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": "2c32995a07b48e16dcaafc728301779b447c776a69f801e281fcde5579068d01",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cf07e5f2d2894cd807b5a1504b06e4111f... | by
refine ⟨fun h ↦ fun _ _ _ ↦ h _ _, fun h ↦ fun i₁ i₂ ↦ ?_⟩
by_cases hi : i₁ = i₂
· simp [hi]
· exact h hi | false | {
"chosen_tokens": 44,
"rejected_tokens": 49,
"token_jaccard": 0.846154,
"token_length_ratio": 1.113636
} | markdown_preamble_trailing | preamble | 4b85b167521b8fa902566e6a1df148fa862875a224b8587e50be733343da03a3 | 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.MetricSpace.Pseudo.Defs
public import Mathlib.Topology.MetricSpace.Isometry
public import Mathlib.Topology.MetricSpace.Dilation
Namespace:
(root)
Lo... | Here is the proof:
by
refine ⟨fun h ↦ fun _ _ _ ↦ h _ _, fun h ↦ fun i₁ i₂ ↦ ?_⟩
by_cases hi : i₁ = i₂
· simp [hi]
· exact h hi | {
"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/MetricSpace",
"family_id": "congruent_iff_pairwise_edist_eq",
"file_id": "mathlib/Mathlib/Topology/MetricSpace/Congruence.lean",
"sample_id": "fce701ee6ff6a6606b14ec7394e9ebfdd374d824dea16656e00b6156e86509d7"
} | train |
{
"chosen_proof_sha256": "124731cfc6276098aa30620798da6c884c01af47d833522f931a5e1fb2b79484",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_algEquiv_matrix_end_mulOpposite R₀ R
have := Module.Finite.equiv e.toLinearEquiv
classical exact ⟨n, hn, _, _, _, .of_surjective
(Matrix.entryLinearMap R₀ _ (0 : Fin n) (0 : Fin n)) fun f ↦ ⟨fun _ _ ↦ f, rfl⟩, ⟨e⟩⟩ | true | {
"chosen_tokens": 82,
"rejected_tokens": 8,
"token_jaccard": 0.111111,
"token_length_ratio": 0.097561
} | unsolved_goals | unsolved_goals | 4b9406d9f25a023a15e20d3040b09065189dc9712c3c396627e5cd1ee5e0a852 | 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.FreeModule.Finite.Basic
public import Mathlib.RingTheory.FiniteLength
public import Mathlib.RingTheory.SimpleModule.Isotypic
public import Mathli... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "82e0fcc22e3556b503f6ac44940706255c58c98e3a2257322c00f2ae0e7388ef",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/SimpleModule",
"family_id": "exists_algequiv_matrix_divisionring_finite",
"file_id": "mathlib/Mathlib/RingTheory/SimpleModule/WedderburnArtin.lean",
"sample_id": "45556f8bd9370bd1415c420e560973ba88e4c44babfc2defddcf5c8f39aa00bb"
} | train |
{
"chosen_proof_sha256": "3f5dfa9a05050a7e8f6e2944c4416e1a85109aff6c3b31b2844a70ac199a91e0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
obtain ⟨C, D, rfl⟩ := hNM
exact hI.of_delete.of_contract | true | {
"chosen_tokens": 17,
"rejected_tokens": 8,
"token_jaccard": 0.1,
"token_length_ratio": 0.470588
} | unsolved_goals | unsolved_goals | 4bb1015705ff0383fe350ad9a560d2b59676bbcb2d21f846ee82f1d691d438f5 | 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.Combinatorics.Matroid.Minor.Contract
Namespace:
Matroid
Local context:
/-
Copyright (c) 2025 Peter Nelson. All rights reserved.
Released under Apache 2.0 lic... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "d8a236761568859e199298ca9787b6884663aa7e010bc5c776ba51f9fbe8a6ae",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/Matroid",
"family_id": "indep",
"file_id": "mathlib/Mathlib/Combinatorics/Matroid/Minor/Order.lean",
"sample_id": "44d2da225f20cd3b6e6276fb821a5a22fe9ec474c9711a39a7cbc160460c406f"
} | train |
{
"chosen_proof_sha256": "c601c0ce6ee21cbe086a05718f7a601ae3b0678e21055794ba74900b0b12e072",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "28d10eb9cccf5e65b337e07ea501309ede... | by
rw [IsPeriodicPt, hco.comp_iterate] at hfg
exact hfg.left_of_comp hg | true | {
"chosen_tokens": 16,
"rejected_tokens": 2,
"token_jaccard": 0.066667,
"token_length_ratio": 0.125
} | invalid_tactic | invalid_tactic | 4bcf3eace62aa9d57d97c0b2a3da967898101501b484f3dbf749f84ad2e00ad8 | 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.Action.Defs
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.Sub.Basic
public import Mathlib.Data.List.Cycle
pub... | by
__oprover_invalid_tactic_15c500daf4b5 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "c2e00198ccdbe96a029065a2b718bc4860980a8bbcc8f358f432886ea3680822",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Dynamics/PeriodicPts",
"family_id": "left_of_comp",
"file_id": "mathlib/Mathlib/Dynamics/PeriodicPts/Defs.lean",
"sample_id": "15c500daf4b584078896ed64949b949708a161dcfff41d36df74a29d2e07383f"
} | train |
{
"chosen_proof_sha256": "0fd4fc28ba9a30a85e46c5d1d3ba799c62c0383a953832f15efb88028c8307ee",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4fe496042d93eaaba67d648ef8798ee607... | by
haveI := Fintype.ofFinite α
induction φ with simp only [BoundedFormula.Realize]
| equal t₁ t₂ =>
rcases term_realize_eq_add_dotProduct t₁ with ⟨k₁, u₁, ht₁⟩
rcases term_realize_eq_add_dotProduct t₂ with ⟨k₂, u₂, ht₂⟩
convert! Nat.isSemilinearSet_setOf_mulVec_eq ![k₁] ![k₂] (.of ![u₁]) (.of ![u₂])
... | false | {
"chosen_tokens": 270,
"rejected_tokens": 277,
"token_jaccard": 0.98,
"token_length_ratio": 1.025926
} | markdown_preamble_trailing | fence | 4bef2db32b2a281b4b0efe919fc7887b2b2ba25599d1a53674926d0efc22183e | 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.ModelTheory.Arithmetic.Presburger.Basic
public import Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic
public import Mathlib.ModelTheory.Definability... | ```lean
by
haveI := Fintype.ofFinite α
induction φ with simp only [BoundedFormula.Realize]
| equal t₁ t₂ =>
rcases term_realize_eq_add_dotProduct t₁ with ⟨k₁, u₁, ht₁⟩
rcases term_realize_eq_add_dotProduct t₂ with ⟨k₂, u₂, ht₂⟩
convert! Nat.isSemilinearSet_setOf_mulVec_eq ![k₁] ![k₂] (.of ![u₁]) (.of ... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "ModelTheory/Arithmetic",
"family_id": "issemilinearset_boundedformula_realize",
"file_id": "mathlib/Mathlib/ModelTheory/Arithmetic/Presburger/Definability.lean",
"sample_id": "a75eaccf6d18b5c2eb5e6918fb406ae54ba35c64367a2abe5f21d18d5b2dac31"
} | train |
{
"chosen_proof_sha256": "c30b62521a4c15bdccd0e80ec4d2a61314f23daa4911bb4ad6a467ce05c1c053",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5fc66937e637b4b0522e166ab9a5794714... | by
refine ⟨fun h I hIX ↦ ?_, fun h ↦ ext_indep rfl fun I (hI : I ⊆ X) ↦ ?_⟩
· rw [← and_iff_left (a := (M.Indep I)) hIX, ← and_iff_left (a := (M'.Indep I)) hIX,
← restrict_indep_iff, h, restrict_indep_iff]
rw [restrict_indep_iff, and_iff_left hI, restrict_indep_iff, and_iff_left hI, h _ hI] | false | {
"chosen_tokens": 83,
"rejected_tokens": 88,
"token_jaccard": 0.891892,
"token_length_ratio": 1.060241
} | markdown_preamble_trailing | trailing | 4c34b82311f97c6c46cdaaedd5fd2836247ee1fb5604b6398410b31906b112c3 | 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.Matroid.Dual
Namespace:
Matroid
Local context:
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as de... | by
refine ⟨fun h I hIX ↦ ?_, fun h ↦ ext_indep rfl fun I (hI : I ⊆ X) ↦ ?_⟩
· rw [← and_iff_left (a := (M.Indep I)) hIX, ← and_iff_left (a := (M'.Indep I)) hIX,
← restrict_indep_iff, h, restrict_indep_iff]
rw [restrict_indep_iff, and_iff_left hI, restrict_indep_iff, and_iff_left hI, h _ hI]
This completes t... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/Matroid",
"family_id": "restrict_eq_restrict_iff",
"file_id": "mathlib/Mathlib/Combinatorics/Matroid/Minor/Restrict.lean",
"sample_id": "ff91d6fc0a99e05c49e2fa58e7af54a13ccacbe396be47e3ff0015e67bd86101"
} | train |
{
"chosen_proof_sha256": "a3707e1c83ef260ac94852111d4f8128b1e6f031babe4d0a83c426fc4ebac9e6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "176932c1d8e508eb9eba7d81a4615dd683... | by
dsimp [Part.fix]
rw [assert_neg h'] | true | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.25
} | circular_dependency | circular_dependency | 4c46308a663d4ae9dced66d37cd2e298ec2e8da1563c90d2c2ccd0f348a4d304 | 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.Part
public import Mathlib.Data.Nat.Find
public import Mathlib.Data.Nat.Upto
public import Mathlib.Data.Stream.Defs
Namespace:
Part
Local context:
/-
Co... | by
exact fix_def | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "7c044083bcd6223612786074b9ffdb47b01f2ebe117fe03e5ddab86c8c885001",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Control",
"family_id": "fix_def",
"file_id": "mathlib/Mathlib/Control/Fix.lean",
"sample_id": "9944f4492e2edc232793a5f7e57320d19cd2217d990686c0e9b0e255481105bf"
} | train |
{
"chosen_proof_sha256": "61e6f75295dcf0ee0f0edfdb232d6f4ff00a1685b5790d5e3eca26ccc61aea57",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [← Module.finrank_mul_finrank ℝ ℂ E, Complex.finrank_real_complex] | false | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.0625,
"token_length_ratio": 0.2
} | by_exact_placeholder | placeholder | 4c4c26977612f1a5bbb792f9cd11ddc329542d8435d99a15e967e0d489fa6b19 | 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.Analysis.Complex.Cardinality
public import Mathlib.LinearAlgebra.Complex.Module
public import Mathlib.LinearAlgebra.F... | 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": "LinearAlgebra/Complex",
"family_id": "finrank_real_of_complex",
"file_id": "mathlib/Mathlib/LinearAlgebra/Complex/FiniteDimensional.lean",
"sample_id": "91a92b19287fd834779cff643bf9db6f489d497152bb848e415ab87fcceac41f"
} | train |
{
"chosen_proof_sha256": "d988ca19062348f5a91c0b5ac55ee8cf0c8bfcea69c1997f85ea9823ef01e86a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
refine ⟨fun h => Set.infinite_prod.1 ?_, ?_⟩
· rw [← image_uncurry_prod] at h
exact h.of_image _
· rintro (⟨hs, b, hb⟩ | ⟨ht, a, ha⟩)
· exact hs.image2_left hb (hfs _ hb)
· exact ht.image2_right ha (hft _ ha) | true | {
"chosen_tokens": 71,
"rejected_tokens": 8,
"token_jaccard": 0.023256,
"token_length_ratio": 0.112676
} | unsolved_goals | unsolved_goals | 4c5597cfca38a7e67b8c5a017fabffb0eee7449caa7da8df58b9c09edb4cbb73 | 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.Finite.Basic
public import Mathlib.Data.Fintype.Prod
public import Mathlib.Data.Fintype.Pi
public import Mathlib.Algebra.Order.Group.Multiset
public i... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "f83a0232fcbd1ba953a3b544ffbe0b503479aef0db0e59df808b3c79db3e7882",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Finite",
"family_id": "infinite_image",
"file_id": "mathlib/Mathlib/Data/Finite/Prod.lean",
"sample_id": "21544d2f51f369dc7014292be7e61bf52982b39514987f4c5a1ef51ecf1523b5"
} | train |
{
"chosen_proof_sha256": "ea63d2911a78e27e3d0efd6769a583eaec219edad30fe20857b3d050d16b7158",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
letI := (IntermediateField.inclusion H.le).toAlgebra
have : IsScalarTower F E L := .of_algebraMap_eq' rfl
refine lt_of_le_of_ne ?_ ?_
· exact Module.finrank_top_le_finrank_of_isScalarTower _ _ _
· exact .symm (mt (eq_of_le_of_finrank_eq' H.le) H.ne) | false | {
"chosen_tokens": 53,
"rejected_tokens": 2,
"token_jaccard": 0.03125,
"token_length_ratio": 0.037736
} | sorry_admit_sorryAx_proof_hole | admit | 4c5e170dc08e386d7a279385ba3a0a0adcb8e4296d64539a4bf17698bdf66fd4 | 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.IntermediateField.Basic
public import Mathlib.FieldTheory.Minpoly.Basic
public import Mathlib.FieldTheory.Tower
public import Mathlib.LinearAlgebra... | 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": "FieldTheory/IntermediateField",
"family_id": "finrank_lt_of_gt",
"file_id": "mathlib/Mathlib/FieldTheory/IntermediateField/Algebraic.lean",
"sample_id": "f611f13c4698d3e99009d189350bca949401d584539fd12a5756b6e5b77a146b"
} | train |
{
"chosen_proof_sha256": "6ec31665b33411506cda64640e21f6774b047c685e35cff9b518e65219ef78f0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cdb77a9f055c59dec5cff52d796012f86c... | by
refine c.isColimit.hom_ext (fun j ↦ ?_)
dsimp
induction j using SuccOrder.limitRecOn with
| isMin j hj =>
obtain rfl := hj.eq_bot
simpa [← cancel_epi c.isoBot.inv] using h₀
| succ j hj hj' =>
apply (c.attachCells j hj).hom_ext
· simpa using hj'
· intro i
simpa only [Category.assoc... | false | {
"chosen_tokens": 129,
"rejected_tokens": 133,
"token_jaccard": 0.951613,
"token_length_ratio": 1.031008
} | missing_wrong_import | import | 4c8218252586fe1977c94c63e6ae6c59ea6d939600c7f2bae42446d1471841a9 | 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.RelativeCellComplex.AttachCells
public import Mathlib.CategoryTheory.MorphismProperty.TransfiniteComposition
Namespace:
HomotopicalAlgebra.R... | import Mathlib.WrongImport
by
refine c.isColimit.hom_ext (fun j ↦ ?_)
dsimp
induction j using SuccOrder.limitRecOn with
| isMin j hj =>
obtain rfl := hj.eq_bot
simpa [← cancel_epi c.isoBot.inv] using h₀
| succ j hj hj' =>
apply (c.attachCells j hj).hom_ext
· simpa using hj'
· intro i
... | {
"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/RelativeCellComplex",
"family_id": "hom_ext",
"file_id": "mathlib/Mathlib/AlgebraicTopology/RelativeCellComplex/Basic.lean",
"sample_id": "96fed0cd3e0fdbf1ac8825b9ea3ac8717dd91342a9fea198dd325c9dbb8823cc"
} | train |
{
"chosen_proof_sha256": "bf3037486d0c6b30f0d1754728261534db0d266ca9de3c9d72286f831a1bc900",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
haveI : IsCentral K D :=
{ out := fun x ↦ show x ∈ Subalgebra.center k D → _ by
simp only [center_eq_bot, mem_bot, Set.mem_range, forall_exists_index]
rintro x rfl
exact ⟨algebraMap k K x, by simp [algebraMap_eq_smul_one, smul_assoc]⟩ }
refine ⟨FaithfulSMul.algebraMap_injective k K, fun x => ... | false | {
"chosen_tokens": 161,
"rejected_tokens": 2,
"token_jaccard": 0.016667,
"token_length_ratio": 0.012422
} | sorry_admit_sorryAx_proof_hole | admit | 4cc110ac059e95ceba606f8cfb21f827284bcf75f9d1a1b58a60b6e24ff05765 | 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.Central.Defs
import Mathlib.Algebra.Module.Torsion.Field
Namespace:
Algebra.IsCentral
Local context:
/-
Copyright (c) 2024 Jujian Zhang. All rights r... | 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": "Algebra/Central",
"family_id": "basefield_essentially_unique",
"file_id": "mathlib/Mathlib/Algebra/Central/Basic.lean",
"sample_id": "53e21ad5e315c317c8084e011afafd42127d23df856e46c6d95998155a1cbf26"
