1450cf564e127aa61be91fa | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Dual.Lemmas
public import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
public import Mathlib.LinearAlgebra.QuadraticForm.Prod
Namespace:
Li... | by
exact OProverMissing.missing_6f6ae3b60a22 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a1cd792e41f8264e0873890de7e6e0c0c697cec886773af634497dc1fb85445c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/QuadraticForm",
"family_id": "separatingleft_dualprod",
"file_id": "mathlib/Mathlib/LinearAlgebra/QuadraticForm/Dual.lean",
"sample_id": "6f6ae3b60a22d3ee8598639d07edd15dc214dc93b32db436d37ba0364ed7ec2a"
} | train |
{
"chosen_proof_sha256": "5fc5fcc6ed6270b255f9132aa985cf7070d173e497c862b80ffbe80e51562de8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "36e6a6a064db8ebbe5935e08c6a7f9aae4... | by
simp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index]
exact forall_comm
-- `simp` normal form is `sized_iUnion`. | true | {
"chosen_tokens": 27,
"rejected_tokens": 2,
"token_jaccard": 0.05,
"token_length_ratio": 0.074074
} | invalid_tactic | invalid_tactic | 04a5fd50bb702262dca90065fecf02aad0d4b646d83b5ae4421cb8cb17361903 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Powerset
public import Mathlib.Order.Interval.Finset.Nat
public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
Namespace:
Set
Local cont... | by
__oprover_invalid_tactic_0f82b3064f5d | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "ccbccaa3b15516808c6e7c4993eb990ce90eaaad1e5262b3004b3cb74c374b7e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Finset",
"family_id": "sized_iunion",
"file_id": "mathlib/Mathlib/Data/Finset/Slice.lean",
"sample_id": "0f82b3064f5d92a16f748b7b934596ecec0fdc7677d78faf22eb29829f183996"
} | train |
{
"chosen_proof_sha256": "e79d2c183534cfffc99b615577f1db04cdc59929d66d45cf7905520be00385d4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2a949889ffca8542db6fbc433458938646... | by
classical
obtain ⟨F₁, hF₁, hAF₁⟩ := hA.sq_covBySMul
obtain ⟨F₂, hF₂, hBF₂⟩ := hB.sq_covBySMul
have := hA.one_le
choose f hf using exists_smul_inter_smul_subset_smul_inv_mul_inter_inv_mul A B
refine ⟨.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1)), ?_, ?_⟩
· calc
(#(.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1))... | false | {
"chosen_tokens": 429,
"rejected_tokens": 436,
"token_jaccard": 0.977011,
"token_length_ratio": 1.016317
} | markdown_preamble_trailing | fence | 04eaba62f4e2be0fad56c2419db1bd1553812e384969072ab892f566a8dcf757 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Subgroup.Pointwise
public import Mathlib.Combinatorics.Additive.CovBySMul
public import Mathlib.Combinatorics.Additive.RuzsaCovering
public impor... | ```lean
by
classical
obtain ⟨F₁, hF₁, hAF₁⟩ := hA.sq_covBySMul
obtain ⟨F₂, hF₂, hBF₂⟩ := hB.sq_covBySMul
have := hA.one_le
choose f hf using exists_smul_inter_smul_subset_smul_inv_mul_inter_inv_mul A B
refine ⟨.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1)), ?_, ?_⟩
· calc
(#(.image₂ f (F₁ ^ (m - 1)) (F₂ ^ ... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/Additive",
"family_id": "pow_inter_pow_covbysmul_sq_inter_sq",
"file_id": "mathlib/Mathlib/Combinatorics/Additive/ApproximateSubgroup.lean",
"sample_id": "5627650f4b1175b07bdab889615a32ef5fc6133ef66c020ba65b75f9c09ed833"
} | train |
{
"chosen_proof_sha256": "9ea2d48ebe00bb73f1eea97bd737a2f6b8f66a4067141833800162c91765c532",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp [GradedMonoid.GOne.one, TensorPower.toTensorAlgebra_tprod] | true | {
"chosen_tokens": 13,
"rejected_tokens": 8,
"token_jaccard": 0.058824,
"token_length_ratio": 0.615385
} | unsolved_goals | unsolved_goals | 05239f75c87635d84490583f3f9a8c44b6f1a90bcc8585dd96bf1b926ab86034 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorAlgebra.Basic
public import Mathlib.LinearAlgebra.TensorPower.Basic
Namespace:
TensorPower
Local context:
/-
Copyright (c) 2021 Eric Wies... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "98aefb3cadd5f1aade6e34aae778184241b0dff9f78ebac25e4dfc51ce971434",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/TensorAlgebra",
"family_id": "totensoralgebra_gone",
"file_id": "mathlib/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean",
"sample_id": "dc8404380687b2e37f6ac350d18b83560147a6545b4e3c6396ca1a3bc04d0633"
} | train |
{
"chosen_proof_sha256": "389a8937a3f2af4740351819b93e71736e399c56154cd81ab2e2e646a0bfda28",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [eq_comm, prod_eq_prod_of_nonempty h, eq_comm, @eq_comm _ eY'] | false | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.2
} | by_exact_placeholder | exact? | 052eb700b25f8527bd38890cecc9319b535d5dc2c6e56d3ff48a7b865dd7a646 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.OpenPartialHomeomorph.Composition
Namespace:
OpenPartialHomeomorph
Local context:
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Rele... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/OpenPartialHomeomorph",
"family_id": "prod_eq_prod_of_nonempty",
"file_id": "mathlib/Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean",
"sample_id": "b0b58afdb4cef1d362a08061526d43713fac33d2fd2c151db38d446b2e4f58cd"
} | train |
{
"chosen_proof_sha256": "0e11e10bbaaaed2dc873b84e165eba0bc5466c4e31acb825aeb49b92a5fe94d8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
-- Let `Z` be a minimal set of prime ideals such that their product is contained in `J`.
obtain ⟨Z₀, hZ₀⟩ := exists_primeSpectrum_prod_le_and_ne_bot_of_domain hNF hI0
obtain ⟨Z, ⟨hZI, hprodZ⟩, h_eraseZ⟩ :=
wellFounded_lt.has_min
{Z | (Z.map asIdeal).prod ≤ I ∧ (Z.map asIdeal).prod ≠ ⊥}
⟨Z₀, hZ₀... | false | {
"chosen_tokens": 346,
"rejected_tokens": 2,
"token_jaccard": 0.008197,
"token_length_ratio": 0.00578
} | sorry_admit_sorryAx_proof_hole | admit | 05511de79a89488a1357faf2a592ef25671df129fd76d0f508cfe5bc6eeb30fa | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
public import Mathlib.RingTheory.DedekindDomain.Basic
public import Mathlib.RingTheory.FractionalIdeal.Inverse
public impo... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/DedekindDomain",
"family_id": "primespectrum",
"file_id": "mathlib/Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean",
"sample_id": "0996acbca200c0dccec01a46d4e81205ed46c98fd950cb9f91af86182dcfbe7d"
} | train |
{
"chosen_proof_sha256": "843286c15dd0e5c3687da59e28ddbf32542077397522d384d533e4f852e034a9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "96285e98a90c2e00636f856577b756ddde... | by
rw [map_comp]
exact hg.trans <| map_mono _ hf | true | {
"chosen_tokens": 14,
"rejected_tokens": 3,
"token_jaccard": 0.133333,
"token_length_ratio": 0.214286
} | circular_dependency | circular_dependency | 05578208c6008563b61054e814ca53c19b5bccf2e619f3de237f20f838328eea | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
public import Mathlib.RingTheory.GradedAlgebra.RingHom
Namespace:
HomogeneousIdeal
Local context:
/-
Copyright (c)... | by
exact irrelevant_le_map_comp | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "fa9e40f9ac772bc5aa5a2780dca6e94d3723a408946d5514dcb8cd1bae663a85",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/GradedAlgebra",
"family_id": "irrelevant_le_map_comp",
"file_id": "mathlib/Mathlib/RingTheory/GradedAlgebra/Homogeneous/Maps.lean",
"sample_id": "2c90f2e32be3da6fb260df3bb94bf0c15d7de35d2f1513e19789e1136caf26c6"
} | train |
{
"chosen_proof_sha256": "de67f9b6451b5db3ba017889f07c8f6879cb98f3fc2736a86445d62c14d1c664",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3d470e7ad773073099cd7eb08ccd0ead6e... | by
rw [Squarefree, not_forall]
exact ⟨0, by simp⟩ | false | {
"chosen_tokens": 14,
"rejected_tokens": 18,
"token_jaccard": 0.75,
"token_length_ratio": 1.285714
} | missing_wrong_import | import | 05ff9d4452c635482cdef1b657e4ec716ce9b8154a9b10402ed87294aeedbfe7 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Coprime.Lemmas
public import Mathlib.RingTheory.Nilpotent.Basic
public import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid
public import M... | import Mathlib.WrongImport
by
rw [Squarefree, not_forall]
exact ⟨0, by simp⟩ | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Squarefree",
"family_id": "not_squarefree_zero",
"file_id": "mathlib/Mathlib/Algebra/Squarefree/Basic.lean",
"sample_id": "ba8a1ee7158309c944c89fb3fff36b668ca7628b981cba7184536316677b8d1f"
} | train |
{
"chosen_proof_sha256": "dfb0cfa54bf916f968be75edf3a87f502ebb8bc73b72d17a24c14c29c6f3e807",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "dfa2375d52b2512c773d6c8fddc24e78da... | by
convert! sup_mem_subalgebra_closure A f g
apply SetLike.ext'
simp | false | {
"chosen_tokens": 12,
"rejected_tokens": 17,
"token_jaccard": 0.705882,
"token_length_ratio": 1.416667
} | markdown_preamble_trailing | preamble | 060d74d0d090822121affa5c6d6fbf46b7f7ce76bc162e83c6a261b674b0c44d | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Subalgebra.Tower
public import Mathlib.Analysis.RCLike.Basic
public import Mathlib.Topology.Algebra.Star.Real
public import Mathlib.Topology.Al... | Here is the proof:
by
convert! sup_mem_subalgebra_closure A f g
apply SetLike.ext'
simp | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/ContinuousMap",
"family_id": "sup_mem_closed_subalgebra",
"file_id": "mathlib/Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean",
"sample_id": "5718bd96cf9f59ca4c83944f21497324dfdb689e13608aa103fc1d5ac63a1635"
} | train |
{
"chosen_proof_sha256": "30015a6154bd2aef8d0848639c98385965744ef4b2fb839692a5ae2632c8bacb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [← List.filterMap_eq_filter]
apply listFilterMap .id
simp only [Primrec₂, Option.guard, decide_eq_true_eq]
exact ite (hf.comp snd) (option_some_iff.mpr snd) (const none) | true | {
"chosen_tokens": 42,
"rejected_tokens": 8,
"token_jaccard": 0.027778,
"token_length_ratio": 0.190476
} | unsolved_goals | unsolved_goals | 061085a06360faa3c0ed20fb10bd3809c69af39fc97327ee2a3024bc727d74da | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Computability.Primrec.Basic
public import Mathlib.Logic.Encodable.Pi
Namespace:
Primrec
Local context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reser... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "754536d407ca07e61abeb3df0dafa3d8e2b6dafeec4230ddccf28b66656c9c91",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Computability/Primrec",
"family_id": "listfilter",
"file_id": "mathlib/Mathlib/Computability/Primrec/List.lean",
"sample_id": "61faa742462a962beb76f3f1545ae04bd1db702ab9662b4c763eedf05c97a9ff"
} | train |
{
"chosen_proof_sha256": "6bde054e8dd9e2e89dc5431f020cd1a2610cdbc6535abeb8845dde9b2fdd1677",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e31719f32ba1c30d133bd4fc148ed4aeca... | by
change
Quotient.map (@funMap L M ps.toStructure n f) Prestructure.fun_equiv (Quotient.finChoice _) =
_
rw [Quotient.finChoice_eq, Quotient.map_mk] | true | {
"chosen_tokens": 37,
"rejected_tokens": 3,
"token_jaccard": 0.035714,
"token_length_ratio": 0.081081
} | unknown_identifier | unknown_identifier | 0618a971244095a66f24ab982d7243a5dab1c42ea87f6f3b0a8a261ecd3336bb | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Quotient
public import Mathlib.ModelTheory.Semantics
