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": "1fd411efd0adcfd9a0a62aeb48dc422a81d7841cde6fbd3cbad674180785c839",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "54b19734bb7d230ea7d5c492620ce8dade... | by
simp +instances only [r_iff_exists, Subtype.exists, exists_prop, OreLocalization.oreEqv,
smul_eq_mul, Submonoid.mk_smul]
constructor
· rintro ⟨u, hu, e⟩
exact ⟨_, mul_mem hu x.2.2, u * y.2, by rw [mul_assoc, mul_assoc, ← e], mul_right_comm _ _ _⟩
· rintro ⟨u, hu, v, e₁, e₂⟩
exact ⟨u, hu, by rw [←... | true | {
"chosen_tokens": 112,
"rejected_tokens": 5,
"token_jaccard": 0.068182,
"token_length_ratio": 0.044643
} | wrong_namespace_qualification | wrong_namespace_qualification | 1825f31b028bc4ce17a9f9a8bb9486506368df20e6716ed439d6960c238c6ead | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
public import Mathlib.Algebra.Group.Submonoid.Operations
public import Mathlib.Algebra.Regular.Basic
public import Math... | by
exact OProverMissing.missing_bdf9d42bef05 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1bcba9fb6cf91d3da91be0032bd1550ab9ea750edd24277ecc68262545d60fb9",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "GroupTheory/MonoidLocalization",
"family_id": "r_iff_oreeqv_r",
"file_id": "mathlib/Mathlib/GroupTheory/MonoidLocalization/Basic.lean",
"sample_id": "bdf9d42bef05adc3f26afabef77136e641ea973a1c33f1f631c7b584f6c81787"
} | train |
{
"chosen_proof_sha256": "b654cbf08d5ade13b5e6184e32b41bf84d2fa139990565902f32727bb1db98ee",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
refine ⟨fun ⟨g, hg⟩ ↦ ?_, fun h ↦ (isCuspForm_iff f).mpr (isZeroAt_of_coeffZero_eq_zero f h)⟩
rw [← hg, qExpansion_coeff_zero one_pos
(ModularFormClass.analyticAt_cuspFunction_zero _ one_pos one_mem_strictPeriods_SL)
(periodic_comp_ofComplex _ one_mem_strictPeriods_SL)]
exact (CuspFormClass.zero_at_inf... | false | {
"chosen_tokens": 58,
"rejected_tokens": 3,
"token_jaccard": 0.090909,
"token_length_ratio": 0.051724
} | by_exact_placeholder | exact? | 1856d90552b32adf5a2c98910bdd2383fef974bed25e093a11cc17ae9d79696f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.NumberTheory.ModularForms.QExpansion
public import Mathlib.NumberTheory.ModularForms.LevelOne.Basic
public import Mathlib.NumberTheory.ModularForms.EisensteinS... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/ModularForms",
"family_id": "iscuspform_iff_coeffzero_eq_zero",
"file_id": "mathlib/Mathlib/NumberTheory/ModularForms/CuspFormSubmodule.lean",
"sample_id": "0eb67f824175dac229b7646063b2648b75a17b30e9821e1a51528362c79353d3"
} | train |
{
"chosen_proof_sha256": "abdd5d1c7dac8f0ca69859030dcc532b57d2b6a28ac7caf0a9109fccea1058bd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "229b30587f745dbc42850d319f503873d6... | by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) := by
rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓ z ⊔ x \ y)) := by ac_rfl
_ = x ⊓ (y \ z ⊔ (x ⊓ z ⊔ x ⊓ y... | true | {
"chosen_tokens": 334,
"rejected_tokens": 2,
"token_jaccard": 0.021739,
"token_length_ratio": 0.005988
} | invalid_tactic | invalid_tactic | 187a34dd603e3dfa90090f73bc875db7f57892ceaf4f46da61b226a333376914 | 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.BooleanAlgebra.Defs
public import Mathlib.Tactic.GRewrite
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
R... | by
__oprover_invalid_tactic_5b70195082df | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b3624a21daba327ae92c5e75facfbd8decf99068f26fa435e8d144b09e10d957",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/BooleanAlgebra",
"family_id": "sdiff_sdiff_right",
"file_id": "mathlib/Mathlib/Order/BooleanAlgebra/Basic.lean",
"sample_id": "5b70195082dfe681d1970751f7a4885c8fdbaa9cac549078cb4a1812112d4d46"
} | train |
{
"chosen_proof_sha256": "c30b62521a4c15bdccd0e80ec4d2a61314f23daa4911bb4ad6a467ce05c1c053",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "88c76701a2939d9e21c6541f82d98c2885... | 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] | true | {
"chosen_tokens": 83,
"rejected_tokens": 5,
"token_jaccard": 0.055556,
"token_length_ratio": 0.060241
} | wrong_namespace_qualification | wrong_namespace_qualification | 18acb6283a9a8bc8c34b90b4872bfac179477d7368a2a006c3cbf8d32846b582 | 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.Dual
Namespace:
Matroid
Local context:
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as de... | by
exact OProverMissing.missing_ff91d6fc0a99 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "98195a6ebe3fbaae305452dab65c75ec48d1f7928a5a8ac7678037b2b7f61276",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "a6d320117b4f0031ed5a5af663cf8c7397e183bff5fdfb69cb2aa2ba76b6c4c2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "40cd2f9260990c60dc526a39078b883362... | by
rw [← mul_right_inj (x : M), mul_inv_cancel, mul_leftInvEquiv_symm] | true | {
"chosen_tokens": 15,
"rejected_tokens": 2,
"token_jaccard": 0.066667,
"token_length_ratio": 0.133333
} | invalid_tactic | invalid_tactic | 18f15646ec5716bd8bc6471ff21fea7d2c095cf9bb7bcec128646ec4dbd5cbb1 | 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.Submonoid.Pointwise
Namespace:
Submonoid
Local context:
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 licen... | by
__oprover_invalid_tactic_a81f25208ce0 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "8814970482573410bd42fe91c8cefa95b69a972141faa1ac27902b4878e1303f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "GroupTheory/Submonoid",
"family_id": "leftinvequiv_symm_eq_inv",
"file_id": "mathlib/Mathlib/GroupTheory/Submonoid/Inverses.lean",
"sample_id": "a81f25208ce01c9c1e94e1a7e16ac4440ac59daf394e99a5ec3de20f35dc63a8"
} | train |
{
"chosen_proof_sha256": "da9b69d9b6dc7fdb8d3935d0065bc697214d7335fb75b8a7df981601b0c50a86",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "928089061b1d0efe8c64d1c8e4d6ab36c6... | by
ext
simpa using (eq_bot_or_eq_top _).symm | true | {
"chosen_tokens": 10,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.3
} | unknown_identifier | unknown_identifier | 1913f79605220401903e5448744ec4bd7b57561660b48894b3a1378d624127a8 | 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 __oprover_missing_c850dd5f6807 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1dc8d8ed96cfa63fdcbe0c2735eef4b06488d2d6f83e501b2dc2c2dd8ae49e05",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Atoms",
"family_id": "univ",
"file_id": "mathlib/Mathlib/Order/Atoms/Finite.lean",
"sample_id": "c850dd5f680796ef8939e552d2ed07dbbf81b4eaedf7784baebc1c5d3cb8013c"
} | train |
{
"chosen_proof_sha256": "882f9cd65c324f45860677174f52bfb0dee89c309337831b30132904e328a1fb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "955d5266a22901504a1cd758ee8e4fc19d... | by
lift s to Finset α using hs
classical
rw [stabilizer_coe_finset, mem_stabilizer_finset_iff_smul_finset_subset, ← Finset.coe_smul_finset,
Finset.coe_subset] | true | {
"chosen_tokens": 24,
"rejected_tokens": 3,
"token_jaccard": 0.047619,
"token_length_ratio": 0.125
} | circular_dependency | circular_dependency | 1927b176490d7255104eb1720cdfa2193cabb331aaa8cf3aee8932929e1a32f0 | 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 mem_stabilizer_set_iff_smul_set_subset | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "4944e8aa27a61f3e74b3aac570b34e0cbd605869d883587860b440c0b809b44e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Algebra/Pointwise",
"family_id": "mem_stabilizer_set_iff_smul_set_subset",
"file_id": "mathlib/Mathlib/Algebra/Pointwise/Stabilizer.lean",
"sample_id": "5ad3f4c1074336dbfd17b6d6041c87d9308d36a6123c6ad702c726184bd0c5fe"
} | train |
{
"chosen_proof_sha256": "4644b39b47ec31a8d61ff06e763a66da7cadd0e1fb2b3309d9d1db8d535d144c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [← sub_eq_zero]; exact hS.eq_zero_of_inner_left 𝕜 (by simpa [inner_sub_left, sub_eq_zero]) | false | {
"chosen_tokens": 21,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.095238
} | sorry_admit_sorryAx_proof_hole | admit | 1935bd5654919074f9c058e7ac6464f42c7ceabdee9597b2f5defc0f6e32db4f | 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.InnerProductSpace.Basic
public import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
Namespace:
(root)
Local context:
/-
Copyright (c) 2019 Zhou... | 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/InnerProductSpace",
"family_id": "dense",
"file_id": "mathlib/Mathlib/Analysis/InnerProductSpace/Continuous.lean",
"sample_id": "6d45385fed9b0ad68c7032cd158ad8e6ed294ff9a6d585f7ef6679ce0eea1fd4"
} | train |
{
"chosen_proof_sha256": "8563be0743a9370b287fc6eea19b728e9532ff758a843c12bb7a89f00678f9fd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [← cons_comp_cycleRange, map_perm, Matrix.vecCons]
simp [Units.smul_def] | false | {
"chosen_tokens": 18,
"rejected_tokens": 5,
"token_jaccard": 0.055556,
"token_length_ratio": 0.277778
} | sorry_admit_sorryAx_proof_hole | sorryAx | 195c39d1d06cd99336255d53d3b3b2299904add93819aa5be185104bdd40c3c5 | 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.Alternating.Curry
public import Mathlib.GroupTheory.Perm.Fin
public import Mathlib.Data.Fin.Parity
Namespace:
AlternatingMap
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": "LinearAlgebra/Alternating",
"family_id": "map_insertnth",
"file_id": "mathlib/Mathlib/LinearAlgebra/Alternating/Uncurry/Fin.lean",
"sample_id": "23133e4cdbd3c18af59ef41f856739915fb4a88cdab3bfe7a080ab4e9717b499"
} | train |
{
"chosen_proof_sha256": "1917f8da7209b00ed86e3ac8e7571325a8372a2d3f50420181edba9e230a5abb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "12e052d74731a6e1b56e362de439a6012d... | by
simp_all [SProd.sprod, product, mem_flatMap, mem_map, Prod.ext_iff, and_left_comm] | true | {
"chosen_tokens": 19,
"rejected_tokens": 3,
"token_jaccard": 0.0625,
"token_length_ratio": 0.157895
} | unknown_identifier | unknown_identifier | 19bfe2c4bc3f9e9a72874f8d7b6fdefa80280a2eaee7103131920ace1ccb936f | 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.List.Defs
public import Mathlib.Tactic.Attr.Core
public import Mathlib.Tactic.Common
Namespace:
List
Local context:
/-
Copyright (c) 2015 Leonardo de Mo... | by
exact __oprover_missing_c126f860a188 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "9a8f321056d918cc559b72d365bfb30014ad9588a4f1ee9ba90fdbe7d61c2180",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/List",
"family_id": "mem_product",
"file_id": "mathlib/Mathlib/Data/List/ProdSigma.lean",
"sample_id": "c126f860a1889f0eed701887869d359f44f2980da9414820754ce91fa0c2db26"
} | train |
{
"chosen_proof_sha256": "a40d38439b063352a0a57202db0bbc83f3e167b29ffee4ace96a68c95b1c7096",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9aff7c65eeea9c95f2ed37bd598fa3d4c3... | by
apply Subtype.ext; funext n
rcases h : s.val n; · rfl
refine absurd ?_ H; exact ⟨⟨_, _, h.symm⟩⟩ | true | {
"chosen_tokens": 36,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.083333
} | unknown_identifier | unknown_identifier | 19d7dfdccc79acb63f890787c64cf1a0ede3e92acc699603daf4b52337a904c2 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.Find
public import Mathlib.Data.Stream.Init
public import Mathlib.Logic.Relator
public import Mathlib.Tactic.Common
public import Batteries.Tactic.Lin... | by
exact __oprover_missing_74e41a4b5bb9 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d277a78876e97e303acf8528c188b44052546aaaf8a3b321d1763a66902a4060",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Seq",
"family_id": "eq_empty_of_not_terminates",
"file_id": "mathlib/Mathlib/Data/Seq/Computation.lean",
"sample_id": "74e41a4b5bb950621cded82f802859d90e618c5ffbeaff846b1d0a22fbcf641d"
} | train |
{
"chosen_proof_sha256": "47f42a3f39b92164ba24c6342d141209f44e0c15b9d67ad24b48bf7df68f4b85",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cca863f431b487d24cfdcc63d33eb97176... | by
have := degree_lt_wf.min_mem _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)
simp_all [uniformizingPolynomial] | true | {
"chosen_tokens": 15,
"rejected_tokens": 5,
"token_jaccard": 0.111111,
"token_length_ratio": 0.333333
} | wrong_namespace_qualification | wrong_namespace_qualification | 1a057fc2644995df354dfa488219ee91dbd732f7e95299ccb4f8634d258295be | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.Finite.Valuation
public import Mathlib.NumberTheory.FunctionField
public import Mathlib.RingTheory.Valuation.Discrete.Basic
Namespace:
RatFunc
Lo... | by
exact OProverMissing.missing_2293cf123c21 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "83c06b073bcf3e0618a7ac8f675b562f071866e8873afde5c632a1cc8fe80d75",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/RatFunc",
"family_id": "uniformizingpolynomial_ne_zero",
"file_id": "mathlib/Mathlib/NumberTheory/RatFunc/Ostrowski.lean",
"sample_id": "2293cf123c21972c1d2af5e35604c06efbc61aa089ba31c3f82c50d912dfbf35"
} | train |
{
"chosen_proof_sha256": "297365a1d8f1a47429f28c814575b3f64d65dc37c4e0b829e52928a4af8f384b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | 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... | false | {
"chosen_tokens": 90,
"rejected_tokens": 3,
"token_jaccard": 0.032258,
"token_length_ratio": 0.033333
} | by_exact_placeholder | placeholder | 1aff229bc63d479ebebb6ab307970995ae1afb5b3944fe76aef5de353a235217 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Action.Pointwise.Finset
public import Mathlib.GroupTheory.QuotientGroup.Defs
public import Mathlib.Order.ConditionallyCompleteLattice.Basic
Name... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Pointwise",
"family_id": "stabilizer_union_eq_left",
"file_id": "mathlib/Mathlib/Algebra/Pointwise/Stabilizer.lean",
"sample_id": "7f59f02f1adc338374d83c01438102e2efac30fffd4f3810fcf815de0d1655df"
} | train |
{
"chosen_proof_sha256": "e0b9bed3a6acbad85454a981928799bf6c6adb0fb6d0013326ad4cea47a0b244",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp only [r_eq_r' S, r', Con.rel_mk] | true | {
"chosen_tokens": 13,
"rejected_tokens": 8,
"token_jaccard": 0.055556,
"token_length_ratio": 0.615385
} | unsolved_goals | unsolved_goals | 1b2701093ed9e2bc39f5a1c97f74b26c79bfcf69fcce92f5a8854ec382843f96 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
public import Mathlib.Algebra.Group.Submonoid.Operations
public import Mathlib.Algebra.Regular.Basic
public import Math... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "94e24a42d1169d605949dda928a07c4171eb9f911df0f2c03338dc723c0cf0ce",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "GroupTheory/MonoidLocalization",
"family_id": "r_iff_exists",
"file_id": "mathlib/Mathlib/GroupTheory/MonoidLocalization/Basic.lean",
"sample_id": "194f0f4367719ea1e6824ec86d27af7e10162a98d826fa13df412b7a41e3d577"
} | train |
{
"chosen_proof_sha256": "6296f9ab27c166fefa518a334c1d06c276278c6742ae2b92e22ac06c24c52565",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6775aa7a9f8509a5e61608d136b1fa3add... | by
conv_rhs => rw [← sum_single a, ← sum_single b]
simp_rw [mul_def, map_finsuppSum, liftNC_single, Finsupp.sum_mul, Finsupp.mul_sum]
refine Finset.sum_congr rfl fun y hy => Finset.sum_congr rfl fun x _hx => ?_
simp [mul_assoc, (h_comm hy).left_comm] | true | {
"chosen_tokens": 59,
"rejected_tokens": 5,
"token_jaccard": 0.051282,
"token_length_ratio": 0.084746
} | wrong_namespace_qualification | wrong_namespace_qualification | 1b800e55e58c2bf813d8b2e51ce54465c3efa3cb636b13eb467e653adb70fd2e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MonoidAlgebra.Defs
Namespace:
MonoidAlgebra
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 licen... | by
exact OProverMissing.missing_c88d0c7a08db | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "2b155782d8d28f9712302bdd48a7ab8ff39d800118237fa6fc0bc0dd2ef8f3f0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/MonoidAlgebra",
"family_id": "liftnc_mul",
"file_id": "mathlib/Mathlib/Algebra/MonoidAlgebra/Lift.lean",
"sample_id": "c88d0c7a08db8e0ee629366fac952f34c781e69b50a4902cc54b66d0c91187ae"
} | train |
{
"chosen_proof_sha256": "5cffe96cad332b72c0ae2797275016448548d2c1b107d88e2f58540117e63993",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f6946cb3c4bfb0278075b43218a3f06272... | by
obtain ⟨a', m⟩ := w.mem_essImage α
obtain ⟨m'⟩ := m
exact ⟨a', ⟨(Iso.toEquiv m').symm.trans Equiv.ulift⟩⟩ | true | {
"chosen_tokens": 38,
"rejected_tokens": 5,
"token_jaccard": 0.125,
"token_length_ratio": 0.131579
} | wrong_namespace_qualification | wrong_namespace_qualification | 1b9dc9f4c97d0b6a8ffbac9f793b15465283301d55cefc78c179fa3545231cde | 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.EssentialImage
public import Mathlib.CategoryTheory.Types.Basic
public import Mathlib.Logic.UnivLE
Namespace:
(root)
Local context:
/-
Copyrig... | by
exact OProverMissing.missing_501b3d364012 | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "0ab2687159bd543ea36b52c22feaf5c779f21a802821f63d591475b8f15e0bb8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory",
"family_id": "univle",
"file_id": "mathlib/Mathlib/CategoryTheory/UnivLE.lean",
"sample_id": "501b3d3640125c7dca7fdd0dca6da79f794dca628efb68f4fea9ebb8533db64a"
} | train |
{
"chosen_proof_sha256": "5e2bc200513a76ad4fb8bd2871f6348a8c6776f4fb4e2960dcaa904707705fee",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
-- Using `have` not `let` since it is only existence of `NormedSpace` structure that we need.
