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": "8269b551fa1e3baafeb590db545d184ccf387498c0689c5533fadd958fae33b7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a2717c01644181afb92dd63d96aa9ca9ce... | by
let adj := mapPullbackAdj g
let U : Over Z := Over.mk (φ₁ ≫ snd f g)
let φ₁' : U ⟶ (pullback g).obj (Over.mk f) := Over.homMk φ₁
let φ₂' : U ⟶ (pullback g).obj (Over.mk f) := Over.homMk φ₂ (by simpa using! h₂.symm)
have : φ₁' = φ₂' := by
apply (adj.homEquiv U _).symm.injective
apply (Over.forget X)... | true | {
"chosen_tokens": 126,
"rejected_tokens": 3,
"token_jaccard": 0.044444,
"token_length_ratio": 0.02381
} | circular_dependency | circular_dependency | b6b5b67a640c29ca60017a12982da20c1225a7e0c371e7c210aba43c10e4a96d | 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.Comma.Over.Pullback
public import Mathlib.CategoryTheory.Monoidal.Cartesian.Basic
public import Mathlib.CategoryTheory.Adjunction.Unique
public ... | by
exact hom_ext | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "9b2785e9207717371a0fe5b97ec4bf30758e3e39f285a35903c6e16d7905009e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/LocallyCartesianClosed",
"family_id": "hom_ext",
"file_id": "mathlib/Mathlib/CategoryTheory/LocallyCartesianClosed/ChosenPullbacksAlong.lean",
"sample_id": "56dfb71d32b180653fe5db01d6101ad74566b666462b27c7889ed2d552e18dc1"
} | train |
{
"chosen_proof_sha256": "018854a18bc4ddad72fb2634603cc3211f19d9c9b481d90eec379823ad5a6440",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
simp only [← zsmul_eq_mul, ofAdd_zsmul, map_zpow] | false | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.181818
} | sorry_admit_sorryAx_proof_hole | sorry | b6d24590cd0e0007427823199b16c7dae66ad56fbff561f6172b968aede15488 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Order.SemiconjSup
Namespace:
CircleDeg1Lift
Local context:
/-
Copyright (c) 2020 Yury Kudryashov. All rig... | by
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": "Dynamics/Circle",
"family_id": "translate_zpow",
"file_id": "mathlib/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean",
"sample_id": "8e6d776a112cd89de7aa1f1a126b48dfead5089fd7abc542320a1969d23d73fc"
} | train |
{
"chosen_proof_sha256": "a139748a86235708cf34b19e320c41795f248777d1dc57cebe0b73665a686a07",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "59f057c902f53c00cf6f43ba76e1799f47... | by
constructor
· exact HallMarriageTheorem.hall_hard_inductive
· rintro ⟨f, hf₁, hf₂⟩ s
rw [← card_image_of_injective s hf₁]
apply card_le_card
intro
rw [mem_image, mem_biUnion]
rintro ⟨x, hx, rfl⟩
exact ⟨x, hx, hf₂ x⟩ | true | {
"chosen_tokens": 54,
"rejected_tokens": 2,
"token_jaccard": 0.033333,
"token_length_ratio": 0.037037
} | invalid_tactic | invalid_tactic | b6f163953875c77810db4e7f37cd191b8b3e17842f477ba1a0309bbdaa18cd09 | 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.Basic
public import Mathlib.Data.Fintype.Powerset
public import Mathlib.Data.Set.Finite.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) ... | by
__oprover_invalid_tactic_9844a70ca9aa | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "42f7ec6e02aae223df7362714fd72ad4d1d2ff79f28a8be2874dcfc3f883c847",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/Hall",
"family_id": "finset",
"file_id": "mathlib/Mathlib/Combinatorics/Hall/Finite.lean",
"sample_id": "9844a70ca9aa095c0c679f810bb6057aadb712cb3e2289c8ec909fdd876bd5a3"
} | train |
{
"chosen_proof_sha256": "b1c7cebb6805403539cf95bc62a5c22e029b7da315a1e90239408f66439637c6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "08841f49db68791c6da3b198f7e39c3a02... | by
rw [superpolynomialDecay_iff_abs_tendsto_zero] at hf ⊢
refine fun z =>
tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds (hf z)
(Eventually.of_forall fun x => abs_nonneg _) (hfg.mono fun x hx => ?_)
calc
|k x ^ z * g x| = |k x ^ z| * |g x| := abs_mul (k x ^ z) (g x)
_ ≤ |k x ^ z| ... | false | {
"chosen_tokens": 123,
"rejected_tokens": 128,
"token_jaccard": 0.911111,
"token_length_ratio": 1.04065
} | markdown_preamble_trailing | trailing | b6f9172b44bf3d8ec067a6d06f65ceb3613e81037a297137dddb5cb860f7b98e | 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.Eval.Defs
public import Mathlib.Analysis.Asymptotics.Lemmas
Namespace:
Asymptotics
Local context:
/-
Copyright (c) 2021 Devon Tuma. All ri... | by
rw [superpolynomialDecay_iff_abs_tendsto_zero] at hf ⊢
refine fun z =>
tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds (hf z)
(Eventually.of_forall fun x => abs_nonneg _) (hfg.mono fun x hx => ?_)
calc
|k x ^ z * g x| = |k x ^ z| * |g x| := abs_mul (k x ^ z) (g x)
_ ≤ |k x ^ z| ... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/Asymptotics",
"family_id": "superpolynomialdecay",
"file_id": "mathlib/Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean",
"sample_id": "f97d07cceec981c168c5f2eca445cd4e95e1981b7df2ab5f563b675d89d0e28f"
} | train |
{
"chosen_proof_sha256": "1c6409609dd76697d622025294c8bc5145bcfe414ffbcf9cd053bd8d9c3b4785",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "986cb62814f148ed04d4c7dd4be579d36d... | by
let w'' : V₂.op ⋙ eB.op.functor ≅ eT.op.functor ⋙ V₃.op := NatIso.op w'
have : (w ≫ₕ w'.hom).op = (w.op ≫ᵥ w''.hom) := by ext; simp [w'']
rw [← guitartExact_op_iff, ← guitartExact_op_iff w,
← vComp_iff_of_equivalences _ _ _ w'', this]
rfl | true | {
"chosen_tokens": 80,
"rejected_tokens": 3,
"token_jaccard": 0.02439,
"token_length_ratio": 0.0375
} | unknown_identifier | unknown_identifier | b718e42fc4cf14eb28d7d82f80e60f7351aac8a778e673c1c1699aa7669d1468 | 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.GuitartExact.Opposite
Namespace:
CategoryTheory.TwoSquare.GuitartExact
Local context:
/-
Copyright (c) 2026 Joël Riou. All rights reserved.