} | train |
{
"chosen_proof_sha256": "ad8808e6ef06989ebffa9f2a94c9e16fc819a971dc924b479659fd09f73634ab",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c0b4f5ee96fca0c0f0a00a8f442ac70878... | by
simp only [Ideal.eq_top_iff_one, ← Subtype.range_val (s := s), ← Set.range_comp,
Finsupp.mem_ideal_span_range_iff_exists_finsupp] at hs' ⊢
choose f hf using hs'
have : Fintype s := hs.fintype
refine ⟨Finsupp.equivFunOnFinite.symm fun i x ↦ f x i, ?_⟩
ext i
simpa [Finsupp.sum_fintype] using hf i | true | {
"chosen_tokens": 72,
"rejected_tokens": 2,
"token_jaccard": 0.021277,
"token_length_ratio": 0.027778
} | invalid_tactic | invalid_tactic | 4d298730cd1f3c306750a768626c598733f8b6b62025ecfb49ff124966ad3d46 | 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.Immersion
Namespace:
AlgebraicGeometry
Local context:
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apac... | by
__oprover_invalid_tactic_92326efcf9c3 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "4b4970a52a3e0acee569902af02fae78f97589e8cc80ebc362c0c1fa7eabdf87",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry",
"family_id": "ideal",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/PointsPi.lean",
"sample_id": "92326efcf9c32173b0a7545e06842af3e1db8144b2ac5b58b5ccfdd3ce10f509"
} | train |
{
"chosen_proof_sha256": "c628935d9750505c8adb2f5d44da9ce697c92da40b87a39fbb8e2a552d65d0b4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
ext; exact geometricallyReduced_iff _ | true | {
"chosen_tokens": 6,
"rejected_tokens": 8,
"token_jaccard": 0.083333,
"token_length_ratio": 1.333333
} | unsolved_goals | unsolved_goals | 4d3752c44c94b818a92521913f032d1b581843e844e98d7627af9a5c36bad379 | 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.Artinian
public import Mathlib.AlgebraicGeometry.Geometrically.Basic
public import Mathlib.AlgebraicGeometry.Morphisms.SchemeTheoreticallyDom... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "05650197539227623cd8139f84f7093968cfbcee6139fd125290d396b2b3f0b9",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Geometrically",
"family_id": "geometricallyreduced",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Geometrically/Reduced.lean",
"sample_id": "8fa0d3b17dbb462bc253c216aedf5600d245ea36ea5fef5dc975ff5afcf7c43e"
} | train |
{
"chosen_proof_sha256": "355d8928efd1f0f2dd702b5a22182d531523711ce4f995520e27017b7c2e8b46",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
· filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy
· intro ε εpos
filter_upwards [Icc_mem_nhdsLT <| show x - ε < x by linarith] with y hy
rw [closedBall_eq_Icc]
exact Icc_subset_Icc hy.1 (by linarith) | false | {
"chosen_tokens": 63,
"rejected_tokens": 2,
"token_jaccard": 0.025641,
"token_length_ratio": 0.031746
} | sorry_admit_sorryAx_proof_hole | sorry | 4d7757c821237c3b682b7375c71b48783fba66866cb20c41a35cafe6acc2426c | 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.Covering.DensityTheorem
public import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
Namespace:
Real
Local context:
/-
Copyright (c) 2022 Sébast... | 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": "MeasureTheory/Covering",
"family_id": "tendsto_icc_vitalifamily_left",
"file_id": "mathlib/Mathlib/MeasureTheory/Covering/OneDim.lean",
"sample_id": "8ad925533b8c56254faf2ab0de8790476e52ea1f36cb7dddd76cb2f24ea5a395"
} | train |
{
"chosen_proof_sha256": "8bb9f6292b5b7cdafeb1af9846ff9f4bfcbbf84b87fb669f9431c4e7c51bed33",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp only [iSup, hf.map_sSup', ← range_comp]
rfl | true | {
"chosen_tokens": 14,
"rejected_tokens": 8,
"token_jaccard": 0.052632,
"token_length_ratio": 0.571429
} | unsolved_goals | unsolved_goals | 4d8bdcb0e4adf3c283cbed0e81e900050e438413613d11a5a28e349e20137752 | 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.Order.ConditionallyCompleteLattice.Basic
public import Mathlib.Order.RelIso.Basic
Namespace:
LeftOrdContinuous
Local context:
/-
Copyright (c) 2020 Yury Kudr... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "74f024850af0ff304abf422d1458554df0119526ff462bd70f18f633e6107c0c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Order",
"family_id": "map_isup",
"file_id": "mathlib/Mathlib/Order/OrdContinuous.lean",
"sample_id": "8b95081cd7f651acb1a47e7a53ca218a713ae886e12ac7935b03bee7470a950f"
} | train |
{
"chosen_proof_sha256": "bc3578532551ba1c8a4369ed77aebd3d2f1775c8fadb93080f656df6af2cd8fe",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
have ⟨φ⟩ := ne
refine h.toIsAlgebraic.of_injective φ φ.injective
|>.normalClosure_eq_iSup_adjoin_of_splits fun x ↦ ?_
rw [← minpoly.algHom_eq _ φ.injective]
apply h.splits | false | {
"chosen_tokens": 41,
"rejected_tokens": 2,
"token_jaccard": 0.033333,
"token_length_ratio": 0.04878
} | sorry_admit_sorryAx_proof_hole | admit | 4db77acf22f2f5746d1fbcc6a12a9ab4f3d2dfae8eacd6623002eeb4be6def61 | 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.SimpleRing.Basic
public import Mathlib.FieldTheory.Normal.Basic
public import Mathlib.Order.Closure
public import Mathlib.LinearAlgebra.FreeModule.F... | 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": "FieldTheory/Normal",
"family_id": "normalclosure_eq_isup_adjoin",
"file_id": "mathlib/Mathlib/FieldTheory/Normal/Closure.lean",
"sample_id": "f6fe57d0752e0cd229648ee764e154901c2c974592574e9f5ce3b35211409c9e"
} | train |
{
"chosen_proof_sha256": "24379af29dd6a4b6bd06c4a471ab0fb06ebf44b4f629deaf2e40a3cfa29440d2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2f3517d963ac31c8db62c299de697bf8f1... | by
simp_rw [← card_univ, ← card_powersetCard]
refine (card_le_card fun s hs ↦ mem_biUnion.2 ⟨#s, ?_⟩).trans card_biUnion_le
exact ⟨mem_Iic.2 (mem_shatterer.1 hs).card_le_vcDim, mem_powersetCard_univ.2 rfl⟩ | false | {
"chosen_tokens": 49,
"rejected_tokens": 54,
"token_jaccard": 0.891892,
"token_length_ratio": 1.102041
} | markdown_preamble_trailing | trailing | 4dd85ced0cc56d70bfa80e250ff52949c905f4f67b753079d05d7492fee7cfbb | 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.SetFamily.Compression.Down
public import Mathlib.Data.Fintype.Powerset
public import Mathlib.Order.Interval.Finset.Nat
public import Mathlib.Alge... | by
simp_rw [← card_univ, ← card_powersetCard]
refine (card_le_card fun s hs ↦ mem_biUnion.2 ⟨#s, ?_⟩).trans card_biUnion_le
exact ⟨mem_Iic.2 (mem_shatterer.1 hs).card_le_vcDim, mem_powersetCard_univ.2 rfl⟩
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": "Combinatorics/SetFamily",
"family_id": "card_shatterer_le_sum_vcdim",
"file_id": "mathlib/Mathlib/Combinatorics/SetFamily/Shatter.lean",
"sample_id": "c9a34ccd888b6cf8e9fb2f8c6ffab0caeeb794e571e1779f7f95f329d2895ac6"
} | train |
{
"chosen_proof_sha256": "c91bd3015c099e4be465fbcb7168b44d0092e9266e1434d35ffc1f51e248e636",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4c4e9dcac1de909b93b46f79fa507e9e65... | by
classical
obtain ⟨I, hIC, hI, hst⟩ := hC.sdiff_eq_sUnion' s hs t ht
refine ⟨.ofErase I (supIndep_iff_pairwiseDisjoint.mpr hI) ?_, ?_⟩
· rw [sup_id_eq_sSup, sSup_eq_sUnion, hst]
· grw [Finpartition.ofErase_parts, Finset.erase_subset, hIC] | true | {
"chosen_tokens": 59,
"rejected_tokens": 3,
"token_jaccard": 0.025641,
"token_length_ratio": 0.050847
} | circular_dependency | circular_dependency | 4def5ada9fde120ec6ba300161a75872547282271659bc6b0aa71be041f58a13 | 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.Pairwise.Lattice
public import Mathlib.MeasureTheory.PiSystem
public import Mathlib.Order.Lattice.Nat
public import Mathlib.Order.Partition.Finpartiti... | by
exact exists_finpartition_sdiff | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "16ca00327996b3a9f09ff593366e17352c36ff58f290e435c3d0d360692ab0c6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory",
"family_id": "exists_finpartition_sdiff",
"file_id": "mathlib/Mathlib/MeasureTheory/SetSemiring.lean",
"sample_id": "befab83d2930909beac8cfad141326e33a75bf337f03f5f78fa5a152c0c13634"
} | train |
{
"chosen_proof_sha256": "55840963317e3a0d749bfa4ab4253cf6959feaf799aad1f633d62acc28d49b26",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "17501c489548fc579d16232355b4d42b56... | by valid) : F.map' i i = 𝟙 _ := F.map_id _ | true | {
"chosen_tokens": 17,
"rejected_tokens": 2,
"token_jaccard": 0.071429,
"token_length_ratio": 0.117647
} | invalid_tactic | invalid_tactic | 4e1057157db780ad8cf263856fa9b13db945c0dc105525f927b69bb1a8f189df | 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.Nat.Defs
public import Mathlib.CategoryTheory.Category.Preorder
public import Mathlib.CategoryTheory.Comma.Arrow
public import Mathlib.Data.Finty... | by
__oprover_invalid_tactic_a1e462f08764 | {
"accepted": false,
"failure_type": "unknown_tactic",
"output_sha256": "4e32ad2f0af4a6e9a38133d76b55b664f63c636db97a765122d4cc21f8be2e4a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/ComposableArrows",
"family_id": "map'_self",
"file_id": "mathlib/Mathlib/CategoryTheory/ComposableArrows/Basic.lean",
"sample_id": "a1e462f08764c6d42623fb769d4457124c492b34cbacfef092d7509dfdd31de7"
} | train |
{
"chosen_proof_sha256": "0c64ae53f4b1abd57d7d7d427b1d8d5ae6a8abade34b2f455335c0dfb08d0184",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [subsingleton_iff_support_subset, PrimeSpectrum.basicOpen_eq_zeroLocus_compl,
disjoint_compl_left_iff, Set.le_iff_subset] | false | {
"chosen_tokens": 15,
"rejected_tokens": 5,
"token_jaccard": 0.0625,
"token_length_ratio": 0.333333
} | sorry_admit_sorryAx_proof_hole | sorryAx | 4e27cddb31488dc9013d8680742a034a762b51f3c1eaecb5c4fb4dec4b1e2b20 | 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.Spectrum.Prime.Topology
public import Mathlib.RingTheory.Support
Namespace:
(root)
Local context:
/-
Copyright (c) 2024 Andrew Yang. All rights re... | 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": "RingTheory/Spectrum",
"family_id": "localizedmodule",
"file_id": "mathlib/Mathlib/RingTheory/Spectrum/Prime/Module.lean",
"sample_id": "f05312ee1b649b954d0ad3d3cedaebdac4a9c0766690dacc1ff7776f9463beba"
} | train |
{
"chosen_proof_sha256": "861cfb34c8727a0d4d437e7097497ea9d3f14804aa184568a47ee591868cf5bf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [Equiv.eq_symm_comp (e := Quiver.symmetrifyStar (φ.obj u))]
ext ⟨v, f | g⟩ <;>
-- Porting note (https://github.com/leanprover-community/mathlib4/issues/10745): was `simp [Quiver.symmetrifyStar]`
simp only [Quiver.symmetrifyStar, Function.comp_apply] <;>
erw [Equiv.sigmaSumDistrib_apply, Equiv.sigm... | true | {
"chosen_tokens": 96,
"rejected_tokens": 8,
"token_jaccard": 0.053571,
"token_length_ratio": 0.083333
} | unsolved_goals | unsolved_goals | 4eb289c4654787f230f26312c46d6db24b48198cd1802389b837ce7e66b898de | 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.Combinatorics.Quiver.Cast
public import Mathlib.Combinatorics.Quiver.Symmetric
public import Mathlib.Data.Sigma.Basic
public import Mathlib.Data.Sum.Basic
publ... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b736ef4ea45ce21f69cf741c022990c9c33b575d17708de1cfe1e04a2dae968e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/Quiver",
"family_id": "prefunctor",
"file_id": "mathlib/Mathlib/Combinatorics/Quiver/Covering.lean",
"sample_id": "87030b969f81b483b5bdfa5c63449397f06437d95959ae3498323790baf68ce1"
} | train |
{
"chosen_proof_sha256": "a0b20474523f493304d2e5c3c4796a60a88d5f5165ad276749d342179c8ed04e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | 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... | false | {
"chosen_tokens": 174,
"rejected_tokens": 2,
"token_jaccard": 0.032258,
"token_length_ratio": 0.011494
} | by_exact_placeholder | by? | 4ec883b8c93f0aeacd0ed82a5be1b2b4ee6b41186b3f77569fdf67528d616959 | 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.Dual.Lemmas
public import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
public import Mathlib.LinearAlgebra.QuadraticForm.Prod
Namespace:
Li... | 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": "LinearAlgebra/QuadraticForm",
"family_id": "separatingleft_dualprod",
"file_id": "mathlib/Mathlib/LinearAlgebra/QuadraticForm/Dual.lean",
"sample_id": "6f6ae3b60a22d3ee8598639d07edd15dc214dc93b32db436d37ba0364ed7ec2a"
} | train |
{
"chosen_proof_sha256": "ed61a64bf86bd3e8070ca4701c17ddc605afcea75c9e8aec1615eb96bd1b85a1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8fdf799834e94c10d8af1bf7c477253a01... | by
induction n with
| zero => simp
| succ n IH =>
rw [iteratedLineDerivOp_succ_left]
exact (tsupport_lineDerivOp_subset (m 0) _).trans (IH <| Fin.tail m) | false | {
"chosen_tokens": 37,
"rejected_tokens": 42,
"token_jaccard": 0.83871,
"token_length_ratio": 1.135135
} | markdown_preamble_trailing | preamble | 4ee16d24cd4b77f035228cc116946fe45150dd58832ba6c6745318ff99d1fdfb | 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.Distribution.SchwartzSpace.Basic
public import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
public import Mathlib.Analysis.Calculus.LineDeriv.Integrat... | Here is the proof:
by
induction n with
| zero => simp
| succ n IH =>
rw [iteratedLineDerivOp_succ_left]
exact (tsupport_lineDerivOp_subset (m 0) _).trans (IH <| Fin.tail m) | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/Distribution",
"family_id": "tsupport_iteratedlinederivop_subset",
"file_id": "mathlib/Mathlib/Analysis/Distribution/SchwartzSpace/Deriv.lean",
"sample_id": "591110f7b37cc2daae15e5485ae2d271b10317dae65d4419218d3c8977316858"
} | train |
{
"chosen_proof_sha256": "20d35803748fd7d6978af67d3ca14fe96ffdbfdea01cb8bdd6440cb3a20e70bb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e02776207e53734b0bc60bad4f0d744549... | by
let nop := @neg_one_pow_div_two_of_three_mod_four
rw [quadratic_reciprocity', pow_mul, nop hp, nop hq, neg_one_mul] <;>
rwa [← Prime.mod_two_eq_one_iff_ne_two Fact.out, odd_of_mod_four_eq_three] | true | {
"chosen_tokens": 35,
"rejected_tokens": 3,
"token_jaccard": 0.035714,
"token_length_ratio": 0.085714
} | circular_dependency | circular_dependency | 4efccae5d697a6e43b27703b33dc396738a7243e7a553dc59835492186548581 | 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.NumberTheory.LegendreSymbol.Basic