have : NormedSpace ℝ E := .restrictScalars ℝ K E
have : FiniteDimensional ℝ E := FiniteDimensional.trans ℝ K E
infer_instance
variable {E} | false | {
"chosen_tokens": 53,
"rejected_tokens": 3,
"token_jaccard": 0.030303,
"token_length_ratio": 0.056604
} | by_exact_placeholder | exact? | 1bb09380576cee7409ea4c91452dc6959355926ed229f9c9de9be57da2e8dbda | 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.Module.FiniteDimension
public import Mathlib.Analysis.RCLike.Basic
public import Mathlib.Topology.Instances.RealVectorSpace
Namespace:
FiniteD... | 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": "Analysis/RCLike",
"family_id": "proper_rclike",
"file_id": "mathlib/Mathlib/Analysis/RCLike/Lemmas.lean",
"sample_id": "6a8e5453b7a4db469efd31b432f7b75acf3e7eec129c2d2a6b546cf031d8a71a"
} | train |
{
"chosen_proof_sha256": "230ad1a22636c034f298a25415dfe821b738da09682e6993796fa2ce84eb0a0a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [← nsmul_one, D.map_smul_of_tower n, map_one_eq_zero, smul_zero] | false | {
"chosen_tokens": 15,
"rejected_tokens": 2,
"token_jaccard": 0.071429,
"token_length_ratio": 0.133333
} | by_exact_placeholder | by? | 1bc307d1b4b2666b956ef06221e5269c0b98aed952e53dcd53d11700c0aa253a | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Derivative
Namespace:
Derivation
Local context:
/-
Copyright (c) 2020 Nicolò Cavalleri... | 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": "RingTheory/Derivation",
"family_id": "map_natcast",
"file_id": "mathlib/Mathlib/RingTheory/Derivation/Basic.lean",
"sample_id": "16bdb0569806200171e13b3e0d9b0d935a38c6f276941a0946af8ff38e8cee10"
} | train |
{
"chosen_proof_sha256": "4d337297249649c3a21f0b564e2b0bbdb408fd0e071471a465068a7173e4de87",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cfc88893dfc4400feb7142404abe76629a... | by
rw [isSymm_iff_basis b, Matrix.IsHermitian.ext_iff, forall_comm]
simp [Eq.comm] | true | {
"chosen_tokens": 20,
"rejected_tokens": 3,
"token_jaccard": 0.058824,
"token_length_ratio": 0.15
} | unknown_identifier | unknown_identifier | 1bd5510092a909f6adfdd0a74c9a302aaba11f78e08015a0b3c9ad2dc39c0db8 | 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.Matrix.PosDef
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as ... | by
exact __oprover_missing_91bf84bd60b2 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a9d3f6a1c0980cf019f6eb771bf669545173be1302c5d6721278e94b70605a5d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/SesquilinearForm",
"family_id": "linearmap",
"file_id": "mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Star.lean",
"sample_id": "91bf84bd60b26a35f7dabcc39cf740009b92e0c19b956edd8d544cfa8e467dbb"
} | train |
{
"chosen_proof_sha256": "5110e356531b5aca6a5cc8c637eddfe770b9e7e43c44858448f93b443e747ce9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6c64e74b2c20ff26c2afb932dbf9dff923... | by
refine ⟨fun ⟨H⟩ ↦ ⟨?_⟩, fun ⟨H⟩ ↦ ⟨?_⟩⟩ <;>
simp only [isCardinalFiltered_aleph0_iff] at * <;>
exact H | false | {
"chosen_tokens": 38,
"rejected_tokens": 43,
"token_jaccard": 0.807692,
"token_length_ratio": 1.131579
} | markdown_preamble_trailing | trailing | 1bd8a7dfa24371e52b3607102934209e9286447ca70cf379b655b95928f2895a | 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.Limits.Filtered
public import Mathlib.CategoryTheory.Limits.Preserves.Filtered
public import Mathlib.CategoryTheory.Limits.Types.Filtered
public... | by
refine ⟨fun ⟨H⟩ ↦ ⟨?_⟩, fun ⟨H⟩ ↦ ⟨?_⟩⟩ <;>
simp only [isCardinalFiltered_aleph0_iff] at * <;>
exact H
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": "CategoryTheory/Presentable",
"family_id": "hascardinalfilteredcolimits_iff_hasfilteredcolimitsofsize",
"file_id": "mathlib/Mathlib/CategoryTheory/Presentable/Finite.lean",
"sample_id": "948388cf0ecec93fe348cbe9fadcd98478b5894a8ffcc1cd67d7ba5e880077f1"
} | train |
{
"chosen_proof_sha256": "60c914c7101fd19d31a3d057e5fece308a5e2edc08549f3280fae32cf5fabb05",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3924443c538f044c69cb7756f1f4c2d5ba... | by
dsimp only [mem_setOf_eq]
rw [← Submodule.map_top, ← iSup_range_single, Submodule.map_iSup]
congr; funext i
rw [← LinearMap.range_comp, diagonal_comp_single, ← range_smul'] | false | {
"chosen_tokens": 36,
"rejected_tokens": 41,
"token_jaccard": 0.846154,
"token_length_ratio": 1.138889
} | markdown_preamble_trailing | trailing | 1c0192c27d304c374cc3035fb149cdc4ebbd070fb58a3eac86060b37f98bb029 | 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.Dimension.LinearMap
public import Mathlib.LinearAlgebra.Matrix.ToLin
Namespace:
Matrix
Local context:
/-
Copyright (c) 2019 Johannes Hölzl. All... | by
dsimp only [mem_setOf_eq]
rw [← Submodule.map_top, ← iSup_range_single, Submodule.map_iSup]
congr; funext i
rw [← LinearMap.range_comp, diagonal_comp_single, ← range_smul']
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Matrix",
"family_id": "range_diagonal",
"file_id": "mathlib/Mathlib/LinearAlgebra/Matrix/Diagonal.lean",
"sample_id": "cf2066a6aac720a3b5d609e8b3fe1b26d84bc8deb53c0ec1b5795944dd568539"
} | train |
{
"chosen_proof_sha256": "76978dc753664885955d4019ab183b19aab997594734243a4e6b99373e308b2b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "30fea064c64881073cd6086113f489a658... | by
simp_rw [isCompactElement_iff_exists_le_sSup_of_le_sSup]
refine ⟨fun h k s hs => ?_, fun h s => ?_⟩
· obtain ⟨t, ⟨hts, htsup⟩⟩ := h s
use t, hts
rwa [← htsup]
· obtain ⟨t, ⟨hts, htsup⟩⟩ := h (sSup s) s (by rfl)
have : sSup s = t.sup id := by
suffices t.sup id ≤ sSup s by apply le_antisymm <... | true | {
"chosen_tokens": 120,
"rejected_tokens": 5,
"token_jaccard": 0.053571,
"token_length_ratio": 0.041667
} | wrong_namespace_qualification | wrong_namespace_qualification | 1c14d8095a58066d8e93466c454c9725166b376161f25f01d0da053085d5ff09 | 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
exact OProverMissing.missing_4ea77ade5bca | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "7a14d292732b6fa82657755401f0f91bf894beabc44c4466e3f22df9195eeed5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/CompactlyGenerated",
"family_id": "issupfinitecompact_iff_all_elements_compact",
"file_id": "mathlib/Mathlib/Order/CompactlyGenerated/Basic.lean",
"sample_id": "4ea77ade5bcaf82f263f63d2e35e31706e2c9a6c7ae719c3b1558eff0efaaa66"
} | train |
{
"chosen_proof_sha256": "9ea2d48ebe00bb73f1eea97bd737a2f6b8f66a4067141833800162c91765c532",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "90d3570afa5eb87a779588a4326f9118d8... | by
simp [GradedMonoid.GOne.one, TensorPower.toTensorAlgebra_tprod] | false | {
"chosen_tokens": 13,
"rejected_tokens": 18,
"token_jaccard": 0.6875,
"token_length_ratio": 1.384615
} | markdown_preamble_trailing | preamble | 1c2700cf986b307f1856ac9930109e2840b0dabcdd028abb973d3051ccbf595b | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorAlgebra.Basic
public import Mathlib.LinearAlgebra.TensorPower.Basic
Namespace:
TensorPower
Local context:
/-
Copyright (c) 2021 Eric Wies... | Here is the proof:
by
simp [GradedMonoid.GOne.one, TensorPower.toTensorAlgebra_tprod] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/TensorAlgebra",
"family_id": "totensoralgebra_gone",
"file_id": "mathlib/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean",
"sample_id": "dc8404380687b2e37f6ac350d18b83560147a6545b4e3c6396ca1a3bc04d0633"
} | train |
{
"chosen_proof_sha256": "dc1780cb3fb0b15d1b902233abc1a076e6c3e1b7b64ffb6ddfc98b9761298fce",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
refine .fromProd (fun a d hd₁ hd₂ c hdc ↦ ?_) (fun b d hd₁ hd₂ c hdc ↦ ?_)
· have e1 : (fun b ↦ (f a, g b)) '' d = {f a} ×ˢ (g '' d) := by aesop
simp_rw [Prod.map_apply, e1]
exact .prod (singleton_nonempty _) (hd₁.image _) isLUB_singleton (hg hd₁ hd₂ hdc)
· have e2 : ((fun a ↦ (f a, g b)) '' d) = (f ''... | true | {
"chosen_tokens": 166,
"rejected_tokens": 8,
"token_jaccard": 0.081633,
"token_length_ratio": 0.048193
} | unsolved_goals | unsolved_goals | 1c3d1a5f7d948f10649ad19223cfeb5d4332e76e441403983b9667aee959f26e | 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.ScottContinuity
public import Mathlib.Order.Bounds.Lattice
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Christopher Hoskin. All rights reserv... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "89a1205b433729915e2a2c0e75dbddfa06b4e7e46a17bf69c69831147495e55f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/ScottContinuity",
"family_id": "scottcontinuous",
"file_id": "mathlib/Mathlib/Order/ScottContinuity/Prod.lean",
"sample_id": "1a6f779a26d6933a1ab93cb5daf06274698653e526a8b75426906873d50dc3c8"
} | train |
{
"chosen_proof_sha256": "228b521f0401eea33d29434cfc850bb07b080dd681e51957a62ad89f2e88ea24",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "200254cdeea524e0fb3be35346d663e1d4... | by rwa [sign_apply, if_neg <| asymm ha, if_pos] | true | {
"chosen_tokens": 13,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.230769
} | unknown_identifier | unknown_identifier | 1c3d33173424041454c9eab467c7619157401f50447ef5bbcb2a735caffa19c2 | 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.GroupWithZero.Defs
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.Algebra.Order.Ring.Defs
public import Mathlib.Tactic.DeriveFintype --... | by
exact __oprover_missing_3edd16c3ca03 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a4d454f625fb5d30fc0fc504aad6499b7d5099f59be59599b7ef7aa51349853a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Sign",
"family_id": "sign_neg",
"file_id": "mathlib/Mathlib/Data/Sign/Defs.lean",
"sample_id": "3edd16c3ca0318c2e5d045f7be510639e5b1ebe6db412620dbe457e127e3ccb4"
} | train |
{
"chosen_proof_sha256": "efb528cee40199b8a7f10c4af81ecc5b602aa05a4991b3b60a5968591ef7ad2c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
dsimp [section_]
rw [assoc, dsimp% X.hom.naturality, comp_id]
exact ed.s'_comp_ε | false | {
"chosen_tokens": 24,
"rejected_tokens": 5,
"token_jaccard": 0.095238,
"token_length_ratio": 0.208333
} | sorry_admit_sorryAx_proof_hole | sorryAx | 1c5e479d9829761933af88ce30d0ec454f6ea70f8735f1685c4b9d9a4639bff5 | 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.Homotopy
public import Mathlib.AlgebraicTopology.AlternatingFaceMapComplex
public import Mathlib.AlgebraicTopology.CechNerve
public import Mat... | 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": "AlgebraicTopology",
"family_id": "section_app_comp_hom_app",
"file_id": "mathlib/Mathlib/AlgebraicTopology/ExtraDegeneracy.lean",
"sample_id": "2c31379e40cbecc4b70db87b9684b86d81f7fb6fcff5b69f3c8ba705b4e20f03"
} | train |
{
"chosen_proof_sha256": "f4dbb4c53a6e3a2d5b935a93c9fc29cc9b719e218918f120b3b408ba8d371698",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
unfold logDeriv
simp [field, Derivation.leibniz_div] | false | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.272727
} | by_exact_placeholder | exact? | 1c70f55398586dcf9a6bd21af6ef17a54558ada3de20a56346dc50824d01493c | 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.Derivation.MapCoeffs
public import Mathlib.FieldTheory.PrimitiveElement
Namespace:
Differential
Local context:
/-
Copyright (c) 2024 Daniel Weber.... | 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": "FieldTheory/Differential",
"family_id": "logderiv_div",
"file_id": "mathlib/Mathlib/FieldTheory/Differential/Basic.lean",
"sample_id": "5a2ece2a22fcf936fd41bf7ec3c1d0ec84ea941a229b765b720040afa6959a7e"
} | train |
{
"chosen_proof_sha256": "c91bd3015c099e4be465fbcb7168b44d0092e9266e1434d35ffc1f51e248e636",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | 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] | false | {
"chosen_tokens": 59,
"rejected_tokens": 3,
"token_jaccard": 0.052632,
"token_length_ratio": 0.050847
} | by_exact_placeholder | exact? | 1c76e5ec838c937b1ff12063d0bdf455a4d7d9ee1d7fb0474b8503c48ae44e4e | 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.Pairwise.Lattice