Re... | by
exact __oprover_missing_de181494cf6d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d997171ef5a3e87589788677fd6325b625cd8c2a7c475773ee3eb52a5adcbff1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/GuitartExact",
"family_id": "hcomp_iff_of_equivalences",
"file_id": "mathlib/Mathlib/CategoryTheory/GuitartExact/HorizontalComposition.lean",
"sample_id": "de181494cf6df2f46264b7108f8b5fe8ade01b1f3c5c162ddf30528ffc055308"
} | train |
{
"chosen_proof_sha256": "85d4d418bf899b74281aa7eeb44015fb0b192ec7ec0ce946edf50b0cafb9b526",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "fbd4866a79e5cecb09bbdac5f941dd35a0... | by
ext
simp [map, hw] | true | {
"chosen_tokens": 8,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.375
} | unknown_identifier | unknown_identifier | b72c6c3b16b380e125223b6b73cc28fa128d0cda88996f80396a8052c1bf35ca | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.LocallyFinsupp
public import Mathlib.Topology.Spectral.Basic
Namespace:
Function.locallyFinsupp
Local context:
/-
Copyright (c) 2026 Raphael Douglas... | by
exact __oprover_missing_ed1374ebe28e | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "bfa530c85cedc13583d1953d314416650ae95521403b5ad879a6ec23bcbd1666",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/LocallyFinsupp",
"family_id": "map_id",
"file_id": "mathlib/Mathlib/Topology/LocallyFinsupp/Pushforward.lean",
"sample_id": "ed1374ebe28e9b99f5e81874724199c0edfb80ce0f4bc23069e0f6178e331a7a"
} | train |
{
"chosen_proof_sha256": "97f9e88375230bfbd4354802c179b8f0d71d6327ad14ee730bd5022e5fe5beee",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ea9df1cb32d99e6e86eeba3223e23a8702... | by
rw [center_eq_bot, Algebra.mem_bot]
simp [eq_comm] | false | {
"chosen_tokens": 13,
"rejected_tokens": 20,
"token_jaccard": 0.846154,
"token_length_ratio": 1.538462
} | markdown_preamble_trailing | fence | b73f437f00185e944294a17517b200429bd355dc1530840c326a1c8e2791e423 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Central.Defs
import Mathlib.Algebra.Module.Torsion.Field
Namespace:
Algebra.IsCentral
Local context:
/-
Copyright (c) 2024 Jujian Zhang. All rights r... | ```lean
by
rw [center_eq_bot, Algebra.mem_bot]
simp [eq_comm]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Central",
"family_id": "mem_center_iff",
"file_id": "mathlib/Mathlib/Algebra/Central/Basic.lean",
"sample_id": "2a418ca608341e871713a18e1c34bfa17c3846a9b9d6a3f1d347ce4be3076bae"
} | train |
{
"chosen_proof_sha256": "ebb83eb68b98b09837a3cd4cf60703741f66ac05dee2a408489caf38cd22110c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
unfold rieszContentAux
gcongr
apply rieszContentAux_image_nonempty | true | {
"chosen_tokens": 6,
"rejected_tokens": 8,
"token_jaccard": 0.083333,
"token_length_ratio": 1.333333
} | unsolved_goals | unsolved_goals | b76d0e0bdc98443cf611a21967f1f245b28de17051d6f6fc551b1be902847cd6 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Measure.Content
public import Mathlib.Topology.ContinuousMap.CompactlySupported
public import Mathlib.Topology.PartitionOfUnity
Namespace:
(root... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "d329a38231e339bade0e65e9fa058419fb0d1ee37f77c506c8627201d3b718ee",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Integral",
"family_id": "rieszcontentaux_mono",
"file_id": "mathlib/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Basic.lean",
"sample_id": "9fd02cd1275b92cba5b435d3686fa58fec1fc7b2cad79d852c9e074db6daabbc"
} | train |
{
"chosen_proof_sha256": "3c1f07b625c847d49cc43a89c05d3139e535c28f43f07276611fd8f7f1db78de",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e9b755e8d18b77c79dddaaf9d3ff11dacc... | by trunc) (hb : b.len ≤ n := by trunc)
(hc : c.len ≤ n := by trunc) :
tr (f ≫ g) = tr f ≫ tr g :=
rfl | true | {
"chosen_tokens": 42,
"rejected_tokens": 5,
"token_jaccard": 0.086957,
"token_length_ratio": 0.119048
} | wrong_namespace_qualification | wrong_namespace_qualification | b774febead84c3f3fd39087cf7c31957af20ee8e9aa3307afcf0801a9b0c4db2 | 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.Category.Preorder
public import Mathlib.CategoryTheory.Opposites
public import Mathlib.Order.Fin.Basic
public import Mathlib.Util.Superscript
N... | by
exact OProverMissing.missing_cb9967aa5d54 | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "04b6c5833103f468e67ace4601ecc004ea5ab4cae53e2bfcbfc47efe49c8e5d1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/SimplexCategory",
"family_id": "hom",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplexCategory/Defs.lean",
"sample_id": "cb9967aa5d54ae74d6b7c334cf672946e1ac267085a38bfbf8b9d07e6fea6e09"
} | train |
{
"chosen_proof_sha256": "a4055cfe4983bb68dd9d6d5e363dc43c3ee390bf3b7daa2b26130a168ae26243",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rcases eq_or_ne p 1 with rfl | hp
· simp
· rcases eq_or_ne n 0 with rfl | hn
· simp
· simp [← padicValNat_eq_emultiplicity_of_ne_one, *] | false | {
"chosen_tokens": 30,
"rejected_tokens": 5,
"token_jaccard": 0.041667,
"token_length_ratio": 0.166667
} | sorry_admit_sorryAx_proof_hole | sorryAx | b79d7015d103273fd5f3d35400c28c04300ba0da8ecbe6f4d59b1284b2679796 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.MaxPowDiv