public import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
Namespace:
legendreSym
Local context:
/-
Copyright... | by
exact quadratic_reciprocity_three_mod_four | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "aeb4065d2b8f3bdc8324d3f4cbe041dfd76be3d5d4cbf8b9d4c96dce7b3ba323",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/LegendreSymbol",
"family_id": "quadratic_reciprocity_three_mod_four",
"file_id": "mathlib/Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean",
"sample_id": "87698f733f1f8800ecece55747b606be522d5d9ae2965cfc2fd1ae226fa9c49c"
} | train |
{
"chosen_proof_sha256": "c198f493b5c0a079bb4ea7d468b89dbbd0202be048c196e631696690e8762509",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
obtain rfl | hμ := eq_zero_or_isProbabilityMeasure μ
· simp
simpa [Pi.mul_def, integral_of_ae_eq_zero_or_one, integral_one_sub_of_ae_eq_zero_or_one, mul_comm,
*] using condVar_of_ae_eq_zero_or_one bot_le hXmeas hX | false | {
"chosen_tokens": 30,
"rejected_tokens": 5,
"token_jaccard": 0.033333,
"token_length_ratio": 0.166667
} | sorry_admit_sorryAx_proof_hole | sorryAx | 4f0317c0938615165c65458c4678ddef1b8f25e96c1522eead7bc4274ca554ab | 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.Integral.Bochner.Basic
public import Mathlib.Probability.CondVar
import Mathlib.Probability.Notation
Namespace:
ProbabilityTheory
Local context... | 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": "Probability/Distributions",
"family_id": "variance_of_ae_eq_zero_or_one",
"file_id": "mathlib/Mathlib/Probability/Distributions/TwoValued.lean",
"sample_id": "f83dd1b97c9a129dc210da5c52474019b307010c9a4d0c19de7a0ae4adc2ce13"
} | train |
{
"chosen_proof_sha256": "fecffccc9af025141c64f561cc932cc9a7f6f3a1d76dd0660a012426d4c05839",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
unfold xgcdAux
exact if_pos rfl | true | {
"chosen_tokens": 6,
"rejected_tokens": 8,
"token_jaccard": 0.083333,
"token_length_ratio": 1.333333
} | unsolved_goals | unsolved_goals | 4f118d6f3c2732cc08b0a9a589b3088369750c36297a779b9c64f6b791b4da31 | 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.Ring.Defs
public import Mathlib.Order.RelClasses
Namespace:
EuclideanDomain
Local context:
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
R... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "50ad3d62bc125ab492492a9f3567c17234fb5eb46e7ef18552c6a12ba608b637",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/EuclideanDomain",
"family_id": "xgcd_zero_left",
"file_id": "mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean",
"sample_id": "c3604586f2ac7368d223dc67d1199e013bd8135ef158fafca76c177f59a92d99"
} | train |
{
"chosen_proof_sha256": "eff3de0495382d14fb1653bae8a0cc2d9bf54b578da62db37b5ffda72bfdcbdf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
simp [← le_bot_iff, not_le_iff_exists] | false | {
"chosen_tokens": 8,
"rejected_tokens": 2,
"token_jaccard": 0.111111,
"token_length_ratio": 0.25
} | sorry_admit_sorryAx_proof_hole | admit | 4f86a9d0017a9602ba65901ac9f93b69381344943d74d1ea9d689b315489c6ec | 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.ObjectProperty.ClosedUnderIsomorphisms
public import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
public import Mathlib.Order.CompleteL... | 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": "CategoryTheory/ObjectProperty",
"family_id": "ne_bot_iff_exists",
"file_id": "mathlib/Mathlib/CategoryTheory/ObjectProperty/CompleteLattice.lean",
"sample_id": "27c82bde5cab40ddeb3cf4fe4ff5388002b1c6d7395440a1ff8ff9c86f617bfd"
} | train |
{
"chosen_proof_sha256": "f18be9dd1c59e0ebc0b61eaff66514bafe594968c74bd37a4cddbaa1b0ea3c89",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9d5cb2b7c8e2dbd23ec1db01a351a3a7f7... | by
rintro _ (rfl | rfl | rfl) _ (rfl | rfl | rfl) h <;>
simp only
[cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h ⊢
· exact ((Ring.neg_one_ne_one_of_char_ne_two hR).symm h).elim
· exact ((Ring.neg_one_ne_one_of_char_ne_two hR) h).elim | true | {
"chosen_tokens": 70,
"rejected_tokens": 3,
"token_jaccard": 0.058824,
"token_length_ratio": 0.042857
} | unknown_identifier | unknown_identifier | 4f997d32f6a9e9c1a1b1a282bf56045dd32333f55af9dd727bd5140b7df4f56c | 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.CharP.Defs
public import Mathlib.Algebra.Group.Fin.Basic
public import Mathlib.Algebra.Ring.ULift
public import Mathlib.Algebra.Ring.Opposite
public im... | by
exact __oprover_missing_197f653d4b36 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "9a7d37ce0fcd0538d172ea0a9f2be0082afe1edc35fbb751a0c4e16175feeb12",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/CharP",
"family_id": "int",
"file_id": "mathlib/Mathlib/Algebra/CharP/Basic.lean",
"sample_id": "197f653d4b36fd6849b8b6ebccef410407e7a70fbd810f7a0a75770cb14b42e2"
} | train |
{
"chosen_proof_sha256": "0a479f8dae478b3bb9603d60f313d74028b2da15cce1a50538dcbe628c6b865b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7710ac4a7094e4ea990b13bcd33210bc3b... | by
refine (QuadraticMap.associated_rightInverse R).injective ?_
ext m₁ m₂ m₁' m₂' m₁'' m₂''
simp only [associated_tmul, QuadraticMap.associated_comp]
exact mul_assoc _ _ _ | true | {
"chosen_tokens": 45,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.066667
} | circular_dependency | circular_dependency | 4fbaf64e58630e632255af040534dde4a60e5559f2af006655541cf23a13a960 | 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.QuadraticForm.TensorProduct
public import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
Namespace:
QuadraticForm
Local context:
/-
Copyrigh... | by
exact tmul_comp_tensorAssoc | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "bfa7f0f43e92f80671448652ab65d12f3f9c80d0e762ccaeff7e1e2176fd5e69",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "576771f2ca0b7fae2b561a23269a331367199c4b2800aa18415fec083cc832ae",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a6cca448626f77dd5c9ef36b63004c7574... | by
rw [liftOfDerivationToSquareZero_apply, map_add, Ideal.Quotient.eq_zero_iff_mem.mpr (d x).prop,
zero_add, Ideal.Quotient.mk_algebraMap] | true | {
"chosen_tokens": 29,
"rejected_tokens": 3,
"token_jaccard": 0.047619,
"token_length_ratio": 0.103448
} | circular_dependency | circular_dependency | 4fdf9fd54987d81d04bd20ee5108d27ba9257b0f1acf56094e413d25cea78119 | 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.Derivation.Basic
public import Mathlib.RingTheory.Ideal.Quotient.Operations
Namespace:
(root)
Local context:
/-
Copyright (c) 2020 Nicolò Cavaller... | by
exact liftOfDerivationToSquareZero_mk_apply | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "ce2bff483618949a061787dc5297ae0e80e0a2ac0e886162d158743dd1da8aad",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Derivation",
"family_id": "liftofderivationtosquarezero_mk_apply",
"file_id": "mathlib/Mathlib/RingTheory/Derivation/ToSquareZero.lean",
"sample_id": "be9b8d29bc8008d9f2006f7c0e2e488c12668f4a1e6f2ba4ffcf6937f3ace739"
} | train |
{
"chosen_proof_sha256": "e26bc9c73cc42d866d4dadb94c4f9430e4046c553d052939bf6c8d1e66df3de3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "68566ca8925ab58d94e66a6a28da0a7a65... | by
refine hK.induction_on ⟨∅, by simp⟩ (fun s t hst ⟨W, Wc, Wo, hKW, hWU⟩ ↦ ?_) ?_ ?_
· use W, Wc, Wo, subset_trans hst hKW, hWU
· intro s t ⟨W₁, Wc₁, Wo₁, hKW₁, hWU₁⟩ ⟨W₂, Wc₂, Wo₂, hKW₂, hWU₂⟩
exact ⟨W₁ ∪ W₂, Wc₁.union Wc₂, Wo₁.union Wo₂, Set.union_subset_union hKW₁ hKW₂,
Set.union_subset hWU₁ hWU₂⟩
... | false | {
"chosen_tokens": 182,
"rejected_tokens": 189,
"token_jaccard": 0.964912,
"token_length_ratio": 1.038462
} | markdown_preamble_trailing | fence | 5007e6af0957866990d0d1cb0704b1e97e392129502d020f571c5b416598cfa1 | 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.Order.Ideal
public import Mathlib.Topology.Sets.Compacts
public import Mathlib.Topology.Sets.OpenCover
public import Mathlib.Topology.Spectral.Hom
Namespace:
... | ```lean
by
refine hK.induction_on ⟨∅, by simp⟩ (fun s t hst ⟨W, Wc, Wo, hKW, hWU⟩ ↦ ?_) ?_ ?_
· use W, Wc, Wo, subset_trans hst hKW, hWU
· intro s t ⟨W₁, Wc₁, Wo₁, hKW₁, hWU₁⟩ ⟨W₂, Wc₂, Wo₂, hKW₂, hWU₂⟩
exact ⟨W₁ ∪ W₂, Wc₁.union Wc₂, Wo₁.union Wo₂, Set.union_subset_union hKW₁ hKW₂,
Set.union_subset hWU₁... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/Spectral",
"family_id": "prespectralspace",
"file_id": "mathlib/Mathlib/Topology/Spectral/Prespectral.lean",
"sample_id": "bad13452563a2e4433cf2964ac3661b1745a1d26c8fd4dce023319c5196a9b1e"
} | train |
{
"chosen_proof_sha256": "74ef8f872938daf0468322013395261e7267b7edbfb838bf7741225d397b7c55",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6f1c9a92ed629452f0151fcb0fc207570a... | by
rw [← card k V]
have : Nat.card V > 0 := Nat.card_pos
lia | true | {
"chosen_tokens": 21,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.095238
} | invalid_tactic | invalid_tactic | 501de5693f27c2c231cba4f85bd073d76ec4407768c35416bf51a367b6292408 | 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.Ring.GeomSum
public import Mathlib.Algebra.GroupWithZero.Units.Fintype
public import Mathlib.Data.Finite.Sum
public import Mathlib.GroupTheory.GroupAct... | by
__oprover_invalid_tactic_de57e8cad85d | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "7b49672f151f29c159ecddc514b6581dfa16438cea634bf1c8b96eff5b787676",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Projectivization",
"family_id": "card",
"file_id": "mathlib/Mathlib/LinearAlgebra/Projectivization/Cardinality.lean",
"sample_id": "de57e8cad85d8b44648cf10f3d51ce2288da48a2805529ae83b12ceae714db81"
} | train |
{
"chosen_proof_sha256": "80d415d211305782b1e1301d4d0be21abb30589a9327984a8f2cfe8e37267cf8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b851594e49436ef442f014bb3282b6289d... | by
have : ∀ᵐ x ∂μ, (f - f') x = 0 := by
apply ae_eq_zero_of_integral_contMDiff_smul_eq_zero I (hf.sub hf')
intro g g_diff g_supp
simp only [Pi.sub_apply, smul_sub]
rw [integral_sub, sub_eq_zero]
· exact h g g_diff g_supp
· exact hf.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_... | true | {
"chosen_tokens": 84,
"rejected_tokens": 5,
"token_jaccard": 0.058824,
"token_length_ratio": 0.059524
} | wrong_namespace_qualification | wrong_namespace_qualification | 5048be38f67859049d4f1f6b9d1c829026ea81ce21f2e8176c34944812f5fb2f | 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.Geometry.Manifold.PartitionOfUnity
public import Mathlib.MeasureTheory.Function.AEEqOfIntegral
Namespace:
(root)
Local context:
/-
Copyright (c) 2023 Sébasti... | by
exact OProverMissing.missing_d55812070999 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "edc83f01ae4243a8c26683a1e1085c57beb5351944fecf5be0206597f9224c42",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Distribution",
"family_id": "ae_eq_of_integral_contmdiff_smul_eq",
"file_id": "mathlib/Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean",
"sample_id": "d55812070999d9cd69897ef7878e60eedf847e5cca72e88cad19720567719cd3"
} | train |
{
"chosen_proof_sha256": "af1766a439a78c83ac802fa03802f5aeae0b2eb140261f6b10e3a3388ff5d5e4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "745a6c18148ff085b3e3109620a641446b... | by
let f (u : R) : R × R := (star u * u, u * star u)
have hf : f ⁻¹' {(1, 1)} = unitary R := by ext u; simp [f, Unitary.mem_iff]
rw [← hf]
exact isClosed_singleton.preimage (by fun_prop) | true | {
"chosen_tokens": 67,
"rejected_tokens": 3,
"token_jaccard": 0.052632,
"token_length_ratio": 0.044776
} | unknown_identifier | unknown_identifier | 5066c446b4d8501ee839f04c639d9355b1bd6bace89459a197d9790e5204d9fb | 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.Star.Unitary
public import Mathlib.Topology.Algebra.Group.Defs
public import Mathlib.Topology.Algebra.Star
public import Mathlib.Topology.Algebra.Monoi... | by
exact __oprover_missing_22f2feedaac2 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f6a33baac10a779348f19609f946f7357f0483c9037871e3032589c244288a1b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Algebra",
"family_id": "isclosed_unitary",
"file_id": "mathlib/Mathlib/Topology/Algebra/Star/Unitary.lean",
"sample_id": "22f2feedaac2189bc2f75b488121e482511637bba540133178964abac118fe65"
} | train |
{
"chosen_proof_sha256": "dab19792c72b6e7680af848aa1b358000086832bd7758b76eb7e5a5172d70aec",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "032679d5999e79cef4404c0d285be7a3e6... | by
grw [Sum.dist, ← le_add_of_nonneg_right dist_nonneg, ← le_add_of_nonneg_left dist_nonneg] | false | {
"chosen_tokens": 15,
"rejected_tokens": 20,
"token_jaccard": 0.75,
"token_length_ratio": 1.333333
} | markdown_preamble_trailing | trailing | 5080f3d935c50a769094e1e5271d4c0c5342b7fa650518ee9305064fc844a802 | 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.Order.ConditionallyCompleteLattice.Group
public import Mathlib.Topology.MetricSpace.Isometry
Namespace:
Metric
Local context:
/-
Copyright (c) 2019 Sébastien... | by
grw [Sum.dist, ← le_add_of_nonneg_right dist_nonneg, ← le_add_of_nonneg_left dist_nonneg]
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": "Topology/MetricSpace",
"family_id": "sum",
"file_id": "mathlib/Mathlib/Topology/MetricSpace/Gluing.lean",
"sample_id": "21c2606d6758e396ea3dedfc301c790b246ae8c817f34e167e84ed4e781597b5"
} | train |
{
"chosen_proof_sha256": "893f9954189be14b6650b8e11183442e164c9e72d10c7e09ce63fc59ebc05f3e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8eea71a463014df8b67da19c424d8a2a9e... | by
ext; simp [equiv] | true | {
"chosen_tokens": 7,
"rejected_tokens": 5,
"token_jaccard": 0.090909,
"token_length_ratio": 0.714286
} | wrong_namespace_qualification | wrong_namespace_qualification | 508c362a4cf6edc0b99aff676d8fe77e006fd29170d9a62b60e76c190c07bfef | 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.Module
public import Mathlib.RingTheory.Finiteness.Basic
Namespace:
Module.fgSystem
Local context:
/-
Copyright (c) 2024 Junyan Xu. All right... | by
exact OProverMissing.missing_e5021bc99d39 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "b207211b5acc4d6caa2214cad1f3e5e32f94e5a9b004b49c3c20cb0bfa6f1ac7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Colimit",
"family_id": "equiv_comp_of",