public import Mathlib.MeasureTheory.PiSystem
public import Mathlib.Order.Lattice.Nat
public import Mathlib.Order.Partition.Finpartiti... | 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": "MeasureTheory",
"family_id": "exists_finpartition_sdiff",
"file_id": "mathlib/Mathlib/MeasureTheory/SetSemiring.lean",
"sample_id": "befab83d2930909beac8cfad141326e33a75bf337f03f5f78fa5a152c0c13634"
} | train |
{
"chosen_proof_sha256": "59b2a49c3792a452986527f1fd02d5e769f33e00ec8381ee8f5c72d68ad2ecd4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5f8fd4810614a2a60bbe889c763a731558... | by
simp [le_antisymm_iff, h x y, h y x]
variable [IsDomain S] [Nontrivial R] | true | {
"chosen_tokens": 22,
"rejected_tokens": 2,
"token_jaccard": 0.066667,
"token_length_ratio": 0.090909
} | invalid_tactic | invalid_tactic | 1c81b906bbab5d4877e330b3d16d118fdf8abef713ba75b6c3319ea41f0cf60e | 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.Normed.Field.WithAbs
public import Mathlib.Analysis.SpecialFunctions.Pow.Real
Namespace:
AbsoluteValue
Local context:
/-
Copyright (c) 2025 Michael ... | by
__oprover_invalid_tactic_99f1e37e3ec1 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "3ea7a7afc7fec1b41400f7ee608ccf41f6101bd37537d05f2e6e1bff003060a7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/AbsoluteValue",
"family_id": "isequiv",
"file_id": "mathlib/Mathlib/Analysis/AbsoluteValue/Equivalence.lean",
"sample_id": "99f1e37e3ec119044c219636cfcc8ffdd76a19fdac55cff796f9c655385230aa"
} | train |
{
"chosen_proof_sha256": "55394e3b94e4002c95f3216c1764d8bfdefb9347ecc72f11001d5f75ebad2dcb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2401688d1603099d05da8c9eb9c2a484d6... | by
rintro x ⟨hf, hg⟩
rw [mem_fixedPoints_iff] at *
rw [comp_apply, hg, hf] | false | {
"chosen_tokens": 22,
"rejected_tokens": 26,
"token_jaccard": 0.789474,
"token_length_ratio": 1.181818
} | missing_wrong_import | import | 1cc62332447619be8ebbf06d08c8d77cd5753ad76027f45bd1d852ad80f89bce | 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.Operations
Namespace:
Function
Local context:
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as descr... | import Mathlib.WrongImport
by
rintro x ⟨hf, hg⟩
rw [mem_fixedPoints_iff] at *
rw [comp_apply, hg, hf] | {
"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": "Dynamics/FixedPoints",
"family_id": "inter_subset_fixedpoints_comp",
"file_id": "mathlib/Mathlib/Dynamics/FixedPoints/Defs.lean",
"sample_id": "65f5358ecacae8480f414a0ec307a37a8131f8d46fb796e1817fa646fac75533"
} | train |
{
"chosen_proof_sha256": "c601c0ce6ee21cbe086a05718f7a601ae3b0678e21055794ba74900b0b12e072",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "451ace8d95c1b634db051c4f50d43d08f0... | by
rw [IsPeriodicPt, hco.comp_iterate] at hfg
exact hfg.left_of_comp hg | false | {
"chosen_tokens": 16,
"rejected_tokens": 23,
"token_jaccard": 0.875,
"token_length_ratio": 1.4375
} | markdown_preamble_trailing | fence | 1cd7d344796e466ec0c7769727f23526cf3aa9c6bd4272e56a276fa4af1e9592 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Action.Defs
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.Sub.Basic
public import Mathlib.Data.List.Cycle
pub... | ```lean
by
rw [IsPeriodicPt, hco.comp_iterate] at hfg
exact hfg.left_of_comp hg
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | 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": "b62df895e5ecbe83fea44bae35972a3514a09e499e02633c6e24648777cc00bf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9a39643b19053974b943a97a575958ebd9... | by
intro contr
obtain ⟨a, b, hab, mcfab⟩ := mcf
specialize contr ![![a, b], ![b, a]]
rw [Fin.sum_univ_two', ← mcfab.left, ← two_nsmul] at contr
have sharp :
2 • ((ω.tt ![![a, b], ![b, a]]).map (fun _ => f ![a, b])).sum <
2 • ((ω.tt ![![a, b], ![b, a]]).map f).sum := by
have half_sharp :
((ω.... | true | {
"chosen_tokens": 456,
"rejected_tokens": 2,
"token_jaccard": 0.012821,
"token_length_ratio": 0.004386
} | invalid_tactic | invalid_tactic | 1ce64bcd27abe28dc787819952db5b0292664c4448ec32b5954cb9cd13a0ba7c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Fin
public import Mathlib.Algebra.Order.BigOperators.Group.Multiset
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Linea... | by
__oprover_invalid_tactic_5e07b58c14b8 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "cf29f216a84d9eebfb60bc8f39a6318a9a8f0b5acda098716b846e43d86446f1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/Optimization",
"family_id": "function",
"file_id": "mathlib/Mathlib/Combinatorics/Optimization/ValuedCSP.lean",
"sample_id": "5e07b58c14b8fda2d13e55aab1c1520c2c2eed27aece7742846d0bae8fae4ed6"
} | train |
{
"chosen_proof_sha256": "3db3fcbd4e98076f3d0533d22872c1a4a27f5605675d9e65936bcaf6671c864f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
simp_rw [← scottContinuousOn_univ] at ⊢
exact .fromProd (fun a ↦ (h₁ a).scottContinuousOn) (fun b ↦ (h₂ b).scottContinuousOn)
(fun a ↦ (h₁ a).monotone) (fun b ↦ (h₂ b).monotone) | false | {
"chosen_tokens": 59,
"rejected_tokens": 3,
"token_jaccard": 0.086957,
"token_length_ratio": 0.050847
} | by_exact_placeholder | placeholder | 1d03a2cb305d0db293e1462157048bc889c4d80c250b9e1954c678bad30b1ec9 | 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.ScottContinuity
public import Mathlib.Order.Bounds.Lattice
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Christopher Hoskin. All rights reserv... | 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": "Order/ScottContinuity",
"family_id": "scottcontinuous",
"file_id": "mathlib/Mathlib/Order/ScottContinuity/Prod.lean",
"sample_id": "5f10fa5c3e8b99e44987fd6f1820e919de6c733d5031ca673f738ff243d9f672"
} | train |
{
"chosen_proof_sha256": "4d337297249649c3a21f0b564e2b0bbdb408fd0e071471a465068a7173e4de87",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [isSymm_iff_basis b, Matrix.IsHermitian.ext_iff, forall_comm]
simp [Eq.comm] | false | {
"chosen_tokens": 20,
"rejected_tokens": 3,
"token_jaccard": 0.058824,
"token_length_ratio": 0.15
} | by_exact_placeholder | placeholder | 1d375f9665e8577d89c6d3f9584fa76b875d53793acc232d7804d58a067049ac | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Matrix.PosDef
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as ... | 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/SesquilinearForm",
"family_id": "linearmap",
"file_id": "mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Star.lean",
"sample_id": "91bf84bd60b26a35f7dabcc39cf740009b92e0c19b956edd8d544cfa8e467dbb"
} | train |
{
"chosen_proof_sha256": "0276c875b5de26026fe85dbf0c34475b3acdef44fb07206bfeb84e06366c9607",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1ca3e2d479b890c0c8ec12f1bf45f7c09e... | by
funext n
induction n with
| zero => rfl
| succ n ih =>
unfold map iterate get
rw [map, get] at ih
rw [iterate]
exact congrArg f ih | false | {
"chosen_tokens": 35,
"rejected_tokens": 42,
"token_jaccard": 0.92,
"token_length_ratio": 1.2
} | markdown_preamble_trailing | fence | 1d3c20ecf7ff84fe6effc00abb1d04b58ba0b4d32e01c964b880c81e429d2a20 | 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.Stream.Defs
public import Mathlib.Logic.Function.Basic
public import Mathlib.Data.Nat.Basic
public import Mathlib.Tactic.Common
Namespace:
Stream'
Local... | ```lean
by
funext n
induction n with
| zero => rfl
| succ n ih =>
unfold map iterate get
rw [map, get] at ih
rw [iterate]
exact congrArg f ih
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Stream",
"family_id": "map_iterate",
"file_id": "mathlib/Mathlib/Data/Stream/Init.lean",
"sample_id": "16839e522d1756068dfaf97c745b531d7271caa8d21dfa9ef1a00b09a41091af"
} | train |
{
"chosen_proof_sha256": "9ee4d424ebdd38c7fae4d49084f0c22e4992fdbfa58721d1092ad5e07b192bfb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "05563c96d23b122c116d8c9bb388bbb8f9... | by
exact div_nonneg (Finset.sum_nonneg fun _ _ => sq_nonneg _) <| sq_nonneg _ | true | {
"chosen_tokens": 18,
"rejected_tokens": 3,
"token_jaccard": 0.133333,
"token_length_ratio": 0.166667
} | circular_dependency | circular_dependency | 1d4cd11e885f5f84e9f0fa61052e1c14c92153f8e1e846bca89df83e6367bb43 | 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.Module.NatInt
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import Mathlib.Combinatorics.SimpleGraph.Density
public import Mathl... | by
exact energy_nonneg | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "7813bf8c7517122bcb478d6a0820bb9199f373c14cb7ccdeefd936aac55eb6ae",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/SimpleGraph",
"family_id": "energy_nonneg",
"file_id": "mathlib/Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean",
"sample_id": "ea7f7ba297977c69513c6abcc1bfadf80c88bd3e61590c3a43e8307b939f76d2"
} | train |
{
"chosen_proof_sha256": "65d184d09b9f9abbbd2d10bfaea51c9a0bf698460f9d72b6fbec0c7ce1a8ad5d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f2679e48bf9685f04472782b58d5cd7288... | by
induction n with
| zero => rfl
| succ n ih =>
rw [replicate_succ]
conv => rhs; rw [replicate_succ]
rw [snoc_cons, ih] | true | {
"chosen_tokens": 31,
"rejected_tokens": 3,
"token_jaccard": 0.047619,
"token_length_ratio": 0.096774
} | unknown_identifier | unknown_identifier | 1dc74bea0ea98e8b2f52ddeedd30f3539cf880cbe2bc01a624dfd8c402c88335 | 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.Vector.Basic
Namespace:
List.Vector
Local context:
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as describe... | by
exact __oprover_missing_2b26d3e25a28 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "5043bf761356b5236c4a9b754def8b442bede0cf7f829e3e9ebd957a2a93827f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Vector",
"family_id": "replicate_succ_to_snoc",
"file_id": "mathlib/Mathlib/Data/Vector/Snoc.lean",
"sample_id": "2b26d3e25a280904d8577d75d632bf5a13763ff21493b59168968dcb1e0a5669"
} | train |
{
"chosen_proof_sha256": "68a591d0520d49abb38f3df228ef3dd797bb020e537249311751475dbc1d52d8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [max_def]; split_ifs with h
exacts [hb h, ha (le_of_not_ge h)] | false | {
"chosen_tokens": 20,
"rejected_tokens": 5,
"token_jaccard": 0.05,
"token_length_ratio": 0.25
} | sorry_admit_sorryAx_proof_hole | sorryAx | 1dcaa35be0155baefef7d63fc386f0a95aeee7283837dfdd74f859ebfcf99410 | 0 | 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 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": "Order/Defs",
"family_id": "max_ind",
"file_id": "mathlib/Mathlib/Order/Defs/LinearOrder.lean",
"sample_id": "44ece7bf253c5f603c55e81415554957d2378036c79d5b76487dc473ce633518"
} | train |
{
"chosen_proof_sha256": "362e79b2088809d4572239c27914a8ae94ced25dc58902293b9373aadce248ad",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
haveI : Fintype ι := Fintype.ofFinite ι
rw [fromDirectSumEquiv, ← fromDFinsuppEquiv_single]
convert! rfl | false | {
"chosen_tokens": 20,
"rejected_tokens": 2,
"token_jaccard": 0.052632,
"token_length_ratio": 0.1
} | by_exact_placeholder | by? | 1e0a4619db9fd7836d4cadc3d561a082f6b82b443e5747c3cf361aa81f6b1176 | 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.DirectSum.Module
public import Mathlib.LinearAlgebra.Multilinear.DFinsupp
Namespace:
MultilinearMap
Local context:
/-
Copyright (c) 2024 Sophie Morel... | 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/Multilinear",
"family_id": "fromdirectsumequiv_lof",
"file_id": "mathlib/Mathlib/LinearAlgebra/Multilinear/DirectSum.lean",
"sample_id": "215bc10c17214a80cdf028f691cd9e0cb3d40b6838bce81a23b452ba150e84be"
} | train |
{
"chosen_proof_sha256": "c46ea331c2028c22d1cfe25ee9b2c63dfdbb45feb5dd1bf4ebf33f7cd796b96a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
refine ⟨fun h z => ?_, fun h n => by simpa only [zpow_natCast] using! h (n : ℤ)⟩