public import Mathlib.RingTheory.Multiplicity
public import Mathlib.Data.Nat.Factors
Namespace:
(root)
Local context:
/-
Copyright (c) 201... | 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": "NumberTheory/Padics",
"family_id": "nat",
"file_id": "mathlib/Mathlib/NumberTheory/Padics/PadicVal/Defs.lean",
"sample_id": "c6bc706d166e6b8c0ec0ad881ad47670431feaffc163f29325cfeefd84fa16cb"
} | train |
{
"chosen_proof_sha256": "de9ab835bffa0c1a69cfad4d6364d7e5552c21f89d63ae535eda5295f7d4c8b1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
refine ⟨reg.1.of_flat_of_isBaseChange hf, ?_⟩
rw [← Ideal.map_ofList]
exact ((hf.map_smul_top_ne_top_iff_of_faithfullyFlat R M _).mpr reg.2.symm).symm | false | {
"chosen_tokens": 40,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.075
} | by_exact_placeholder | placeholder | b7ff13dd209914bcbf221e15d0f8369a53eecd92df7333c9a68b2b3af283a676 | 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.Flat.FaithfullyFlat.Basic
public import Mathlib.RingTheory.Flat.Localization
public import Mathlib.RingTheory.Regular.RegularSequence
Namespace:
Ri... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Regular",
"family_id": "isregular",
"file_id": "mathlib/Mathlib/RingTheory/Regular/Flat.lean",
"sample_id": "3a6d8b994748d4e07066dd4b842c5e18a4a267de9cc4efed1e2d52e8ad6a24cb"
} | train |
{
"chosen_proof_sha256": "f144c9feaed254a6570ff3fe42b4f3dd16c597e6f392545a7ebcf73899e7bd04",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "94202eba3f813b725b97a5df43ca926106... | by
classical exact computable_iff_re_compl_re | false | {
"chosen_tokens": 4,
"rejected_tokens": 11,
"token_jaccard": 0.666667,
"token_length_ratio": 2.75
} | markdown_preamble_trailing | fence | b81d208dce261020ed2eb76033189af03c4c0320bd5b6066ba7c9fb6a07263cc | 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.PartrecCode
Namespace:
ComputablePred
Local context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 licen... | ```lean
by
classical exact computable_iff_re_compl_re
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Computability",
"family_id": "computable_iff_re_compl_re",
"file_id": "mathlib/Mathlib/Computability/RE.lean",
"sample_id": "c5c582457298af65c0d36d7c0a352351e64ed55551e90cd77c5b767876e6c965"
} | train |
{
"chosen_proof_sha256": "c785440b2e793b1a2afd030ccba707286c555ec39b3feca32e7d614cb913da7e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "541065cf7e2b0c48add8ba4df93b11dbe3... | by
let ⟨z, hz⟩ := (F.map (algebraMap R ℤ_[p])).evalSubFactor x y
simp only [Polynomial.eval_map_algebraMap] at hz
calc
‖F.aeval x - F.aeval y‖ = ‖z‖ * ‖x - y‖ := by simp [hz]
_ ≤ 1 * ‖x - y‖ := by gcongr; apply PadicInt.norm_le_one
_ = ‖x - y‖ := by simp | true | {
"chosen_tokens": 90,
"rejected_tokens": 2,
"token_jaccard": 0.02381,
"token_length_ratio": 0.022222
} | invalid_tactic | invalid_tactic | b82fa13b5408bbe18e684f3c2a0674479166b6c3f6b00b4e9df065c636d48285 | 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.Identities
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.NumberTheory.Padics.PadicIntegers
public import Mathlib... | by
__oprover_invalid_tactic_7734458b7065 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "1bad10184d63d65f42ce81a74a1abb9dc755a9ce7e098270e7f99299d9735c30",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Padics",
"family_id": "padic_polynomial_dist",
"file_id": "mathlib/Mathlib/NumberTheory/Padics/Hensel.lean",
"sample_id": "7734458b70658c76e232f731e838552dc282d9587d72ef9af109c7d77dc4c28f"
} | train |
{
"chosen_proof_sha256": "86252bc5f575ff25cd53bd8cf62a6e1c8e93248c81bafebff2879ba682468c22",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "baab1cc1fa864e751c82e0367d07242c8d... | by
apply toFinsupp_injective
ext x
simp_rw [toFinsupp_equivMapDomain, Finsupp.equivMapDomain_apply, toFinsupp_mapDomain,
Finsupp.mapDomain_equiv_apply] | true | {
"chosen_tokens": 19,
"rejected_tokens": 3,
"token_jaccard": 0.058824,
"token_length_ratio": 0.157895
} | unknown_identifier | unknown_identifier | b8c4a9c305c0d37ff4b15303c8d780a8ac3603404b40b49e1c711ec5d1bfd217 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.SkewMonoidAlgebra.Basic
public import Mathlib.Algebra.Module.BigOperators
public import Mathlib.Algebra.Algebra.Equiv
Namespace:
SkewMonoidAlgebra
Lo... | by
exact __oprover_missing_24bec75d4c99 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "916a87c2160d1adf9e6e74cbb5d23af3e5484db541dd1f01a62bee5a45aa9bf1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/SkewMonoidAlgebra",
"family_id": "equivmapdomain_eq_mapdomain",
"file_id": "mathlib/Mathlib/Algebra/SkewMonoidAlgebra/Lift.lean",
"sample_id": "24bec75d4c990cfaa562f6c11f1e1ed4950885ceeb40f6408a0260da05969bf6"
} | train |
{
"chosen_proof_sha256": "73fbc60842c9edf00faf984e5736df9245f2ebd275179f441c2133f339aab829",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
by_cases H : ∃ s : Finset S, Nonempty (Basis s R S)
· rw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some]
· simp [trace_eq_zero_of_not_exists_basis R H, finrank_eq_zero_of_not_exists_basis_finset H] | false | {