Namespace:
FirstOrder.Language
Local context:
/-
Copyright (c) 2022 Aaron Anderson. All rig... | by
exact __oprover_missing_6c18e3a6e36f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f3a9f2a24581b149d0f9b691dc031fc933d078bb9d82d773d53581f517ce5bd8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "ModelTheory",
"family_id": "funmap_quotient_mk",
"file_id": "mathlib/Mathlib/ModelTheory/Quotients.lean",
"sample_id": "6c18e3a6e36fcd3e1caadefef8b6d28d63c691769c6b3ffbae3c86bf10ef7481"
} | train |
{
"chosen_proof_sha256": "9863212e81e0182629bd15d927cf3f594339d8438f634ee6f333a33c3de5d6eb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
conv_rhs => simp [← h.commute_iff, Commute.symm_iff (a := e), commute_iff_eq, ← mem_commutant_iff] | false | {
"chosen_tokens": 24,
"rejected_tokens": 3,
"token_jaccard": 0.045455,
"token_length_ratio": 0.125
} | by_exact_placeholder | placeholder | 0666a5e08b451e9ce6f99af41733fd943daa44e5ac2b70d7428f09ae8d10e4d8 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.CStarAlgebra.Classes
public import Mathlib.Analysis.InnerProductSpace.Adjoint
Namespace:
VonNeumannAlgebra
Local context:
/-
Copyright (c) 2022 Kim ... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/VonNeumannAlgebra",
"family_id": "isidempotentelem",
"file_id": "mathlib/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean",
"sample_id": "b3ad2032874ba6a0b6f50397e2b035a5757e486348575d6dc25170f7fefc69ec"
} | train |
{
"chosen_proof_sha256": "03e21e37518786f0e8f480f42a07a2927cdb9237ca44f9d215146d88cd8ce9cd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1721659186d255075a58ab2c96c00145a2... | by
classical
obtain ⟨s, hs⟩ := hI
rw [← adjoin_monomial_eq_reesAlgebra, ← hs]
use s.image (monomial 1)
rw [Finset.coe_image]
change
_ =
Algebra.adjoin R
(Submodule.map (monomial 1 : R →ₗ[R] R[X]) (Submodule.span R ↑s) : Set R[X])
rw [Submodule.map_span, Algebra.adjoin_s... | false | {
"chosen_tokens": 83,
"rejected_tokens": 88,
"token_jaccard": 0.911111,
"token_length_ratio": 1.060241
} | markdown_preamble_trailing | preamble | 06dc353dcc2bade935863497c4cfbb148bf7871adc6cd4ad051486cab904ed46 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Ideal.BigOperators
public import Mathlib.RingTheory.FiniteType
Namespace:
(root)
Local context:
/-
Copyright (c) 2022 Andrew Yang. All rights rese... | Here is the proof:
by
classical
obtain ⟨s, hs⟩ := hI
rw [← adjoin_monomial_eq_reesAlgebra, ← hs]
use s.image (monomial 1)
rw [Finset.coe_image]
change
_ =
Algebra.adjoin R
(Submodule.map (monomial 1 : R →ₗ[R] R[X]) (Submodule.span R ↑s) : Set R[X])
rw [Submodule.map_spa... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory",
"family_id": "reesalgebra",
"file_id": "mathlib/Mathlib/RingTheory/ReesAlgebra.lean",
"sample_id": "7f63bf8b080e808117284c406a2db8ef2f14a5c00152c31fb40b5d94668e2b76"
} | train |
{
"chosen_proof_sha256": "66e860b3b879ffe015e76af344d6b2852af86ab51c1939b65912c04b3badbf97",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
refine ⟨fun h => ?_, fun h => h.symm ▸ mulSingle_one i⟩
rw [← mulSingle_eq_same i x, h, one_apply] | false | {
"chosen_tokens": 30,
"rejected_tokens": 3,
"token_jaccard": 0.041667,
"token_length_ratio": 0.1
} | by_exact_placeholder | placeholder | 06f4b1a5a05acacd143cae3af399d7ba0f61702a697dbafab2a9718292c4e543 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Notation.Pi.Defs
public import Mathlib.Logic.Function.Basic
Namespace:
Pi
Local context:
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Rele... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Notation",
"family_id": "mulsingle_eq_one_iff",
"file_id": "mathlib/Mathlib/Algebra/Notation/Pi/Basic.lean",
"sample_id": "6c05ada6fb867e067b57945bee363d3116da5202db3c69d06f6b508b9b741204"
} | train |
{
"chosen_proof_sha256": "0c7a4b44bba5c20da7810399c16de1e098a5c39bdee940dd765e6735e6e9d672",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "130996c934f7e3dc9a1f69e7437ed491cd... | by
simp [IsSelfAdjoint.star_eq, ← sub_eq_add_neg, (CFC.sqrt_nonneg (1 - a ^ 2 : A)).isSelfAdjoint] | true | {
"chosen_tokens": 27,
"rejected_tokens": 5,
"token_jaccard": 0.08,
"token_length_ratio": 0.185185
} | wrong_namespace_qualification | wrong_namespace_qualification | 0703b97fe34fc512203f8faaeb1e06c6a252bbb272b6c832827dc51a489746cc | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary
public import Mathlib... | by
exact OProverMissing.missing_af27a105e43a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "910d805da3f33b7ad1594723f71f7d85d99ffb7daf7416900d5275e49dd9c46f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/CStarAlgebra",
"family_id": "selfadjoint",
"file_id": "mathlib/Mathlib/Analysis/CStarAlgebra/Unitary/Span.lean",
"sample_id": "af27a105e43a072189a2b49287038416eed37dc52c7226788734b79f2baf138e"
} | train |
{
"chosen_proof_sha256": "d20eb1ad8941b8e34a808b1966083034fd1512741d71702f610e71a800689ea8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [← single_zero_one, orderTop_single one_ne_zero, WithTop.coe_eq_zero] | true | {
"chosen_tokens": 13,
"rejected_tokens": 8,
"token_jaccard": 0.055556,
"token_length_ratio": 0.615385
} | unsolved_goals | unsolved_goals | 07c76a63455edc8852677c642af5e6de4077c2e679b530966ab8dae1e572e2d7 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Subalgebra.Lattice
public import Mathlib.Algebra.GroupWithZero.Regular
public import Mathlib.Algebra.Module.BigOperators
public import Mathlib.... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b4e59b5d70d3037f13266d6f3c34244679154076146ca872ca947e693cb6e8a2",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/HahnSeries",
"family_id": "ordertop_one",
"file_id": "mathlib/Mathlib/RingTheory/HahnSeries/Multiplication.lean",
"sample_id": "ac967f3faf553306a5c24145a1c9caf628fb00eae9e83bd29d882cbc48ac201d"
} | train |
{
"chosen_proof_sha256": "930122cf7815506e3d0570b55467812e4caec3bf1c598928f36877dea55601fa",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f6f3ce4e022ec29b5bf2237824915ea627... | by
simpa only [h.comp_eq] using bijOn_fixedPoints_comp f g | true | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.25
} | unknown_identifier | unknown_identifier | 07f520b641c5b18e3b4e90eab8fb5dab450d77fb31707558fe45ff66c8fb599f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.End
public import Mathlib.Data.Set.Function
public import Mathlib.Dynamics.FixedPoints.Defs
Namespace:
Function
Local context:
/-
Copyright (c)... | by
exact __oprover_missing_636a9fcc3e8a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "702594ef4837066088b7ae06ce9809d9afcf03ff374ea8d6139efd4bd4f8ba5c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Dynamics/FixedPoints",
"family_id": "commute",
"file_id": "mathlib/Mathlib/Dynamics/FixedPoints/Basic.lean",
"sample_id": "636a9fcc3e8a95e1374f41b3672c8c890f7576b9c9fa7d5f0caae9002e2a7ade"
} | train |
{
"chosen_proof_sha256": "2c01520ab51bc25fbe7980ca7811c6f4fd2614f49e05e36aaa18fc8557eeaddd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "71ba16669017a6fd8a1b9efc9f308604b8... | by
rw [modEq_comm]
simp_rw [modEq_iff_eq_add_zsmul, QuotientAddGroup.eq_iff_sub_mem, AddSubgroup.mem_zmultiples_iff,
eq_sub_iff_add_eq', eq_comm] | true | {
"chosen_tokens": 21,
"rejected_tokens": 2,
"token_jaccard": 0.0625,
"token_length_ratio": 0.095238
} | invalid_tactic | invalid_tactic | 08e596b83bd67565d7054379b9b8267b62ec6f2c76149ecb9585b1ecbe650c43 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.ModEq
public import Mathlib.GroupTheory.QuotientGroup.Defs
public import Mathlib.Algebra.Group.Subgroup.ZPowers.Basic
Namespace:
AddCommGroup
L... | by
__oprover_invalid_tactic_f57b40da55b4 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "2f57cca57a4dcb8b94248dc8a844a6b0ce98685f58004564fb111d1f4119b100",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "GroupTheory/QuotientGroup",
"family_id": "modeq_iff_eq_mod_zmultiples",
"file_id": "mathlib/Mathlib/GroupTheory/QuotientGroup/ModEq.lean",
"sample_id": "f57b40da55b42c3f0a2ab8a4512907a91408e6b8713cf272d9f3041b94ceb710"
} | train |
{
"chosen_proof_sha256": "b82467ef621f5a21aaae8eaa62dc55df65f51c9c9619a918a2bac8a08b7ed281",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1979aaf06c49a077cffd0cbacc14afdba7... | by
delta Irrational
contrapose!
rintro ⟨⟨rx, rfl⟩, ⟨ry, rfl⟩⟩
exact ⟨rx + ry, cast_add rx ry⟩ | false | {
"chosen_tokens": 29,
"rejected_tokens": 34,
"token_jaccard": 0.75,
"token_length_ratio": 1.172414
} | markdown_preamble_trailing | preamble | 09f879999edc81c7f7930cb98b2b558018a3026a99ebfe4af6f059f43c356d46 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Rat
public import Mathlib.Data.Nat.Prime.Int
public import Mathlib.Data.Rat.Sqrt
public import Mathlib.Analysis.Real.Sqrt
public import Mathlib... | Here is the proof:
by
delta Irrational
contrapose!
rintro ⟨⟨rx, rfl⟩, ⟨ry, rfl⟩⟩
exact ⟨rx + ry, cast_add rx ry⟩ | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/Real",
"family_id": "add_cases",
"file_id": "mathlib/Mathlib/NumberTheory/Real/Irrational.lean",
"sample_id": "d0d0f38b9e3e428b2ae38db46c5384471141fd719dfd1fd904da355d5cb234fc"
} | train |
{
"chosen_proof_sha256": "e0cdaf3a4b43a4f8fc2adc6c6eee964e632fc1aa70e6d8ebca268efd1191ebba",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "56b98b81c35c609877ff6bd3d13a65d067... | by
rw [Presieve.EffectiveEpimorphic.iff_forall_isSheafFor_yoneda] at hR
rw [← isSheafFor_comp_uliftFunctor_iff]
refine Presieve.isSheafFor_iso (F ⋙ uliftFunctor.{v}).uliftYonedaReprXIso ?_
dsimp only [uliftYoneda, Functor.comp_obj, Functor.whiskeringRight_obj_obj]
rw [isSheafFor_comp_uliftFunctor_iff]
exact... | true | {
"chosen_tokens": 53,
"rejected_tokens": 7,
"token_jaccard": 0.147059,
"token_length_ratio": 0.132075
} | circular_dependency | circular_dependency | 0a52e07b7cf84d666234eba2ae7ee1a32165323848b5c641186745b70b5e6d89 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Sites.SheafOfTypes
public import Mathlib.CategoryTheory.EffectiveEpi.Basic
Namespace:
CategoryTheory
Local context:
/-
Copyright (c) 2023 Adam... | by
exact Presieve.EffectiveEpimorphic.isSheafFor_of_isRepresentable | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "b171306b2273d77e4ffbabf0661f638c1a433e1edb980c40120f8d5ac80db39f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Sites",
"family_id": "presieve",
"file_id": "mathlib/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean",
"sample_id": "3c58404f57528d0f09544fa0d6422c11989243f1a83384b09f5595af7aef1431"
} | train |
{
"chosen_proof_sha256": "5b4476fdc1c0c0f51888b9b0f5b1dfa03fd37018bf6998a42fadc18efa75285e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f2d3f78b3cc280c53094018699cfb4d812... | by
rw [quasicategory_iff_hasLiftingProperty.{u} _ terminalIsTerminal, innerFibration_iff]
exact ⟨fun h _ _ _ ⟨i, h0, hn⟩ ↦ h h0 hn,
fun h _ _ h0 hn ↦ h _ (horn_ι_mem_innerHornInclusions h0 hn)⟩ | false | {
"chosen_tokens": 49,
"rejected_tokens": 53,
"token_jaccard": 0.9,
"token_length_ratio": 1.081633
} | missing_wrong_import | import | 0a5ee35651cecb0719c3fb631c61bfd0f3b663e500e036ec908c65f8c59a54df | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicTopology.Quasicategory.Basic
Namespace:
SSet
Local context:
/-
Copyright (c) 2026 Jack McKoen. All rights reserved.