by_cases! hz : 0 ≤ z
· unfold Tendsto
lift z to ℕ using hz
simpa using! h z
· have : Tendsto (fun a => k a ^ z) l (𝓝 0) :=
Tendsto.comp (tendsto_zpow_atTop_zero hz) hk
have h : Tendsto f l (𝓝 0) := by simpa u... | true | {
"chosen_tokens": 110,
"rejected_tokens": 8,
"token_jaccard": 0.070175,
"token_length_ratio": 0.072727
} | unsolved_goals | unsolved_goals | 1e2a719b0fb45f2cd1076752524941609195fe373f3a27b5c43fa1bd4d6d026c | 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.Eval.Defs
public import Mathlib.Analysis.Asymptotics.Lemmas
Namespace:
Asymptotics
Local context:
/-
Copyright (c) 2021 Devon Tuma. All ri... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "06e7f55fce5af1b527b2a553dd0083585710e63c5f85144bcb171b8a1acee043",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Asymptotics",
"family_id": "superpolynomialdecay_iff_zpow_tendsto_zero",
"file_id": "mathlib/Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean",
"sample_id": "00521cab1e8d5624effd7f9080f88a2a7c2a46ecd1ff08ed8fe772073ff48814"
} | train |
{
"chosen_proof_sha256": "c8dca49869410f2bacfcd9d441c978765c739555c35836f4fba20bb9b5e2825f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | 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 | false | {
"chosen_tokens": 34,
"rejected_tokens": 2,
"token_jaccard": 0.05,
"token_length_ratio": 0.058824
} | by_exact_placeholder | by? | 1e59313f3ff1b4f456767c5d0490ded72de17d8073d50c54083166b6b35bdcc9 | 0 | 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? | {
"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/Defs",
"family_id": "compare_le_iff_le",
"file_id": "mathlib/Mathlib/Order/Defs/LinearOrder.lean",
"sample_id": "34c67d0984704f562f600f9107ee26cdff7c7034435d16dae2284506bc3e8668"
} | train |
{
"chosen_proof_sha256": "38de9a26cb91dd01aaa9b7d35dd028817eb092a585a923cb7e7b911ff5d759bb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "573f1aa7ff0f0816637422072f791f0e7b... | by
refine fun _ _ => ⟨1, by norm_num, 1, by norm_num, ?_⟩
filter_upwards [] with x
simp | false | {
"chosen_tokens": 26,
"rejected_tokens": 31,
"token_jaccard": 0.772727,
"token_length_ratio": 1.192308
} | markdown_preamble_trailing | preamble | 1e73e4e7d2dec99b3134a197be1fba0088f8938bdd15892f4bc9f7a2e8e75495 | 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.SpecialFunctions.Pow.Real
public import Mathlib.Algebra.Order.ToIntervalMod
public import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Al... | Here is the proof:
by
refine fun _ _ => ⟨1, by norm_num, 1, by norm_num, ?_⟩
filter_upwards [] with x
simp | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Computability/AkraBazzi",
"family_id": "growspolynomially_const",
"file_id": "mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean",
"sample_id": "836ef9c3cd55ae2e3f293b4d27ae834545f003ae5a97617e40dffc59a7b9fdae"
} | train |
{
"chosen_proof_sha256": "958795982b90c7f479780dff2a7eb96698a38b055a8fc856a14fd71baa43ed6a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ef64a7044dcbaec85a782fa2c5e1d2c9e9... | by rw [IsLUB.sSup_eq (h d₁ d₂ (isLUB_iff_sSup_eq.mpr rfl))]
mpr h _ d₁ d₂ _ hda := by rw [isLUB_iff_sSup_eq, ← (h d₁ d₂), IsLUB.sSup_eq hda]
alias ⟨ScottContinuous.map_sSup, ScottContinuous.of_map_sSup⟩ :=
scottContinuous_iff_map_sSup | false | {
"chosen_tokens": 61,
"rejected_tokens": 68,
"token_jaccard": 0.933333,
"token_length_ratio": 1.114754
} | markdown_preamble_trailing | fence | 1eaccc5fe0295642d0577697157bdc7b3afbcb0e1338c4617610da8dd3771e37 | 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.ScottContinuity.Prod
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license ... | ```lean
by rw [IsLUB.sSup_eq (h d₁ d₂ (isLUB_iff_sSup_eq.mpr rfl))]
mpr h _ d₁ d₂ _ hda := by rw [isLUB_iff_sSup_eq, ← (h d₁ d₂), IsLUB.sSup_eq hda]
alias ⟨ScottContinuous.map_sSup, ScottContinuous.of_map_sSup⟩ :=
scottContinuous_iff_map_sSup
``` | {
"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/ScottContinuity",
"family_id": "scottcontinuous_iff_map_ssup",
"file_id": "mathlib/Mathlib/Order/ScottContinuity/Complete.lean",
"sample_id": "7885120c783d836f7907adab08dd0290a06a8a8cf35af5f387b7e22d740bb408"
} | train |
{
"chosen_proof_sha256": "3f65d3f4792abc1e42ea24793b4621de9bedda45327f9e9eb8d4aee686a05437",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9eff011adcb66f66cc2b152b3ad39ee8a1... | by
rw [← LinearOrderedCommGroupWithZero.wellFoundedOn_setOf_ge_gt_iff_nonempty_discrete_of_ne_zero
one_ne_zero, ← Set.wellFoundedOn_range]
classical
refine ⟨fun h ↦ (h.mapsTo Subtype.val ?_).mono' (by simp), fun h ↦ (h.mapsTo ?_ ?_).mono' ?_⟩
· rintro ⟨_, x, rfl⟩
simp only [← Subtype.coe_le_coe, OneMemC... | true | {
"chosen_tokens": 173,
"rejected_tokens": 3,
"token_jaccard": 0.032258,
"token_length_ratio": 0.017341
} | unknown_identifier | unknown_identifier | 1ec96dfc2868a805008ab528b4fca80b826a4802121f5b52e24b34e5451f7e0d | 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.GroupTheory.ArchimedeanDensely
public import Mathlib.RingTheory.Valuation.ValuationRing
Namespace:
Valuation.Integers
Local context:
/-
Copyright (c) 2024 Ya... | by
exact __oprover_missing_45bf8f219b5d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "b0015ebf930c50bb357b947c5ab7810c6341b7b3049c08cf0ec977dd527f9716",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Valuation",
"family_id": "wellfounded_gt_on_v_iff_discrete_mrange",
"file_id": "mathlib/Mathlib/RingTheory/Valuation/Archimedean.lean",
"sample_id": "45bf8f219b5d1572a7b7bfc82220b21da6d9c0dadf354c8db833eeedf98c284d"
} | train |
{
"chosen_proof_sha256": "6217ca4bcfee77224af82f13005a67bfc164dca4064e34175a675cb4920a7928",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "afe49b555821b24ebf64bf6c1b67f3d1c0... | by
simp only [List.getD_eq_getElem?_getD]
exact option_getD.comp₂ list_getElem? (const _) | false | {
"chosen_tokens": 21,
"rejected_tokens": 26,
"token_jaccard": 0.826087,
"token_length_ratio": 1.238095
} | markdown_preamble_trailing | trailing | 1f13755f9fac8b85082b7bf833e02c7718221274488e29bf8af25ff2ef8411be | 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.Computability.Primrec.Basic
public import Mathlib.Logic.Encodable.Pi
Namespace:
Primrec
Local context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reser... | by
simp only [List.getD_eq_getElem?_getD]
exact option_getD.comp₂ list_getElem? (const _)
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": "Computability/Primrec",
"family_id": "list_getd",
"file_id": "mathlib/Mathlib/Computability/Primrec/List.lean",
"sample_id": "c422540f094e4df2b71b39ea0c0e9f6334865968fac379ba2de62eb381a0a862"
} | train |
{
"chosen_proof_sha256": "4824a182947d6de23921d345ff1615bb9443926fe4268ae7e79b135cafd2e987",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4b36ce42ac987a500d5fe55c650474a5f3... | by
rintro X ⟨hX⟩
have := hX.prop_diag_obj
simp only [isCardinalPresentable_iff] at this ⊢
exact isCardinalPresentable_of_isColimit _ hX.isColimit κ hJ | true | {
"chosen_tokens": 27,
"rejected_tokens": 3,
"token_jaccard": 0.08,
"token_length_ratio": 0.111111
} | unknown_identifier | unknown_identifier | 1f38ce0889a5bdc99d3d42c5b2b55ceab67814f70f03d2a9bde4d97b6b753ef4 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Types.Filtered
public import Mathlib.CategoryTheory.Limits.Yoneda
public import Mathlib.CategoryTheory.Presentable.Basic
public import Ma... | by
exact __oprover_missing_f97b91f107eb | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "3c80c5bff675cb62613aab6c8932c9203aa877d0968e76ea1b36500aeae22287",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Presentable",
"family_id": "isclosedundercolimitsofshape_iscardinalpresentable",
"file_id": "mathlib/Mathlib/CategoryTheory/Presentable/Limits.lean",
"sample_id": "f97b91f107eb7b4f9d321215e0984f13daac29b819bde8fd09eea122778dc081"
} | train |
{
"chosen_proof_sha256": "1e7bb9669789dabc1d0b4a44d7bd38a7c58136fdfc801571fd7b5e8b4e1c7606",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
conv_lhs => rw [← (mateEquiv adj₁ adj₂).right_inv α]
exact (mateEquiv_counit adj₁ adj₂ ((mateEquiv adj₁ adj₂).symm α) d) | false | {
"chosen_tokens": 38,
"rejected_tokens": 2,
"token_jaccard": 0.047619,
"token_length_ratio": 0.052632
} | by_exact_placeholder | by? | 1f3ae663cf661ed6e010b892bb65315befc9ab321fc23dc63f8133c1190a666b | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Adjunction.Basic
public import Mathlib.CategoryTheory.Functor.TwoSquare
public import Mathlib.CategoryTheory.HomCongr
Namespace:
CategoryTheory... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Adjunction",
"family_id": "mateequiv_counit_symm",
"file_id": "mathlib/Mathlib/CategoryTheory/Adjunction/Mates.lean",
"sample_id": "4c02a391b7ac277a192dd766770fc91e8893ee8cde8243ce667618594d9c0e76"
} | train |
{
"chosen_proof_sha256": "aa83c7abdbb25c29bbdc8a05c929751eafeeab66442c5c16368b86eecdc282c9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2abedb34a57507348c7123447df9943ccb... | by
rw [← image_singleton, preimage_image_mk_eq_mul]
simp | true | {
"chosen_tokens": 9,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.555556
} | wrong_namespace_qualification | wrong_namespace_qualification | 1f41ea2a813c362b4242c16209efe3475213597187c51d3590d30ac759880597 | 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.Quotient
public import Mathlib.Algebra.Group.Action.Opposite
public import Mathlib.Algebra.Group.Subgroup.MulOpposite
public import Mathlib.GroupTheory... | by
exact OProverMissing.missing_423b546f69a4 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "cd1f9ec868d11cffd7a9f773bdf2cabea557d39df2809266924c65586cafea5c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "GroupTheory/Coset",
"family_id": "preimage_mk_one",
"file_id": "mathlib/Mathlib/GroupTheory/Coset/Defs.lean",
"sample_id": "423b546f69a419e2fd6b478d8c8a356a73c2e1fde067e823b879f61db25761a8"
} | train |
{
"chosen_proof_sha256": "2038792152bb22c0a5ae377d16d2f16e94b117cceae2db86d9ede3e4d520885b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7437481b96e36485900fd3de7576c86e05... | by
rw [← cancel_epi ((shiftFunctorZero _ A).inv.app X), Iso.inv_hom_id_app,
pullbackShiftFunctorZero_inv_app, assoc, Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app]
rfl | true | {
"chosen_tokens": 35,
"rejected_tokens": 3,
"token_jaccard": 0.041667,
"token_length_ratio": 0.085714
} | unknown_identifier | unknown_identifier | 1f9a6b2a01e16f0b64bd55904d72ea4772ab017d64f8062d26f7c5cc9958ef95 | 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.Shift.Adjunction
public import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
Namespace:
CategoryTheory
Local context:
/-
Copyright (c) 20... | by
exact __oprover_missing_59fa1cfe920c | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f3cb0b77a3862415cb35d2403d03bcc5001d152d5ea51afc3f6b83026e1ec580",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Shift",
"family_id": "pullbackshiftfunctorzero_hom_app",
"file_id": "mathlib/Mathlib/CategoryTheory/Shift/Pullback.lean",
"sample_id": "59fa1cfe920c44354539a08a73d5bce71392689b4372837276159b664aba9fb8"
} | train |
{
"chosen_proof_sha256": "7098d4ede92516c6303541ad23377423cb7f62e04005055a6e4b359a31869d3c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
intro ⟨h1, h2⟩; use y.drop x.length, h2
nth_rw 1 [← List.take_append_drop x.length y]
simpa [-List.take_append_drop, List.prefix_iff_eq_take, hl] using h1
mpr := by simp +contextual [pullSub] | false | {
"chosen_tokens": 53,
"rejected_tokens": 5,
"token_jaccard": 0.027778,
"token_length_ratio": 0.09434
} | sorry_admit_sorryAx_proof_hole | sorryAx | 1fe42d65f919c227f84d3279c6a8e793a998ceb09e4dffe3c82b2cc424b86b0a | 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.CompleteLattice.SetLike
Namespace:
Descriptive.Tree
Local context:
/-
Copyright (c) 2024 Sven Manthe. All rights reserved.