"chosen_tokens": 44,
"rejected_tokens": 5,
"token_jaccard": 0.033333,
"token_length_ratio": 0.113636
} | sorry_admit_sorryAx_proof_hole | sorryAx | b8dac7a9c3bfc09124cc7900fa12053fcd575d1ac69d45cd874a7f41c6c748d7 | 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.FiniteDimensional.Lemmas
public import Mathlib.LinearAlgebra.Matrix.BilinearForm
public import Mathlib.LinearAlgebra.Trace
Namespace:
Algebra
L... | 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/Trace",
"family_id": "trace_algebramap",
"file_id": "mathlib/Mathlib/RingTheory/Trace/Defs.lean",
"sample_id": "46dbe9e477ddbb7606c3f29bdeda37a0f7a2816233719f84ab33136a865943b2"
} | train |
{
"chosen_proof_sha256": "cf1663d691843ffb2d1c18b62f9a504b9533198687278010c9c58b7251f1dcfb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
dsimp only [LiftR, LiftR']
apply exists_iff_exists_of_mono F (f' rr _ _) (g' rr _ _)
· ext i ⟨x, _⟩ : 2
cases i <;> rfl
· intros
simp +unfoldPartialApp only [map_map, TypeVec.comp]
apply iff_of_eq -- Switch to `eq` so we can use `ext`
congr <;> ext i ⟨x, _⟩ <;> cases i <;> rfl | false | {
"chosen_tokens": 88,
"rejected_tokens": 2,
"token_jaccard": 0.02,
"token_length_ratio": 0.022727
} | sorry_admit_sorryAx_proof_hole | sorry | b941c1372f365304d19704d50fb8b535a936c642d675e131b99e199262bfcb80 | 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.Fin.Fin2
public import Mathlib.Data.TypeVec
public import Mathlib.Logic.Equiv.Defs
Namespace:
MvFunctor
Local context:
/-
Copyright (c) 2018 Jeremy Avig... | 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": "Control/Functor",
"family_id": "liftr_rellast_iff",
"file_id": "mathlib/Mathlib/Control/Functor/Multivariate.lean",
"sample_id": "8b1fcb5d375f7d0c76ac8b5dc5be9785fa4e3630fe7326a206b1b90b1962024a"
} | train |
{
"chosen_proof_sha256": "be5d4d87008e815276be22f67ac5016b50a8ac860e36c4ba227914947bb79f69",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9e9aa61e2c3db4106475af3ddd5d69cd0b... | by
ext <;> simp; ext <;> simp <;> (rw [← Category.assoc]; congr 1; simp) | true | {
"chosen_tokens": 29,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.068966
} | invalid_tactic | invalid_tactic | b991e633bebf6edb4a973f8fb08af9ee000d38f9a597157f3174592c6fbcc5f3 | 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.Subobject.Lattice
public import Mathlib.CategoryTheory.Monoidal.Braided.Basic
public import Mathlib.CategoryTheory.Dialectica.Basic
Namespace:
... | by
__oprover_invalid_tactic_9e0d38f02344 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "8bd4050442c3bf81f26fc58474f5899953cfe9f977479513cb5426a40fc6b434",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Dialectica",
"family_id": "tensorhom_comp_tensorhom",
"file_id": "mathlib/Mathlib/CategoryTheory/Dialectica/Monoidal.lean",
"sample_id": "9e0d38f023447d9d9affb59a11c3c96b5f0f51e0a6c3e310742ed531f794b9ce"
} | train |
{
"chosen_proof_sha256": "ec2ccc87cb1b0580e84707f89c57da330dcbae57aa0cfba902f545e3a992cb11",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704302e78285e7c22e2b57a7665d68d1ff... | by
by_contra h
exact absurd Valuation.IsEquiv.refl (h ▸ adicValuation_not_isEquiv_infty_valuation p) | true | {
"chosen_tokens": 16,
"rejected_tokens": 3,
"token_jaccard": 0.133333,
"token_length_ratio": 0.1875
} | circular_dependency | circular_dependency | b99af4c1e895c6e0995a2ce65a9c6cbbc5ccd50cac1d1ed1965da32543179bb0 | 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 adicValuation_ne_inftyValuation | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "de869dc7a0655d59fd0eb33feb975f83bd34fd14d0f3eb1d872c1845a26a0f2d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/RatFunc",
"family_id": "adicvaluation_ne_inftyvaluation",
"file_id": "mathlib/Mathlib/NumberTheory/RatFunc/Ostrowski.lean",
"sample_id": "d885a32647abb9166c82b5317470ebda2b4ddd48a31b71c34892848d64e4f116"
} | train |
{
"chosen_proof_sha256": "078c3ba86f93f2f5c0b7638f81f2db59b5ef54f0de3d73360e15d4f2f0ce975a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
simp only [normThreshold_zero]
calc a / (1 - √2)
_ ≤ 0 := div_nonpos_of_nonneg_of_nonpos ha_pos.le (by simp)
_ < a := ha_pos | false | {
"chosen_tokens": 32,
"rejected_tokens": 3,
"token_jaccard": 0.038462,
"token_length_ratio": 0.09375
} | by_exact_placeholder | placeholder | b9b2b746a4ee88a63ebcba3858f8858eb66a34ff85967d99c5e71504fba8ab8d | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecificLimits.ArithmeticGeometric
public import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
public import Mathlib.MeasureTheor... | 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": "Probability/Distributions",
"family_id": "lt_normthreshold_zero",
"file_id": "mathlib/Mathlib/Probability/Distributions/Fernique.lean",
"sample_id": "5f55f4262f31682a3e12d8625b11abf778bb7c39ed9ebef5aef18cf9e8a8d94d"
} | train |
{
"chosen_proof_sha256": "ea338c8c6deda19356f39b2f9dc6bfce64075bf47da045c1a31eb2b19f1fe070",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5bfe544231c20197189c9a0b8a98816ff3... | by
refine ⟨fun h => ?_, fun _ => inferInstance⟩
contrapose! h
infer_instance | false | {