"file_id": "mathlib/Mathlib/Algebra/Colimit/Finiteness.lean",
"sample_id": "e5021bc99d39f6a7ba2ff9754e2b8736041ba7f4cf828fd50b3785d42ee868a1"
} | train |
{
"chosen_proof_sha256": "42ee086aa0660c3672751c6ea4362a4dc2000a25c82c17c805dafacc7e49fe69",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "893c0b78dc997ad25e432d8ba1bc777ff1... | by
have eq₁ : _ = (π₁ i, j, k).base := PreservesPullback.iso_hom_fst (forget C) _ _
have eq₂ : _ = (π₂ i, j, k).base := PreservesPullback.iso_hom_snd (forget C) _ _
rw [← eq₁, ← eq₂, TopCat.coe_comp, Set.image_comp, TopCat.coe_comp, Set.preimage_comp,
Set.image_preimage_eq]
· simp only [forget_obj, forget_m... | false | {
"chosen_tokens": 105,
"rejected_tokens": 110,
"token_jaccard": 0.911111,
"token_length_ratio": 1.047619
} | markdown_preamble_trailing | trailing | 508fb31ef39832290414b5d6c1198fa4437874eb06ea8c1e960b0ad0d2a9a960 | 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.Gluing
public import Mathlib.Geometry.RingedSpace.OpenImmersion
public import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits
Namespace:
... | by
have eq₁ : _ = (π₁ i, j, k).base := PreservesPullback.iso_hom_fst (forget C) _ _
have eq₂ : _ = (π₂ i, j, k).base := PreservesPullback.iso_hom_snd (forget C) _ _
rw [← eq₁, ← eq₂, TopCat.coe_comp, Set.image_comp, TopCat.coe_comp, Set.preimage_comp,
Set.image_preimage_eq]
· simp only [forget_obj, forget_m... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Geometry/RingedSpace",
"family_id": "pullback_base",
"file_id": "mathlib/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean",
"sample_id": "141687aaa229bc38944346a4f56eb62c28baec7cf1b9f600772bfcea49a063a4"
} | train |
{
"chosen_proof_sha256": "242a43583e27fd27079936e1ce2e8cce7893c073e1ef4300e30f594b3e61646a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ba42f23de79f77ef3eb81eed5bddcbb4d1... | by
rw [← Ideal.one_eq_top, ← Ideal.add_eq_sup, Ideal.add_eq_one_iff] at h
have ⟨y, hy, z, hz, eq⟩ := h
have ⟨p, hp⟩ := (mem_ideal_map_adjoin ..).mp hy
have ⟨w, hw⟩ := Ideal.mem_span_singleton.mp hz
have ⟨q, hq⟩ := adjoin_eq_exists_aeval R x w
use (1 - p - X * q).reverse
have : (1 - p - X * q).coeff 0 - 1 ... | false | {
"chosen_tokens": 179,
"rejected_tokens": 183,
"token_jaccard": 0.958333,
"token_length_ratio": 1.022346
} | missing_wrong_import | import | 509ece37aaf27539757a138558ca77205771841097aae099161baa2e44a99382 | 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.RingDivision
public import Mathlib.RingTheory.Adjoin.Polynomial.Basic
public import Mathlib.RingTheory.Ideal.Maps
Namespace:
Algebra
Local... | import Mathlib.WrongImport
by
rw [← Ideal.one_eq_top, ← Ideal.add_eq_sup, Ideal.add_eq_one_iff] at h
have ⟨y, hy, z, hz, eq⟩ := h
have ⟨p, hp⟩ := (mem_ideal_map_adjoin ..).mp hy
have ⟨w, hw⟩ := Ideal.mem_span_singleton.mp hz
have ⟨q, hq⟩ := adjoin_eq_exists_aeval R x w
use (1 - p - X * q).reverse
have : (... | {
"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": "RingTheory/Polynomial",
"family_id": "exists_aeval_invof_eq_zero_of_idealmap_adjoin_sup_span_eq_top",
"file_id": "mathlib/Mathlib/RingTheory/Polynomial/Ideal.lean",
"sample_id": "91739cefd9b24c38b935a570789d9ce5a71a9f8d6f001da414f1877193c9705f"
} | train |
{
"chosen_proof_sha256": "b03c7a526376dec68a09b72eb773aacc63927789fd0d788813f105124df25955",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "18fb5557e856be8fbe154a34d1876d0a93... | by
have : Algebra.IsSeparable (FractionRing A) (FractionRing R₂) := by
refine Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv A K).symm.toRingEquiv
(FractionRing.algEquiv R₂ F₂).symm.toRingEquiv ?_
ext x
exact IsFractionRing.algEquiv_commutes (FractionRing.algEquiv A K).symm
(Fra... | false | {
"chosen_tokens": 260,
"rejected_tokens": 265,
"token_jaccard": 0.953488,
"token_length_ratio": 1.019231
} | markdown_preamble_trailing | preamble | 513a28ed81dfc6bb4498c8dec54b7f6f4271deb8f50776ab29ec9b546796e7d2 | 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.LinearDisjoint
public import Mathlib.RingTheory.DedekindDomain.Different
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Xavier Roblot. Al... | Here is the proof:
by
have : Algebra.IsSeparable (FractionRing A) (FractionRing R₂) := by
refine Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv A K).symm.toRingEquiv
(FractionRing.algEquiv R₂ F₂).symm.toRingEquiv ?_
ext x
exact IsFractionRing.algEquiv_commutes (FractionRing.algEquiv A... | {
"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/DedekindDomain",
"family_id": "submodule",
"file_id": "mathlib/Mathlib/RingTheory/DedekindDomain/LinearDisjoint.lean",
"sample_id": "a86a08c70d72b421064dab529cf5e5d4ac90c62963744456fb1461e7c0e42c96"
} | train |
{
"chosen_proof_sha256": "b310f14e4f1b5c03181254cbcff43ceec8a366fb3cce3389a0bcba30d3ef880d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
by_cases h0 : p = 0
· simp [h0]
by_cases h_top : p = ∞
· simp only [h_top, eLpNorm, eLpNormEssSup, ENNReal.top_ne_zero, if_true,
if_false]
apply essSup_eq
exact h.comp (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)
simp only [eLpNorm_eq_eLpNorm' h0 h_top, eLpNorm', one_div]
congr 1
... | true | {
"chosen_tokens": 81,
"rejected_tokens": 8,
"token_jaccard": 0.044444,
"token_length_ratio": 0.098765
} | unsolved_goals | unsolved_goals | 51a3806fd13a2e5f6fe0a800f5a88efa7311e04ccac58eebaeb4934dc4df794f | 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.HasLaw
public import Mathlib.Probability.Moments.Variance
public import Mathlib.MeasureTheory.Function.UniformIntegrable
Namespace:
ProbabilityThe... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "4bda3c70154c73fd82f4f61e82204ebe6803b1d53f5220b246c409dcdfc6f123",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability",
"family_id": "elpnorm_eq",
"file_id": "mathlib/Mathlib/Probability/IdentDistrib.lean",
"sample_id": "9a2df3c5de7a292cbf62d67b3b00525a929d8c851243d6ce0af4fd45360425d2"
} | train |
{
"chosen_proof_sha256": "2fe6742b90ce6b6b787aeae843d8b7476e8ec34c6e8c54b97436e93c09e1ac78",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1144fa410a8b26bda3677497f6231039c4... | by
have : IsNoetherianRing A := Algebra.FiniteType.isNoetherianRing R A
rw [finite_iff_isArtinianRing, isArtinianRing_iff_isNoetherianRing_krullDimLE_zero,
and_iff_right this] | true | {
"chosen_tokens": 22,
"rejected_tokens": 5,
"token_jaccard": 0.090909,
"token_length_ratio": 0.227273
} | circular_dependency | circular_dependency | 51fe28b37bf58dba6860541937fc17444f1e588d8b5508e9c9e40ba714a1badb | 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.HopkinsLevitzki
public import Mathlib.RingTheory.Jacobson.Ring
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Andrew Yang. All rights rese... | by
exact Module.finite_iff_krullDimLE_zero | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "446cda6c7cb6df039c0674d67b8d800b3e4b8e9491e8d8acb1a599cae761bc37",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "RingTheory/Jacobson",
"family_id": "module",
"file_id": "mathlib/Mathlib/RingTheory/Jacobson/Artinian.lean",
"sample_id": "9edf83ad0b7479676e136f7279c23e835e46a349748bddabb519ed9ff699acc7"
} | train |
{
"chosen_proof_sha256": "ed61a64bf86bd3e8070ca4701c17ddc605afcea75c9e8aec1615eb96bd1b85a1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
induction n with
| zero => simp
| succ n IH =>
rw [iteratedLineDerivOp_succ_left]
exact (tsupport_lineDerivOp_subset (m 0) _).trans (IH <| Fin.tail m) | true | {
"chosen_tokens": 37,
"rejected_tokens": 8,
"token_jaccard": 0.03125,
"token_length_ratio": 0.216216
} | unsolved_goals | unsolved_goals | 5224d1e183f7982dd854bbb50f206fa8491ea1bcccb1939de480c55142208494 | 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.Distribution.SchwartzSpace.Basic
public import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
public import Mathlib.Analysis.Calculus.LineDeriv.Integrat... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "1c2bcbc35545db0f2f6b4e75cfecda1cfd94a3928875ff4e6ccbfd85b55e07d8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Distribution",
"family_id": "tsupport_iteratedlinederivop_subset",
"file_id": "mathlib/Mathlib/Analysis/Distribution/SchwartzSpace/Deriv.lean",
"sample_id": "591110f7b37cc2daae15e5485ae2d271b10317dae65d4419218d3c8977316858"
} | train |
{
"chosen_proof_sha256": "0cde5a74deeec0335214ee6e0227e2c1e15f0001cffe80b1335cabf0968874c8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a97e119ca71797897224fbe3e5e9ebae5d... | by
have : (p.map C).LiesOver p := ⟨IsMaximal.eq_of_le inferInstance IsPrime.ne_top' le_comap_map⟩
simp [height_eq_height_add_of_liesOver_of_hasGoingDown p] | true | {
"chosen_tokens": 28,
"rejected_tokens": 3,
"token_jaccard": 0.04,
"token_length_ratio": 0.107143
} | unknown_identifier | unknown_identifier | 523ee8122292988d06872dc4be90f95604e24a4d23992a2c82fc82806d487187 | 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.FieldDivision
public import Mathlib.RingTheory.KrullDimension.PID
public import Mathlib.RingTheory.LocalRing.ResidueField.Fiber
public impor... | by
exact __oprover_missing_80af364b54e4 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c1b4bda3c312a33e4bb71a0bb74c8b18b61cae9034c6143889889e43465aa846",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/KrullDimension",
"family_id": "height_map_c",
"file_id": "mathlib/Mathlib/RingTheory/KrullDimension/Polynomial.lean",
"sample_id": "80af364b54e4f9d5bc6e01d40bc9d559f947616a68e18a5600b526c2e866a061"
} | train |
{
"chosen_proof_sha256": "b9ea88f535ce93e2572a2904d87bf732a9a01a29fb1bf1dec81d4ec155726243",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
simp only [← orderOf_pos_iff, ← Nat.card_zpowers, Nat.card_pos_iff, ← SetLike.coe_sort_coe,
nonempty_coe_sort, Nat.card_pos_iff, Set.finite_coe_iff, OneMemClass.coe_nonempty, true_and] | false | {
"chosen_tokens": 37,
"rejected_tokens": 3,
"token_jaccard": 0.045455,
"token_length_ratio": 0.081081
} | by_exact_placeholder | exact? | 5258275ba68f60994e8b5c1eed692ec4cb604e5e395032867b6165db5a829107 | 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.ZMod.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in th... | 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": "Data/ZMod",
"family_id": "finite_zpowers",
"file_id": "mathlib/Mathlib/Data/ZMod/QuotientGroup.lean",
"sample_id": "6e268d2dae8bb1ebce0d32eba74344e7b2814801568b87e7044f89199cff8536"
} | train |
{
"chosen_proof_sha256": "d0709903e2751d33af1193290b412e76b1fe0c171376fb60a72f4abe9461ccc4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0a2ef5fc379692cad4911bcf497a95e52e... | by
rw [sub_eq_add_neg, ofCauchy_add, ofCauchy_neg]
rfl | true | {
"chosen_tokens": 10,
"rejected_tokens": 3,
"token_jaccard": 0.090909,
"token_length_ratio": 0.3
} | unknown_identifier | unknown_identifier | 52834c2e64957ca128d71c166d2b4387966bc39f47d04a22fa5642552572c755 | 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.CauSeq.Completion
public import Mathlib.Algebra.Order.Ring.Rat
public import Mathlib.Data.Rat.Cast.Defs
Namespace:
Real
Local context:
/-
Copyr... | by
exact __oprover_missing_ea366b04c54f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "b63fe3f7ff6028433c2f93702001151932c6a3fb64fa4b184c3875f9a106972b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Real",
"family_id": "ofcauchy_sub",
"file_id": "mathlib/Mathlib/Data/Real/Basic.lean",
"sample_id": "ea366b04c54fa2d802b07a4ed0e9ad19edd8f2e0e916dddae3f37ee74b249b0d"
} | train |
{
"chosen_proof_sha256": "228e4c456772f8dbd4f2347154e877aedbab995a4c162aadcd1eff84790d42d5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [mul_app']
exact x.naturality (y.app X) | false | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.133333,
"token_length_ratio": 0.2
} | by_exact_placeholder | exact? | 52a9226718fc7c6774c376dd924f81969629626d589c802fe6b6bff2cda43cc3 | 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.Endomorphism
Namespace:
CategoryTheory.CatCenter
Local context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.... | 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": "CategoryTheory/Center",
"family_id": "mul_app",
"file_id": "mathlib/Mathlib/CategoryTheory/Center/Basic.lean",
"sample_id": "d61440398863b603b3185bd9be1a7f02c573d18ffa317ffa26f3c3cf5dacdb05"
} | train |
{
"chosen_proof_sha256": "f8d69be5788c6d89c35f089a74bce8cf26bd128b40b0aa5d2082d7e6f83ba04a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ae2b1c6974f9fb9803b46bf7d64af5cfa3... | by
rw [Int.natCast_dvd]
exact Int.Prime.dvd_pow hp h | true | {
"chosen_tokens": 15,
"rejected_tokens": 2,
"token_jaccard": 0.076923,
"token_length_ratio": 0.133333
} | invalid_tactic | invalid_tactic | 52bc4acd2e279911accb7a19f4ef162c012319ac160a6bc5dc1e15fb958e4400 | 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.EuclideanDomain.Basic
public import Mathlib.Algebra.EuclideanDomain.Int
public import Mathlib.Algebra.GCDMonoid.Nat
public import Mathlib.Data.Nat.Prim... | by
__oprover_invalid_tactic_45175a44a36d | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "2bb370ac6231fd8fe46ed4dc588244fd044b09ec2d7224d9f3475a579711a364",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Int",
"family_id": "int",
"file_id": "mathlib/Mathlib/RingTheory/Int/Basic.lean",
"sample_id": "45175a44a36d53ed6e01fe3580b4a35a13cfe7dd1d0d93468e34de78ad24b899"
} | train |
{
"chosen_proof_sha256": "d23fb76aeeed9353afe60037f9d9b0091fa23e08b64b45bbb0d2d34201990525",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1311b25412146444bec8792249c758390d... | by
rw [loopOfHom, ← Category.assoc, IsIso.comp_inv_eq, Category.id_comp]
rcases H with H | H
· rw [treeHom_eq T (Path.cons default ⟨Sum.inl e, H⟩), homOfPath]
rfl
· rw [treeHom_eq T (Path.cons default ⟨Sum.inr e, H⟩), homOfPath]
simp only [IsIso.inv_hom_id, Category.comp_id, Category.assoc, treeHom] | true | {
"chosen_tokens": 86,
"rejected_tokens": 2,
"token_jaccard": 0.025641,
"token_length_ratio": 0.023256
} | invalid_tactic | invalid_tactic | 52cb3a03eef21b5c80156e080db996a9619f1f0326fcb3a580794eac2b9565b9 | 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.Action
public import Mathlib.Combinatorics.Quiver.Arborescence