Released under Apache 2.0 li... | 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": "SetTheory/Descriptive",
"family_id": "mem_pullsub_long",
"file_id": "mathlib/Mathlib/SetTheory/Descriptive/Tree.lean",
"sample_id": "2b9acd22df8b3e0b46d29d9433ec7adfa97c3ca8da662c2868b5f6dac8fb61d2"
} | train |
{
"chosen_proof_sha256": "9d69af709cb1a99e4edeafcd19d4f6a19e8a85efb124f30d847acee7dc29b2e3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
suffices ∀ x : FreeMonoid α, (xs.map FreeMonoid.of).foldl (· * ·) x = x * ofList xs by
simpa [← List.foldl_map, lift_mk_eq_foldl, toFreeMonoid, lift] using this (FreeMonoid.of x)
induction xs with grind [ofList_nil, ofList_cons] | false | {
"chosen_tokens": 60,
"rejected_tokens": 2,
"token_jaccard": 0.027778,
"token_length_ratio": 0.033333
} | sorry_admit_sorryAx_proof_hole | sorry | 20375be39ef088f14861660f5ae3c3b62767d1c461e58188c9048f680527ffed | 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.FreeMonoid.Basic
public import Mathlib.Algebra.Free
public import Mathlib.Algebra.Group.WithOne.Basic
public import Mathlib.Algebra.Group.Units.Basic
p... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/FreeMonoid",
"family_id": "tofreemonoid_mk_eq_cons",
"file_id": "mathlib/Mathlib/Algebra/FreeMonoid/FreeSemigroup.lean",
"sample_id": "b0369a5a54c56e81b58aafefa0901fd2dbc75fdc809ee1103b64d4a18103c12d"
} | train |
{
"chosen_proof_sha256": "ab45fc24fe8d81ce6bc79891c129b27df70dce3fefc1109e3aecbeb52c81794e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
ext; simp [transvection.apply] | false | {
"chosen_tokens": 9,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.555556
} | sorry_admit_sorryAx_proof_hole | sorryAx | 2043a201ba3d1f11258029b2d9f795706ce9c63283af85f00f7795c6c0b1babb | 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.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup
public import Mathlib.LinearAlgebra.Charpoly.BaseChange
public import Mathlib.LinearAlgebra.Dual.BaseChan... | 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": "LinearAlgebra/Transvection",
"family_id": "congr",
"file_id": "mathlib/Mathlib/LinearAlgebra/Transvection/Basic.lean",
"sample_id": "0ccbfff2e32bc19e9550d2514637413d97b726466dde45498d6c8f3060c8d074"
} | train |
{
"chosen_proof_sha256": "a24cc2dfc6b516924da62f552de8528961f8853b68b3464a392b74c6594e45ef",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "33c5a04717df2d635bf7df2da69f293f07... | by
ext x
apply (IsIntegralClosure.equiv A (integralClosure A L₃) L₃ B₃).symm.injective
ext
simp [galRestrict', Subalgebra.algebraMap_eq] | true | {
"chosen_tokens": 33,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.151515
} | wrong_namespace_qualification | wrong_namespace_qualification | 2058139325b159e410505b33a24c651c8d2222d2a7f153f2d9afde493a2c6027 | 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.DedekindDomain.IntegralClosure
public import Mathlib.RingTheory.RingHom.Finite
public import Mathlib.RingTheory.Localization.LocalizationLocalizatio... | by
exact OProverMissing.missing_2f3292c8af1f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "06db84e7e262dcb197086ffa15bf778fe4416ab799bf00209254707991705d98",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/IntegralClosure",
"family_id": "galrestrict'_comp",
"file_id": "mathlib/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean",
"sample_id": "2f3292c8af1fa98f0dfb39f23e5916bd9e2de4337e8d92b7857cea3205fe336c"
} | train |
{
"chosen_proof_sha256": "2d27cea8f3ca8caf10e9d8775b5b6885d300093e2302453f512d2957b166201a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
let : Field (A ⧸ p) := Ideal.Quotient.field p
rw [Ideal.isRadical_iff_quotient_reduced]
exact Algebra.FormallyUnramified.isReduced_of_field (A ⧸ p) (B ⧸ p.map (algebraMap A B)) | false | {
"chosen_tokens": 45,
"rejected_tokens": 5,
"token_jaccard": 0.071429,
"token_length_ratio": 0.111111
} | sorry_admit_sorryAx_proof_hole | sorryAx | 206751964446c8c6e0b9dbc6e1fa0ecb1f3304df04fa5d0296f19d8754ed6347 | 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.PurelyInseparable.Basic
public import Mathlib.RingTheory.Artinian.Ring
public import Mathlib.RingTheory.Unramified.Finite
public import Mathlib.Rin... | 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/Unramified",
"family_id": "isradical_map_ismaximal",
"file_id": "mathlib/Mathlib/RingTheory/Unramified/Field.lean",
"sample_id": "4e9ba5114733e3b96be27c5b4d01de8367929d2bf280e4296d00c4e5e9fb55bb"
} | train |
{
"chosen_proof_sha256": "ba5868d82bf97e5ff1d70a97d543dc36082af3a6a7d0e0038bdd9326c14ca5d6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
simp only [CompleteLattice.isCompactElement_iff_exists_le_iSup_of_le_iSup,
Finset.sup_eq_iSup] at h ⊢
intro ι s hb
replace hb : (b : α) ≤ iSup ((↑) ∘ s) := le_trans hb <| (coe_iSup s) ▸ le_refl _
obtain ⟨t, ht⟩ := h ι ((↑) ∘ s) hb
exact ⟨t, (by simpa using ht : (b : α) ≤ _)⟩ | false | {
"chosen_tokens": 84,
"rejected_tokens": 5,
"token_jaccard": 0.065217,
"token_length_ratio": 0.059524
} | sorry_admit_sorryAx_proof_hole | sorryAx | 2099a0bcb20f7c4581b35ce061769025425b83b33f4b376cfcc46ae6810157d8 | 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.CompleteLatticeIntervals
public import Mathlib.Order.CompactlyGenerated.Basic
Namespace:
Set.Iic
Local context:
/-
Copyright (c) 2024 Oliver Nash. All ... | 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": "Order/CompactlyGenerated",
"family_id": "iscompactelement",
"file_id": "mathlib/Mathlib/Order/CompactlyGenerated/Intervals.lean",
"sample_id": "c5320067570b9bfc10d7ea602085cbc08d55ba3d758c561b8444485698919964"
} | train |
{
"chosen_proof_sha256": "32e319c4110fd1b6339d57505a0712e65f79dbff575ad63d342ca445e0185468",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "828fcf5f665efc678113e458ba63e19ba9... | by
simp [← HomologicalComplex.comp_f]
/-- The homotopy equivalence from `X.chainComplex R` to `X.normalizedChainComplex R`. -/
noncomputable def homotopyEquivNormalizedChainComplex :
HomotopyEquiv (X.chainComplex R) (X.normalizedChainComplex R) :=
SimplicialObject.Splitting.homotopyEquivNondegComplex _ | true | {
"chosen_tokens": 55,
"rejected_tokens": 5,
"token_jaccard": 0.057143,
"token_length_ratio": 0.090909
} | wrong_namespace_qualification | wrong_namespace_qualification | 20a91c87f68c4d8fc3929aba655bcd39d928258f55b95422c2e4fe0be42f4460 | 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.QuasiIso
public import Mathlib.AlgebraicTopology.SimplicialSet.Homology.Basic
public import Mathlib.AlgebraicTopology.SimplicialSet.Splitting
... | by
exact OProverMissing.missing_5f583e779a0f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "e5a05192a5d6bf57cb49b99f7154208699208d23a0b9f2f05dcc74314645821f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/SimplicialSet",
"family_id": "tonormalizedchaincomplex_f_fromnormalizedchaincomplex_f",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplicialSet/Homology/Nondegenerate.lean",
"sample_id": "5f583e779a0f586168eb801c8f41ed46b5920677578a5ac3171bab04bd27faca"
} | train |
{
"chosen_proof_sha256": "d39e8a584cb9cadcbcf3f438cba249cdb73b9b66b3d2c3c40bd6185edf5ea6c9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "af7cda194f325da0ba382a80db632aee5a... | by
have h : f = 𝟙 j := by cat_disch
rw [h]
simp | true | {
"chosen_tokens": 16,
"rejected_tokens": 3,
"token_jaccard": 0.0625,
"token_length_ratio": 0.1875
} | unknown_identifier | unknown_identifier | 20e11165a19c0a6466b2cc097857d16b2a8587d4d5e3f4ea88b300d2325e2b2f | 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.Pi.Basic
Namespace:
CategoryTheory.Discrete
Local context:
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 ... | by
exact __oprover_missing_75d34468da20 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ac48af2a61ec57ee92967b946c9b267c9f73e5b417dbfe008e0417dd7ec6cb65",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Discrete",
"family_id": "functor_map_id",
"file_id": "mathlib/Mathlib/CategoryTheory/Discrete/Basic.lean",
"sample_id": "75d34468da2034a80fca64c4e5869a12c798eeacfe03a8618a187bb63ae913dc"
} | train |
{
"chosen_proof_sha256": "96180be036a5aa5c1b27a5d6a0c89b53acf0cf1ee820da9c2ba106ab55ce193e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
unfold traverse
simp only [Functor.map_map, fmap_def, map_comp_coe_apply, Multiset.fmap_def, ←
Multiset.map_traverse] | false | {
"chosen_tokens": 23,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.130435
} | by_exact_placeholder | exact? | 20ef6d184262595939cc8ae8b399edad7ce94caebe09ccf683b11902780870ff | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Control.AlternativeMonad
public import Mathlib.Data.Finset.Lattice.Union
public import Mathlib.Data.Finset.NAry
public import Mathlib.Data.Multiset.Functor
... | 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/Finset",
"family_id": "map_traverse",
"file_id": "mathlib/Mathlib/Data/Finset/Functor.lean",
"sample_id": "1daa9f465ccaa4038e45b7b45c48af664510fd77f6ef72028805211ad957a3be"
} | train |
{
"chosen_proof_sha256": "8b8c47751a3cd16893a65f53a169d429e22504bc42c9299e12e8db816e1165e6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "eabbb9ba32c0200e9777fe721b3dc25c57... | 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)) _) | true | {
"chosen_tokens": 76,
"rejected_tokens": 3,
"token_jaccard": 0.047619,
"token_length_ratio": 0.039474
} | unknown_identifier | unknown_identifier | 210fee043b3d7eee34bcd396fde23a67c62b1147e37b172c11ed187622d8a575 | 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.Multiset.Fintype
public import Mathlib.RingTheory.AdjoinRoot
public import Mathlib.RingTheory.Polynomial.RationalRoot
public import Mathlib.RingTheory.Int... | by
exact __oprover_missing_a55937381545 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a1e17c2cc733a9cc8252fd4f77b1d99a6ea1797b79d584c683d338809894d0d7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "ddb596461bd0c4bd75d4fb0cc2c7a82f17820be10b8f87f0c46c723e2b555f0f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
have hg : DifferentiableAt ℂ (fun z ↦ log (1 - z)) 0 := by
have : 1 - 0 ∈ slitPlane := (sub_zero (1 : ℂ)).symm ▸ one_mem_slitPlane
fun_prop
have : (fun z ↦ log (1 - z)) =O[𝓝 0] id := by
simpa only [sub_zero, log_one] using! hg.isBigO_sub
exact this.comp_summable hsum | true | {
"chosen_tokens": 80,
"rejected_tokens": 8,
"token_jaccard": 0.090909,
"token_length_ratio": 0.1
} | unsolved_goals | unsolved_goals | 2170087dc9172d1aca007e714c75aa73bcb5efa102fecaf611bb1fac2c7800f4 | 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.Normed.Module.FiniteDimension
public import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
public import Mathlib.LinearAlgebra.Complex.FiniteDimen... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "87d3fb4739a3ef97859cdbda2fe4d2d5a0363863e1ebc7bd39b18affa66c29bb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/EulerProduct",
"family_id": "summable",
"file_id": "mathlib/Mathlib/NumberTheory/EulerProduct/ExpLog.lean",
"sample_id": "fe21963d2aeeba6cb61599867ca841b158d2155ae815d66b4331eaa8e5ce4e3b"
} | train |
{
"chosen_proof_sha256": "ccb01e73a5d93776c5ed6830bc49ca880d5f04d2137609fc522a9ded57d8bf1c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "be074731ee1e86ad6f0c254a1e86d8fd22... | 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... | true | {
"chosen_tokens": 86,
"rejected_tokens": 3,
"token_jaccard": 0.021277,
"token_length_ratio": 0.034884
} | unknown_identifier | unknown_identifier | 2188c73f5289f0ffaf8f47240824366172c03ed1b754ed46e6eb1cc578e0f200 | 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.Quotient
public import Mathlib.CategoryTheory.Sites.Hypercover.One
public import Mathlib.CategoryTheory.Filtered.Basic
Namespace:
CategoryTheor... | by
exact __oprover_missing_6c405ac24568 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "82567c60cf45e02cc4f6dc664bdb48a87f306a767686ce7592ffd5e9512f9343",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "543d94c4b8beec8d8219800c1a531bf78be218b6d235d8e934ed1722e2e75902",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
refine measure_mono_null (fun x hx ↦ ?_)
(Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet _
(isClosed_closure (s := s)).measurableSet)
by_cases h : x ∈ closure s <;>
simp only [mem_compl_iff, mem_setOf, h, not_false_eq_true, indicator_of_notMem,
indicator_of_mem, Pi.one_apply]
· r... | true | {
"chosen_tokens": 91,
"rejected_tokens": 8,
"token_jaccard": 0.055556,
"token_length_ratio": 0.087912
} | unsolved_goals | unsolved_goals | 218bc6adf15ff3c66ce3f2201e3d3d27307ece96351373cc78eb1849b1aa5b29 | 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.Normed.Order.UpperLower
public import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
public import Mathlib.Topology.Order.DenselyOrdered
Names... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e3c55b818770a3ddaef15bbc762355387e1adddb238521610a9a0ca65bcf1929",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Order",
"family_id": "isupperset",
"file_id": "mathlib/Mathlib/MeasureTheory/Order/UpperLower.lean",
"sample_id": "4561b215385a78d721dfe36030198c0459e29f4db8031b7c27e613ae90c4a078"
} | train |
{
"chosen_proof_sha256": "576771f2ca0b7fae2b561a23269a331367199c4b2800aa18415fec083cc832ae",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [liftOfDerivationToSquareZero_apply, map_add, Ideal.Quotient.eq_zero_iff_mem.mpr (d x).prop,
zero_add, Ideal.Quotient.mk_algebraMap] | false | {
"chosen_tokens": 29,
"rejected_tokens": 3,
"token_jaccard": 0.047619,
"token_length_ratio": 0.103448
} | by_exact_placeholder | exact? | 2194b225115fd1e4dd66bcc6f9d049a994339361c2e4aa4cd0d920fec25041c0 | 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.Derivation.Basic