"chosen_tokens": 18,
"rejected_tokens": 25,
"token_jaccard": 0.875,
"token_length_ratio": 1.388889
} | markdown_preamble_trailing | fence | b9fa642f52de2754fde4a9b671585515fa8d7d2d66365b41a5e19d88ff075baa | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.Sets.Closeds
public import Mathlib.Topology.QuasiSeparated
Namespace:
TopologicalSpace.Compacts
Local context:
/-
Copyright (c) 2020 Floris van Door... | ```lean
by
refine ⟨fun h => ?_, fun _ => inferInstance⟩
contrapose! h
infer_instance
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/Sets",
"family_id": "subsingleton_iff",
"file_id": "mathlib/Mathlib/Topology/Sets/Compacts.lean",
"sample_id": "7c30455a13ea31c0fdc5ee48caccc8329851df034a9e411c0a2bb1d61bbf2a76"
} | train |
{
"chosen_proof_sha256": "f30223a701d84b8cf4910174a2ebd64c60a3db6a916ec5748abb240a8d7ca7c4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "43783c386dd24adc229bb59f8ae2723958... | by
simpa using (toInt_le_toInt val j i).not | false | {
"chosen_tokens": 11,
"rejected_tokens": 16,
"token_jaccard": 0.6875,
"token_length_ratio": 1.454545
} | markdown_preamble_trailing | preamble | ba1984bf89c8eafef1003d1127aea540aa84ac6d65bfcfea3a2bdb6643f6aec5 | 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.Data.List.Pairwise
public import Batteries.Tactic.GeneralizeProofs
public import Mathlib.Tactic.Order.CollectFacts
Namespace:
Mathlib.Tactic.Order.ToInt
Lo... | Here is the proof:
by
simpa using (toInt_le_toInt val j i).not | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Tactic/Order",
"family_id": "toint_lt_toint",
"file_id": "mathlib/Mathlib/Tactic/Order/ToInt.lean",
"sample_id": "766f3502274c6a836f440a41e6af60197e8da15e11cc99e0e09ed8fb000b1581"
} | train |
{
"chosen_proof_sha256": "90db882f63b60882a9cc2eecf39977196f2e1c4d889d00ccb138dc84cbbbe60a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [trop_add_def, trop_eq_iff_eq_untrop]
simp [min_eq_iff] | true | {
"chosen_tokens": 11,
"rejected_tokens": 8,
"token_jaccard": 0.066667,
"token_length_ratio": 0.727273
} | unsolved_goals | unsolved_goals | ba33d24dc0116f27d944d1e8497aec7bfb137f3caf6d1693c47877b5e98f2f98 | 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.AddGroupWithTop
public import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
public import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
public impor... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "be90f8317b7681069fbf6dcb01f7547f3e588d84187009adc6ff5af03651bf7c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "e0cdaf3a4b43a4f8fc2adc6c6eee964e632fc1aa70e6d8ebca268efd1191ebba",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [Presieve.EffectiveEpimorphic.iff_forall_isSheafFor_yoneda] at hR
rw [← isSheafFor_comp_uliftFunctor_iff]
refine Presieve.isSheafFor_iso (F ⋙ uliftFunctor.{v}).uliftYonedaReprXIso ?_
dsimp only [uliftYoneda, Functor.comp_obj, Functor.whiskeringRight_obj_obj]
rw [isSheafFor_comp_uliftFunctor_iff]
exact... | false | {
"chosen_tokens": 53,
"rejected_tokens": 3,
"token_jaccard": 0.090909,
"token_length_ratio": 0.056604
} | by_exact_placeholder | exact? | ba8621d3d67f773bc682b52493cdd35150c112ea226d68d336d2c9a3e05d07e7 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Sites.SheafOfTypes
public import Mathlib.CategoryTheory.EffectiveEpi.Basic
Namespace:
CategoryTheory
Local context:
/-
Copyright (c) 2023 Adam... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Sites",
"family_id": "presieve",
"file_id": "mathlib/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean",
"sample_id": "3c58404f57528d0f09544fa0d6422c11989243f1a83384b09f5595af7aef1431"
} | train |
{
"chosen_proof_sha256": "94092e3c9ae7aafc8bfd4b060b03f1f7bf7ca38bd35810157e3b4c1cdb58864b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
ext1
apply map_C
-- mixing the two monad structures | true | {
"chosen_tokens": 11,
"rejected_tokens": 8,
"token_jaccard": 0.0625,
"token_length_ratio": 0.727273
} | unsolved_goals | unsolved_goals | ba8c7f42ae5f0b2e50502f2385a7de1e35a2b25cba009ee491e5fd8a02c6676f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MvPolynomial.Rename
public import Mathlib.Algebra.MvPolynomial.Variables
Namespace:
MvPolynomial
Local context:
/-
Copyright (c) 2020 Johan Commelin,... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "a973c13228eede58ab54b527611548b7f5b3fa375abd09aaa93bb492d70fa0ff",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/MvPolynomial",
"family_id": "map_comp_c",
"file_id": "mathlib/Mathlib/Algebra/MvPolynomial/Monad.lean",
"sample_id": "bd9359a574dcaca31acf15cd4ed4c49c4962b2ea1c2de798364f401e03500718"
} | train |
{
"chosen_proof_sha256": "2e876053dd524313a985f03953e4b880c7de87fe925bf70214a79832334fbbdd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "fc2d6d9e9f6a1af7e7b952f5f1068ccb99... | by
obtain ⟨F, hF⟩ := hc.exists_eq_range (hn.image _)
simp only [Set.ext_iff, Quotient.forall, mem_image, mem_range] at hF
simp_rw [Quotient.eq_mk_iff_out] at hF
have hF' : ∀ n : ℕ, (F n).out ∈ K := by
intro n
obtain ⟨P, hP1, hP2⟩ := (hF (F n).out).2 ⟨n, Setoid.refl _⟩
-- Porting note: fix hP2 becaus... | true | {
"chosen_tokens": 480,
"rejected_tokens": 2,