public import Mathlib.Combinatorics.Quiver.ConnectedComponent
public import Mathl... | by
__oprover_invalid_tactic_0e496e19d816 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "f68b07bd56d372d9894bc57d82f97ca9532d120d110f8d7c9cc4d1e69fdd6d61",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "GroupTheory/FreeGroup",
"family_id": "loopofhom_eq_id",
"file_id": "mathlib/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean",
"sample_id": "0e496e19d8169f840fb42721b4ba0ad691cd3e68e48df661f9f6fc16f635b298"
} | train |
{
"chosen_proof_sha256": "cc4265fa4a32e1dda64a7dec7b903dcc9cceebb2f55f6e8ee131b8ddefaf2f73",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
have : Ideal.span ({a} : Set α) ≠ ⊤ := by
intro H
rw [Ideal.span_singleton_eq_top] at H
contradiction
rcases Ideal.exists_le_maximal _ this with ⟨I, Imax, H⟩
use I, Imax
apply H
apply Ideal.subset_span
exact Set.mem_singleton a | true | {
"chosen_tokens": 58,
"rejected_tokens": 8,
"token_jaccard": 0.095238,
"token_length_ratio": 0.137931
} | unsolved_goals | unsolved_goals | 52eb69386f60e79de25ff7a01786e71f7738c4184e3634ead432a31f5b27a0c3 | 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.Maximal
Namespace:
(root)
Local context:
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as describe... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "51ac8d578beefa9bf07d61413ab6a1e0d23403b5550f471af68947946066092c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Ideal",
"family_id": "exists_max_ideal_of_mem_nonunits",
"file_id": "mathlib/Mathlib/RingTheory/Ideal/Nonunits.lean",
"sample_id": "55440f62ede7bbdc77ead5e942a883e7750663cc7506c8d69af849e8b683ebab"
} | train |
{
"chosen_proof_sha256": "911bfcaad6d4d642cf91de9a824237ea9c93ccf06c46a0091532a586f961081e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "81cc30a11f3d20483daf75e9fda2f6c76a... | by
simp_rw [SetLike.ext_iff, mem_stabilizer_set]
refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_right ha⟩
have : 1 * MulOpposite.op a ∈ s := (h 1).2 s.one_mem
simpa only [op_smul_eq_mul, SetLike.mem_coe, one_mul] using! this | true | {
"chosen_tokens": 63,
"rejected_tokens": 5,
"token_jaccard": 0.044444,
"token_length_ratio": 0.079365
} | wrong_namespace_qualification | wrong_namespace_qualification | 52fe0a921ac477e6baf6a461b12b8f9c6f79ea55ff93ab783b749cdf305cb8fa | 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.Action.Pointwise.Finset
public import Mathlib.GroupTheory.QuotientGroup.Defs
public import Mathlib.Order.ConditionallyCompleteLattice.Basic
Name... | by
exact OProverMissing.missing_61d172a6c1c4 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "2104f8b3dee36930aa9cd4a99df06874328a55ffe1c940e0b2b2835ece121ddb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Algebra/Pointwise",
"family_id": "stabilizer_subgroup_op",
"file_id": "mathlib/Mathlib/Algebra/Pointwise/Stabilizer.lean",
"sample_id": "61d172a6c1c40adedfc16e781f60f95fac4d1f6ec88c11c9a5b57c8a1f496302"
} | train |
{
"chosen_proof_sha256": "a3c59538351f137c4bd7f08ab998d351346f182e72d5bba3bd7e2722c4d90e7a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
apply injective_of_mono (isoFinYonedaComponents F X).hom
simp only [Iso.inv_hom_id_apply]
ext x
rw [isoFinYonedaComponents_hom_apply]
simp only [← Functor.map_comp_apply, ← op_comp, CompHausLike.const_comp,
← isoFinYonedaComponents_hom_apply, Iso.inv_hom_id_apply, Function.comp_apply] | false | {
"chosen_tokens": 49,
"rejected_tokens": 5,
"token_jaccard": 0.030303,
"token_length_ratio": 0.102041
} | sorry_admit_sorryAx_proof_hole | sorryAx | 530922ee25ebeffbda7f97ba384cd69792eddc059c2e9498949d61ad54d53219 | 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.Condensed.Discrete.LocallyConstant
public import Mathlib.Condensed.Equivalence
public import Mathlib.Topology.Category.LightProfinite.Extend
Namespace:
Conden... | 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": "Condensed/Discrete",
"family_id": "isofinyonedacomponents_inv_comp",
"file_id": "mathlib/Mathlib/Condensed/Discrete/Colimit.lean",
"sample_id": "10fa2791fbfc644d39bf2d81c08a78f073daf00278ede35b73bb7ef9624d8ac7"
} | train |
{
"chosen_proof_sha256": "3b000d981c87ecc371d9ed638c299bf8ec24374d90b8958428a050b3089d276d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rcases ofSeq_surjective x with ⟨f, rfl⟩
rw [tendsto_ofSeq, (nhds_basis_Ioo _).tendsto_right_iff]
simp_rw [Set.mem_Ioo, eventually_and, ← ofSeq_lt_ofSeq]
refine ⟨fun H ↦ ⟨fun s hs ↦ ?_, fun s hs ↦ ?_⟩, fun H ⟨s, t⟩ ⟨hs, ht⟩ ↦ ⟨?_, ?_⟩⟩
· obtain ⟨t, ht⟩ := exists_gt r
exact (H ⟨s, t⟩ ⟨hs, ht⟩).1.le
· o... | false | {
"chosen_tokens": 186,
"rejected_tokens": 2,
"token_jaccard": 0.018868,
"token_length_ratio": 0.010753
} | sorry_admit_sorryAx_proof_hole | admit | 531ecd16b44945ecff81e82a8a5a3340ce2baba21949d3356aa7b98b12c13554 | 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.Ring.StandardPart
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Order.Filter.FilterProduct
Namespace:
Hyperreal
Loc... | 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": "Analysis/Real",
"family_id": "tendsto_iff_forall",
"file_id": "mathlib/Mathlib/Analysis/Real/Hyperreal.lean",
"sample_id": "c8ff660e3cb075cd21a289a8fe0e908d06e3a660d3865e6edea15898913062b8"
} | train |
{
"chosen_proof_sha256": "9c89a7a96cac539d3af0940d447c59a7152af477d925ba54bdc0d7f1377d3647",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
ext; simp [mem_coe, mem_extended_iff] | false | {
"chosen_tokens": 9,
"rejected_tokens": 2,
"token_jaccard": 0.1,
"token_length_ratio": 0.222222
} | sorry_admit_sorryAx_proof_hole | admit | 536b1a01a00eba9f5a204b631097a0450da83bb199cf458f6ca4240c91a063be | 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.DedekindDomain.Ideal.Lemmas
Namespace:
FractionalIdeal
Local context:
/-
Copyright (c) 2024 James Sundstrom. All rights reserved.
Released under A... | 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/FractionalIdeal",
"family_id": "coe_extended_eq_span",
"file_id": "mathlib/Mathlib/RingTheory/FractionalIdeal/Extended.lean",
"sample_id": "70071016d6f956ee807fdebd39b7e38611eccc815e374ddf4ad147924380a172"
} | train |
{
"chosen_proof_sha256": "a27f59db44cea34ebdac75d20e4acc76ed165c06147e346a2eafa82d1ac01d9a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "683dd16c2c5ea5c93eba3da8de5686a9da... | by
obtain ⟨β, _, sgn, g, hg, hβ, hf⟩ := exists_signed_sum s f
refine
⟨β ⊕ (Fin (n - ∑ i ∈ s, (f i).natAbs)), inferInstance, Sum.elim sgn 0,
Sum.elim g (Classical.arbitrary (Fin (n - Finset.sum s fun i => Int.natAbs (f i)) → α)),
?_, by simp [hβ, h], fun a ha => by simp [hf _ ha]⟩
rintro (b | b) ... | false | {
"chosen_tokens": 126,
"rejected_tokens": 130,
"token_jaccard": 0.947368,
"token_length_ratio": 1.031746
} | missing_wrong_import | import | 53a001df2e55db9be0ad446cc73bb9431d900d0eaacc9d8b20a523274ffa527b | 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.GroupWithZero.Units.Lemmas
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import Mathlib.Algebra.Order.Ring.Cast
public import Ma... | import Mathlib.WrongImport
by
obtain ⟨β, _, sgn, g, hg, hβ, hf⟩ := exists_signed_sum s f
refine
⟨β ⊕ (Fin (n - ∑ i ∈ s, (f i).natAbs)), inferInstance, Sum.elim sgn 0,
Sum.elim g (Classical.arbitrary (Fin (n - Finset.sum s fun i => Int.natAbs (f i)) → α)),
?_, by simp [hβ, h], fun a ha => 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": "Data/Sign",
"family_id": "exists_signed_sum",
"file_id": "mathlib/Mathlib/Data/Sign/Basic.lean",
"sample_id": "3236ac1bb6f48e9d26b20fb63c3f2eb332af2bf7a5e2f318caa01e95dce53a89"
} | train |
{
"chosen_proof_sha256": "2bb1a47c41a1654399bf9970e1dd3f153a3e08560ed360131ac9536f9ce93d74",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
induction s using Finset.cons_induction with
| empty => simp
| cons a s ha ihs =>
simp_rw [Finset.sum_cons, ihs]
refine iSup_add_iSup fun i j ↦ (hf i j).imp fun k hk ↦ ?_
gcongr
exacts [(hk a).1, (hk _).2] | false | {
"chosen_tokens": 63,
"rejected_tokens": 5,
"token_jaccard": 0.047619,
"token_length_ratio": 0.079365
} | sorry_admit_sorryAx_proof_hole | sorryAx | 53a41a1a9f07a7a5fb844ca8707cc4a803777bfb25edfee76742d86bf37de416 | 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.Ring.Finset
public import Mathlib.Algebra.BigOperators.WithTop
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import... | 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": "Data/ENat",
"family_id": "sum_isup",
"file_id": "mathlib/Mathlib/Data/ENat/BigOperators.lean",
"sample_id": "d14998e2d236d5892c9ed492b6063a072a0ec193e356245fa60104974e67b50c"
} | train |
{
"chosen_proof_sha256": "b0a03b5526b62c3517381ba5457b6ebaafee0e4da110c931ab1a6ee0269bdd8b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
cases p
rw [← ofFractionRing_smul] | true | {
"chosen_tokens": 8,
"rejected_tokens": 8,
"token_jaccard": 0.071429,
"token_length_ratio": 1
} | unsolved_goals | unsolved_goals | 53deb89327ca4871dc01d4ad970146679a1e338876eca5a53f77558e95ae14e2 | 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.CharP.Algebra
public import Mathlib.FieldTheory.RatFunc.Defs
public import Mathlib.RingTheory.Polynomial.Content
public import Mathlib.RingTheory.Algeb... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "de9878b2bdbca9d755e0906ee6cc2f48bbd71f044a7ac728855c52af26d751a4",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/RatFunc",
"family_id": "tofractionring_smul",
"file_id": "mathlib/Mathlib/FieldTheory/RatFunc/Basic.lean",
"sample_id": "5b0c69fc729fb329df53e747032d70f91d7a865fc26c9ee5566a2b68840871a7"
} | train |
{
"chosen_proof_sha256": "06f47103349d9ae4b5fa9c701936477e3308e59432a0c09cdb18eb8a9e2de388",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
contrapose!; exact parts_eq_empty_iff | false | {
"chosen_tokens": 6,
"rejected_tokens": 2,
"token_jaccard": 0.142857,
"token_length_ratio": 0.333333
} | sorry_admit_sorryAx_proof_hole | admit | 53df40ab2673253f5c170c53ccd444d6ff208859f1e5d21d2b9acb337a68a58b | 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.Group.Finset.Basic
public import Mathlib.Data.Finset.Pairwise
public import Mathlib.Data.Finset.Preimage
public import Mathlib.Data.Fintyp... | 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": "Order/Partition",
"family_id": "parts_nonempty_iff",
"file_id": "mathlib/Mathlib/Order/Partition/Finpartition.lean",
"sample_id": "ea19c96b1833a290f3ad4a2c8868771d2cf2aee1459ba1b113ef3ef0cf9c4a4c"
} | train |
{
"chosen_proof_sha256": "82e688aad755eac6f657be89199e8e6fa7e63dacb10bb688605e04b2c9309872",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "63fa03af2984e0af65d6d9483e8e9d588a... | by
suffices IsSquare (-x) → x = 0 by aesop
exact fun hc ↦ le_antisymm (nonpos_of_neg_nonneg (IsSquare.nonneg hc)) h
mpr := IsSquare.nonneg
alias ⟨_root_.IsSquare.of_nonneg, _⟩ := nonneg_iff_isSquare | true | {
"chosen_tokens": 45,
"rejected_tokens": 2,
"token_jaccard": 0.032258,
"token_length_ratio": 0.044444
} | invalid_tactic | invalid_tactic | 54207e52c7379db8f6b3476978c04053753edc60d1b4596f809603dfbb42caac | 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.Degree.Domain
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.Algebra.Ring.Semireal.Defs
public import Mathlib.Tact... | by
__oprover_invalid_tactic_d3b554a7490c | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "2f24929c52e2499bdb8c260c48f34f98dcf32c92bf85d06536fa8e20bd1afe07",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/IsRealClosed",
"family_id": "nonneg_iff_issquare",
"file_id": "mathlib/Mathlib/FieldTheory/IsRealClosed/Basic.lean",
"sample_id": "d3b554a7490cfc938221d5d7fff9423a8ad834bad53b65c4db2638f83535b57e"
} | train |
{
"chosen_proof_sha256": "f4a24f2bb7a6f0211b6b657b7e7dccc3a2dcf7c7a859099b25ae3a21d107e6f6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
simpa only [← hammingDist_zero_right, hf]
using! hammingDist_comp_le_hammingDist f (y := fun _ ↦ 0) | false | {
"chosen_tokens": 21,
"rejected_tokens": 3,
"token_jaccard": 0.043478,
"token_length_ratio": 0.142857
} | by_exact_placeholder | exact? | 54259e3ecfb005214b7de48f5f4540bea5b1d90ad5807fc619aa08d9dd82d3ae | 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.Normed.Group.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as d... | 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": "InformationTheory",
"family_id": "hammingnorm_comp_le_hammingnorm",
"file_id": "mathlib/Mathlib/InformationTheory/Hamming.lean",
"sample_id": "955e34b4be7edd56dadecaa66b57be2059d1e36951adf123ec19009bee070aac"
} | train |
{
"chosen_proof_sha256": "0c3061002fd6080e1b0c2c6f335b3c8a50d21cd02337f3272c2365c7c29242da",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0ba20bcd7c6a0c24655c4caacb4b814585... | by
obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (NeZero.ne n)
intro i
simpa using (h i).injective.ne (by decide : (0 : Fin 3) ≠ 1) | true | {
"chosen_tokens": 42,
"rejected_tokens": 5,
"token_jaccard": 0.0625,
"token_length_ratio": 0.119048
} | circular_dependency | circular_dependency | 54811d683d74bd35a80c35c2a04d342258fb9658f8fa224bd40aeedc81e73173 | 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.Convex.Between
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.Tactic.Continuity
Namespace:
Polygon
Local context:
/-
Copyright (c) 20... | by
exact HasNondegenerateVertices.hasNondegenerateEdges | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "9f2d654676167592250311d4b0a82e279b0df8ca10cddf034881b7271bd0eef0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Geometry/Polygon",
"family_id": "hasnondegeneratevertices",
"file_id": "mathlib/Mathlib/Geometry/Polygon/Basic.lean",
"sample_id": "1a320ce7ba94d641553130a5996d892f469c6c053d11bc8e9c32a512ac9b6215"
} | train |
{
"chosen_proof_sha256": "01a9a5bebe8404a63c3e4dd6c8ed6a63ca7ab9f3512e93006d4e0432218d9a02",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c4b589473919a5f5307e8873d42d412d03... | by
let ⟨⟨κ, b⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [(equivFreeAlgebra b).toLinearEquiv.rank_eq, FreeAlgebra.rank_eq, mk_list_eq_sum_pow,
Basis.mk_eq_rank'' b] | false | {
"chosen_tokens": 47,
"rejected_tokens": 51,
"token_jaccard": 0.896552,
"token_length_ratio": 1.085106