Released under Apache 2.0 licens... | import Mathlib.WrongImport
by
rw [quasicategory_iff_hasLiftingProperty.{u} _ terminalIsTerminal, innerFibration_iff]
exact ⟨fun h _ _ _ ⟨i, h0, hn⟩ ↦ h h0 hn,
fun h _ _ h0 hn ↦ h _ (horn_ι_mem_innerHornInclusions h0 hn)⟩ | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/Quasicategory",
"family_id": "quasicategory_iff_innerfibration",
"file_id": "mathlib/Mathlib/AlgebraicTopology/Quasicategory/InnerFibration.lean",
"sample_id": "3681767d9aba4e16656d0211cddaff6efe2e454378f2ebc6b91e0ff3858dab3f"
} | train |
{
"chosen_proof_sha256": "db5676c2a4baaaa8601683c727e173d0b3b0fe87aa92feb60828f3a50296fa53",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2679434db9e07f43c58fe299cdd22e9633... | by
have hp : ((f.support : Set G).smulAntidiagonal (Function.support x) p).Finite :=
Set.SMulAntidiagonal.finite_of_finite_fst f.support.finite_toSet x.support p
set s : Set (G × P) := ↑(Finset.SMulAntidiagonal p hp)
have h₁ : s.InjOn Prod.fst := fun _ h₁ _ h₂ h ↦ by
rw [Finset.mem_coe, mem_smulAntidiagon... | true | {
"chosen_tokens": 224,
"rejected_tokens": 3,
"token_jaccard": 0.014493,
"token_length_ratio": 0.013393
} | unknown_identifier | unknown_identifier | 0a6e443d072d5bc8ede2abaa1c80c0f4fafce1fd08c414a38f1bc5be8fb249ab | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MonoidAlgebra.Defs
public import Mathlib.Data.Finset.SMulAntidiagonal
Namespace:
MonoidAlgebra
Local context:
/-
Copyright (c) 2025 Scott Carnahan. A... | by
exact __oprover_missing_ba002fff735a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "4b27115d1268c1c427d90643d07e532a568da181ed0905bcd4e8b2dd2373cc44",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Algebra/MonoidAlgebra",
"family_id": "smul_apply_mulaction",
"file_id": "mathlib/Mathlib/Algebra/MonoidAlgebra/PointwiseSMul.lean",
"sample_id": "ba002fff735a1eceaabf1a8e0dbcc41b8470b3783bbd4a79e63a536436ef24f3"
} | train |
{
"chosen_proof_sha256": "8a67a77843a75ba583cb43ac0e535963aaf96ee59db334d6516f871f6e4da8bd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "09f07bfd57e55788ddf24145640ae69665... | by
rw [← @isBounded_univ, isBounded_induced, image_univ] | true | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.272727
} | unknown_identifier | unknown_identifier | 0a78cfc513926591aed12ca62d4a6439f8a4e598a136a02008a0952b537d8ca8 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.TypeTags.Basic
public import Mathlib.Topology.Bornology.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2022 Yury Kudryashov. All right... | by
exact __oprover_missing_9f9a70e78a68 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "796a0d5b85da265f6eb593b01f9e4e208d0fada9915216503476680da4badb66",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Bornology",
"family_id": "boundedspace_induced_iff",
"file_id": "mathlib/Mathlib/Topology/Bornology/Constructions.lean",
"sample_id": "9f9a70e78a68c5afe844cbabb734e8a42046f9373491df56a598c7ff77b730eb"
} | train |
{
"chosen_proof_sha256": "c8b268100380170538174b78ca7f2045ab5992f487a935451a91e1cbc0d99b37",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
have h₁ : a i < supOfSeq a :=
Nat.lt_add_one_iff.mpr (le_max_of_le_right <| Finset.le_sup (by simp))
have h₂ : supOfSeq a ≤ (i + 1) * (supOfSeq a)! + 1 :=
le_trans (self_le_factorial _) (le_trans (Nat.le_mul_of_pos_left (supOfSeq a)! (succ_pos i))
(le_add_right _ _))
simpa only [coprimes] using l... | false | {
"chosen_tokens": 87,
"rejected_tokens": 2,
"token_jaccard": 0.02439,
"token_length_ratio": 0.022989
} | by_exact_placeholder | by? | 0a8487e37be145dbfc31c1579c657fe78361881899b9b1eedfba5163aa857af1 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.ModEq
public import Mathlib.Data.Nat.ChineseRemainder
public import Mathlib.Data.Nat.Prime.Defs
public import Mathlib.Data.Nat.Pairing
public import M... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Logic/Godel",
"family_id": "coprimes_lt",
"file_id": "mathlib/Mathlib/Logic/Godel/GodelBetaFunction.lean",
"sample_id": "81a0159374a335acfc3a3be57440b1fb6c4f215c8d52062ffeefc8554afc5a68"
} | train |
{
"chosen_proof_sha256": "a329996be1763b313924738bf153d1074388b636f52e211888a4d4f3a26dea07",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a0d831455a2a6ecae7a67822f37fbbc5fc... | by
simpa using w
-- It would be nice to mark this as `aesop cases` if
-- https://github.com/leanprover-community/aesop/issues/59
-- is resolved. | true | {
"chosen_tokens": 44,
"rejected_tokens": 2,
"token_jaccard": 0.032258,
"token_length_ratio": 0.045455
} | invalid_tactic | invalid_tactic | 0a8e903b47393fd7ef91b9f2c671e4129acdd81baabebf6f649eae768f838692 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Logic.Equiv.Defs
public import Mathlib.Tactic.MkIffOfInductiveProp
public import Mathlib.Tactic.PPWithUniv
Namespace:
(root)
Local context:
/-
Copyright (c) ... | by
__oprover_invalid_tactic_1f97d0501e04 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "74a9dff03c20ebc22e4a21a286c8f7662a7f74a8f96225fa1f95e61c1c8d2fb6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Logic/Small",
"family_id": "shrink",
"file_id": "mathlib/Mathlib/Logic/Small/Defs.lean",
"sample_id": "1f97d0501e044f00fde760711297ef657cb86ce42d441db370fdf596ff19fb0c"
} | train |
{
"chosen_proof_sha256": "49bdfcf01bbb1a8c4c2fb91482006ee7e27f5a1dbdbae3ae54b463c762ee66e9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
ext X; simp [Equivalence.unit] | false | {
"chosen_tokens": 10,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.3
} | by_exact_placeholder | placeholder | 0ac5e9a770e587ea1d3b7656c21bc964e951f5fcf37004453b8fc03ae50b47ef | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Adjunction.Basic
public import Mathlib.CategoryTheory.Yoneda
public import Mathlib.CategoryTheory.Opposites
Namespace:
CategoryTheory.Adjunctio... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Adjunction",
"family_id": "rightop_eq",
"file_id": "mathlib/Mathlib/CategoryTheory/Adjunction/Opposites.lean",
"sample_id": "65e0c46ab80804e8a5cd071bf84cbfc579f19ac01cda5e58e506b75975df165e"
} | train |
{
"chosen_proof_sha256": "4bc1dfb496047d2c2f53f0ca55577975a296dcb639bf7ace11b6e910a7a97f44",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5d913d38e4c8561f809e8527a3507548e3... | by
rw [← SetLike.mem_coe, coe_commutant]
rfl | true | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.272727
} | circular_dependency | circular_dependency | 0b553713d409ec94f1445d12ac0178f733a55ace1591583196d71be797f241a7 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.CStarAlgebra.Classes
public import Mathlib.Analysis.InnerProductSpace.Adjoint
Namespace:
VonNeumannAlgebra
Local context:
/-
Copyright (c) 2022 Kim ... | by
exact mem_commutant_iff | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "54af599effc2a1b7c5096aef5feeec82537207f5daca0259a03dd76057e88671",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/VonNeumannAlgebra",
"family_id": "mem_commutant_iff",
"file_id": "mathlib/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean",
"sample_id": "452212369db5a32584e53a4244413eb62217af8a259514599237694d17941ea9"
} | train |
{
"chosen_proof_sha256": "e5473258af32213de8f9e9baf793bc2d791875a7c5122f73158f25e7055ff270",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
simpa only [sub_eq_add_neg] using!
(f.commute_add_nat n).inverses_right (Equiv.addRight _).right_inv (Equiv.addRight _).left_inv | false | {
"chosen_tokens": 32,
"rejected_tokens": 5,
"token_jaccard": 0.086957,
"token_length_ratio": 0.15625
} | sorry_admit_sorryAx_proof_hole | sorryAx | 0be81a42db189b271d14e206ad1626fd3414d8540d79efd35868f814691f6636 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Order.SemiconjSup
Namespace:
CircleDeg1Lift
Local context:
/-
Copyright (c) 2020 Yury Kudryashov. All rig... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Dynamics/Circle",
"family_id": "commute_sub_nat",
"file_id": "mathlib/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean",
"sample_id": "bff817808a5cc69dcd74325b621cd3b889cc91e9bdb2f5fd6f80c7083b9ec6d9"
} | train |
{
"chosen_proof_sha256": "73aa3a7c73d07e3e5e0d2c9a0dc0fab61b3371c9b2e729b99cd088422d654f85",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "dc3039cff6d4052f266ec1fe01fc169549... | by
rcases Finite.exists_equiv_fin β with ⟨n, ⟨e⟩⟩
classical exact .of_equiv (Set.range (e ∘ f)) (Equiv.ofInjective _ (e.injective.comp H)).symm
-- see Note [lower instance priority] | true | {
"chosen_tokens": 53,
"rejected_tokens": 5,
"token_jaccard": 0.078947,
"token_length_ratio": 0.09434
} | wrong_namespace_qualification | wrong_namespace_qualification | 0c04056bc242789491cf4239f93203ab8a94034db86100bb40bfbb56c8706dd1 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.List.NodupEquivFin
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
R... | by
exact OProverMissing.missing_ba99dd8f6392 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "fcde561416e510a9e2b0cad44babb3c67f4c02e586f2f88c2dc1275f420c0434",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Fintype",
"family_id": "finite",
"file_id": "mathlib/Mathlib/Data/Fintype/EquivFin.lean",
"sample_id": "ba99dd8f6392afa104bfc6ac9136b281cb75f223a9c829f9749723664ecb6a1c"
} | train |
{
"chosen_proof_sha256": "8b5027e289eea500a3bcfb155a81791b1a51bc31d201ba8ef228cee84d9c2f28",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simp [IsIdempotentElem, ← star_mul]
alias ⟨_, IsIdempotentElem.star⟩ := IsIdempotentElem.star_iff | false | {
"chosen_tokens": 20,
"rejected_tokens": 2,
"token_jaccard": 0.058824,
"token_length_ratio": 0.1
} | by_exact_placeholder | by? | 0c6b1d877bcc3fc76fa790db5be247d036677f9b6071fc297fb6d3808318556f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Star.SelfAdjoint
public import Mathlib.Algebra.Module.Basic
public import Mathlib.Algebra.Module.Equiv.Defs
public import Mathlib.Algebra.Module.Linear... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Star",
"family_id": "isidempotentelem",
"file_id": "mathlib/Mathlib/Algebra/Star/Module.lean",
"sample_id": "d2e80bfd09f7c8ffd630b3233e0baf00abf1e879c638b96895f9ef4ded614180"
} | train |
{
"chosen_proof_sha256": "c55c96a734207c6a61a412e0cab744c2c07b34209605ac6a4b05c1a6bd545ce9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f264b4d3af83d4231d10d5975f60cb548c... | by
rw [tprod_apply, map_list_prod, List.map_ofFn]
simp_rw [Function.comp_def, toDirectSum_ι]
rw [DirectSum.list_prod_ofFn_of_eq_dProd]
apply DirectSum.of_eq_of_gradedMonoid_eq
rw [GradedMonoid.mk_list_dProd]
rw [TensorPower.list_prod_gradedMonoid_mk_single] | false | {
"chosen_tokens": 42,
"rejected_tokens": 47,
"token_jaccard": 0.851852,
"token_length_ratio": 1.119048
} | markdown_preamble_trailing | trailing | 0caff8188aeaebb78e5afc923133859529113710cced5a8a8bae09f050d8f218 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorAlgebra.Basic
public import Mathlib.LinearAlgebra.TensorPower.Basic
Namespace:
TensorAlgebra
Local context:
/-
Copyright (c) 2021 Eric Wi... | by
rw [tprod_apply, map_list_prod, List.map_ofFn]
simp_rw [Function.comp_def, toDirectSum_ι]
rw [DirectSum.list_prod_ofFn_of_eq_dProd]
apply DirectSum.of_eq_of_gradedMonoid_eq