public import Mathlib.RingTheory.Ideal.Quotient.Operations
Namespace:
(root)
Local context:
/-
Copyright (c) 2020 Nicolò Cavaller... | 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": "RingTheory/Derivation",
"family_id": "liftofderivationtosquarezero_mk_apply",
"file_id": "mathlib/Mathlib/RingTheory/Derivation/ToSquareZero.lean",
"sample_id": "be9b8d29bc8008d9f2006f7c0e2e488c12668f4a1e6f2ba4ffcf6937f3ace739"
} | train |
{
"chosen_proof_sha256": "5e09f885c765d1667ba798f4a88bd6a2385014bfb40b892ea398481ff81d2e8d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3ad3f81ff3635c43ae0eded3174baf2b8d... | by
-- The key is that φ preserves labels, so edges labelled 's' stay labelled 's'
let e : v ⟶ ι s • v := ⟨s, rfl⟩
-- φ.map e is an edge from φ.obj v, and its label is preserved
have h := (φ.map e).property
-- This says: `ι (φ.map e).val • φ.obj v = φ.obj (ι s • v)`
-- We need to show `(φ.map e).val = s`
h... | true | {
"chosen_tokens": 216,
"rejected_tokens": 5,
"token_jaccard": 0.040541,
"token_length_ratio": 0.023148
} | wrong_namespace_qualification | wrong_namespace_qualification | 22033c151df62556097d54d858c083343cdfdcd18fd0202c69ec560ff57021bd | 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.Group.Basic
public import Mathlib.Combinatorics.Quiver.Covering
public import Mathlib.Combinatorics.Qui... | by
exact OProverMissing.missing_abc072079305 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "41d517f223c4bfd5fec91ffe89df47ad54a9cee072c5ce2e9c3a8d291d4b2aec",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/Quiver",
"family_id": "map_smul_of_comp_labelling_eq",
"file_id": "mathlib/Mathlib/Combinatorics/Quiver/Schreier.lean",
"sample_id": "abc072079305390f0bef511c2c60314512d25bfb1b8f705b8e28581cf1707079"
} | train |
{
"chosen_proof_sha256": "4f2754f3c1709f7213213c711ba2777cb527cb9fa5e9c5f811685e4d0a510864",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "aaa0680a89a9aa6b80dccb48330d64d6e7... | by
rw [← IsLocalization.mk'_one (M := K[X]⁰) (FractionRing K[X]) p, ← mk_coe_def, Submonoid.coe_one] | false | {
"chosen_tokens": 32,
"rejected_tokens": 37,
"token_jaccard": 0.84,
"token_length_ratio": 1.15625
} | markdown_preamble_trailing | trailing | 22ae332d660e412f06421002ab5b39535de5d5f6a3c80a61fa6812141c61f8b4 | 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.Basic
public import Mathlib.Algebra.Ring.NonZeroDivisors
public import Mathlib.RingTheory.Localization.FractionRing
Namespace:
RatFunc
Loc... | by
rw [← IsLocalization.mk'_one (M := K[X]⁰) (FractionRing K[X]) p, ← mk_coe_def, Submonoid.coe_one]
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": "FieldTheory/RatFunc",
"family_id": "mk_one",
"file_id": "mathlib/Mathlib/FieldTheory/RatFunc/Defs.lean",
"sample_id": "2ba6bee601213da996a0d324b595d9dabc6794f2b60ae27ff16595b74c0086ad"
} | train |
{
"chosen_proof_sha256": "6dfab6bad543a2d9c97fdd7bad870518b9b6feb74bbecb3fcb1c2ddbf6b41bd4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
have : IsReflexive R N := e.isReflexive_of_equiv_dual_of_isReflexive
suffices
(p.map (e.symm : Dual R M →ₗ[R] N)).dualAnnihilator.map (e.flip.symm : Dual R N →ₗ[R] M) =
(p.dualCoannihilator.map (e.flip : M →ₗ[R] Dual R N)).map (e.flip.symm : Dual R N →ₗ[R] M)
from (Submodule.map_injective_of_in... | false | {
"chosen_tokens": 135,
"rejected_tokens": 2,
"token_jaccard": 0.02381,
"token_length_ratio": 0.014815
} | by_exact_placeholder | by? | 22e313d13272ce019dfc3eb44b147564aa4726759d0b7ea0dac62e5bd66d7058 | 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
Namespace:
Submodule
Local context:
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as de... | 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/PerfectPairing",
"family_id": "map_dualcoannihilator_linearequiv_flip",
"file_id": "mathlib/Mathlib/LinearAlgebra/PerfectPairing/Basic.lean",
"sample_id": "44eed7b983007fa4c1d9c73cb7fb1872722dad2bf7f08f9f9ca305f717904a0a"
} | train |
{
"chosen_proof_sha256": "255334aceee88256ccde5808751019138f069b260876b485bc11ea482a1aed8d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8a92de4cf09f54437fffe352359b2fe17d... | by
intro b hb
have hg' := hg b hb
obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hg'⟩ := hg'
refine ⟨c₁, hc₁_mem, c₂, hc₂_mem, ?_⟩
filter_upwards [hg', (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg, hfg]
with x hx₁ hx₂ hx₃
intro u hu
rw [hx₂ u hu.1, hx₃]
exact hx₁ u hu | false | {
"chosen_tokens": 95,
"rejected_tokens": 99,
"token_jaccard": 0.928571,
"token_length_ratio": 1.042105
} | missing_wrong_import | import | 2335bedc644e23a0e929bce581ada7640866b3e1884652288ea617cc065a13f8 | 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.SpecialFunctions.Pow.Real
public import Mathlib.Algebra.Order.ToIntervalMod
public import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Al... | import Mathlib.WrongImport
by
intro b hb
have hg' := hg b hb
obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hg'⟩ := hg'
refine ⟨c₁, hc₁_mem, c₂, hc₂_mem, ?_⟩
filter_upwards [hg', (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg, hfg]
with x hx₁ hx₂ hx₃
intro u hu
rw [hx₂ u hu.1, hx₃]
exact hx₁ u ... | {
"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": "Computability/AkraBazzi",
"family_id": "congr_of_eventuallyeq",
"file_id": "mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean",
"sample_id": "04f65d5b50a300cc6227733efdcd3f39e044ddd8767d10f8047fdbe408fc2df7"
} | train |
{
"chosen_proof_sha256": "46daf5b2e7bf4d90e3013b1a193808b8c73aa5defa9de5956e0736e8e4908309",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b1b68db34cb48a6039b6507dde69182127... | by
apply of_span_eq_top_target _ hs
rintro - ⟨i, rfl⟩
exact .equiv <| (IsLocalization.algEquiv (.powers <| s i) _ (T i)).symm |>.restrictScalars R | false | {
"chosen_tokens": 42,
"rejected_tokens": 47,
"token_jaccard": 0.848485,
"token_length_ratio": 1.119048
} | markdown_preamble_trailing | preamble | 239a92bc3dca9067c2fa51e48aac57dbb7a02d1c7d477729df4282490bd80097 | 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.Finiteness.FiniteTypeLocal
public import Mathlib.RingTheory.Localization.Away.AdjoinRoot
Namespace:
Algebra.FinitePresentation
Local context:
/-
C... | Here is the proof:
by
apply of_span_eq_top_target _ hs
rintro - ⟨i, rfl⟩
exact .equiv <| (IsLocalization.algEquiv (.powers <| s i) _ (T i)).symm |>.restrictScalars R | {
"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/Finiteness",
"family_id": "of_span_eq_top_target_of_islocalizationaway",
"file_id": "mathlib/Mathlib/RingTheory/Finiteness/FinitePresentationLocal.lean",
"sample_id": "9439016a6f4e6a57c85536640298da69f3a589358dd7d886fc77db2d3daea612"
} | train |
{
"chosen_proof_sha256": "2933327708396b2202fe5681ecb1091bcb221fad36d86c22e4ed4f768e81a87c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d58f1e0ee25611d5386590a468f4a79172... | by
ext <;> simp | false | {
"chosen_tokens": 6,
"rejected_tokens": 11,
"token_jaccard": 0.545455,
"token_length_ratio": 1.833333
} | markdown_preamble_trailing | preamble | 24208e0adf1f8ddd0ac8dac34ac4834a4ae66d40044f9c3eee4e3696e49f4a73 | 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.Bicategory.LocallyDiscrete
public import Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo
Namespace:
CategoryTheory.Pseudofunctor... | Here is the proof:
by
ext <;> simp | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Bicategory",
"family_id": "map_id_map",
"file_id": "mathlib/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean",
"sample_id": "77f925e6d7fb9f855b12d8cba8f5cb36cbebbc68e7684132b1a6af65dd6fe312"
} | train |
{
"chosen_proof_sha256": "f9aa1f2b0c3ec5667a46cb932e6917f3da85ddecbe4e4201b9fff6f973d74358",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
obtain ⟨a, b, hij⟩ | ⟨a, b, hab⟩ := h
· exact Lex.left _ _ (hr _ _ hij)
· exact Lex.right _ _ (hs _ _ _ hab) | false | {
"chosen_tokens": 46,
"rejected_tokens": 5,
"token_jaccard": 0.12,
"token_length_ratio": 0.108696
} | sorry_admit_sorryAx_proof_hole | sorryAx | 242bc8e63e5959bf2431ff03014f6c585fb6037bb738514c6a4bc1d0bb00d08a | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Logic.Function.Defs
public import Mathlib.Order.Defs.Unbundled
public import Batteries.Logic
Namespace:
Sigma
Local context:
/-
Copyright (c) 2021 Yaël Dilli... | 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/Sigma",
"family_id": "lex",
"file_id": "mathlib/Mathlib/Data/Sigma/Lex.lean",
"sample_id": "ca4d4f9887ee3a892e6506dc6023e4d2417fcaf8ede1273a6e1bb19013900ef6"
} | train |
{
"chosen_proof_sha256": "da9b69d9b6dc7fdb8d3935d0065bc697214d7335fb75b8a7df981601b0c50a86",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
ext
simpa using (eq_bot_or_eq_top _).symm | false | {
"chosen_tokens": 10,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.3
} | by_exact_placeholder | placeholder | 244021c0c967282e0f564e4ca4fc44557fc485ce1fe12759dc70e9ea9bec6cb5 | 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... | 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": "Order/Atoms",
"family_id": "univ",
"file_id": "mathlib/Mathlib/Order/Atoms/Finite.lean",
"sample_id": "c850dd5f680796ef8939e552d2ed07dbbf81b4eaedf7784baebc1c5d3cb8013c"
} | train |
{
"chosen_proof_sha256": "447b3ac2404a7fdae48d1d41ba080e9c5434ee1fd77cdc32e3d7e6b39eb7aa1d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
refine ⟨fun h ↦ ?_, fun h ↦ h.units_smul w⟩
convert! h.units_smul (fun i ↦ (w i)⁻¹)
simp [funext_iff] | true | {
"chosen_tokens": 37,
"rejected_tokens": 8,
"token_jaccard": 0.033333,
"token_length_ratio": 0.216216
} | unsolved_goals | unsolved_goals | 245ea563a30fcf5dc95495367db27739788352c8c6bc6e7f5623f0a4a17a6a43 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Fin
public import Mathlib.LinearAlgebra.LinearIndependent.Defs
public import Mathlib.Logic.Equiv.Fin.Rotate
Namespace:
(root)
Local cont... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "fb2973245bb56815e051c21d24450521ae551b5d9f39883d1ff1e55acbd6736f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/LinearIndependent",
"family_id": "linearindependent",
"file_id": "mathlib/Mathlib/LinearAlgebra/LinearIndependent/Basic.lean",
"sample_id": "b38b14c96b38afbba7c3259cf4c4e7b4c145e1b2ac6c153a770feeac3a7076d5"
} | train |
{
"chosen_proof_sha256": "8da08d0db4cb5d343e5a5a407377b94360b60c72eeb504f2655147b8545bacfb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
cases isEmpty_or_nonempty α <;> simp [FreeCommRing] | true | {
"chosen_tokens": 11,
"rejected_tokens": 8,
"token_jaccard": 0.058824,
"token_length_ratio": 0.727273
} | unsolved_goals | unsolved_goals | 2503ee875e5ead31a29d5772dd8a0d1b9662b0bc149536c48b10d5b8c012d855 | 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.FreeAbelianGroup.Finsupp
public import Mathlib.Algebra.Ring.TransferInstance
public import Mathlib.Data.Finsupp.Fintype
public import Mathlib.Data.ZMod... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "89889556b4425adfa9c730e30d09ec974755f2bba8657580a1f9b42b33e524ec",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "bb5b4eccf3c9035f221ecef92ea8d7966625741540da04e726a173d5b6907beb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
apply Functor.ext; rotate_left
· intro X
ext i
specialize h i
have := congr_obj h X
simpa
· intro X Y g
funext i
specialize h i
have := congr_hom h g
simpa | false | {
"chosen_tokens": 37,
"rejected_tokens": 2,
"token_jaccard": 0.045455,
"token_length_ratio": 0.054054
} | sorry_admit_sorryAx_proof_hole | sorry | 253126cef318b090d400d45200ee703aa9a04b71caffb4c24bab1dc6eae042b8 | 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.EqToHom
public import Mathlib.CategoryTheory.NatIso
public import Mathlib.CategoryTheory.Products.Basic
Namespace:
Functor
Local context:
/-
C... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Pi",
"family_id": "pi_ext",
"file_id": "mathlib/Mathlib/CategoryTheory/Pi/Basic.lean",
"sample_id": "f1e96e086d107cbb628f9333e3a0d90e5f01c368a40a55b24a768acd38a2bd9b"
} | train |
{
"chosen_proof_sha256": "7b7262505c003dbad9c3aa96a2764e75ade60fa912085a50b5740b06b8e012c5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b01e50f28aa56d2ac7f58b9ce851e208ce... | by
intro H
refine fermatLastTheoremThree_of_three_dvd_only_c (fun a b c hc ha hb ⟨x, hx⟩ hcoprime h ↦ ?_)
refine H a b c 1 (by simp [hc]) (fun hdvd ↦ ha ?_) (fun hdvd ↦ hb ?_) ?_ ?_ ?_
· rwa [← Ideal.norm_dvd_iff (hζ.prime_norm_toInteger_sub_one_of_prime_ne_two' (by decide)),
hζ.norm_toInteger_sub_one_of_... | true | {
"chosen_tokens": 144,
"rejected_tokens": 3,
"token_jaccard": 0.040816,
"token_length_ratio": 0.020833
} | circular_dependency | circular_dependency | 25337fbdcf5bb30be3c010180143c4b5c9a5d4b6e4b4017cb8464eb4d36b235d | 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.FLT.Basic
public import Mathlib.NumberTheory.NumberField.Cyclotomic.PID
public import Mathlib.NumberTheory.NumberField.Cyclotomic.Three
public imp... | by
exact FermatLastTheoremForThree_of_FermatLastTheoremThreeGen | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "c60aba24f3a207097e7a17518d4dfb7e37cc24baba00072a5990f1e9cd88fef8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/FLT",
"family_id": "fermatlasttheoremforthree_of_fermatlasttheoremthreegen",
"file_id": "mathlib/Mathlib/NumberTheory/FLT/Three.lean",
"sample_id": "a8d869c19b5e1d1d76d3c28a6dc188be2ec2a1a13841d22f5d203f9a92b23e39"
} | train |
{
"chosen_proof_sha256": "5ca120de9531c196478bb85a2839de944f04348088dc99c356e217a11b64504f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0e4bac0eeddb51f4e6c61f8570f09dd4bd... | by
induction xs generalizing ys i with
| nil => cases h₂
| cons x' xs xs_ih =>
cases i
· simp only [length_cons, lt_add_iff_pos_left, add_pos_iff, Nat.lt_add_one, or_true,
getElem?_eq_getElem, getElem_cons_zero, Option.some.injEq] at h₂