"token_jaccard": 0.00813,
"token_length_ratio": 0.004167
} | invalid_tactic | invalid_tactic | babbafd82167d7331ee873a548c626c87490edf0232c96aced02e84e9ec17d3e | 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.ModelTheory.FinitelyGenerated
public import Mathlib.ModelTheory.PartialEquiv
public import Mathlib.ModelTheory.Bundled
public import Mathlib.Algebra.Order.Arch... | by
__oprover_invalid_tactic_3b9738526fb6 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "186804f81a80cd23fad2ab389897abc2b41bd66a6b84eebbd01e80703c0fe2e6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "ModelTheory",
"family_id": "exists_cg_is_age_of",
"file_id": "mathlib/Mathlib/ModelTheory/Fraisse.lean",
"sample_id": "3b9738526fb66d231f0474bd9878a45063a3e22be465efe04b4fbbe9d62f823a"
} | train |
{
"chosen_proof_sha256": "faf6bc027aba61a3300a23f2d313e28567ba4f3a90cdcf2560518c60032b7b01",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6ae87415fab9e244b6175127e9ef4c306d... | by
classical
let ι := Module.Free.ChooseBasisIndex R S
let b : Module.Basis ι R S := Module.Free.chooseBasis R S
rw [trace_eq_matrix_trace b, trace_eq_matrix_trace (basisQuotient b), AddMonoidHom.map_trace]
congr 1
ext i j
simp only [leftMulMatrix_apply, coe_lmul_eq_mul, LinearMap.toMatrix_apply,
basi... | true | {
"chosen_tokens": 75,
"rejected_tokens": 3,
"token_jaccard": 0.022222,
"token_length_ratio": 0.04
} | unknown_identifier | unknown_identifier | baf5be089a505559b27c1b7048d7d84f877d57647bd2a87ab88a7e598c8b726d | 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.Dvr
public import Mathlib.RingTheory.IntegralClosure.IntegralRestrict
public import Mathlib.RingTheory.LocalRing.Quotient
public impo... | by
exact __oprover_missing_bb2be5794bc2 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "4643ca815847487ea26f9ea8a49e064039017395f5b7511ba2d77bbddd81251a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Trace",
"family_id": "algebra",
"file_id": "mathlib/Mathlib/RingTheory/Trace/Quotient.lean",
"sample_id": "bb2be5794bc2bef47d57ee2509f8e94d0f35879b91907a617a92bba8e4b1af69"
} | train |
{
"chosen_proof_sha256": "077e58b8ebfdb691115134cf203f73b4ed07755ad550ac73f5ae401c07fbed75",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3eed304e6af690a7ca4f95af9814fe1128... | by
refine Measure.ext_of_charFunDual ?_
ext L
simp_rw [charFunDual_map, charFunDual_prod, charFunDual_eq_of_forall_strongDual_eq_zero hμ,
← Complex.exp_add]
rw [← add_div, ← add_div, ← neg_add, ← neg_add]
congr 3
norm_cast
have h1 : (L.comp (.rotation θ)).comp (.inl ℝ E E)
= Real.cos θ • L.comp ... | true | {
"chosen_tokens": 294,
"rejected_tokens": 2,
"token_jaccard": 0.012195,
"token_length_ratio": 0.006803
} | invalid_tactic | invalid_tactic | bb0130ecbc03cebd9bfba730d07b889eef7c5fee21e086371a61ad0340cfe66c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Probability.Distributions.Fernique
public import Mathlib.Probability.Distributions.Gaussian.Basic
Namespace:
ProbabilityTheory.IsGaussian
Local context:
/-
C... | by
__oprover_invalid_tactic_bf8d6d5204ce | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "1371d5e65e7a8aebb3663c8a104dac65b000417f0973bf58769265992e607958",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability/Distributions",
"family_id": "map_rotation_eq_self_of_forall_strongdual_eq_zero",
"file_id": "mathlib/Mathlib/Probability/Distributions/Gaussian/Fernique.lean",
"sample_id": "bf8d6d5204ce084429052d3f687b1b4007057ecf29fbdc62aab2b93e39615bf4"
} | train |
{
"chosen_proof_sha256": "0fd4fc28ba9a30a85e46c5d1d3ba799c62c0383a953832f15efb88028c8307ee",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "da1265a9cce4836c0845ed096f6cb5e87e... | by
haveI := Fintype.ofFinite α
induction φ with simp only [BoundedFormula.Realize]
| equal t₁ t₂ =>
rcases term_realize_eq_add_dotProduct t₁ with ⟨k₁, u₁, ht₁⟩
rcases term_realize_eq_add_dotProduct t₂ with ⟨k₂, u₂, ht₂⟩
convert! Nat.isSemilinearSet_setOf_mulVec_eq ![k₁] ![k₂] (.of ![u₁]) (.of ![u₂])
... | true | {
"chosen_tokens": 270,
"rejected_tokens": 2,
"token_jaccard": 0.010101,
"token_length_ratio": 0.007407
} | invalid_tactic | invalid_tactic | bb79219269b47c8edd64208cda5bd1f1747975ee4ba5807eb914825a7ed14faf | 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.ModelTheory.Arithmetic.Presburger.Basic
public import Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic
public import Mathlib.ModelTheory.Definability... | by
__oprover_invalid_tactic_a75eaccf6d18 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "5014888971e89aa8983f1efe16af9621e1da499311453b86b4f77297386d5a70",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "ModelTheory/Arithmetic",
"family_id": "issemilinearset_boundedformula_realize",
"file_id": "mathlib/Mathlib/ModelTheory/Arithmetic/Presburger/Definability.lean",
"sample_id": "a75eaccf6d18b5c2eb5e6918fb406ae54ba35c64367a2abe5f21d18d5b2dac31"
} | train |
{
"chosen_proof_sha256": "eb4847d4ec18aeaffa781e4a9bdcbca4898991cdee673fe5996e2fd2c98cb1bf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [Function.Injective] at hv
push Not at hv
rcases hv with ⟨i₁, i₂, heq, hne⟩
exact f.map_eq_zero_of_eq v heq hne
/-!