rw [GradedMonoid.mk_list_dProd]
rw [TensorPower.list_prod_gradedMonoid_mk_single]
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/TensorAlgebra",
"family_id": "todirectsum_tensorpower_tprod",
"file_id": "mathlib/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean",
"sample_id": "4d49f41524ca78215a8298981fc8795e5f250086d26cc6a75dc11a9339f59a8c"
} | train |
{
"chosen_proof_sha256": "153be3b3963b7b4e295fef95384346f85509c50a1d50e691a207cccee178ce99",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "95e884eecf0883ef717211cc0723527aed... | by
constructor
· rw [Presheaf.isLocallySurjective_iff_whisker_forget (J := extensiveTopology C)]
exact fun h _ ↦
surjective_of_isLocallySurjective_sheaf_of_types (Functor.whiskerRight f (forget D)) h
· exact fun a ↦
Presheaf.isLocallySurjective_of_surjective _ _ (fun _ ↦ a _) | false | {
"chosen_tokens": 49,
"rejected_tokens": 56,
"token_jaccard": 0.933333,
"token_length_ratio": 1.142857
} | markdown_preamble_trailing | fence | 0d1fd8531ff775156e1d6e9cb157b3759f5bb0b9fbdf51afe9265aa6b6e8a259 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
public import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
public import Mathlib.CategoryTheory.Sites.... | ```lean
by
constructor
· rw [Presheaf.isLocallySurjective_iff_whisker_forget (J := extensiveTopology C)]
exact fun h _ ↦
surjective_of_isLocallySurjective_sheaf_of_types (Functor.whiskerRight f (forget D)) h
· exact fun a ↦
Presheaf.isLocallySurjective_of_surjective _ _ (fun _ ↦ a _)
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Sites",
"family_id": "extensivetopology",
"file_id": "mathlib/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean",
"sample_id": "fffbd570a9bc968d3091c7ef16d1e7aecf0edf42a3c5fce61027b019b7b6b538"
} | train |
{
"chosen_proof_sha256": "741b44ba22accbbd70bb07bc169d65e562975536f8496b4d492a4c4b786fdd83",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7efc732133ef1a6bdb2e645313e4807fc5... | by
classical
lift N to {N : Submodule R P // N.FG} using hN
apply_fun (Module.fgSystem.equiv R P).symm.toLinearMap.rTensor M at eq
apply_fun directLimitLeft _ _ at eq
simp_rw [← LinearMap.rTensor_comp_apply, ← (LinearEquiv.eq_toLinearMap_symm_comp _ _).mpr
(Module.fgSystem.equiv_comp_of N), directLimitLef... | false | {
"chosen_tokens": 101,
"rejected_tokens": 106,
"token_jaccard": 0.928571,
"token_length_ratio": 1.049505
} | markdown_preamble_trailing | trailing | 0d9684eeb4b924392c1cf04413710a5e4332c159efdbd6a9769a41d394954fcb | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Colimit.Finiteness
public import Mathlib.LinearAlgebra.TensorProduct.DirectLimit
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Junyan Xu. Al... | by
classical
lift N to {N : Submodule R P // N.FG} using hN
apply_fun (Module.fgSystem.equiv R P).symm.toLinearMap.rTensor M at eq
apply_fun directLimitLeft _ _ at eq
simp_rw [← LinearMap.rTensor_comp_apply, ← (LinearEquiv.eq_toLinearMap_symm_comp _ _).mpr
(Module.fgSystem.equiv_comp_of N), directLimitLef... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Colimit",
"family_id": "submodule",
"file_id": "mathlib/Mathlib/Algebra/Colimit/TensorProduct.lean",
"sample_id": "6c6463defa0ee8680fe4b3b439ae453214eb49f285173f4fb331b4b3b6e3c669"
} | train |
{
"chosen_proof_sha256": "8563be0743a9370b287fc6eea19b728e9532ff758a843c12bb7a89f00678f9fd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [← cons_comp_cycleRange, map_perm, Matrix.vecCons]
simp [Units.smul_def] | true | {
"chosen_tokens": 18,
"rejected_tokens": 8,
"token_jaccard": 0.05,
"token_length_ratio": 0.444444
} | unsolved_goals | unsolved_goals | 0da39cfbc840edc20c3c262711a5f19d80075ff165598bd1d86f47a4d5a05a1a | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Alternating.Curry
public import Mathlib.GroupTheory.Perm.Fin
public import Mathlib.Data.Fin.Parity
Namespace:
AlternatingMap
Local context:
/-
... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "414ffd38143406d29403ac97d39105771ea951770a10afa1be9dcbaecdef4717",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Alternating",
"family_id": "map_insertnth",
"file_id": "mathlib/Mathlib/LinearAlgebra/Alternating/Uncurry/Fin.lean",
"sample_id": "23133e4cdbd3c18af59ef41f856739915fb4a88cdab3bfe7a080ab4e9717b499"
} | train |
{
"chosen_proof_sha256": "7aefead92a5663164517806ecfcb8b153d186d51071288da4cb411af6d57db41",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e479d953b87af298df6f7cefb47da40ed2... | by
ext v
simp [alternatizeUncurryFin_apply, ← map_insertNth]
variable {S : Type*} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] | true | {
"chosen_tokens": 32,
"rejected_tokens": 5,
"token_jaccard": 0.038462,
"token_length_ratio": 0.15625
} | wrong_namespace_qualification | wrong_namespace_qualification | 0db1c93e6172899c3e5cca6d143d76e4e9f9a0d2546c4cc028be9853ab6fc386 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Alternating.Curry
public import Mathlib.GroupTheory.Perm.Fin
public import Mathlib.Data.Fin.Parity
Namespace:
AlternatingMap
Local context:
/-
... | by
exact OProverMissing.missing_c860402497f1 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "7a492ef398182a1fda647614b5a804310589449f64085d07b3becbc74516bbc2",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Alternating",
"family_id": "alternatizeuncurryfin_curryleft",
"file_id": "mathlib/Mathlib/LinearAlgebra/Alternating/Uncurry/Fin.lean",
"sample_id": "c860402497f1b4dc3bdc93021a1e3e3c1ea1267241369ebbfdacfe5570fa2d4c"
} | train |
{
"chosen_proof_sha256": "7b7262505c003dbad9c3aa96a2764e75ade60fa912085a50b5740b06b8e012c5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
intro H
refine fermatLastTheoremThree_of_three_dvd_only_c (fun a b c hc ha hb ⟨x, hx⟩ hcoprime h ↦ ?_)
refine H a b c 1 (by simp [hc]) (fun hdvd ↦ ha ?_) (fun hdvd ↦ hb ?_) ?_ ?_ ?_
· rwa [← Ideal.norm_dvd_iff (hζ.prime_norm_toInteger_sub_one_of_prime_ne_two' (by decide)),
hζ.norm_toInteger_sub_one_of_... | false | {
"chosen_tokens": 144,
"rejected_tokens": 3,
"token_jaccard": 0.0625,
"token_length_ratio": 0.020833
} | by_exact_placeholder | exact? | 0db391223a64e0ebea4439977f721c7b19ac386e18bcef7133e4c6935c3471d3 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.NumberTheory.FLT.Basic
public import Mathlib.NumberTheory.NumberField.Cyclotomic.PID
public import Mathlib.NumberTheory.NumberField.Cyclotomic.Three
public imp... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/FLT",
"family_id": "fermatlasttheoremforthree_of_fermatlasttheoremthreegen",
"file_id": "mathlib/Mathlib/NumberTheory/FLT/Three.lean",
"sample_id": "a8d869c19b5e1d1d76d3c28a6dc188be2ec2a1a13841d22f5d203f9a92b23e39"
} | train |
{
"chosen_proof_sha256": "5710d934b0159d30d30972bfc5b7ddfd3eb8cd27b8c73a16ac7a346cdcff98c4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cd30468868389860cfba3fec091f109642... | by
rfl
variable (φ : A →ₐc[R] B) | false | {
"chosen_tokens": 15,
"rejected_tokens": 20,
"token_jaccard": 0.75,
"token_length_ratio": 1.333333
} | markdown_preamble_trailing | trailing | 0dd952312495f24c6b1343e1a91489e0d6647c9097cd6a24460b0899e08ab85f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Coalgebra.Hom
public import Mathlib.RingTheory.Bialgebra.Basic
Namespace:
BialgHom
Local context:
/-
Copyright (c) 2024 Amelia Livingston. All rig... | by
rfl
variable (φ : A →ₐc[R] B)
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Bialgebra",
"family_id": "toalghom_tolinearmap",
"file_id": "mathlib/Mathlib/RingTheory/Bialgebra/Hom.lean",
"sample_id": "519e1718b1b4803bcd063f65500a31317c48e550e0d65b418a9de8968712bef5"
} | train |
{
"chosen_proof_sha256": "f77254043cd1457d306845776290c0242267d27c173b195a06e4b8478f5ec277",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [Set.eq_univ_iff_forall]
intro x
rw [Set.mem_iUnion]
exact 𝒰.exists_eq x | true | {
"chosen_tokens": 20,
"rejected_tokens": 8,
"token_jaccard": 0.052632,
"token_length_ratio": 0.4
} | unsolved_goals | unsolved_goals | 0ddbd7c1276c002874d5a8f0a44004d9e5a50f9c034b2c486e6c8933ddcaac9b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Sites.MorphismProperty
public import Mathlib.CategoryTheory.MorphismProperty.Limits
Namespace:
AlgebraicGeometry.Scheme
Local context:
/-
C... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "8b989a86843ca5ae3710db04a49f1153f8ddd365f27f0d927511a31e0a169f4c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Cover",
"family_id": "cover",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Cover/MorphismProperty.lean",
"sample_id": "3d805ac3a07f605169cd17d09107c30e012f4d0f06e8571fe5f44b26cb0ba912"
} | train |
{
"chosen_proof_sha256": "f1c76c9c21cec19c82ec1c29aff36596a09a140cc79532bb60d64518174b61b0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6f9d3bcbb3b89d6ec55e4c066a99b47682... | by
convert! LinearMap.iSupIndep_map _ hv (iSupIndep_range_lsingle ι R R)
ext; simp [mem_span_singleton] | true | {
"chosen_tokens": 20,
"rejected_tokens": 5,
"token_jaccard": 0.090909,
"token_length_ratio": 0.25
} | circular_dependency | circular_dependency | 0df90959052c85b7da770b80286a359296eb4c0dc02f03b95af008fccb619e05 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fin.Tuple.Reflection
public import Mathlib.LinearAlgebra.Dual.Defs
public import Mathlib.LinearAlgebra.Finsupp.SumProd
public import Mathlib.LinearAlgebra... | by
exact LinearIndependent.iSupIndep_span_singleton | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "246bb52515751dbe99ec85d48ee54869e6232628148a728aeddd74895968305a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/LinearIndependent",
"family_id": "linearindependent",
"file_id": "mathlib/Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean",
"sample_id": "2f46fd8127f7a4707cce4dc2e25df1a6780c736d06d1699ff4e93a369a16eb74"
} | train |
{
"chosen_proof_sha256": "426d456c7ecb6c1bdb729eb16ba1456459d7dc0d59c4a4d027a77d78ce78493f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
simp_rw [← Nat.not_odd_iff_even]
constructor <;> intro h
· intro _ w ho
have := (w.three_le_chromaticNumber_of_odd_loop ho).trans h.chromaticNumber_le
norm_cast
· apply colorable_iff_forall_connectedComponents.2
intro c
obtain ⟨_, hv⟩ := c.nonempty_supp
use fun a ↦ Fin.ofNat 2 (c.connecte... | false | {
"chosen_tokens": 212,
"rejected_tokens": 2,
"token_jaccard": 0.013889,
"token_length_ratio": 0.009434
} | sorry_admit_sorryAx_proof_hole | sorry | 0dfdab81ce49e67057fbe7cd8872948ee26472252bc15c8d7db5de09649af203 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Combinatorics.SimpleGraph.Bipartite
public import Mathlib.Combinatorics.SimpleGraph.Circulant
public import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
p... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/SimpleGraph",
"family_id": "two_colorable_iff_forall_loop_even",
"file_id": "mathlib/Mathlib/Combinatorics/SimpleGraph/Coloring/Constructions.lean",
"sample_id": "56f2b093801a2a41875bf30e118429a5cc73ba8976a1adb97741fabeeedbed7b"
} | train |
{
"chosen_proof_sha256": "9fa7af5d48ddb6db2529f448df0e30aa81c953ad207f9980f8c0f3df88f4e48e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "421e4acb4fbfc76c1e4546e0b6c98c0caf... | by
intro V