} | missing_wrong_import | import | 548bcfaee18e6dd3b4b8ee64866abc636af2388cf1a8ad327758a817c3dcc7d1 | 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.FreeAlgebra
Namespace:
TensorAlgebra
Local context:
/-
Copyright (c) 2023 Eric Wieser. ... | import Mathlib.WrongImport
by
let ⟨⟨κ, b⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [(equivFreeAlgebra b).toLinearEquiv.rank_eq, FreeAlgebra.rank_eq, mk_list_eq_sum_pow,
Basis.mk_eq_rank'' b] | {
"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": "LinearAlgebra/TensorAlgebra",
"family_id": "rank_eq",
"file_id": "mathlib/Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean",
"sample_id": "d99fd6fb9456101e1d28de19e801f40e15b75e0d01ea3c483f56e794b32e60bf"
} | train |
{
"chosen_proof_sha256": "c8dca49869410f2bacfcd9d441c978765c739555c35836f4fba20bb9b5e2825f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6af85baa030133d662458b29f177a94490... | by
cases h : compare a b
· simpa using le_of_lt <| compare_lt_iff_lt.1 h
· simpa using le_of_eq <| compare_eq_iff_eq.1 h
· simpa using compare_gt_iff_gt.1 h | true | {
"chosen_tokens": 34,
"rejected_tokens": 5,
"token_jaccard": 0.090909,
"token_length_ratio": 0.147059
} | wrong_namespace_qualification | wrong_namespace_qualification | 54a2ae9b9ca155ce8342a440d78938aab02cdb6bacfbc58687c308e52f85bd11 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Classes.Order
public import Batteries.Tactic.Trans
public import Mathlib.Data.Ordering.Basic
public import Mathlib.Tactic.ExtendDoc
public import Mathlib.Tac... | by
exact OProverMissing.missing_34c67d098470 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "2e1909660ac264933960bb0f0fde3367ca57c38790ec68461a6a41e4dc566e37",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Defs",
"family_id": "compare_le_iff_le",
"file_id": "mathlib/Mathlib/Order/Defs/LinearOrder.lean",
"sample_id": "34c67d0984704f562f600f9107ee26cdff7c7034435d16dae2284506bc3e8668"
} | train |
{
"chosen_proof_sha256": "8da08d0db4cb5d343e5a5a407377b94360b60c72eeb504f2655147b8545bacfb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "31e49ca21c3f68d25f9457b6c613ab9945... | by
cases isEmpty_or_nonempty α <;> simp [FreeCommRing] | false | {
"chosen_tokens": 11,
"rejected_tokens": 16,
"token_jaccard": 0.6875,
"token_length_ratio": 1.454545
} | markdown_preamble_trailing | preamble | 54f40f60542f8940ca6cd1683a8e551e9299ecef939a1e02bf77583ef0dcc655 | 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.FreeAbelianGroup.Finsupp
public import Mathlib.Algebra.Ring.TransferInstance
public import Mathlib.Data.Finsupp.Fintype
public import Mathlib.Data.ZMod... | Here is the proof:
by
cases isEmpty_or_nonempty α <;> simp [FreeCommRing] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "SetTheory/Cardinal",
"family_id": "mk_freecommring",
"file_id": "mathlib/Mathlib/SetTheory/Cardinal/Free.lean",
"sample_id": "24a5ad421e4a7eee0e5feabb06c49c3c686c8f596032a3cb50682135e2ce5773"
} | train |
{
"chosen_proof_sha256": "6124388c54e3601f5b392168c48f6a150dee4519aaf6dbd646732ffc8061300b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d59287e3947f846b3ad368db2e503ad465... | by
ext x <;> simp | true | {
"chosen_tokens": 7,
"rejected_tokens": 3,
"token_jaccard": 0.111111,
"token_length_ratio": 0.428571
} | circular_dependency | circular_dependency | 5508cb0c2a2461fbfe8be45a8f7d49c26f5c692cf0a886d19f4e484c4fc86e2b | 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.RootSystem.Basic
public import Mathlib.LinearAlgebra.RootSystem.Defs
Namespace:
RootPairing.Hom
Local context:
/-
Copyright (c) 2024 Scott Carn... | by
exact comp_id | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "9da21032eee273c05bf700eee8814e0d8c29e06eddb9e1b87b1ca60f25501287",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/RootSystem",
"family_id": "comp_id",
"file_id": "mathlib/Mathlib/LinearAlgebra/RootSystem/Hom.lean",
"sample_id": "84f48056d33f497341acedeaa75d75fead8d9c6cb4f1d4df07a45458a9009e00"
} | train |
{
"chosen_proof_sha256": "a956e0393455f70ac3f0634d07a69fa1a653d99212eafe809fbf92b858839bc8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
constructor
· induction fs generalizing t with
| nil =>
simp only [sequence, mem_pure, imp_self, forall₂_nil_left_iff, exists_eq_left, Set.pure_def,
singleton_subset_iff, traverse_nil]
| cons b fs ih =>
intro ht
rcases mem_seq_iff.1 ht with ⟨u, hu, v, hv, ht⟩
rcases mem_ma... | false | {
"chosen_tokens": 127,
"rejected_tokens": 2,
"token_jaccard": 0.016393,
"token_length_ratio": 0.015748
} | sorry_admit_sorryAx_proof_hole | admit | 554f1933643c5a10f33fcba815ead4f78471ee9fda92bad0578fb8484c07e2de | 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.Control.Traversable.Instances
public import Mathlib.Order.Filter.Map
Namespace:
Filter
Local context:
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserv... | 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": "Order/Filter",
"family_id": "mem_traverse",
"file_id": "mathlib/Mathlib/Order/Filter/ListTraverse.lean",
"sample_id": "a1347b35a1da940b0cf7bb9fa19ec8fb67a45c4aec52bd9f946f2c2f04d77b83"
} | train |
{
"chosen_proof_sha256": "e13f4e0fbff74fe329154bb27fe6656b950cc8d302959791efca88cd4237a5ab",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
refine ⟨fun h => ?_, fun ⟨h₁, h₂⟩ => Equiv.antisymm h₁ h₂⟩
obtain - | ⟨h₁, h₂⟩ := h
· simp [Lists'.Subset.refl]
· exact ⟨h₁, h₂⟩ | false | {
"chosen_tokens": 56,
"rejected_tokens": 2,
"token_jaccard": 0.074074,
"token_length_ratio": 0.035714
} | by_exact_placeholder | by? | 55ac925c075dda9902643abf20a731e3b1d24eb2aa212a85281012e74a5c4069 | 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.Group.Nat
public import Mathlib.Algebra.Order.Monoid.NatCast
public import Mathlib.Algebra.Ring.Nat
public import Mathlib.Data.Sigma.Basic
public... | 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": "SetTheory",
"family_id": "islist_of_mem",
"file_id": "mathlib/Mathlib/SetTheory/Lists.lean",
"sample_id": "727321c81fb49fb44e787715ddf3c0aafbedfa69ef549affbe7e565a44f0be51"
} | train |
{
"chosen_proof_sha256": "f4dbb4c53a6e3a2d5b935a93c9fc29cc9b719e218918f120b3b408ba8d371698",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "da2067aad04e85cd3a30fda8e64f9e051d... | by
unfold logDeriv
simp [field, Derivation.leibniz_div] | true | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.272727
} | circular_dependency | circular_dependency | 56752ee8d3d9ab9b8d9bee9c069009f17c220a55ecf516756c69f42ea72c3212 | 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.Derivation.MapCoeffs
public import Mathlib.FieldTheory.PrimitiveElement
Namespace:
Differential
Local context:
/-
Copyright (c) 2024 Daniel Weber.... | by
exact logDeriv_div | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "eb653610184ca293a54e6fc6b94ba94bc9bca30812cf9eab257e3854b221bd61",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/Differential",
"family_id": "logderiv_div",
"file_id": "mathlib/Mathlib/FieldTheory/Differential/Basic.lean",
"sample_id": "5a2ece2a22fcf936fd41bf7ec3c1d0ec84ea941a229b765b720040afa6959a7e"
} | train |
{
"chosen_proof_sha256": "e79d2c183534cfffc99b615577f1db04cdc59929d66d45cf7905520be00385d4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1928d6ae6dc61f8cdfa63dfcb56b1f379f... | 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))... | true | {
"chosen_tokens": 429,
"rejected_tokens": 2,
"token_jaccard": 0.011628,
"token_length_ratio": 0.004662
} | invalid_tactic | invalid_tactic | 5692b1e8ac29d5c1a49e338e52e72fc95aef0a9c746d45866b05ddd94030b6da | 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.Subgroup.Pointwise
public import Mathlib.Combinatorics.Additive.CovBySMul
public import Mathlib.Combinatorics.Additive.RuzsaCovering
public impor... | by
__oprover_invalid_tactic_5627650f4b11 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "669e6640dcf03570b34e698c51aae8f7b19206fbeb47941dfd0b6b1a8af24901",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "f30223a701d84b8cf4910174a2ebd64c60a3db6a916ec5748abb240a8d7ca7c4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simpa using (toInt_le_toInt val j i).not | true | {
"chosen_tokens": 11,
"rejected_tokens": 8,
"token_jaccard": 0.058824,
"token_length_ratio": 0.727273
} | unsolved_goals | unsolved_goals | 56ae7c939929edcd14e39dfeb778b8bc5dad11b61816f1feba6cdb641f85ba88 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Data.List.Pairwise
public import Batteries.Tactic.GeneralizeProofs
public import Mathlib.Tactic.Order.CollectFacts
Namespace:
Mathlib.Tactic.Order.ToInt
Lo... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "a9c18c9a47d73bd29dc8909c0fc1d328f2ce5842f7dcfa21a0df684a9389fc7c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/Order",
"family_id": "toint_lt_toint",
"file_id": "mathlib/Mathlib/Tactic/Order/ToInt.lean",
"sample_id": "766f3502274c6a836f440a41e6af60197e8da15e11cc99e0e09ed8fb000b1581"
} | train |
{
"chosen_proof_sha256": "cff2f50ea6340f1c0d647e96a5ab7bcbcc74041793ed6f6cea3ea0d685f9013d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "abe2f0c7fc30ae281c7ff189951bcaa472... | by
intro s hs
induction hs with
| base t t_mem => exact self_subset_generateSetAlgebra (h t_mem)
| empty => exact isSetAlgebra_generateSetAlgebra.empty_mem
| compl t _ t_mem => exact isSetAlgebra_generateSetAlgebra.compl_mem t_mem
| union t u _ _ t_mem u_mem => exact isSetAlgebra_generateSetAlgebra.union_me... | true | {
"chosen_tokens": 51,
"rejected_tokens": 5,
"token_jaccard": 0.103448,
"token_length_ratio": 0.098039
} | wrong_namespace_qualification | wrong_namespace_qualification | 56bf80222f5ef214403484d2405998564d1ad9edd7200a06255af78ff1960025 | 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.Finite.Prod
public import Mathlib.MeasureTheory.SetSemiring
Namespace:
MeasureTheory
Local context:
/-
Copyright (c) 2024 Etienne Marion. All rights res... | by
exact OProverMissing.missing_865430ae7ad7 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "060547565e9a4dc8913d6bb4c86217cd0dd07ec697d6f20db2a4b4b343a100de",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory",
"family_id": "generatesetalgebra_mono",
"file_id": "mathlib/Mathlib/MeasureTheory/SetAlgebra.lean",
"sample_id": "865430ae7ad7a3af9b967fd8cc81590df7b116ddc79d82d6edeca41aba827e49"
} | train |
{
"chosen_proof_sha256": "f11a99acaff28e0866b0cf9264b8d87a92a312c57ace97d7c08c3310cdd9276a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [Finset.offDiag_filter_lt_eq_filter_le]
conv_rhs => rw [← Finset.sum_subtype_eq_sum_filter]
refine (Finset.sum_equiv Sym2.sortEquiv.symm ?_ ?_).symm
all_goals aesop | false | {
"chosen_tokens": 35,
"rejected_tokens": 3,
"token_jaccard": 0.086957,
"token_length_ratio": 0.085714
} | by_exact_placeholder | exact? | 56c6ef6628ceaf35cf25b4584a3da3371fa8d2b58dd4c39b0364e2a1a833eaa0 | 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.Finset.Sym
public import Mathlib.Data.Sym.Sym2.Order
public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
Namespace:
Finset
Local context:
/-
C... | 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": "Algebra/BigOperators",
"family_id": "sum_sym2_filter_not_isdiag",
"file_id": "mathlib/Mathlib/Algebra/BigOperators/Sym.lean",
"sample_id": "5013414c846a303e2d7ae6cf16e87bd0d86e0af5364ad267930855e6631a1174"
} | train |
{
"chosen_proof_sha256": "693a609014e39cacfdbaa86bb0c471973c1d57e2dd7fdea8e3f828ddfc7bc6a5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "01d2f765d25d3c26ead89a2832e8a488c1... | by
refine RelEmbedding.wellFounded_iff_isEmpty.mpr ⟨fun a => ?_⟩
suffices sSup (Set.range a) ∈ Set.range a by
obtain ⟨n, hn⟩ := Set.mem_range.mp this
have h' : sSup (Set.range a) < a (n + 1) := by
change _ > _
simp [← hn, a.map_rel_iff]
apply lt_irrefl (a (n + 1))
apply l... | true | {
"chosen_tokens": 141,
"rejected_tokens": 2,
"token_jaccard": 0.016393,
"token_length_ratio": 0.014184
} | invalid_tactic | invalid_tactic | 56c6ef9aa5a7837ac1fd6948ef0de2a3b32e959c44781abf6df08e9c1bc19ff5 | 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.Order.Atoms
public import Mathlib.Order.OrderIsoNat
public import Mathlib.Order.RelIso.Set
public import Mathlib.Order.SupClosed
public import Mathlib.Order.Su... | by
__oprover_invalid_tactic_535e482a4908 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e8d8ba2400f522352e252c3b72c364f3ff83333fd3670dffa56dc5c6ddc78702",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/CompactlyGenerated",
"family_id": "issupclosedcompact",
"file_id": "mathlib/Mathlib/Order/CompactlyGenerated/Basic.lean",
"sample_id": "535e482a49088acaa3ff943705a11086429404a3e1464c08016c546a7309ccb5"
} | train |
{
"chosen_proof_sha256": "7fb3cc1775db12a3f434a99b04b7c0a0f79496406a0eb39713c9f5d39c18a786",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8e9095fa0fd358aa442905acbee67ff7e4... | by
rw [sInter_eq_biInter]
exact MeasurableSet.biInter hs h | true | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.166667,
"token_length_ratio": 0.272727
} | unknown_identifier | unknown_identifier | 56ccdf872a2b782017832b60c771462302e196e514f9d1320284f884cb30ccc1 | 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 __oprover_missing_b9d6520e23cf | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a5248173b206fd8a8128faa1c226b843bc44e38d1e25b58dc3db20a5782224c3",
"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": "b9d6520e23cf7ef508708d905f3af667f91b01c1501bdfb215036dc08f82f360"
} | train |
{
"chosen_proof_sha256": "59cc5684ec22bcbe1a47c40b86884f4ebd0253cacb27e5dc09b1539a4441b454",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
simpa [ChainComplex.of.d] using ((inhomogeneousChains A).d_comp_d (n + 2) (n + 1) n) | false | {
"chosen_tokens": 29,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.103448
} | by_exact_placeholder | placeholder | 575b8df59cf6c8a9c846a363125d9bfbd4a38754ce5172f69ed33cc0a19b8ac4 | 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.Homology.ConcreteCategory
public import Mathlib.RepresentationTheory.Coinvariants
public import Mathlib.RepresentationTheory.Homological.Resolution
pub... | 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": "RepresentationTheory/Homological",