### Algebraic structure inherited from `MultilinearMap`
`AlternatingMap` carries the same `AddCommMonoid`, `AddCommGroup`, and `Module` structure
as `MultilinearMap`
-/ | false | {
"chosen_tokens": 72,
"rejected_tokens": 2,
"token_jaccard": 0.022222,
"token_length_ratio": 0.027778
} | sorry_admit_sorryAx_proof_hole | admit | bbd08c8041043ff9aab35fab5c7558026f3ac43ef4bcdb9bcd4b34c2e95e31c2 | 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.Perm.Sign
public import Mathlib.LinearAlgebra.LinearIndependent.Defs
public import Mathlib.LinearAlgebra.Multilinear.Basis
Namespace:
AlternatingM... | 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/Alternating",
"family_id": "map_eq_zero_of_not_injective",
"file_id": "mathlib/Mathlib/LinearAlgebra/Alternating/Basic.lean",
"sample_id": "cb8c4de23317146f19f32df2485e1f047d44025176348b35088a93dfd2c0286c"
} | train |
{
"chosen_proof_sha256": "0e429534a623f3db5dd3586328da30692723c2a4071b25aa658c931a2d458bab",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [commutator_eq_closure G, ← image_commutatorSet_closureCommutatorRepresentatives, ←
MonoidHom.map_closure, ← commutator_eq_closure]
exact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _))
variable [Finite (commutatorSet G)] | false | {
"chosen_tokens": 42,
"rejected_tokens": 2,
"token_jaccard": 0.038462,
"token_length_ratio": 0.047619
} | by_exact_placeholder | by? | bbe0c936d5dd335ffed48a312d30820c94b95e37b50ffa71159130c1e048121a | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Subgroup.Finite
public import Mathlib.GroupTheory.Commutator.Basic
public import Mathlib.GroupTheory.Rank
public import Mathlib.GroupTheory.Index... | 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": "GroupTheory/Commutator",
"family_id": "card_commutator_closurecommutatorrepresentatives",
"file_id": "mathlib/Mathlib/GroupTheory/Commutator/Finite.lean",
"sample_id": "a6146b935412bf1cd41ece91f946c0daf117bb2ef301024873391e1ed5ceafb2"
} | train |
{
"chosen_proof_sha256": "72e566f5060686e112fed795e07f0f9fff6e410e86e784e0d5fb3063e2798458",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
nontriviality α using Subsingleton.elim a 0
obtain rfl | h := eq_or_ne a 0
· rw [normUnit_zero, zero_mul, normUnit_zero]
· rw [normUnit_mul h (Units.ne_zero _), normUnit_coe_units, mul_inv_eq_one] | false | {
"chosen_tokens": 42,
"rejected_tokens": 3,
"token_jaccard": 0.03125,
"token_length_ratio": 0.071429
} | by_exact_placeholder | placeholder | bc5f9da74c02a2b11357755fc031c6cb6ef4973068a7f5f70e8aae67afe8827c | 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.Ring.Associated
Namespace:
(root)
Local context:
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as desc... | 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/GCDMonoid",
"family_id": "normunit_mul_normunit",
"file_id": "mathlib/Mathlib/Algebra/GCDMonoid/Basic.lean",
"sample_id": "80e10db2175365293b28a61fb2baadedb77fa1adfc56606eaf56f02d6e239f96"
} | train |
{
"chosen_proof_sha256": "6d7fe2ba7791e82259c7c7dac10b1de23d8b4e925a2fa271084af7b3f600bee5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a83c5a1c18e47c4bbeb57a883865f74ecb... | by
by_cases hf2 : f = 0; · rw [hf2]; exact dvd_zero _
rw [factor, dif_pos (WfDvdMonoid.exists_irreducible_factor hf1 hf2)]
exact (Classical.choose_spec <| WfDvdMonoid.exists_irreducible_factor hf1 hf2).2 | true | {
"chosen_tokens": 45,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.066667
} | circular_dependency | circular_dependency | bc608ad4d5a66b18b53b40c1b888e5aa05ac5994a98ac6f7db04cf7380ef70ab | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.CharP.Algebra
public import Mathlib.FieldTheory.SplittingField.IsSplittingField
public import Mathlib.RingTheory.Algebraic.Basic
Namespace:
Polynomial... | by
exact factor_dvd_of_not_isUnit | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "8d31474c75b26fc5f278031a6533a2fa4b1d763ed175e5ac03bd90c29cf3b730",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/SplittingField",
"family_id": "factor_dvd_of_not_isunit",
"file_id": "mathlib/Mathlib/FieldTheory/SplittingField/Construction.lean",
"sample_id": "3b9bf8a1b9ae9ed64027deecfa9b2b28510490bbc0204132c8e750b77e3c634e"
} | train |
{
"chosen_proof_sha256": "4f2754f3c1709f7213213c711ba2777cb527cb9fa5e9c5f811685e4d0a510864",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "365ae057ef87f87f7e7b270aae3a76feaf... | by
rw [← IsLocalization.mk'_one (M := K[X]⁰) (FractionRing K[X]) p, ← mk_coe_def, Submonoid.coe_one] | true | {
"chosen_tokens": 32,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.15625
} | wrong_namespace_qualification | wrong_namespace_qualification | bc7671ac1f486d74cdb9990ce7055b7c8579c96e4a860d4aec3212d83bcac40f | 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.Basic
public import Mathlib.Algebra.Ring.NonZeroDivisors
public import Mathlib.RingTheory.Localization.FractionRing
Namespace:
RatFunc
Loc... | by
exact OProverMissing.missing_2ba6bee60121 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "bc391f7444b504cd9d191cc20dda70df6909688d088f573ebc8144353ac8c3d0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "05a30ac01794d34e26abf0bb89eb49ff18aa58a6481d8190121651504b0314f8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rcases eq_or_ne n 0 with rfl | hn₀; · simp
rcases eq_or_ne p 1 with rfl | hp₁; · simp
simpa [*] using pow_dvd_iff_le_padicValNat (k := 1) hp₁ hn₀ |>.symm |>.not | false | {
"chosen_tokens": 48,
"rejected_tokens": 2,
"token_jaccard": 0.03125,
"token_length_ratio": 0.041667
} | by_exact_placeholder | by? | bc849a97d22dfe72805687731b79141d216761d9e734b17525b0c9f768097bf7 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.MaxPowDiv
public import Mathlib.RingTheory.Multiplicity
public import Mathlib.Data.Nat.Factors
Namespace:
padicValNat
Local context:
/-
Copyright (c... | 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": "NumberTheory/Padics",
"family_id": "eq_zero_iff",
"file_id": "mathlib/Mathlib/NumberTheory/Padics/PadicVal/Defs.lean",
"sample_id": "b28f2d047f3f02792ce315aca654c65c6b846562c70579c8ee832efe4c79dcf3"
} | train |
{
"chosen_proof_sha256": "e064b140ea88edff320b5c2da73ccde8cd1103d2a90b6e31a432655ee413c425",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "049aeda8e9db27c47b238863ebb28fa2fd... | by