have : ∀ x : V.1, ∃ (i : ι) (r : Γ(X, V.1)) (rU : Γ(X, U i)),
X.basicOpen r = X.basicOpen rU ∧ x.1 ∈ X.basicOpen r := by
intro ⟨x, hxV⟩
obtain ⟨i, hi⟩ := TopologicalSpace.Opens.mem_iSup.mp (hU.ge (Set.mem_univ x))
exact ⟨i, exists_basicOpen_le_affine_inter V.2 (U i).2 _ ⟨hxV, hi⟩⟩
cho... | true | {
"chosen_tokens": 413,
"rejected_tokens": 3,
"token_jaccard": 0.020833,
"token_length_ratio": 0.007264
} | unknown_identifier | unknown_identifier | 0e6113fde0cb11fe0b8f319c0a645ee92621c85bf22ef1c0c1383732c04168fa | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
public import Mathlib.AlgebraicGeometry.Properties
public import Mathlib.Tactic.DepRewrite
Namespace:
AlgebraicGeomet... | by
exact __oprover_missing_66542818e479 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1df6a5ea50ccab30bd25357159721a0a1062edd2e2b068cbc46c48a875124846",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/IdealSheaf",
"family_id": "le_of_isup_eq_top",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/Basic.lean",
"sample_id": "66542818e479c89882f725bfc7d9605b37b594439204873e38030237ce086893"
} | train |
{
"chosen_proof_sha256": "be2b6932dda4e0b3c56c94860ddf8d756e2f7715633872f9e4a77499f7decf69",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c0c4b5e544c8ef5b36acb2b62220ff6fea... | by
simp only [cmpUsing, Ordering.ite_eq_gt_distrib, if_false_right, and_true, if_false_left,
and_iff_right_iff_imp, reduceCtorEq]
exact fun hba hab ↦ (irrefl a) (_root_.trans hab hba) | true | {
"chosen_tokens": 36,
"rejected_tokens": 3,
"token_jaccard": 0.074074,
"token_length_ratio": 0.083333
} | circular_dependency | circular_dependency | 0f302e6100c5e60c2551c6013829b2a049adbd1cfccb2ef6ca9406a84d706fcb | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Ordering.Basic
public import Mathlib.Order.Defs.Unbundled
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Microsoft Corporation. All rights reser... | by
exact cmpUsing_eq_gt | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "8e309914c85f0f9214cffcb1369d67da64de4623628add3a1611ca74f64b7744",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Ordering",
"family_id": "cmpusing_eq_gt",
"file_id": "mathlib/Mathlib/Data/Ordering/Lemmas.lean",
"sample_id": "0700ede7788b015baae01a8e4aa3cd674212af76354c06e79abb4f1dca42ce86"
} | train |
{
"chosen_proof_sha256": "e934aed5577882b0e1846e82a16eff1d5d3efe027757ba1c1868dea83273eb47",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2d684b3c442a8ef210bbe9338a5474c025... | by
obtain ⟨T, hT₁, hT₂⟩ := has_injective_coseparator (op G) ((isSeparator_op_iff _).2 hG)
exact ⟨unop T, inferInstance, (isSeparator_unop_iff _).2 hT₂⟩ | false | {
"chosen_tokens": 42,
"rejected_tokens": 46,
"token_jaccard": 0.888889,
"token_length_ratio": 1.095238
} | missing_wrong_import | import | 0f3c8391133da095103e258e525a42d307bc35201e9b0693aac86734df32de84 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Abelian.Subobject
public import Mathlib.CategoryTheory.Limits.EssentiallySmall
public import Mathlib.CategoryTheory.Preadditive.Injective.Basic
... | import Mathlib.WrongImport
by
obtain ⟨T, hT₁, hT₂⟩ := has_injective_coseparator (op G) ((isSeparator_op_iff _).2 hG)
exact ⟨unop T, inferInstance, (isSeparator_unop_iff _).2 hT₂⟩ | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Generator",
"family_id": "has_projective_separator",
"file_id": "mathlib/Mathlib/CategoryTheory/Generator/Abelian.lean",
"sample_id": "a0442559c44ff5e1900294674d91b30edd8758751f47d7b1327b13d6af184bd6"
} | train |
{
"chosen_proof_sha256": "002c7d89e4368c0249a9cb4cfc7e30d967572004732ef0400a272686c72ce645",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "af3b95b0732be5f55a3324fb2e3c5ad5c5... | by
simpa [basis_repr_apply] using ιMultiDual_apply_nondiag R n b t s hst.symm | true | {
"chosen_tokens": 16,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.1875
} | unknown_identifier | unknown_identifier | 0ff750a82ba5bd53fc54a12e80f647294feaafce2008f9ddcd471768014fccbd | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.ExteriorPower.Basic
public import Mathlib.LinearAlgebra.ExteriorPower.Pairing
public import Mathlib.RingTheory.Finiteness.Subalgebra
public impor... | by
exact __oprover_missing_8c1abff1d102 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d0800bfe131a7b35e86e7a8d89ea4dcb0eb72c8916d94f85532caed9306d0fc7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/ExteriorPower",
"family_id": "basis_repr_ne",
"file_id": "mathlib/Mathlib/LinearAlgebra/ExteriorPower/Basis.lean",
"sample_id": "8c1abff1d102ab8663f817566b5f1e0679aded535e3b61cc03bd6dcfd4532f76"
} | train |
{
"chosen_proof_sha256": "05a30ac01794d34e26abf0bb89eb49ff18aa58a6481d8190121651504b0314f8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "956162824dd70572aa71bea9f0c8aee184... | by
rcases eq_or_ne n 0 with rfl | hn₀; · simp
rcases eq_or_ne p 1 with rfl | hp₁; · simp
simpa [*] using pow_dvd_iff_le_padicValNat (k := 1) hp₁ hn₀ |>.symm |>.not | true | {
"chosen_tokens": 48,
"rejected_tokens": 5,
"token_jaccard": 0.058824,
"token_length_ratio": 0.104167
} | wrong_namespace_qualification | wrong_namespace_qualification | 101b9f31af38d3bb25030b452279db35540d353bee016b62e03a68ff0fb175e3 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.MaxPowDiv
public import Mathlib.RingTheory.Multiplicity
public import Mathlib.Data.Nat.Factors
Namespace:
padicValNat
Local context:
/-
Copyright (c... | by
exact OProverMissing.missing_b28f2d047f3f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1c902c86f1e0b64c65fd4be7eca2c3e35b5812cc9a34f1b5e6a751b13e0c94d3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Padics",
"family_id": "eq_zero_iff",
"file_id": "mathlib/Mathlib/NumberTheory/Padics/PadicVal/Defs.lean",
"sample_id": "b28f2d047f3f02792ce315aca654c65c6b846562c70579c8ee832efe4c79dcf3"
} | train |
{
"chosen_proof_sha256": "335297b55a581bb6fe6dcb38f704182637c77f8f85b5c081f025c29aaa5ebb61",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [← Option.some_inj]
simp [finSuccAboveOrderIso, finSuccAboveEquiv, OrderIso.symm] | true | {
"chosen_tokens": 18,
"rejected_tokens": 8,
"token_jaccard": 0.05,
"token_length_ratio": 0.444444
} | unsolved_goals | unsolved_goals | 102f4f29c2e5e7606343e16a8f2e16caeb3bda5ce52f3a2743cde60da4663a62 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Logic.Equiv.Fin.Basic
public import Mathlib.Order.Fin.Basic
public import Mathlib.Order.PiLex
public import Mathlib.... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "26272c5b978e7bc20d7fbe034534b0d95cfe66a7458381f8324928d807d8d750",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Fin",
"family_id": "finsuccaboveorderiso_symm_apply_last",
"file_id": "mathlib/Mathlib/Order/Fin/Tuple.lean",
"sample_id": "c092795e5fd0e22d540c502fd1b4fd017f05beade7817be0597a69f2c004e7d6"
} | train |
{
"chosen_proof_sha256": "60c914c7101fd19d31a3d057e5fece308a5e2edc08549f3280fae32cf5fabb05",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c3ebc26cd68e8f84cca3c5c39ecb58bd11... | by
dsimp only [mem_setOf_eq]
rw [← Submodule.map_top, ← iSup_range_single, Submodule.map_iSup]
congr; funext i
rw [← LinearMap.range_comp, diagonal_comp_single, ← range_smul'] | true | {
"chosen_tokens": 36,
"rejected_tokens": 5,
"token_jaccard": 0.08,
"token_length_ratio": 0.138889
} | wrong_namespace_qualification | wrong_namespace_qualification | 103c790c5979157196aa0b654513e4ee65a14337f977ea4a4938c113b6f058c1 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Dimension.LinearMap
public import Mathlib.LinearAlgebra.Matrix.ToLin
Namespace:
Matrix
Local context:
/-
Copyright (c) 2019 Johannes Hölzl. All... | by
exact OProverMissing.missing_cf2066a6aac7 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "e9673935d7b2cea4000578bcc71a46f2a8ae67be458f335b8f76d526c1ddb947",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Matrix",
"family_id": "range_diagonal",
"file_id": "mathlib/Mathlib/LinearAlgebra/Matrix/Diagonal.lean",
"sample_id": "cf2066a6aac720a3b5d609e8b3fe1b26d84bc8deb53c0ec1b5795944dd568539"
} | train |
{
"chosen_proof_sha256": "41ba997d60350b8410456fab7fd86644a15e80166bdc8dbee3ebfacac2920661",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
induction l with
| nil => simp only [List.not_mem_nil, false_and, exists_false] at hl
| cons x l ih =>
simp only [List.mem_cons, exists_eq_or_imp] at hl
rcases hl with h | hal
· simpa only [List.map_cons, List.prod_cons] using I.mul_mem_right _ _ h
· simpa only [List.map_cons, List.prod_cons] u... | true | {
"chosen_tokens": 84,
"rejected_tokens": 8,
"token_jaccard": 0.023256,
"token_length_ratio": 0.095238
} | unsolved_goals | unsolved_goals | 104721a19b73f9cbc1177310e6714afccb50a3e37cc85f233cec7bd70fc5bc15 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Congruence.BigOperators
public import Mathlib.RingTheory.TwoSidedIdeal.Basic
Namespace:
TwoSidedIdeal
Local context:
/-
Copyright (c) 2024 Jujian ... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "713e8111a6705ca467e58514aca0c28acd47aab4443715540f78cb003c041f78",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/TwoSidedIdeal",
"family_id": "listprod_mem",
"file_id": "mathlib/Mathlib/RingTheory/TwoSidedIdeal/BigOperators.lean",
"sample_id": "ab793cebc21f139d2eab524640ed552866c6f6bd28b0f8c110edadda18cfb9c9"
} | train |
{
"chosen_proof_sha256": "741b44ba22accbbd70bb07bc169d65e562975536f8496b4d492a4c4b786fdd83",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "92d3d3be6de007f629396e1b5b63d1e606... | by
classical
lift N to {N : Submodule R P // N.FG} using hN
apply_fun (Module.fgSystem.equiv R P).symm.toLinearMap.rTensor M at eq
apply_fun directLimitLeft _ _ at eq
simp_rw [← LinearMap.rTensor_comp_apply, ← (LinearEquiv.eq_toLinearMap_symm_comp _ _).mpr
(Module.fgSystem.equiv_comp_of N), directLimitLef... | true | {
"chosen_tokens": 101,
"rejected_tokens": 5,
"token_jaccard": 0.055556,
"token_length_ratio": 0.049505
} | wrong_namespace_qualification | wrong_namespace_qualification | 106dcfa1524001859755232ec9a2c321a174a4198acec8fe8f17060acc300c8b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Colimit.Finiteness
public import Mathlib.LinearAlgebra.TensorProduct.DirectLimit
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Junyan Xu. Al... | by
exact OProverMissing.missing_6c6463defa0e | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d2feac7966fa052974ec08df95436627672438dcdc40f075ce1a5862c3d0834a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Colimit",
"family_id": "submodule",
"file_id": "mathlib/Mathlib/Algebra/Colimit/TensorProduct.lean",
"sample_id": "6c6463defa0ee8680fe4b3b439ae453214eb49f285173f4fb331b4b3b6e3c669"
} | train |
{
"chosen_proof_sha256": "238c36aa3d5b83b0f47d94bdb4d2f15b92abc10f86ad4c7efd31a50fddc87c83",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f5318c3795c3dc2787b18f8ea59670daa7... | by
rw [IsTranscendenceBasis, maximal_iff, Subtype.range_val]; rfl | true | {
"chosen_tokens": 13,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.230769
} | unknown_identifier | unknown_identifier | 10e4df5eafaf1e015c2551b512e073b7473e6bced07b93f99c50cb165d59c14d | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MvPolynomial.CommRing