"family_id": "inhomogeneouschains",
"file_id": "mathlib/Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean",
"sample_id": "dbf6ffda00f48d6763a755f6b1baea6be9639bc7a19fe67ae570fed274326119"
} | train |
{
"chosen_proof_sha256": "407de8aea2c8c096a4b6c72b70c997e3bb00015ef6148a6a35ed2aaaed552283",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b0eb9dca5eadf93208ff9a93aeaae01b29... | by
obtain ⟨T', e, h'⟩ := hF T hT
exact (ShortComplex.exact_iff_of_iso
(F.mapShortComplex.mapIso ((shortComplexOfDistTriangleIsoOfIso e hT)))).2 h' | true | {
"chosen_tokens": 36,
"rejected_tokens": 5,
"token_jaccard": 0.12,
"token_length_ratio": 0.138889
} | wrong_namespace_qualification | wrong_namespace_qualification | 57654030e90807aefc68d361d835d2671f8d382a1dbe9aa740bc461c3db6ab3b | 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.Homology.ShortComplex.Exact
public import Mathlib.CategoryTheory.Shift.ShiftSequence
public import Mathlib.CategoryTheory.Triangulated.Functor
public i... | by
exact OProverMissing.missing_99a463e8b538 | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "ce8f3ce39a285957bb0c08bd803f77516ad2c33a66ea13ba64029dc0e6154946",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Triangulated",
"family_id": "ishomological",
"file_id": "mathlib/Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean",
"sample_id": "99a463e8b538cc1129c05262ca34214f41e74d8600f9e3aa84ec10439902b1f8"
} | train |
{
"chosen_proof_sha256": "8b8c47751a3cd16893a65f53a169d429e22504bc42c9299e12e8db816e1165e6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
classical
induction s using Finset.induction generalizing j with
| empty => simp [coeff_one, apply_ite, isIntegral_zero, isIntegral_one]
| insert a s has IH =>
rw [Finset.prod_insert has, coeff_mul]
exact IsIntegral.sum _ fun i hi ↦ .mul (H _ (by simp) _) (IH (fun _ _ ↦ H _ (by aesop)) _) | false | {
"chosen_tokens": 76,
"rejected_tokens": 2,
"token_jaccard": 0.02381,
"token_length_ratio": 0.026316
} | sorry_admit_sorryAx_proof_hole | sorry | 57a2913553354a4249f36a6e43e5e817282ab87c0ae7bbac929161f04d7a3889 | 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.Multiset.Fintype
public import Mathlib.RingTheory.AdjoinRoot
public import Mathlib.RingTheory.Polynomial.RationalRoot
public import Mathlib.RingTheory.Int... | 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": "RingTheory/Polynomial",
"family_id": "isintegral_coeff_prod",
"file_id": "mathlib/Mathlib/RingTheory/Polynomial/IsIntegral.lean",
"sample_id": "a559373815454d0166c5dab8fbbe7db64a117d1cd999728a36144f94e03e6450"
} | train |
{
"chosen_proof_sha256": "31a2975efba554cce04bfa852066668eb27b98da82a9d33652d228e97ecec653",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e1358bd4ad7cf3e79ebde6f12a407b8ab6... | by
rw [removeFactor, natDegree_divByMonic _ (monic_X_sub_C _), natDegree_map, natDegree_X_sub_C] | true | {
"chosen_tokens": 16,
"rejected_tokens": 5,
"token_jaccard": 0.058824,
"token_length_ratio": 0.3125
} | wrong_namespace_qualification | wrong_namespace_qualification | 580b736f64034e0c9dde6c8163e1d78aad7ef24d4073614005343df6c3f7334a | 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.CharP.Algebra
public import Mathlib.FieldTheory.SplittingField.IsSplittingField
public import Mathlib.RingTheory.Algebraic.Basic
Namespace:
Polynomial... | by
exact OProverMissing.missing_7e8c17610a3b | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c4e097d2e5270f07c3fc95d5c3727ad68846f1a480964a231e43128455b3d092",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/SplittingField",
"family_id": "natdegree_removefactor",
"file_id": "mathlib/Mathlib/FieldTheory/SplittingField/Construction.lean",
"sample_id": "7e8c17610a3baa53fbdd954115a7d7f9ea7c655e0a0e548683cb4072c1194356"
} | train |
{
"chosen_proof_sha256": "17f240f800994d7074febbde9e3f6dc2b5012103ed8b58582f7bf051f16e73b0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "71b5431732be50bdd6b70dfcf19109b211... | by
intro b₁ b₂ h
simpa only [and_true, eq_self_iff_true, mk_inj] using h | true | {
"chosen_tokens": 18,
"rejected_tokens": 2,
"token_jaccard": 0.0625,
"token_length_ratio": 0.111111
} | invalid_tactic | invalid_tactic | 5867f91fb5358c488a96554e8f024260fb11a11ac630cd3f2b05d8c8c663b0d4 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Lean.PrettyPrinter.Delaborator.Builtins
public import Mathlib.Logic.Function.Defs
public import Mathlib.Logic.Function.Iterate
public import Mathlib.Tactic.Inhabit
pub... | by
__oprover_invalid_tactic_dd825554f724 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "3b0b53c6afff56ef38aa0c0f1ea09f08eef30e10813f7c00134bb70bfb00fd41",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Prod",
"family_id": "mk_left_injective",
"file_id": "mathlib/Mathlib/Data/Prod/Basic.lean",
"sample_id": "dd825554f724a499bc7ed68c1db6d9b3cab6e04c69fa70bc2acb5b885250233e"
} | train |
{
"chosen_proof_sha256": "badd85084558f6da2abee0803d9cfe41d37ed7236c401bd880c419ea46047156",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "37ef8d929e9bcfe001f6766d7178835316... | by
rw [lt_univ']
constructor
· rintro ⟨β, e⟩
exact ⟨#β, lift_mk_eq.{u, _, v + 1}.2 e⟩
· rintro ⟨c, hc⟩
exact ⟨⟨c.out, lift_mk_eq.{u, _, v + 1}.1 (hc.trans (congr rfl c.mk_out.symm))⟩⟩ | false | {
"chosen_tokens": 76,
"rejected_tokens": 81,
"token_jaccard": 0.897436,
"token_length_ratio": 1.065789
} | markdown_preamble_trailing | trailing | 587b1e609a8abb0fe56a4eb42f42ab21da66f44e1cff5588bbf034106c5df7ab | 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.SetTheory.Ordinal.Basic
Namespace:
Cardinal
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as de... | by
rw [lt_univ']
constructor
· rintro ⟨β, e⟩
exact ⟨#β, lift_mk_eq.{u, _, v + 1}.2 e⟩
· rintro ⟨c, hc⟩
exact ⟨⟨c.out, lift_mk_eq.{u, _, v + 1}.1 (hc.trans (congr rfl c.mk_out.symm))⟩⟩
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/Ordinal",
"family_id": "small_iff_lift_mk_lt_univ",
"file_id": "mathlib/Mathlib/SetTheory/Ordinal/Univ.lean",
"sample_id": "73b36450d7beabc8b542d5bd6e54c7d1928d98c7ccc3d417ffbb153021170e98"
} | train |
{
"chosen_proof_sha256": "38de9a26cb91dd01aaa9b7d35dd028817eb092a585a923cb7e7b911ff5d759bb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
refine fun _ _ => ⟨1, by norm_num, 1, by norm_num, ?_⟩
filter_upwards [] with x
simp | true | {
"chosen_tokens": 26,
"rejected_tokens": 8,
"token_jaccard": 0.043478,
"token_length_ratio": 0.307692
} | unsolved_goals | unsolved_goals | 58d5a99c0c6eff27aa92b8e126798666f1d41493e3a14476bb8c35181062a4cb | 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
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "2695022722712bd1e5816bd8eef6d9b823f970978310953830938acd5f1e77de",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Computability/AkraBazzi",
"family_id": "growspolynomially_const",
"file_id": "mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean",
"sample_id": "836ef9c3cd55ae2e3f293b4d27ae834545f003ae5a97617e40dffc59a7b9fdae"
} | train |
{
"chosen_proof_sha256": "af1766a439a78c83ac802fa03802f5aeae0b2eb140261f6b10e3a3388ff5d5e4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
let f (u : R) : R × R := (star u * u, u * star u)
have hf : f ⁻¹' {(1, 1)} = unitary R := by ext u; simp [f, Unitary.mem_iff]
rw [← hf]
exact isClosed_singleton.preimage (by fun_prop) | false | {
"chosen_tokens": 67,
"rejected_tokens": 2,
"token_jaccard": 0.026316,
"token_length_ratio": 0.029851
} | sorry_admit_sorryAx_proof_hole | sorry | 595b3d9c7858b19deac7542cf57063f8dab1aeea11d5f2f7c27de5567129ef61 | 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.Unitary
public import Mathlib.Topology.Algebra.Group.Defs
public import Mathlib.Topology.Algebra.Star
public import Mathlib.Topology.Algebra.Monoi... | 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": "Topology/Algebra",
"family_id": "isclosed_unitary",
"file_id": "mathlib/Mathlib/Topology/Algebra/Star/Unitary.lean",
"sample_id": "22f2feedaac2189bc2f75b488121e482511637bba540133178964abac118fe65"
} | train |
{
"chosen_proof_sha256": "76fdca8112ad5b228e441ce2f0fc6042386b4c4cdeafde082a85932a16c29411",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "207dc58f8ca84bc6887d256453cd927a01... | by
simp_rw [← toDual_covBy_toDual_iff (α := α)] at hfin ⊢
exact exists_covby_infinite_Ici_of_infinite_Ici (α := αᵒᵈ) ha hfin | false | {
"chosen_tokens": 25,
"rejected_tokens": 32,
"token_jaccard": 0.9,
"token_length_ratio": 1.28
} | markdown_preamble_trailing | fence | 598816a90977e25addaa7a53face0a23876802e74962e119601ee902534a0f74 | 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.Set.Finite.Lattice
public import Mathlib.Order.Atoms
public import Mathlib.Order.Interval.Finset.Defs
public import Mathlib.Order.Preorder.Finite
Namespa... | ```lean
by
simp_rw [← toDual_covBy_toDual_iff (α := α)] at hfin ⊢
exact exists_covby_infinite_Ici_of_infinite_Ici (α := αᵒᵈ) ha hfin
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/Atoms",
"family_id": "exists_covby_infinite_iic_of_infinite_iic",
"file_id": "mathlib/Mathlib/Order/Atoms/Finite.lean",
"sample_id": "8ceba723b8cfca101595c85d0f256daa590d35c274c45371de9577614b0b3c30"
} | train |
{
"chosen_proof_sha256": "61f4608135cc80db3b144a8f0b385332ee17b58f1c351db640ceccbb2a9d7cc8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "fe4564cce88cf91d6bce5cf6a3f2448a73... | by
ext x; grind [eq_one_or_toFreeMonoid x] | true | {
"chosen_tokens": 9,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.333333
} | circular_dependency | circular_dependency | 599ba9e85a7fad7dc5f4665c704b9f0eb621d4f70f2227db14af4acdbe8dd7b7 | 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.FreeMonoid.Basic
public import Mathlib.Algebra.Free
public import Mathlib.Algebra.Group.WithOne.Basic
public import Mathlib.Algebra.Group.Units.Basic
p... | by
exact range_toFreeMonoid | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "e4f37dc6e55fa6e45fa46154a37c990cc00f400ad3da1e6afa6d451ec054f775",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Algebra/FreeMonoid",
"family_id": "range_tofreemonoid",
"file_id": "mathlib/Mathlib/Algebra/FreeMonoid/FreeSemigroup.lean",
"sample_id": "6a4d757b91c5500aa6ea19c37733ffcdf411c7c5b0b8869100541b0d79adff6c"
} | train |
{
"chosen_proof_sha256": "3d02c82bbfbe829ee560bc14711178f8e05f9f6e82051c0e8f4a63b14e8d38c0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
let φ (i : ι) := separatingFamily c (fun (_ : Unit) ↦ X) ⟨⟨⟩, i⟩
refine isSeparator_of_isColimit_cofan
(isSeparating_separatingFamily c (X := fun (_ : Unit) ↦ X) (by simpa using! hX))
(c := Cofan.mk (∐ φ) (fun ⟨_, i⟩ ↦ Sigma.ι φ i)) ?_
exact IsColimit.ofWhiskerEquivalence
(Discrete.equivalence (E... | false | {
"chosen_tokens": 104,
"rejected_tokens": 2,
"token_jaccard": 0.022222,
"token_length_ratio": 0.019231
} | sorry_admit_sorryAx_proof_hole | admit | 59f24280146f01a5bef105407893c9e7bb3155520b272ec67591552315f0746c | 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.Homology.Double
public import Mathlib.Algebra.Homology.HomologicalComplexLimits
public import Mathlib.CategoryTheory.Generator.Basic
Namespace:
Homolo... | 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": "CategoryTheory/Generator",
"family_id": "isseparator_coproduct_separatingfamily",
"file_id": "mathlib/Mathlib/CategoryTheory/Generator/HomologicalComplex.lean",
"sample_id": "595e91c0da49b46a91c3741f0089b51f2875fc10f72a63c9a56e6dda6a7ca6c2"
} | train |
{
"chosen_proof_sha256": "3dc6c6b95959f83835d9501fcfa554ee77808794954cb31229e8e1de18edb21b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
have : NeZero (Nat.card G) := ⟨Nat.card_pos.ne'⟩
have hord := HasEnoughRootsOfUnity.natCard_rootsOfUnity M (Nat.card G)
let e := (IsCyclic.monoidHom_mulEquiv_rootsOfUnity G Mˣ).some
exact ⟨e.trans (rootsOfUnityUnitsMulEquiv M (Nat.card G)) |>.trans (mulEquivOfCyclicCardEq hord)⟩ | false | {
"chosen_tokens": 68,
"rejected_tokens": 3,
"token_jaccard": 0.0625,
"token_length_ratio": 0.044118
} | by_exact_placeholder | placeholder | 5a67ea9e6f37cf3f1f1c0badc524bb182372931ea352913157ee9b608a998911 | 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.RootsOfUnity.PrimitiveRoots
Namespace:
(root)
Local context:
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 l... | 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/RootsOfUnity",
"family_id": "iscyclic",
"file_id": "mathlib/Mathlib/RingTheory/RootsOfUnity/EnoughRootsOfUnity.lean",
"sample_id": "3f67b4910954a3acaaa3087e7993d00a824bcfea76d08a647dfda9454dd07897"
} | train |
{
"chosen_proof_sha256": "d02b7bd3abbc9b6e64a37811efb4eb652c5a1b34ba47dca636dd232d861f2cb3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0f11cb72d8fdf85708f09496e1d4ec940b... | by
ext
simpa [Units.ext_iff] using pow_card_sub_one_eq_one (Units.ne_zero _) | false | {
"chosen_tokens": 16,
"rejected_tokens": 21,
"token_jaccard": 0.777778,
"token_length_ratio": 1.3125
} | markdown_preamble_trailing | trailing | 5a750ad150523f2dee0ece8f31d0d5f35b75f887961d579a9e841a7d8779141c | 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.RootsOfUnity.EnoughRootsOfUnity
public import Mathlib.FieldTheory.Finite.Basic
Namespace:
ZMod
Local context:
/-
Copyright (c) 2025 Yakov Pechersk... | by
ext
simpa [Units.ext_iff] using pow_card_sub_one_eq_one (Units.ne_zero _)
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/ZMod",
"family_id": "rootsofunity_eq_top",
"file_id": "mathlib/Mathlib/RingTheory/ZMod/Torsion.lean",
"sample_id": "f90ebae3e238223bb942229ba9072bef620bae81190f54e359d35b11e6164e6b"
} | train |
{
"chosen_proof_sha256": "297365a1d8f1a47429f28c814575b3f64d65dc37c4e0b829e52928a4af8f384b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3aefbf7570c01386e517f76b1bd54bb6df... | by
refine le_antisymm ?_ ?_
· calc
stabilizer G (s ∪ t)
≤ stabilizer G (s ∪ t) ⊓ stabilizer G t := by simpa
_ ≤ stabilizer G ((s ∪ t) \ t) := stabilizer_inf_stabilizer_le_stabilizer_sdiff
_ = stabilizer G s := by rw [union_sdiff_cancel_right]; simpa [← disjoint_iff_inter_eq_empty]
· calc... | true | {
"chosen_tokens": 90,
"rejected_tokens": 3,
"token_jaccard": 0.032258,
"token_length_ratio": 0.033333