subst h₂
cases ys
· cases h₁
· simp
... | true | {
"chosen_tokens": 129,
"rejected_tokens": 5,
"token_jaccard": 0.048387,
"token_length_ratio": 0.03876
} | circular_dependency | circular_dependency | 269750fb8edad59304aa811ce6a0dbbd33afc5b6c48f9ad4f1b728523d361996 | 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.MLList.Basic
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.ZeroLEOne
public import Mathlib.Data.DFinsupp.Defs
public... | by
exact List.applyId_zip_eq | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "79ffa88d073191f8ba5409d33573682fd570e65b0c7a9d766971052451f60a0c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Testing/Plausible",
"family_id": "list",
"file_id": "mathlib/Mathlib/Testing/Plausible/Functions.lean",
"sample_id": "e2cd3c1a8076b3af47b98c32f101f8a48aa6a72e4acab38020fc2356c4738a6b"
} | train |
{
"chosen_proof_sha256": "4fe5d002e1432ce371569b334eb570afebb4011df3c1ab6e1eb29382c5b0d1c6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ee5d0df3c7f8bc043a92caea4286a93e5e... | by
haveI := Nonempty.map f ‹_›
haveI := Nonempty.map g ‹_›
exact (map_injective.and map_surjective).trans and_and_and_comm | false | {
"chosen_tokens": 29,
"rejected_tokens": 34,
"token_jaccard": 0.826087,
"token_length_ratio": 1.172414
} | markdown_preamble_trailing | trailing | 26a9f6314cb5e3df27f4d4e9cb0a3d7933c47d635727e22a495a91e9b104fb51 | 0 | 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
haveI := Nonempty.map f ‹_›
haveI := Nonempty.map g ‹_›
exact (map_injective.and map_surjective).trans and_and_and_comm
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Prod",
"family_id": "map_bijective",
"file_id": "mathlib/Mathlib/Data/Prod/Basic.lean",
"sample_id": "5d67e5e7b9950556ea8bd8cdfa3767dbe10a6562717cb86050afc39cdaf6a4b7"
} | train |
{
"chosen_proof_sha256": "4e24363fb7ba779b1d49eb8030dc7b2f8941ee6210796f291a186ebbe93a5464",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "bffda7eb658c234a9b6c2ed0b066a47fa2... | by
apply complementedLattice_of_sSup_atoms_eq_top
have : ∀ i ∈ s, ∃ t : Set α, f i = sSup t ∧ ∀ a ∈ t, IsAtom a := fun i hi ↦ by
replace h := complementedLattice_iff_isAtomistic.mp (h i hi)
obtain ⟨u, hu, hu'⟩ := eq_sSup_atoms (⊤ : Iic (f i))
refine ⟨(↑) '' u, ?_, ?_⟩
· replace hu : f i = ↑(sSup u) ... | true | {
"chosen_tokens": 244,
"rejected_tokens": 3,
"token_jaccard": 0.02439,
"token_length_ratio": 0.012295
} | unknown_identifier | unknown_identifier | 26c97243878f536132b113f5de48c4a6850fadfc42945681a20d71f40d507235 | 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.CompleteLatticeIntervals
public import Mathlib.Order.CompactlyGenerated.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2024 Oliver Nash. All r... | by
exact __oprover_missing_31d3eaad908c | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "fcef7b034e506b6423a928364b9961e54d0c1d5f7a0d6df53b0d61e0af74f6e1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/CompactlyGenerated",
"family_id": "complementedlattice_of_complementedlattice_iic",
"file_id": "mathlib/Mathlib/Order/CompactlyGenerated/Intervals.lean",
"sample_id": "31d3eaad908c26db9fa8cd7a18a465f80cc33c5d4402214fcc63790030c8fb99"
} | train |
{
"chosen_proof_sha256": "f4a24f2bb7a6f0211b6b657b7e7dccc3a2dcf7c7a859099b25ae3a21d107e6f6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "956561a8c5c16a4f630b857e9ae4145c51... | by
simpa only [← hammingDist_zero_right, hf]
using! hammingDist_comp_le_hammingDist f (y := fun _ ↦ 0) | true | {
"chosen_tokens": 21,
"rejected_tokens": 3,
"token_jaccard": 0.043478,
"token_length_ratio": 0.142857
} | circular_dependency | circular_dependency | 26dfce741a44216871b84c3e29ec976fc17785b97e7ca021f994dae056a5ae6d | 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.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 hammingNorm_comp_le_hammingNorm | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "ac917cfe9c75c6e989bb977b96027860dd8dde27ca91cc532283cf877b46aa19",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "5e809a8a5b6dc52f86458c7ec1ba0de189d9ba286d3abe6189f2eb23181c4aa7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "72e24772baf3562fa19ff8d7c94a207bf1... | by
refine Set.ext fun R ↦ ⟨fun ⟨g, hg⟩ ↦ hg ▸ ⟨hQ.smul g, hP.smul g⟩, fun h ↦ ?_⟩
have : R.IsPrime := h.1
obtain ⟨g, hg⟩ := exists_smul_of_under_eq A B G Q R (hP.over.symm.trans h.2.over)
exact ⟨g, hg.symm⟩ | false | {
"chosen_tokens": 80,
"rejected_tokens": 84,
"token_jaccard": 0.926829,
"token_length_ratio": 1.05
} | missing_wrong_import | import | 26ee24b7c91c431d3213ef463003a93c54a9723cb87d5a4868e3d1751b40c22d | 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.Invariant.Defs
public import Mathlib.RingTheory.IntegralClosure.IntegralRestrict
public import Mathlib.RingTheory.LocalRing.ResidueField.Ideal
Name... | import Mathlib.WrongImport
by
refine Set.ext fun R ↦ ⟨fun ⟨g, hg⟩ ↦ hg ▸ ⟨hQ.smul g, hP.smul g⟩, fun h ↦ ?_⟩
have : R.IsPrime := h.1
obtain ⟨g, hg⟩ := exists_smul_of_under_eq A B G Q R (hP.over.symm.trans h.2.over)
exact ⟨g, hg.symm⟩ | {
"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/Invariant",
"family_id": "orbit_eq_primesover",
"file_id": "mathlib/Mathlib/RingTheory/Invariant/Basic.lean",
"sample_id": "94279022a062f44341d707de6cd3452c8f04400967480dbebc54e8c3a85912a9"
} | train |
{
"chosen_proof_sha256": "75232f0c801e33fc6a2f2b42c9fbde78aef8e1d36674a56a7714ae3434228c02",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8129b6b03a4962538cfee55f286a700211... | by
simp only [(v.filterAt_basis_closedBall x).mem_iff, ← and_imp, subset_def, mem_setOf] | false | {
"chosen_tokens": 20,
"rejected_tokens": 25,
"token_jaccard": 0.809524,
"token_length_ratio": 1.25
} | markdown_preamble_trailing | trailing | 27dcd539c7528a16bab0e42c10877179eb044a1b2367e5446fd31b51e65116e0 | 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.Measure.AbsolutelyContinuous
Namespace:
VitaliFamily
Local context:
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released unde... | by
simp only [(v.filterAt_basis_closedBall x).mem_iff, ← and_imp, subset_def, mem_setOf]
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": "MeasureTheory/Covering",
"family_id": "mem_filterat_iff",
"file_id": "mathlib/Mathlib/MeasureTheory/Covering/VitaliFamily.lean",
"sample_id": "419897952885489eef42100d7198d985d0ed456c5f3c123051bf15c5b6fac476"
} | train |
{
"chosen_proof_sha256": "95de6bb494dcefbe243626eb340736b29bf3b291e155dac01267e899f1bb69c2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b30174f1e4c5cb8ee3644e42068a798768... | by
rw [of_eq, Prefunctor.comp_assoc, Prefunctor.comp_assoc, Functor.toPrefunctor_comp]
dsimp [lift]
rw [Quotient.lift_spec, Paths.lift_spec, Quiver.Symmetrify.lift_spec] | true | {
"chosen_tokens": 37,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.081081
} | unknown_identifier | unknown_identifier | 27fb21e5b556ddac7ba755431ec6bda6f95a299bda9187fccb1e81541d6bde26 | 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.Groupoid
public import Mathlib.CategoryTheory.PathCategory.Basic
Namespace:
Quiver.FreeGroupoid
Local context:
/-
Copyright (c) 2022 Rémi Bott... | by
exact __oprover_missing_c031c9432732 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "450640791efd04eabba4f685a05e93e504e4ea3c544156655c5431e6c20d8721",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Groupoid",
"family_id": "lift_spec",
"file_id": "mathlib/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean",
"sample_id": "c031c9432732d05cd237126cbcac6a473645115eab86e0eb5195fb77c4512d50"
} | train |
{
"chosen_proof_sha256": "eb3d60d437f15cfc69d1d37a56f9a9b73ef1279b8f1ffb7d0327dabbc53ce4fd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
ext
simp only [alternatingMapLinearEquiv_comp_ιMulti, LinearMap.compAlternatingMap_apply,
LinearMap.coe_comp, comp_apply, alternatingMapLinearEquiv_apply_ιMulti]
/-! Functoriality of the exterior powers. -/
variable (n) in
/-- The linear map between `n`th exterior powers induced by a linear map between the... | false | {
"chosen_tokens": 110,
"rejected_tokens": 2,
"token_jaccard": 0.018868,
"token_length_ratio": 0.018182
} | sorry_admit_sorryAx_proof_hole | admit | 27fd89cbbb71a9ef083392658abdb1c67642492f7c6ce3190a2e6b1491328683 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Module.Presentation.Basic
public import Mathlib.LinearAlgebra.ExteriorAlgebra.OfAlternating
Namespace:
exteriorPower
Local context:
/-
Copyright (c) ... | 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": "LinearAlgebra/ExteriorPower",
"family_id": "alternatingmaplinearequiv_comp",
"file_id": "mathlib/Mathlib/LinearAlgebra/ExteriorPower/Basic.lean",
"sample_id": "b5eed30bb36819b24fa9a164628ee2a2c0338152109146fc3f9ca3ebec531d4e"
} | train |
{
"chosen_proof_sha256": "eeea18d1418cb641b169bfeb8bce930a2742ccdb2c216d63523e2c130f059c37",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cbe42b39bb30027dd1ac1ec3aabbd6e486... | by
obtain ⟨χ, hχ⟩ := Finset.rado_selection_subtype (β := β) (fun s ↦ g s s.finite_toSet)
refine ⟨χ, fun s hs ↦ ?_⟩
obtain ⟨t, ht, hst⟩ := hχ hs.toFinset
simp only [Set.Finite.toFinset_subset] at ht
exact ⟨t, by simp_all⟩ | false | {
"chosen_tokens": 70,
"rejected_tokens": 75,
"token_jaccard": 0.902439,
"token_length_ratio": 1.071429
} | markdown_preamble_trailing | trailing | 281dbc449cbb0b535f74a149deb0c06cde610f3d57c008e7bef30087774b47f1 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
import Mathlib.Topology.Compactness.Compact
public import Mathlib.Data.Set.Finite.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Bhavik Mehta. All rights reserved.
R... | by
obtain ⟨χ, hχ⟩ := Finset.rado_selection_subtype (β := β) (fun s ↦ g s s.finite_toSet)
refine ⟨χ, fun s hs ↦ ?_⟩
obtain ⟨t, ht, hst⟩ := hχ hs.toFinset
simp only [Set.Finite.toFinset_subset] at ht
exact ⟨t, by simp_all⟩
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",
"family_id": "set",
"file_id": "mathlib/Mathlib/Combinatorics/Compactness.lean",
"sample_id": "88630b1e90f48b536f58e1419d7c32cb382c083b0e23e75073e0b1e0fdf80749"
} | train |
{
"chosen_proof_sha256": "0276c875b5de26026fe85dbf0c34475b3acdef44fb07206bfeb84e06366c9607",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "36e832bb8cc9ec1ccec45966728b66a2c4... | by
funext n
induction n with
| zero => rfl
| succ n ih =>
unfold map iterate get
rw [map, get] at ih
rw [iterate]
exact congrArg f ih | true | {
"chosen_tokens": 35,
"rejected_tokens": 2,
"token_jaccard": 0.041667,
"token_length_ratio": 0.057143
} | invalid_tactic | invalid_tactic | 2821bb255314f9ee038db5ea697ede39bca49b53562eaa0159e2eba953c39b59 | 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.Stream.Defs
public import Mathlib.Logic.Function.Basic
public import Mathlib.Data.Nat.Basic
public import Mathlib.Tactic.Common
Namespace:
Stream'
Local... | by
__oprover_invalid_tactic_16839e522d17 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "f8b9f546e80bddd9ebbe91538ecae622fc54e7250a98abf6d6d8156c75444397",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Stream",
"family_id": "map_iterate",
"file_id": "mathlib/Mathlib/Data/Stream/Init.lean",
"sample_id": "16839e522d1756068dfaf97c745b531d7271caa8d21dfa9ef1a00b09a41091af"
} | train |
{
"chosen_proof_sha256": "3dd577e7ce2be81d92ba051dd678c134f2a6bcae0a357014a7b7228cadb929c6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "872abf858e904265f85b85d943b70513c7... | by
refine strictMonoOn_of_lt_add_one ⟨?_⟩ fun n _ _ hn ↦
pow_ssubset_pow_succ_of_pow_ne_closure hX₁ hX hn
rintro - - n hn m ⟨-, hmn⟩ hm
apply hn
obtain rfl | hm₀ := m.eq_zero_or_pos
· simp [eq_comm (a := (1 : Set _)), coe_set_eq_one, -Set.subset_singleton_iff,
hX.coe.not_subset_singleton] at hm
· ... | true | {
"chosen_tokens": 138,
"rejected_tokens": 3,
"token_jaccard": 0.016129,
"token_length_ratio": 0.021739
} | unknown_identifier | unknown_identifier | 2828126f12b3797d60c2acd2a09b7f61c43d10d82ed21da7d2021cf5e54597ce | 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.Algebra.Group.Subgroup.Pointwise
public import Mathlib.Data.Nat.SuccPred
Namespace:
Finset
Local ... | by
exact __oprover_missing_1a64a2331eda | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "51bb6274b82ac391cf9ea275c995259c64685cf2d84285860d04e4f8c5ab170d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Geometry/Group",
"family_id": "pow_right_strictmonoon",
"file_id": "mathlib/Mathlib/Geometry/Group/Growth/LinearLowerBound.lean",
"sample_id": "1a64a2331eda0da0fbaef78bea3f8bc0d6cc844d45d3bd42a1481a8c40fbd0be"
} | train |
{
"chosen_proof_sha256": "76e3cedf98327795879092160501512043429427fee2f5b0e827ab58125ee1b5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f4641d98f5a96bd0f077eb90f92ed674d3... | by
rw [IsSimplyConnected, simply_connected_iff_loops_nullhomotopic,
← isPathConnected_iff_pathConnectedSpace]
refine .and .rfl ⟨fun h x p hp ↦ ?_, fun h x p ↦ ?_⟩
· lift x to s using by simpa using hp 0
rcases h x {
toFun := fun t ↦ ⟨p t, hp t⟩
source' := by simp
target' := by simp
}... | false | {
"chosen_tokens": 153,
"rejected_tokens": 158,
"token_jaccard": 0.929825,
"token_length_ratio": 1.03268
} | markdown_preamble_trailing | trailing | 28bfaea17662d50921c524d94deb2540779e3255131b3c10b4164a04d7348200 | 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.FundamentalGroupoid.FundamentalGroup
public import Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps
public import Mathlib.Topology.H... | by
rw [IsSimplyConnected, simply_connected_iff_loops_nullhomotopic,
← isPathConnected_iff_pathConnectedSpace]
refine .and .rfl ⟨fun h x p hp ↦ ?_, fun h x p ↦ ?_⟩
· lift x to s using by simpa using hp 0
rcases h x {
toFun := fun t ↦ ⟨p t, hp t⟩
source' := by simp
target' := by simp
}... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/FundamentalGroupoid",