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as ... | by
exact __oprover_missing_d594da141648 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "741ac458cf0fb72b5cdcc8beb0e6c5f9a9cf275a0f754b54b2d5a91028025efa",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/AlgebraicIndependent",
"family_id": "istranscendencebasis_iff_maximal",
"file_id": "mathlib/Mathlib/RingTheory/AlgebraicIndependent/Defs.lean",
"sample_id": "d594da1416489b60a7466e1d8f39b122bd55afcaa513107653ab63deea58dfe7"
} | train |
{
"chosen_proof_sha256": "054bd373a1fa3d5f15bd0fe1ba1c34023be8febf1ced40b818a6af0feb340b77",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6e36103246bee59d4bc6e678583d099025... | by
simp only [dp_def]
rw [mkAlgHom_rel (A := MvPolynomial (ℕ × M) R) R Rel.add, map_sum,
Finset.sum_congr rfl (fun k _ ↦ by rw [_root_.map_mul])] | false | {
"chosen_tokens": 45,
"rejected_tokens": 50,
"token_jaccard": 0.861111,
"token_length_ratio": 1.111111
} | markdown_preamble_trailing | preamble | 10f7cd4dd7f69120be1fc6fe09923db551ae345f65e7adbac643f35c79b467ef | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MvPolynomial.Eval
public import Mathlib.Algebra.RingQuot
public import Mathlib.RingTheory.DividedPowers.Basic
Namespace:
DividedPowerAlgebra
Local co... | Here is the proof:
by
simp only [dp_def]
rw [mkAlgHom_rel (A := MvPolynomial (ℕ × M) R) R Rel.add, map_sum,
Finset.sum_congr rfl (fun k _ ↦ by rw [_root_.map_mul])] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/DividedPowerAlgebra",
"family_id": "dp_add",
"file_id": "mathlib/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean",
"sample_id": "1f3bfea624fb86d6c8b755b05ec1c6fe8d68cd178710d19c21a20e4947546990"
} | train |
{
"chosen_proof_sha256": "ff1ae62e27377baaefea54f50e782d5e8647a507a59b2f78b75ee566cf2b6bd5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0e4d7ccbd372ec11fb03813b285ac23f53... | by
rw [iff_comp_injective]
intro B _ _ I hI f₁ f₂ e
ext x
have : f₁ x - f₂ x ∈ I := by
simpa [Ideal.Quotient.mk_eq_mk_iff_sub_mem] using AlgHom.congr_fun e x
have := Polynomial.eval_add_of_sq_eq_zero ((minpoly K x).map (algebraMap K B)) (f₂ x)
(f₁ x - f₂ x) (show (f₁ x - f₂ x) ^ 2 ∈ ⊥ from hI ▸ Ideal.... | true | {
"chosen_tokens": 187,
"rejected_tokens": 5,
"token_jaccard": 0.027027,
"token_length_ratio": 0.026738
} | wrong_namespace_qualification | wrong_namespace_qualification | 11153394b2fd3826377a26f1f7235603afc3f98ca5e0f5c7b795565ee3da6084 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.PurelyInseparable.Basic
public import Mathlib.RingTheory.Artinian.Ring
public import Mathlib.RingTheory.Unramified.Finite
public import Mathlib.Rin... | by
exact OProverMissing.missing_5c3559b3cf2e | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "eabd019117069e68452051f85a3dd05fc138657a1ad98f95e1f18b007985ebb6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Unramified",
"family_id": "of_isseparable",
"file_id": "mathlib/Mathlib/RingTheory/Unramified/Field.lean",
"sample_id": "5c3559b3cf2ee9a241bb81b7e23213b68dd9f8307cc2252bb99a753e0c04c076"
} | train |
{
"chosen_proof_sha256": "47f42a3f39b92164ba24c6342d141209f44e0c15b9d67ad24b48bf7df68f4b85",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "71edde365de57e77fa976ba82cd9d031cd... | by
have := degree_lt_wf.min_mem _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)
simp_all [uniformizingPolynomial] | false | {
"chosen_tokens": 15,
"rejected_tokens": 20,
"token_jaccard": 0.789474,
"token_length_ratio": 1.333333
} | markdown_preamble_trailing | trailing | 111f39b0ab4fcde985d09b101e8807657290d9354b21627c95d755f6e40945d4 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.Finite.Valuation
public import Mathlib.NumberTheory.FunctionField
public import Mathlib.RingTheory.Valuation.Discrete.Basic
Namespace:
RatFunc
Lo... | by
have := degree_lt_wf.min_mem _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)
simp_all [uniformizingPolynomial]
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/RatFunc",
"family_id": "uniformizingpolynomial_ne_zero",
"file_id": "mathlib/Mathlib/NumberTheory/RatFunc/Ostrowski.lean",
"sample_id": "2293cf123c21972c1d2af5e35604c06efbc61aa089ba31c3f82c50d912dfbf35"
} | train |
{
"chosen_proof_sha256": "82d206dfe82328407b70971415868cac0e9eb31b2da76433329c285b45739547",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
apply Linear.comp_smul | true | {
"chosen_tokens": 5,
"rejected_tokens": 8,
"token_jaccard": 0.090909,
"token_length_ratio": 1.6
} | unsolved_goals | unsolved_goals | 113bf5fcc78d83b283b1abd59164a20a5b7d6f38b4351b823bdf98f6140cf016 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Algebra.Module.TransferInstance
public import Mathlib.Algebra.Group.Invertible.Defs
public import Mathlib.Algebra.Mo... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "cf1dfd0c153f9c26912a120a41a2059d682480ccdab5e991e58d2753dc498894",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Linear",
"family_id": "comp_units_smul",
"file_id": "mathlib/Mathlib/CategoryTheory/Linear/Basic.lean",
"sample_id": "f2d217efd8dc67e5b8a5a62c60104ed7db1fe692e856a2299afd32cbd082efda"
} | train |
{
"chosen_proof_sha256": "0be2d8fff9f4e0668c3d36339336ba1af35cd1dd0eb1d09d433511d22d232ffa",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp [← h₀.ordConnected_iff_of_bdd (OrderBot.bddBelow I) h₂, ordConnected_iff_disjoint_Ioo_empty] | true | {
"chosen_tokens": 19,
"rejected_tokens": 8,
"token_jaccard": 0.043478,
"token_length_ratio": 0.421053
} | unsolved_goals | unsolved_goals | 11d9a304193cb818fdea7f52225abf8e7b39b0f3f55e11ef2325d44d8810d343 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Int.ConditionallyCompleteOrder
public import Mathlib.Data.Int.Interval
public import Mathlib.Data.Int.SuccPred
public import Mathlib.Order.Lattice.Nat
Na... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "227e47caac3491472fbc0a591a4991f7bc68b507907afe0ed6bc45240bc4a7d5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Interval",
"family_id": "set",
"file_id": "mathlib/Mathlib/Order/Interval/Set/OrdConnectedLinear.lean",
"sample_id": "43e80cf9725299057626e124ada0c7fa7af7a66a7d45d1cc2cc1eef0f94a344c"
} | train |
{
"chosen_proof_sha256": "e9b602b11908c749a9de1394766214188bf5ab4658bf596bbcd37cd55e75b8a2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [← S.toNatSubmodule_toAddSubmonoid, ← Submodule.fg_iff_addSubmonoid_fg,
Submodule.fg_iff_exists_fin_linearMap]
exact exists_congr fun n => ⟨fun ⟨f, hf⟩ => ⟨f, hf ▸ LinearMap.range_toAddSubmonoid _⟩,
fun ⟨f, hf⟩ => ⟨f.toNatLinearMap, Submodule.toAddSubmonoid_inj.mp <|
hf ▸ LinearMap.range_toAddSu... | false | {
"chosen_tokens": 68,
"rejected_tokens": 2,
"token_jaccard": 0.032258,
"token_length_ratio": 0.029412
} | sorry_admit_sorryAx_proof_hole | admit | 11e9b3779fc382981c845ea6beaa9b846ec218c67277ebbf1f72ccb81e2f4499 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Module.Congruence.Defs
public import Mathlib.LinearAlgebra.Basis.Cardinality
public import Mathlib.LinearAlgebra.DFinsupp
public import Mathlib.LinearA... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Finiteness",
"family_id": "addsubmonoid",
"file_id": "mathlib/Mathlib/RingTheory/Finiteness/Cardinality.lean",
"sample_id": "e2bfab7a5a1b1912f82cb1a9e10696dbb413a79fc583a7da1b4cad05b058c943"
} | train |
{
"chosen_proof_sha256": "cf1663d691843ffb2d1c18b62f9a504b9533198687278010c9c58b7251f1dcfb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "52db5e0abeb3da4c111971ea0930014c39... | by
dsimp only [LiftR, LiftR']
apply exists_iff_exists_of_mono F (f' rr _ _) (g' rr _ _)
· ext i ⟨x, _⟩ : 2
cases i <;> rfl
· intros
simp +unfoldPartialApp only [map_map, TypeVec.comp]
apply iff_of_eq -- Switch to `eq` so we can use `ext`
congr <;> ext i ⟨x, _⟩ <;> cases i <;> rfl | true | {
"chosen_tokens": 88,
"rejected_tokens": 3,
"token_jaccard": 0.019608,
"token_length_ratio": 0.034091
} | unknown_identifier | unknown_identifier | 123694f69ca9d7aa457964871adf8f032cb3113d42db59793c500fd1e7849745 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fin.Fin2
public import Mathlib.Data.TypeVec
public import Mathlib.Logic.Equiv.Defs
Namespace:
MvFunctor
Local context:
/-
Copyright (c) 2018 Jeremy Avig... | by
exact __oprover_missing_8b1fcb5d375f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "760005ab7ad9dcc50777246fcb558f4d3a74328ebf40f905fea69392731e06fc",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Control/Functor",
"family_id": "liftr_rellast_iff",
"file_id": "mathlib/Mathlib/Control/Functor/Multivariate.lean",
"sample_id": "8b1fcb5d375f7d0c76ac8b5dc5be9785fa4e3630fe7326a206b1b90b1962024a"
} | train |
{
"chosen_proof_sha256": "732ed3f932374a610e819e8e805afcfdf95b627321f8ae3cc0670b9127888807",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
obtain ⟨n, rfl⟩ | ⟨n, rfl⟩ := Int.even_or_odd n
· simp [zpow_add, ← mul_zpow, Int.negOnePow_even _ (Even.add_self n)]
· rw [Int.negOnePow_odd _ (by exact odd_two_mul_add_one n)]
simp [Units.smul_def, zpow_add, Int.two_mul, ← mul_zpow] | false | {
"chosen_tokens": 66,
"rejected_tokens": 3,
"token_jaccard": 0.060606,
"token_length_ratio": 0.045455
} | by_exact_placeholder | placeholder | 1238e57de9bec4706c122e43ebc803cddbb30dcae8333b00b62191a945d561b2 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Center.Preadditive
public import Mathlib.Algebra.Ring.NegOnePow
Namespace:
CategoryTheory.CatCenter
Local context:
/-
Copyright (c) 2025 Joël ... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Center",
"family_id": "app_neg_one_zpow",
"file_id": "mathlib/Mathlib/CategoryTheory/Center/NegOnePow.lean",
"sample_id": "5b7bc29de3804ac7bf45079e461f320c398aef01effbb7aab20e4f7fde575a9c"
} | train |
{
"chosen_proof_sha256": "36011e8dfa494a26101903f706aa37335eb673289a71da56c05014c7180fb544",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
induction n generalizing g with
| zero =>
funext (x : Fin 1)
simp [diagonalSuccIsoTensorTrivial, diagonalOneIsoLeftRegular, Subsingleton.elim x 0,
ofMulAction_V, types_tensorObj_def, types_tensorUnit_def]
| succ n hn =>
funext x
induction x using Fin.cases with
| zero => simp; rfl
... | false | {
"chosen_tokens": 91,
"rejected_tokens": 2,
"token_jaccard": 0.022222,
"token_length_ratio": 0.021978
} | by_exact_placeholder | by? | 12ad3b5272bffd369a46fe99108f471e7d23d4015468668e32a31487e34afd36 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Fin
public import Mathlib.CategoryTheory.Monoidal.Linear
public import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory
public import... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Action",
"family_id": "diagonalsuccisotensortrivial_inv_hom_apply",
"file_id": "mathlib/Mathlib/CategoryTheory/Action/Monoidal.lean",
"sample_id": "e1e83a15a2543ab2c7734d45b0801536ee9f8edb36b69899defb9d3f1b0cfbea"
} | train |
{
"chosen_proof_sha256": "f3a8fc7a4995882f5a643c316d6ec43e868e48edcba54745dfaa0649321d1a67",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "87307a21217d56836522ac81f517e1c61d... | by
have hb' : b.Dom := Part.dom_iff_mem.mpr ⟨a.get ha, hb⟩
rwa [← eq_get_iff_mem hb', eq_iff_of_dom ha hb'] at hb | false | {
"chosen_tokens": 33,
"rejected_tokens": 38,
"token_jaccard": 0.862069,