} | unknown_identifier | unknown_identifier | 5abaf2c83647f1f13938464dbaf7ab107edacae50cbc77734f8fbc94e91a50f7 | 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.Action.Pointwise.Finset
public import Mathlib.GroupTheory.QuotientGroup.Defs
public import Mathlib.Order.ConditionallyCompleteLattice.Basic
Name... | by
exact __oprover_missing_7f59f02f1adc | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f8abfe531cdcfbd7bd118f7e9f07557252723c897c758308d7c834717df2a13a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Algebra/Pointwise",
"family_id": "stabilizer_union_eq_left",
"file_id": "mathlib/Mathlib/Algebra/Pointwise/Stabilizer.lean",
"sample_id": "7f59f02f1adc338374d83c01438102e2efac30fffd4f3810fcf815de0d1655df"
} | train |
{
"chosen_proof_sha256": "4854fca9d1b2dda857b810abc21527817558defad53eaaee4f2088331645faca",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "dd3f7e3a83fee5c50a2179510fc132b93f... | by cases x; rfl
nonrec def ReaderT.callCC {ε} [MonadCont m] {α β : Type _}
(f : Label α (ReaderT ε m) β → ReaderT ε m α) : ReaderT ε m α :=
ReaderT.mk (fun r => callCC fun f' => (f <| ReaderT.mkLabel _ f').run r) | false | {
"chosen_tokens": 72,
"rejected_tokens": 76,
"token_jaccard": 0.925,
"token_length_ratio": 1.055556
} | missing_wrong_import | import | 5ac172c0558f3c2505c7931bbe57e9e7be51241d86f42eac1f3acb82ee65583d | 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.Control.Monad.Basic
public import Mathlib.Control.Monad.Writer
public import Mathlib.Control.Lawful
public import Batteries.Tactic.Congr
public import Batterie... | import Mathlib.WrongImport
by cases x; rfl
nonrec def ReaderT.callCC {ε} [MonadCont m] {α β : Type _}
(f : Label α (ReaderT ε m) β → ReaderT ε m α) : ReaderT ε m α :=
ReaderT.mk (fun r => callCC fun f' => (f <| ReaderT.mkLabel _ f').run r) | {
"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": "Control/Monad",
"family_id": "readert",
"file_id": "mathlib/Mathlib/Control/Monad/Cont.lean",
"sample_id": "063710952a3a4d125df5a3e180296e39817ba1f9ff1c0061323053fb46b49fc4"
} | train |
{
"chosen_proof_sha256": "6bde054e8dd9e2e89dc5431f020cd1a2610cdbc6535abeb8845dde9b2fdd1677",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
change
Quotient.map (@funMap L M ps.toStructure n f) Prestructure.fun_equiv (Quotient.finChoice _) =
_
rw [Quotient.finChoice_eq, Quotient.map_mk] | false | {
"chosen_tokens": 37,
"rejected_tokens": 3,
"token_jaccard": 0.035714,
"token_length_ratio": 0.081081
} | by_exact_placeholder | placeholder | 5afa3c2bcb3183ecfcee1dc1c3ed85e621d5c8e7bc9146ef287b40d90f4da266 | 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.Fintype.Quotient
public import Mathlib.ModelTheory.Semantics
Namespace:
FirstOrder.Language
Local context:
/-
Copyright (c) 2022 Aaron Anderson. All rig... | 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": "ModelTheory",
"family_id": "funmap_quotient_mk",
"file_id": "mathlib/Mathlib/ModelTheory/Quotients.lean",
"sample_id": "6c18e3a6e36fcd3e1caadefef8b6d28d63c691769c6b3ffbae3c86bf10ef7481"
} | train |
{
"chosen_proof_sha256": "e6c90bca07c73d1bcb9479a5b66cf4f9a197038e47590f39f131d2b92e478d48",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2e55761302eb4d18256d8d74dab0d8cb06... | by
by_contra! h
refine ((hfin.biUnion (t := Set.Ici) (by simpa using h)).subset (fun b hb ↦ ?_)).not_infinite
(ha.sdiff (Set.finite_singleton a))
obtain ⟨x, hax, hxb⟩ := ((show a ≤ b from hb.1).lt_of_ne (Ne.symm hb.2)).exists_covby_le
exact Set.mem_biUnion hax hxb | true | {
"chosen_tokens": 87,
"rejected_tokens": 3,
"token_jaccard": 0.042553,
"token_length_ratio": 0.034483
} | circular_dependency | circular_dependency | 5b1dc0853c16d2801f3f94dfca8599dc0e21f8d01ed7b4b0d9a6ed410dfe3738 | 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.Finite.Lattice
public import Mathlib.Order.Atoms
public import Mathlib.Order.Interval.Finset.Defs
public import Mathlib.Order.Preorder.Finite
Namespa... | by
exact exists_covby_infinite_Ici_of_infinite_Ici | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "11c797f08da7c4ce94e8fa8979a94d78bb4ba6c8e635ffeeda9f6940b78b6258",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Atoms",
"family_id": "exists_covby_infinite_ici_of_infinite_ici",
"file_id": "mathlib/Mathlib/Order/Atoms/Finite.lean",
"sample_id": "7b81765a8031ad51037f4259b92e28a5ea92d5e155ac346ebd82828d508f44f4"
} | train |
{
"chosen_proof_sha256": "1927463d606bf10bea7c2ddb5c31b00ff037d766b129f9620262d1b12ca44ebf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "908ac39382a7f493cb7b06dfb923c7c2bb... | by
rw [← tsum_dirichletSummand χ hs]
apply eulerProduct_completely_multiplicative <| summable_dirichletSummand χ hs | true | {
"chosen_tokens": 15,
"rejected_tokens": 2,
"token_jaccard": 0.071429,
"token_length_ratio": 0.133333
} | invalid_tactic | invalid_tactic | 5b21ff78d3f57636f8c33e569c1fdea243ce383b9e43965f1207010b8d10763e | 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.NumberTheory.EulerProduct.ExpLog
public import Mathlib.NumberTheory.LSeries.Dirichlet
Namespace:
(root)
Local context:
/-
Copyright (c) 2023 Michael Stoll. A... | by
__oprover_invalid_tactic_f7a04a0bdfb9 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "666a68473ce8afb834a0807101d07c13568e348b411fd8f66be7e4f83b04e0c5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/EulerProduct",
"family_id": "dirichletcharacter",
"file_id": "mathlib/Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean",
"sample_id": "f7a04a0bdfb9132901b240788f15f017596751bca9984db5e9bf9d3c080b9f4c"
} | train |
{
"chosen_proof_sha256": "ad208d86006761f39c52958101642c0dd5169f84bd3c5136aba050921aea4aea",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d3597b3bf39cf226418d617a055f2bd47b... | by
let ⟨g, hg⟩ := (ContinuousLinearMap.id 𝕜 S).exist_extension_of_finiteDimensional_range
exact ⟨g, DFunLike.congr_fun hg.symm⟩ | false | {
"chosen_tokens": 28,
"rejected_tokens": 35,
"token_jaccard": 0.909091,
"token_length_ratio": 1.25
} | markdown_preamble_trailing | fence | 5b2435560eb218eecc313172d6c6754c5996fc8be5109235d738d204df01ab5b | 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.Convex.Cone.Extension
public import Mathlib.Analysis.LocallyConvex.AbsConvexOpen
public import Mathlib.Analysis.LocallyConvex.WeakDual
public import M... | ```lean
by
let ⟨g, hg⟩ := (ContinuousLinearMap.id 𝕜 S).exist_extension_of_finiteDimensional_range
exact ⟨g, DFunLike.congr_fun hg.symm⟩
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/LocallyConvex",
"family_id": "submodule",
"file_id": "mathlib/Mathlib/Analysis/LocallyConvex/HahnBanach.lean",
"sample_id": "0659aa452ee5b4142dc99d160a8e44415976c14c3bb5194948dedcb030d02219"
} | train |
{
"chosen_proof_sha256": "ccb01e73a5d93776c5ed6830bc49ca880d5f04d2137609fc522a9ded57d8bf1c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
refine Multifork.IsLimit.hom_ext hc fun a ↦ ?_
have heq := d.condition ⟨⟨(f.s₀ a), (g.s₀ a)⟩, H.H a⟩
simp only [multicospanIndex_right, multicospanShape_fst, multicospanIndex_left,
multicospanIndex_fst, multicospanShape_snd, multicospanIndex_snd] at heq
simp [-Homotopy.wl, -Homotopy.wr, ← H.wl, ← H.wr, r... | false | {
"chosen_tokens": 86,
"rejected_tokens": 3,
"token_jaccard": 0.021277,
"token_length_ratio": 0.034884
} | by_exact_placeholder | placeholder | 5b9661add22f5b32cbc10c679eee56a9c11d0fa02f33ae7cc26700c037f5c8f0 | 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.Quotient
public import Mathlib.CategoryTheory.Sites.Hypercover.One
public import Mathlib.CategoryTheory.Filtered.Basic
Namespace:
CategoryTheor... | 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/Sites",
"family_id": "homotopy",
"file_id": "mathlib/Mathlib/CategoryTheory/Sites/Hypercover/Homotopy.lean",
"sample_id": "6c405ac245686730f2e5f7fda35bbf73c516cea158bccf92ad5f323956571fcb"
} | train |
{
"chosen_proof_sha256": "9117fb79a21990bf1e7a81654768056689d985c1af2446daa090c5cc9faf03cd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rcases eq_or_lt_of_le hj with h | h
· rw [h]
· exact T.row_weak h cell | false | {
"chosen_tokens": 20,
"rejected_tokens": 3,
"token_jaccard": 0.117647,
"token_length_ratio": 0.15
} | by_exact_placeholder | placeholder | 5bcb2aae0b28e1d06d12f60b24640fc7447a50ff412c0a47d7fb58d037c6cc83 | 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.Young.YoungDiagram
Namespace:
SemistandardYoungTableau
Local context:
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under ... | 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": "Combinatorics/Young",
"family_id": "row_weak_of_le",
"file_id": "mathlib/Mathlib/Combinatorics/Young/SemistandardTableau.lean",
"sample_id": "3890101cc6d52342ce69e19715aee46bb30db6af8bbb140e8a5fac3718968504"
} | train |
{
"chosen_proof_sha256": "a57d660deed4ffd6ef566ef7d9ca6c07159f27d62499692c71e882d53f3bfa34",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "de470aca7ab0cd9ed863886604fdb71bf8... | by
grind [= lt_iff_le_not_ge] | true | {
"chosen_tokens": 6,
"rejected_tokens": 2,
"token_jaccard": 0.142857,
"token_length_ratio": 0.333333
} | invalid_tactic | invalid_tactic | 5c15b65eedaa892cb703a0fc9844f59e52ef7c50a28686f8d6cc9e9e89ae7e63 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Tactic.Alias
public import Batteries.Tactic.Trans
public import Mathlib.Tactic.ExtendDoc
public import Mathlib.Tactic.ToDual
Namespace:
(root)
Local contex... | by
__oprover_invalid_tactic_211d293f722f | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "d8f151fa4ec2b750bec1866298c99eefc7bf66ba484608549f30dba80ed0671e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Defs",
"family_id": "lt_iff_gt_iff_le_iff_ge",
"file_id": "mathlib/Mathlib/Order/Defs/PartialOrder.lean",
"sample_id": "211d293f722fe1130c34a732f32615615b914ff3488d05c85b24b7fea5b7d189"
} | train |
{
"chosen_proof_sha256": "b9ea88f535ce93e2572a2904d87bf732a9a01a29fb1bf1dec81d4ec155726243",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "823540ed3b37d6e59c7f7ee773f32c63d1... | by
simp only [← orderOf_pos_iff, ← Nat.card_zpowers, Nat.card_pos_iff, ← SetLike.coe_sort_coe,
nonempty_coe_sort, Nat.card_pos_iff, Set.finite_coe_iff, OneMemClass.coe_nonempty, true_and] | true | {
"chosen_tokens": 37,
"rejected_tokens": 3,
"token_jaccard": 0.045455,
"token_length_ratio": 0.081081
} | circular_dependency | circular_dependency | 5c2e3529ea383f616c73414b201f2ff8a6e4b12f18eb2b249e40028fbfb85847 | 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.ZMod.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in th... | by
exact finite_zpowers | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "70805a4f2bee6a55777f2380d9825fd257154326fdf94ed8b1a65ba5cd01258d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Data/ZMod",
"family_id": "finite_zpowers",
"file_id": "mathlib/Mathlib/Data/ZMod/QuotientGroup.lean",
"sample_id": "6e268d2dae8bb1ebce0d32eba74344e7b2814801568b87e7044f89199cff8536"
} | train |
{
"chosen_proof_sha256": "9eee49c5561e216892ae126cb44a6a373a2be5f508e4256a2d5f533053f0d013",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [(ofSet s hs).trans_ofSet hs']
ext <;> simp [hs'.interior_eq] | false | {
"chosen_tokens": 22,
"rejected_tokens": 2,
"token_jaccard": 0.052632,
"token_length_ratio": 0.090909
} | by_exact_placeholder | by? | 5c3de58afe1bc0fb8014575e3a99600bbd0769625ceba6abef3278da576c9fa9 | 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.IsImage
Namespace:
OpenPartialHomeomorph
Local context:
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released... | 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": "Topology/OpenPartialHomeomorph",
"family_id": "ofset_trans_ofset",
"file_id": "mathlib/Mathlib/Topology/OpenPartialHomeomorph/Composition.lean",
"sample_id": "3e916153cf2d028f0fe2211091c80661d43ef75952801649ac6582ce0158842e"
} | train |
{
"chosen_proof_sha256": "af8967d58c125507dfdd8a870d8e2d61acb151c1a146b922c5673f800fa162b8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4d42ffb30d467171f90d25061bdb6019ce... | by
rw [mem_disjSum, or_iff_right]
· simp only [inr.injEq, exists_eq_right]
rintro ⟨a, _, ha⟩
exact inl_ne_inr ha | true | {
"chosen_tokens": 28,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.071429
} | invalid_tactic | invalid_tactic | 5c4929242b552524a8bb8a73b2337ead51dc600282646d90930fe92afd69316f | 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.Group.Multiset
Namespace:
Multiset
Local context:
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as... | by
__oprover_invalid_tactic_2f8dba92a8be | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "513d375df8fc0d79246cb01048d64efb2270107cc87fdd011c65e50a5ab53b99",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Multiset",
"family_id": "inr_mem_disjsum",
"file_id": "mathlib/Mathlib/Data/Multiset/Sum.lean",
"sample_id": "2f8dba92a8be5eab58edb05807e69424a9fcd6a33e1126c48358e2bed2791279"
} | train |
{
"chosen_proof_sha256": "3dc6c6b95959f83835d9501fcfa554ee77808794954cb31229e8e1de18edb21b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "02bca982f4b83936374ddc4eb72ca5b4ec... | by
have : NeZero (Nat.card G) := ⟨Nat.card_pos.ne'⟩
have hord := HasEnoughRootsOfUnity.natCard_rootsOfUnity M (Nat.card G)
let e := (IsCyclic.monoidHom_mulEquiv_rootsOfUnity G Mˣ).some
exact ⟨e.trans (rootsOfUnityUnitsMulEquiv M (Nat.card G)) |>.trans (mulEquivOfCyclicCardEq hord)⟩ | true | {
"chosen_tokens": 68,
"rejected_tokens": 3,
"token_jaccard": 0.0625,
"token_length_ratio": 0.044118
} | unknown_identifier | unknown_identifier | 5c5b9b3968b04975c62ae33e51c9b7c0ba91798b1d2e0cd380fcb6b7988f59ba | 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.RootsOfUnity.PrimitiveRoots
Namespace:
(root)
Local context:
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 l... | by
exact __oprover_missing_3f67b4910954 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ff66b6e65fee72c5bdf3480afd10653bb3e53e9e7c60648332ced037f0ae6ea3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/RootsOfUnity",
"family_id": "iscyclic",
"file_id": "mathlib/Mathlib/RingTheory/RootsOfUnity/EnoughRootsOfUnity.lean",
"sample_id": "3f67b4910954a3acaaa3087e7993d00a824bcfea76d08a647dfda9454dd07897"
} | train |
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