"family_id": "issimplyconnected_iff_exists_homotopy_refl_forall_mem",
"file_id": "mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean",
"sample_id": "856d27b4c3ec825fbdd58bf9dd767cc218a09754dd1106fc2031d94545f79fef"
} | train |
{
"chosen_proof_sha256": "c337966156f7fa0b44c661383f2baec35f6094ce1c36181a113cc67c17fd2ac2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
by_cases c <;> simp [*] | false | {
"chosen_tokens": 10,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.2
} | by_exact_placeholder | by? | 28c74d95d1f9cd44be4bb9036ba080299b3ffef4bcab6f7adbf1713af2c1f491 | 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.Ordering.Basic
public import Mathlib.Order.Defs.Unbundled
Namespace:
Ordering
Local context:
/-
Copyright (c) 2017 Microsoft Corporation. All rights res... | 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": "Data/Ordering",
"family_id": "ite_eq_gt_distrib",
"file_id": "mathlib/Mathlib/Data/Ordering/Lemmas.lean",
"sample_id": "4771507a2edaa490165ad6dbbb8bbee190b6099c2dfd923f1e5c8b47e63ba2b7"
} | train |
{
"chosen_proof_sha256": "340281fbf0a45d34c3a34d2f4a61a24c355dc2f03e29d91afa10e3fb584bf2fd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3fd6053780e5eb00bf5e0f3cec4bde111b... | by
intro a b h
simp only [LT.lt, Finsupp.lex_def]
simp only [ofLex_toLex, Nat.lt_eq]
use a
constructor
· intro d hd
simp only [Finsupp.single_eq_of_ne hd.ne, Finsupp.single_eq_of_ne (hd.trans h).ne]
· simp [h.ne'] | true | {
"chosen_tokens": 61,
"rejected_tokens": 3,
"token_jaccard": 0.032258,
"token_length_ratio": 0.04918
} | unknown_identifier | unknown_identifier | 28fd13829bb27468a9ae9f5e9535fe3acb144b75ec6ad7302e14a04ef6c06872 | 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.Finsupp.Order
public import Mathlib.Data.DFinsupp.Lex
public import Mathlib.Data.Finsupp.ToDFinsupp
Namespace:
Finsupp
Local context:
/-
Copyright (c) 2... | by
exact __oprover_missing_3b594b8efd71 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "0e47a2a545dfed109ecf5e553bc0319f688e317d34d6d13da4ec88a015a1e011",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Finsupp",
"family_id": "lex",
"file_id": "mathlib/Mathlib/Data/Finsupp/Lex.lean",
"sample_id": "3b594b8efd7138c1edf9cf4beb8848fba944bf82946b75971c431aea36b7d8df"
} | train |
{
"chosen_proof_sha256": "9c5374a4a76445b21ceb5288576943f3dfabe49605e7c1eedae05176a4a561c7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [← isLocallySurjective_iff_epi']
exact LightCondensed.isLocallySurjective_iff_locallySurjective_on_lightProfinite _ f | false | {
"chosen_tokens": 12,
"rejected_tokens": 2,
"token_jaccard": 0.076923,
"token_length_ratio": 0.166667
} | sorry_admit_sorryAx_proof_hole | sorry | 293d1c19c91b7f2d834e7f66976ec97948e88a4fc64b98b947b9485f2d7cfdb3 | 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.Limits.Shapes.SequentialProduct
public import Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit
public import Mathlib.Condensed.Light.Functo... | 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": "Condensed/Light",
"family_id": "epi_iff_locallysurjective_on_lightprofinite",
"file_id": "mathlib/Mathlib/Condensed/Light/Epi.lean",
"sample_id": "cb787666449dd4a52c1a3d0c14107bfed4df4b3d94d031fd1a98bc977a411bf3"
} | train |
{
"chosen_proof_sha256": "ce223990efe19a3952beb8c48c08eec52484b0eabebe4090119bef4f0e984746",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ebf2287be0dff80d8527d98a13ffae0d07... | by simp [mk, ← ofCauchy_add] | true | {
"chosen_tokens": 8,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.625
} | wrong_namespace_qualification | wrong_namespace_qualification | 2955e439973cf0e54f62fbb48c3368e4917d01fa3c8b1ca0c7212e8a25f36fc7 | 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 OProverMissing.missing_210dd76ae67a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1361893137bf51dcbae79e0dd23223aed20c8d91ae57d717422c61a8df3efe13",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Real",
"family_id": "mk_add",
"file_id": "mathlib/Mathlib/Data/Real/Basic.lean",
"sample_id": "210dd76ae67ac01eff7a888fdcf07f7b44a5b4586ca30f9a119d28a997965242"
} | train |
{
"chosen_proof_sha256": "de0e9cf3147279392578300fd27bc7777015480616482bce0bced9bd64ff6f36",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [NatTrans.ext_iff]
simp [le_def, Set.subset_def, ConcreteCategory.hom_ext_iff, funext_iff] | false | {
"chosen_tokens": 21,
"rejected_tokens": 5,
"token_jaccard": 0.052632,
"token_length_ratio": 0.238095
} | sorry_admit_sorryAx_proof_hole | sorryAx | 295ae110ea111122ef18bd440b3ae2b287b254623a0ac2b5d21783ab2ecebf6f | 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.Limits.Shapes.Equalizers
public import Mathlib.CategoryTheory.Subfunctor.Image
Namespace:
CategoryTheory.Subfunctor
Local context:
/-
Copyrigh... | 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": "CategoryTheory/Subfunctor",
"family_id": "range_le_equalizer_iff",
"file_id": "mathlib/Mathlib/CategoryTheory/Subfunctor/Equalizer.lean",
"sample_id": "7c2cc263cb08cd0ae7b3b6f1e5d4cbec38b9a07797c8b210ba27cbac169d95dc"
} | train |
{
"chosen_proof_sha256": "f6f5bbf3230230f3a0428b8083f535c36ff227e3eca17b3d7dbf4a73cb1bc39d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [weaklyEtale_iff, weaklyEtale_iff, Module.Flat.ulift_left_iff, Module.Flat.ulift_right_iff]
congr!
conv_rhs => rw [← RingHom.Flat.ulift_iff.{u₁, u₂}]
rw [TensorProduct.lmul'_ulift, AlgHom.toRingHom_eq_coe, AlgHom.comp_toRingHom]
exact RingHom.Flat.comp_iff_of_bijective_right (Equiv.bijective _) | true | {
"chosen_tokens": 66,
"rejected_tokens": 8,
"token_jaccard": 0.02439,
"token_length_ratio": 0.121212
} | unsolved_goals | unsolved_goals | 298cd7b25b0ffcbd5188ba9b9b27f37f11d09a876331551754ccfa878d7978ab | 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.RingHom.Flat
public import Mathlib.RingTheory.Etale.Basic
public import Mathlib.RingTheory.Smooth.Flat
Namespace:
Algebra.WeaklyEtale
Local contex... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "21e84bd0a80657de1d30789cdb6932b85962cc35f335ed13c74e1d1a7824e0c7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Etale",
"family_id": "ulift_iff",
"file_id": "mathlib/Mathlib/RingTheory/Etale/Weakly.lean",
"sample_id": "ea51963a627d887e53897ff9660cd0137a6ca3fccf9a879fe29c6140902dfe65"
} | train |
{
"chosen_proof_sha256": "0d95bb02e9e9a9b98bdd4fbdf7abc16ec6edaa269b3beda79e93adb84be4b936",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5dd2d45a5990b3cf1553e9a7690924d8cc... | by
by_cases h : k = 0
· simp [h]
exact (isPrimitiveRoot_exp k h).card_primitiveRoots | false | {
"chosen_tokens": 20,
"rejected_tokens": 27,
"token_jaccard": 0.894737,
"token_length_ratio": 1.35
} | markdown_preamble_trailing | fence | 29ddd831492637e0a029b97532214282686237c74cad53e65a4e8dad36b4a2f0 | 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.SpecialFunctions.Complex.Log
public import Mathlib.RingTheory.Int.Basic
public import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
public import Mat... | ```lean
by
by_cases h : k = 0
· simp [h]
exact (isPrimitiveRoot_exp k h).card_primitiveRoots
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/RootsOfUnity",
"family_id": "card_primitiveroots",
"file_id": "mathlib/Mathlib/RingTheory/RootsOfUnity/Complex.lean",
"sample_id": "32b115c85276c93388c62a93ae6422efd911c7d0d06efd966d995ffe632b53e9"
} | train |
{
"chosen_proof_sha256": "475307141c3a1cbf5e416ce7878a4f3fd1a2e5cb9833e295c816f2bfea6746a8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simp [Fintype.subtype_card, Finset.card_univ] | false | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.181818
} | by_exact_placeholder | by? | 29fa3bda623fc371960080f054aee6783eafa62672d775562bf40f579f6b0950 | 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.Powerset
public import Mathlib.Data.Fintype.EquivFin
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
... | 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": "Data/Fintype",
"family_id": "fintype",
"file_id": "mathlib/Mathlib/Data/Fintype/Powerset.lean",
"sample_id": "8718a7560479506834cd432f1d1bbbcdc14349fe8b10b51da3d79c17a3f32f95"
} | train |
{
"chosen_proof_sha256": "83fc6e2c4e62e7f25a2d9e2ffddabb7d5875dc78574f118790a50383327ea178",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
obtain ⟨n, hn : ∀ m, n ≤ m → LinearMap.ker (f ^ n) = LinearMap.ker (f ^ m)⟩ :=
monotone_stabilizes_iff_noetherian.mpr inferInstance f.iterateKer
refine eventually_atTop.mpr ⟨n, fun m hm ↦ disjoint_iff.mpr ?_⟩
rw [← hn _ hm, Submodule.eq_bot_iff]
rintro - ⟨hx, ⟨x, rfl⟩⟩
apply pow_map_zero_of_le hm
rep... | false | {
"chosen_tokens": 142,
"rejected_tokens": 5,
"token_jaccard": 0.033898,
"token_length_ratio": 0.035211
} | sorry_admit_sorryAx_proof_hole | sorryAx | 2a3bb5be608940175f348a1e168fe6c221ca05e0c1856d0932fde4e42006f85e | 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.Filter.AtTopBot.Basic
public import Mathlib.RingTheory.Finiteness.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2018 Mario Carneiro, Kevin Bu... | 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/Noetherian",
"family_id": "module",
"file_id": "mathlib/Mathlib/RingTheory/Noetherian/Defs.lean",
"sample_id": "30d6aae9bfaa67acea75dfbb91ae0fa520b66296064e76c798d32752d9029399"
} | train |
{
"chosen_proof_sha256": "00de7867ae9086f2cf9c529fdfa5d87816a8332564eca883373884a7662e8cc7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "774c673a4deeecd41f98949f24cac76430... | by
rw [rowLen, Nat.lt_find_iff]
push Not
exact ⟨fun h _ hmj => μ.up_left_mem (by rfl) hmj h, fun h => h _ (by rfl)⟩ | true | {
"chosen_tokens": 38,
"rejected_tokens": 2,
"token_jaccard": 0.04,
"token_length_ratio": 0.052632
} | invalid_tactic | invalid_tactic | 2a59de97015c1f1b510526fc5ab69a98b769bd65ede3133fcecd438a6e365b5c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finset.Preimage
public import Mathlib.Data.Finset.Prod
public import Mathlib.Data.SetLike.Basic
public import Mathlib.Order.UpperLower.Basic
Namespace:
Y... | by
__oprover_invalid_tactic_7d66ee4f35ab | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "14f5edffae279320cebdad57bda0f496e1ad8d439457a621bc0f327193d068cd",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/Young",
"family_id": "mem_iff_lt_rowlen",
"file_id": "mathlib/Mathlib/Combinatorics/Young/YoungDiagram.lean",
"sample_id": "7d66ee4f35ab6035f8dacc7a5cf8575522e26c3c67988759a6c4ebffd4318014"
} | train |
{
"chosen_proof_sha256": "90db882f63b60882a9cc2eecf39977196f2e1c4d889d00ccb138dc84cbbbe60a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [trop_add_def, trop_eq_iff_eq_untrop]
simp [min_eq_iff] | false | {
"chosen_tokens": 11,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.454545
} | sorry_admit_sorryAx_proof_hole | sorryAx | 2a85e4f910706c9db96e2f403364c74758f2445b5ca5e43864246d349f960c76 | 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.AddGroupWithTop
public import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
public import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
public impor... | 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": "Algebra/Tropical",
"family_id": "add_eq_iff",
"file_id": "mathlib/Mathlib/Algebra/Tropical/Basic.lean",
"sample_id": "4ed39d9450f827e035b841d5a8626fbdbc747601015b7cb760148514ffb7149a"
} | train |
{
"chosen_proof_sha256": "9a5efd93ec86ca2b78ac7e681d48ea70e75ceb2212f655294f2322b5b525cad8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "461385a2074d603323a793c95b869b667b... | by
apply isCyclic_of_card_dvd_prime (p := 2)
simpa [card_perm] using factorial_dvd_factorial hα | true | {
"chosen_tokens": 16,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.1875
} | circular_dependency | circular_dependency | 2ab73f6131c86d470dc932cd10d46dfeadc70036eec08a87d447e6256ec412f8 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Perm
public import Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic
public import Mathlib.SetTheory.Cardinal.Finite
Namespace:
Equiv.Perm
Local c... | by
exact isCyclic_of_card_le_two | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "9155e4475d081d0fb00927bdf2189664c909f701deba7ff97f1e82d130d74d32",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Finite",
"family_id": "iscyclic_of_card_le_two",
"file_id": "mathlib/Mathlib/Data/Finite/Perm.lean",
"sample_id": "aad3a3baeb6dacb034a541f0080f4a6bbc685cc200a408a1414f15d0a2cae0b7"
} | train |
{
"chosen_proof_sha256": "3fc07c2c4acb95dfe246db6ffb6277493987fabf1c2b630f887c08fb23242d47",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1ab3415365954ac703a89b50238d1e7749... | by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.toShHom
g.toShHom)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← TopCat.coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
rw [imageBasicOp... | true | {
"chosen_tokens": 75,
"rejected_tokens": 3,
"token_jaccard": 0.054054,
"token_length_ratio": 0.04
} | unknown_identifier | unknown_identifier | 2ac05a5727ef4c514e115bbf3c8edae488f8e3e0c0319f738144097a4be32887 | 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.RingedSpace.LocallyRingedSpace
public import Mathlib.Algebra.Category.Ring.Constructions
public import Mathlib.Geometry.RingedSpace.OpenImmersion
publ... | by
exact __oprover_missing_879654c433cf | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ea7254f8bddf0769d3a937fa65e7658170ea2f8ac3e535c08e68a10c753d8f82",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Geometry/RingedSpace",
"family_id": "imagebasicopen_image_open",
"file_id": "mathlib/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean",
"sample_id": "879654c433cff299cfe89ef259733d4899e296266c7758cbde162533660dcf70"
} | train |
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