"token_length_ratio": 1.151515
} | markdown_preamble_trailing | trailing | 12b5ce0e040d5ad4af11f2224d32439feaac81dee4904790968fd1ebb58a4cfa | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Notation.Defs
public import Mathlib.Data.Set.Subsingleton
public import Mathlib.Logic.Equiv.Defs
Namespace:
Part
Local context:
/-
Copyright (c) 2017... | by
have hb' : b.Dom := Part.dom_iff_mem.mpr ⟨a.get ha, hb⟩
rwa [← eq_get_iff_mem hb', eq_iff_of_dom ha hb'] at hb
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data",
"family_id": "eq_of_mem",
"file_id": "mathlib/Mathlib/Data/Part.lean",
"sample_id": "28eb45ede6284ff2c969d0ba34870b558934c40303d056e5740787b98d7dcd28"
} | train |
{
"chosen_proof_sha256": "5bd7c7167db7f215623bed49eb0f0085e5528c13934e7559ad6540e992b06a8a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b950a5480671beab0f3b98bc2ebc1a58c3... | by
letI := h𝕜.rclike
exact hs.mono_field | true | {
"chosen_tokens": 11,
"rejected_tokens": 5,
"token_jaccard": 0.25,
"token_length_ratio": 0.454545
} | wrong_namespace_qualification | wrong_namespace_qualification | 12da3d0165d4f2605052966862ff9ecba9709b056f9679cc74d75845448b03ca | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.RCLike.Basic
public import Mathlib.Topology.Instances.RealVectorSpace
public import Mathlib.Analysis.Calculus.TangentCone.Real
Namespace:
(root)
Loc... | by
exact OProverMissing.missing_2e71596933c2 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d43b23fba541f580915d0cbf2955ba12b24bd443cf53c79df4cb87516f00df92",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/RCLike",
"family_id": "uniquediffwithinat",
"file_id": "mathlib/Mathlib/Analysis/RCLike/TangentCone.lean",
"sample_id": "2e71596933c22369bea1ce7d19358deda1acb57a1b511e60283fae72b0ed1051"
} | train |
{
"chosen_proof_sha256": "5b8c55401ccfdd0326db1a35194aa3affc02ada3be7e2075a756a2a2bba77cb4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "499fbc700dadc344361ad4af367c82f433... | by
rw [isCoskeletal_iff]
exact isRightKanExtension_iff_isIso ((coskAdj n).unit.app X)
((coskAdj n).counit.app _) (𝟙 _) ((coskAdj n).left_triangle_components X) | true | {
"chosen_tokens": 42,
"rejected_tokens": 5,
"token_jaccard": 0.142857,
"token_length_ratio": 0.119048
} | wrong_namespace_qualification | wrong_namespace_qualification | 134f85183e27480fb30c0967287e04534edbed39b6600d743b5aa5e4f98543ae | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicTopology.SimplicialObject.Basic
public import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction
public import Mathlib.CategoryTheory.Functor.KanE... | by
exact OProverMissing.missing_fad9f1ea8d66 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ef01acce0d3b29fbce563bc5624051edb1cac56e5b580d086e3141e806c29aa6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/SimplicialObject",
"family_id": "iscoskeletal_iff_isiso",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplicialObject/Coskeletal.lean",
"sample_id": "fad9f1ea8d6654621705b02879012779787dbbf5d7900dc540f712fce41505c9"
} | train |
{
"chosen_proof_sha256": "054bd373a1fa3d5f15bd0fe1ba1c34023be8febf1ced40b818a6af0feb340b77",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp only [dp_def]
rw [mkAlgHom_rel (A := MvPolynomial (ℕ × M) R) R Rel.add, map_sum,
Finset.sum_congr rfl (fun k _ ↦ by rw [_root_.map_mul])] | true | {
"chosen_tokens": 45,
"rejected_tokens": 8,
"token_jaccard": 0.055556,
"token_length_ratio": 0.177778
} | unsolved_goals | unsolved_goals | 136a283650b81dc17a88ee08c13f7cf55dc2feffafa957322ff8bbbe501b8b07 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MvPolynomial.Eval
public import Mathlib.Algebra.RingQuot
public import Mathlib.RingTheory.DividedPowers.Basic
Namespace:
DividedPowerAlgebra
Local co... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "8411547c3325da7e440e5445c9a3eeb534ad0c850ae122ee86bc61793e60967f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/DividedPowerAlgebra",
"family_id": "dp_add",
"file_id": "mathlib/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean",
"sample_id": "1f3bfea624fb86d6c8b755b05ec1c6fe8d68cd178710d19c21a20e4947546990"
} | train |
{
"chosen_proof_sha256": "58196917b1a24b2be60c95c2b5d60e57c3e3c717c1a775b2f741e63bbfde64ca",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "97ab846b1a668a1e9d99cd8860bda5a20a... | by
have h := lintegral_condCDF ((dirac Unit.unit).prod μ) x
simpa only [fst_prod, prod_prod, measure_univ, one_mul, lintegral_dirac] using! h | true | {
"chosen_tokens": 33,
"rejected_tokens": 2,
"token_jaccard": 0.037037,
"token_length_ratio": 0.060606
} | invalid_tactic | invalid_tactic | 13986a6c4fe41e20089c2362cfe5dc3dd9de737a83594672fa21f71ee4ef1fe8 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Probability.Kernel.Disintegration.CondCDF
public import Mathlib.Tactic.CrossRefAttribute
Namespace:
ProbabilityTheory
Local context:
/-
Copyright (c) 2023 Ré... | by
__oprover_invalid_tactic_41dfd3889602 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "a610e84eb9224551be0aa8be53bc3bee6efd049bdfc7e68b15ca1fe7c4347e26",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability",
"family_id": "ofreal_cdf",
"file_id": "mathlib/Mathlib/Probability/CDF.lean",
"sample_id": "41dfd38896021be841aea499e4a561d7890d2e4973f5f2dd9b0701af439b41dd"
} | train |
{
"chosen_proof_sha256": "d0f64a8fe631e03ccedfddafbaffe13122e5547990701f69b60f4ed018241b86",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3c4a724f529b48895bc878479987d95522... | by
ext f : 1
apply (Matrix.stdBasis _ _ _).ext
intro ⟨i, j⟩
simp only [LinearMap.coe_comp, LinearMap.coe_mk, AddHom.coe_mk, Function.comp_apply, map_sum,
map_smul, stdBasis_eq_single, LinearMap.coe_sum, Finset.sum_apply,
LinearMap.smul_apply, LinearMap.id_coe, id_eq]
ext k l
simp [sum_apply, Matrix.... | true | {
"chosen_tokens": 84,
"rejected_tokens": 5,
"token_jaccard": 0.042553,
"token_length_ratio": 0.059524
} | wrong_namespace_qualification | wrong_namespace_qualification | 13e26e708e6845ee7441fb127715c21882228aa2b8dcba9ce2e6ab0b196599c3 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Azumaya.Defs
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Yunzhou Xie. All righ... | by
exact OProverMissing.missing_fbfa2ca2104f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "72a57f4ee1b877a803a09a0288ff6deccbb80cf8b538686339ee535cd1b804a7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Azumaya",
"family_id": "alghom",
"file_id": "mathlib/Mathlib/Algebra/Azumaya/Matrix.lean",
"sample_id": "fbfa2ca2104fb09021e16e7f3542762a6ce83fb6ca653dc82a4df2bcb75e0375"
} | train |
{
"chosen_proof_sha256": "bdd21497f8131a5f5a6c2c2e48cfeb6b69583b6be9858ce150b32d0a5afe6448",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [← Representation.IntertwiningMap.toLinearMap_apply, d_eq]; rfl | false | {
"chosen_tokens": 14,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.214286
} | by_exact_placeholder | exact? | 14a196523e8eee2c5a254a4d0ea875d01e109b19f7641888f9fac3859f033a2c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Category.ModuleCat.Adjunctions
public import Mathlib.AlgebraicTopology.ExtraDegeneracy
public import Mathlib.CategoryTheory.Abelian.Ext
public import M... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RepresentationTheory/Homological",
"family_id": "d_apply",
"file_id": "mathlib/Mathlib/RepresentationTheory/Homological/Resolution.lean",
"sample_id": "2f1e37c134e39703c11252b834c1e076724924538d8af83dbd0bc1c8dab12e37"
} | train |
{
"chosen_proof_sha256": "7e80ec87feb47441a23ea0489b04ea9d8954472b614c08ce067659d50bc27e69",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "38130104b915c0d587ef920e495ec4fb83... | by
rw [hasBasis_self, (hasBasis_nhdsSet _).forall_iff (by grind)]
intro U ⟨hU, h_KU⟩
obtain ⟨L, hL, hKL, hLU⟩ := exists_compact_between hK hU h_KU
exact ⟨L, by rwa [← subset_interior_iff_mem_nhdsSet], hL, hLU⟩ | true | {
"chosen_tokens": 53,
"rejected_tokens": 5,
"token_jaccard": 0.090909,
"token_length_ratio": 0.09434
} | wrong_namespace_qualification | wrong_namespace_qualification | 1557b918bebf25efc27a3f1b3820e53c93492fdf3d0e3a4430dd2097015b629e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.Compactness.Compact
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as... | by
exact OProverMissing.missing_ec0e8191257e | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f776b64abd4ab228e7bc3fa509451301acd66378d871d6391ea3729236523b9d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Compactness",
"family_id": "iscompact",
"file_id": "mathlib/Mathlib/Topology/Compactness/LocallyCompact.lean",
"sample_id": "ec0e8191257eb538c96b568fc34a540eaf43a4e996b7fbafb697f19a5d8137e8"
} | train |
{
"chosen_proof_sha256": "fda78651fadb9c3fc00301a4f0a7e0c199693b04a98a2e407d36dcf722f12255",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "50baa089f2e40a96f16f849fe21aaf0cb1... | by
let e := Fintype.equivFin ι
obtain ⟨i₀, i₁, ne, h⟩ := h.exists_approx_aux (Fintype.card ι) hε hb fun x y ↦ A x (e.symm y)
refine ⟨i₀, i₁, ne, fun k ↦ ?_⟩
convert! h (e k) <;> simp only [e.symm_apply_apply] | true | {
"chosen_tokens": 78,
"rejected_tokens": 2,
"token_jaccard": 0.022727,
"token_length_ratio": 0.025641
} | invalid_tactic | invalid_tactic | 15a0c8eda23c1afbc71fd926b46f04b76f2ac80f3afffc3668c0a8d69ca39f2e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Real.Basic
public import Mathlib.Combinatorics.Pigeonhole
public import Mathlib.Algebra.Order.AbsoluteValue.Euclidean
Namespace:
AbsoluteValue.IsAdmissib... | by
__oprover_invalid_tactic_fc015e197e7e | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e45111bb0b93cea36e589f696c3c74e8b459c24eba96686ecf4c7310d6898d34",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/ClassNumber",
"family_id": "exists_approx",
"file_id": "mathlib/Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean",
"sample_id": "fc015e197e7e22d0acabad316890661898d1a5fa61e903e26832dbe043d9fa2c"
} | train |
{
"chosen_proof_sha256": "7f711a73112f71461efde233fe8c7cca840501a653c0391902a7be231d8cd075",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f0926250948255762b99b2b1a18aec1ff0... | by
induction l with
| nil => simp
| cons head tail ih => simp [← ih, smul_mul_smul_comm, pow_succ'] | true | {
"chosen_tokens": 23,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.130435
} | unknown_identifier | unknown_identifier | 15adc09976b9d60d2ce9fd6d471b2cbd8ead9b311b77ef0e696213326487a4fa | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Finprod
public import Mathlib.Algebra.GroupWithZero.Action.Defs
public import Mathlib.Algebra.Order.Group.Multiset
public import Mathlib.D... | by
exact __oprover_missing_6a93f461afc6 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "70f686ce2128aff900747b363091f223e238069ba1df9cbcf34186c6033e3bea",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Algebra/BigOperators",
"family_id": "smul_prod",
"file_id": "mathlib/Mathlib/Algebra/BigOperators/GroupWithZero/Action.lean",
"sample_id": "6a93f461afc68c4564b026d64e62d3228a07b26e1dd6066974bf51a8514702ac"
} | train |
{
"chosen_proof_sha256": "0a479f8dae478b3bb9603d60f313d74028b2da15cce1a50538dcbe628c6b865b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
refine (QuadraticMap.associated_rightInverse R).injective ?_
ext m₁ m₂ m₁' m₂' m₁'' m₂''
simp only [associated_tmul, QuadraticMap.associated_comp]
exact mul_assoc _ _ _ | false | {
"chosen_tokens": 45,
"rejected_tokens": 3,
"token_jaccard": 0.12,
"token_length_ratio": 0.066667
} | by_exact_placeholder | exact? | 15d4793de2886ba8abb09fbae0ec61951e6f8040568683fe4191bd5e3aeddb43 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
public import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
Namespace:
QuadraticForm
Local context:
/-
Copyrigh... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/QuadraticForm",
"family_id": "tmul_comp_tensorassoc",
"file_id": "mathlib/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean",
"sample_id": "2b9e2ca58e134ad4578da49d74b391a4ef05b73b213f399f3d3257947965fc7d"
} | train |
{
"chosen_proof_sha256": "79e94f24c2f6775d53dff748682f833bc2540c933d9b7a7fd1509ddae3499adc",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6397f9eeed335e5c4d5285b3d24ecfded6... | by
rw [← ofSupportFinite_coe (f := a) (hf := Set.toFinite _),
prod_congr (fun r _ ↦ rfl), ← coeff_linearCombination_X_pow]
simp [linearCombination_apply, sum_of_support_subset (s := Finset.univ)] | true | {
"chosen_tokens": 44,
"rejected_tokens": 5,
"token_jaccard": 0.0625,
"token_length_ratio": 0.113636
} | wrong_namespace_qualification | wrong_namespace_qualification | 161bf6cc6a74be59df96303ddd5bf9baac644f0485c1e3a30d9221fb2bc03556 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MvPolynomial.Basic
public import Mathlib.Data.Nat.Choose.Multinomial
Namespace:
MvPolynomial
Local context:
/-
Copyright (c) 2026 Antoine Chambert-Lo... | by
exact OProverMissing.missing_0fc65d800093 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "5c13b1a19a8f2289b412aabb782fb06dae214f3342ccbdeef2bf8f64b0ce43bf",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/MvPolynomial",
"family_id": "coeff_linearcombination_x_pow_of_fintype",
"file_id": "mathlib/Mathlib/Algebra/MvPolynomial/Coeff.lean",
"sample_id": "0fc65d8000935f5344eab88fab4e2c3713d4a97efe4538dc3faf285b863f943a"
} | train |
{
"chosen_proof_sha256": "5e809a8a5b6dc52f86458c7ec1ba0de189d9ba286d3abe6189f2eb23181c4aa7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7f909b761cf539f37fcd85869fee376cc5... | by
refine Set.ext fun R ↦ ⟨fun ⟨g, hg⟩ ↦ hg ▸ ⟨hQ.smul g, hP.smul g⟩, fun h ↦ ?_⟩
have : R.IsPrime := h.1
obtain ⟨g, hg⟩ := exists_smul_of_under_eq A B G Q R (hP.over.symm.trans h.2.over)
exact ⟨g, hg.symm⟩ | true | {
"chosen_tokens": 80,
"rejected_tokens": 3,
"token_jaccard": 0.051282,
"token_length_ratio": 0.0375
} | circular_dependency | circular_dependency | 1645b26b14e94d51a6518bacd842c0ae67a8c4ef1d36e8dcb9fea9c07faaf1bb | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Invariant.Defs
public import Mathlib.RingTheory.IntegralClosure.IntegralRestrict
public import Mathlib.RingTheory.LocalRing.ResidueField.Ideal
Name... | by
exact orbit_eq_primesOver | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "106f678c679aa863ad967512be9c635f35b53268a547623947663476ac597609",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Invariant",
"family_id": "orbit_eq_primesover",
"file_id": "mathlib/Mathlib/RingTheory/Invariant/Basic.lean",
"sample_id": "94279022a062f44341d707de6cd3452c8f04400967480dbebc54e8c3a85912a9"
} | train |
{
"chosen_proof_sha256": "a2f7906395a8fceee2188be53f90d4ed089166e4d7b933b3ba61e10fb185f1d5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2518e676d38db0b863404107b061157ad0... | by
split_ifs with h
exacts [hs h, ht h] | true | {
"chosen_tokens": 12,
"rejected_tokens": 5,
"token_jaccard": 0.071429,
"token_length_ratio": 0.416667
} | circular_dependency | circular_dependency | 166d368d8d70810ca361a38fb07d0a15efaf31c4eca46313f823dae9fb24e3ba | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Set.Countable
public import Mathlib.Order.ConditionallyCompleteLattice.Basic
public import Mathlib.Tactic.CrossRefAttribute
public import Mathlib.Tactic.F... | by
exact MeasurableSet.ite | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "7633c5feba12763231f58ac18145c871d09aea68d2e42caf7b72e6733a566bdf",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/MeasurableSpace",
"family_id": "measurableset",
"file_id": "mathlib/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean",
"sample_id": "aa6161864a0c5f5bc6339fe001dfcdf0b66c0b5bdb3cf1678dcb5f82be308aa0"
} | train |
{
"chosen_proof_sha256": "586f1604059baac7ae4d22c8ed3b6c941cba1beaead4cb818f84462ea38efbce",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f99fb33905c3e834807b289faba5aa4e5d... | by
simp only [nonempty_iff_ne_empty, Ne, sumLexLift_eq_empty, not_and_or, exists_prop, not_forall] | true | {
"chosen_tokens": 16,
"rejected_tokens": 5,
"token_jaccard": 0.0625,
"token_length_ratio": 0.3125
} | wrong_namespace_qualification | wrong_namespace_qualification | 168419567e71cd2ce3231e9a917cf68f6a7546013ba74a7f1f96962eb22381fd | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finset.Sum
public import Mathlib.Data.Sum.Order
public import Mathlib.Order.Interval.Finset.Defs
Namespace:
Finset
Local context:
/-
Copyright (c) 2022 ... | by
exact OProverMissing.missing_dce4fa550c06 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "5d88000b3f5335cf008faace2e19656483624637da8c8d1c74d5eb157d098efb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Sum",
"family_id": "sumlexlift_nonempty",
"file_id": "mathlib/Mathlib/Data/Sum/Interval.lean",
"sample_id": "dce4fa550c062aaa1e0e5d299822fa3b4591439eeb6e8fb744a8c9d9b1a2391e"
} | train |
{
"chosen_proof_sha256": "ace5857b04711e3f6e5ec35f7db5255eacbf0110e349e471daaf3e18ca693ed4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "21693a2d3819a5c289e31cd4b3eeb30894... | by
by_cases! hn : n = 0
· rw [pow_eq_top_iff.mpr (Or.inr hn)]
exact isUnit_of_subsingleton _
exact (isUnit_mk_pow_iff_notMem I hn).mpr hx | true | {
"chosen_tokens": 33,
"rejected_tokens": 7,
"token_jaccard": 0.107143,
"token_length_ratio": 0.212121
} | circular_dependency | circular_dependency | 1701f83daf9a8ccb237893bb8c124eb99499fe170657b2a99d0d067ab5fddc54 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Ideal.Quotient.Operations
public import Mathlib.RingTheory.Nilpotent.Lemmas
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Oliver Nash. Al... | by
exact Ideal.Quotient.isUnit_mk_pow_of_notMem | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "fd4afe14d32008c27559fa53b80c4501450857db78d8170a348af7e96bc29b7c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Ideal",
"family_id": "ideal",
"file_id": "mathlib/Mathlib/RingTheory/Ideal/Quotient/Nilpotent.lean",
"sample_id": "22290c022264f0f12d52570adb279eb6ca130b8e7ce84838db022df1840fc31a"
} | train |
{
"chosen_proof_sha256": "ff5b125214a3c976ea99dae2474b0a2f292cf08626af894b2e62bc6e985e11b4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "519681ab2bbfd1d17a2a2e4441b416bdb4... | by
apply freeYonedaHomEquiv.injective
simp only [freeYonedaHomEquiv_comp, Equiv.apply_symm_apply]
variable (C) | true | {
"chosen_tokens": 18,
"rejected_tokens": 5,
"token_jaccard": 0.1,
"token_length_ratio": 0.277778
} | wrong_namespace_qualification | wrong_namespace_qualification | 173c12566e6c511e91d7e5214d20bdfe9ec289cde116dad6c7e888bb87b9700d | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Generator.Basic
public import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
Namespace:
CategoryTheory.Presheaf
Local context:
/-
Copyrig... | by
exact OProverMissing.missing_2c8f4c46b465 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d5b5f26625a749abdd13b6d8dd0df4e409a58b18a87268059c904774c3314726",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Generator",
"family_id": "freeyonedahomequiv_symm_comp",
"file_id": "mathlib/Mathlib/CategoryTheory/Generator/Presheaf.lean",
"sample_id": "2c8f4c46b465c3b61a9d9f12832774c7822ae516e1e560bc50ecc34f4e2ef998"
} | train |
{
"chosen_proof_sha256": "b2e750b3c5b8740fb444ea49e43055beae89260e7b9e80c29b949ec289033978",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d4463edd748beb11f3b39ffe113ee78d6b... | by
rw [of_apply, ← lift_symm_apply, Equiv.symm_apply_apply] | true | {
"chosen_tokens": 12,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.166667
} | invalid_tactic | invalid_tactic | 174bd753ceebdb7cc5ef3da50a9be98699ff246dd327bc4146ac464206699635 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.SkewMonoidAlgebra.Basic
public import Mathlib.Algebra.Module.BigOperators
public import Mathlib.Algebra.Algebra.Equiv
Namespace:
SkewMonoidAlgebra
Lo... | by
__oprover_invalid_tactic_f0978345095e | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "c16d93f2a95c3a2b1da59d485dd74fee64886ba975bdce9c6e9238e1ddb922a2",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/SkewMonoidAlgebra",
"family_id": "lift_of",
"file_id": "mathlib/Mathlib/Algebra/SkewMonoidAlgebra/Lift.lean",
"sample_id": "f0978345095eddbdb23e72c7878138dd691ea3f6ff7d3034237e21db83bc2237"
} | train |
{
"chosen_proof_sha256": "187dbb6cbe73173dbea1f8f380afdbd3c2e979ea26a53fbec596e63a5f1c929e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "804a5cbe713c4c438d0140658a7f1c0cb5... | by by_cases a = 0 <;> grind [zero_le] | true | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.25
} | circular_dependency | circular_dependency | 175439ef0c42bbbe1e8bac772f544e44cc3241661700538c492707e4cdfa2ee7 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.Finite.Basic
public import Mathlib.RingTheory.Valuation.Basic
Namespace:
FiniteField
Local context:
/-
Copyright (c) 2026 María Inés de Frutos-Fe... | by
exact valuation_algebraMap_le_one | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "4f13116cf6bbd60f2b97058bfd97798f3a29901b6ed86c01536641ccd5c882e9",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/Finite",
"family_id": "valuation_algebramap_le_one",
"file_id": "mathlib/Mathlib/FieldTheory/Finite/Valuation.lean",
"sample_id": "9f7eb7bcc0d3153a05f5019eda72791728388ea6466ec87e305f6457e18d50ee"
} | train |
{
"chosen_proof_sha256": "8390e404c259cefd040f4b8484fc4ca8346f13b8985655df805e82d4eca600cf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "62dbb2c1f7ca5d01d29e0e066c6ef072aa... | by
ext x
induction x <;> rfl | true | {
"chosen_tokens": 9,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.555556
} | wrong_namespace_qualification | wrong_namespace_qualification | 1771f005f27ba38ac9053aa45d62343058b7b4ca393d378927bcad73a781926b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Basic
public import Mathlib.Algebra.Group.Equiv.Defs
public import Mathlib.Algebra.Group.WithOne.Defs
Namespace:
WithOne
Local context:
/-
Copy... | by
exact OProverMissing.missing_923ffa343160 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "6210317329d985e44b4f599f05b36c7e58246f42db347bc8803db118dd7be3e6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Algebra/Group",
"family_id": "mapmulhom_id",
"file_id": "mathlib/Mathlib/Algebra/Group/WithOne/Basic.lean",
"sample_id": "923ffa343160dc678018bf65a4585befc8d6632387419cd5ee47dffbd11b76e4"
} | train |
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