rw [← Semigroup.mem_center_iff]
exact Iff.rfl | false | {
"chosen_tokens": 12,
"rejected_tokens": 16,
"token_jaccard": 0.785714,
"token_length_ratio": 1.333333
} | missing_wrong_import | import | bca5f429ade416255804bd341308e92162e9bc5de5739947c3f4417f57265e2a | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Submonoid.Operations
public import Mathlib.GroupTheory.Subsemigroup.Center
Namespace:
Submonoid
Local context:
/-
Copyright (c) 2021 Eric Wiese... | import Mathlib.WrongImport
by
rw [← Semigroup.mem_center_iff]
exact Iff.rfl | {
"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": "GroupTheory/Submonoid",
"family_id": "mem_center_iff",
"file_id": "mathlib/Mathlib/GroupTheory/Submonoid/Center.lean",
"sample_id": "0329a22051730a8fdf389196512cdc9c856c4238f5a4d017fdd705ece3a68b4f"
} | train |
{
"chosen_proof_sha256": "ab45fc24fe8d81ce6bc79891c129b27df70dce3fefc1109e3aecbeb52c81794e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
ext; simp [transvection.apply] | true | {
"chosen_tokens": 9,
"rejected_tokens": 8,
"token_jaccard": 0.066667,
"token_length_ratio": 0.888889
} | unsolved_goals | unsolved_goals | bcb47f6a7eea62ba1dfe11277d6e7a300459db2548c614fc36ddd3244a07f846 | 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.GroupAction.SubMulAction.OfFixingSubgroup
public import Mathlib.LinearAlgebra.Charpoly.BaseChange
public import Mathlib.LinearAlgebra.Dual.BaseChan... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "6d2f6b138ae4587fe51c1217fff3c7de8aa90926a5011edd654f0604a9ecb3c2",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "c1ea19ea49c645e697b6d5466464113860db944a733b14a16a41636e7325b4ec",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ef6b60ad5c0052ed74848bbc4c25cc9c35... | by
obtain ⟨x, rfl⟩ := x.mk_surjective
obtain ⟨y, rfl⟩ := y.mk_surjective
obtain rfl : x = y := h
rfl | false | {
"chosen_tokens": 30,
"rejected_tokens": 35,
"token_jaccard": 0.777778,
"token_length_ratio": 1.166667
} | markdown_preamble_trailing | preamble | bce75a3a5cd6933a3f1d3b90e8aea812f77f1c801138699125b60340ba43b96d | 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.SimplicialObject.Basic
public import Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal
public import Mathlib.AlgebraicTopology.SimplicialSet... | Here is the proof:
by
obtain ⟨x, rfl⟩ := x.mk_surjective
obtain ⟨y, rfl⟩ := y.mk_surjective
obtain rfl : x = y := h
rfl | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/SimplicialSet",
"family_id": "ext",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplicialSet/HomotopyCat.lean",
"sample_id": "3065ddea640d150a719f7ca546348a960da065f54e8ab9a9d7476b502ae8f9af"
} | train |
{
"chosen_proof_sha256": "cb2db471013d02747d266722ac3b318ad74596861a15ff8911bcdfb53d010405",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [GradedAlgebra.proj_apply, decompose_symm_of, Equiv.apply_symm_apply] | false | {
"chosen_tokens": 13,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.153846
} | sorry_admit_sorryAx_proof_hole | admit | bcfefc018fa695ae83ee562279b82fb0022637279091aee12ebc6749d3be166c | 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.Algebra
public import Mathlib.Algebra.DirectSum.Decomposition
public import Mathlib.Algebra.DirectSum.Internal
public import Mathlib.Algebra.... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/GradedAlgebra",
"family_id": "gradedalgebra",
"file_id": "mathlib/Mathlib/RingTheory/GradedAlgebra/Basic.lean",
"sample_id": "7730bf9984af8e1d55b7e5d3174ec963362e837effdd837bd031f3763c30060e"
} | train |
{
"chosen_proof_sha256": "930122cf7815506e3d0570b55467812e4caec3bf1c598928f36877dea55601fa",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
simpa only [h.comp_eq] using bijOn_fixedPoints_comp f g | false | {
"chosen_tokens": 12,
"rejected_tokens": 2,
"token_jaccard": 0.076923,
"token_length_ratio": 0.166667
} | sorry_admit_sorryAx_proof_hole | sorry | bd959678112e1e5907efe9d8f175cd78349bc5212734c961e8bb653c333d14f5 | 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.End
public import Mathlib.Data.Set.Function
public import Mathlib.Dynamics.FixedPoints.Defs
Namespace:
Function
Local context:
/-
Copyright (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": "Dynamics/FixedPoints",
"family_id": "commute",
"file_id": "mathlib/Mathlib/Dynamics/FixedPoints/Basic.lean",
"sample_id": "636a9fcc3e8a95e1374f41b3672c8c890f7576b9c9fa7d5f0caae9002e2a7ade"
} | train |
{
"chosen_proof_sha256": "b276acfbf4317cef3c8ddc23d0eda431624fd9c228809a52c5678c7b8d9bd426",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "02daabd48e88e8b51fc3ee4d773fb0f3f9... | by
refine ENNReal.eq_sub_of_add_eq hg_fin ?_
rw [← lintegral_add_right' _ hg]
exact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx) | false | {
"chosen_tokens": 28,
"rejected_tokens": 35,
"token_jaccard": 0.925926,
"token_length_ratio": 1.25
} | markdown_preamble_trailing | fence | bda2c8ba66549de33f1c14eee7675f3d997f91806e754da90cfd4f5f14ef2889 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Integral.Lebesgue.Add
Namespace:
MeasureTheory
Local context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache ... | ```lean
by
refine ENNReal.eq_sub_of_add_eq hg_fin ?_
rw [← lintegral_add_right' _ hg]
exact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx)
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Integral",
"family_id": "lintegral_sub",
"file_id": "mathlib/Mathlib/MeasureTheory/Integral/Lebesgue/Sub.lean",
"sample_id": "79c5185e6f28d9b4a517c1d0991b691ef5318525bdf3a4fee785bfeb02cb287e"
} | train |
{
"chosen_proof_sha256": "158b158a558370a66bb2762fb6673634c41f0d235bb36659662b90d5aa0cb875",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
contrapose! +distrib
rw [Set.infinite_image2 hfs hft]
grind only [Set.Infinite.nonempty] | false | {
"chosen_tokens": 22,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.090909
} | by_exact_placeholder | by? | bdbf38fa8875d55cfe3efa84fe278244e878cd5831329cde87918a0006064b0e | 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.Basic
public import Mathlib.Data.Fintype.Prod
public import Mathlib.Data.Fintype.Pi
public import Mathlib.Algebra.Order.Group.Multiset
public i... | 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/Finite",
"family_id": "finite_image",
"file_id": "mathlib/Mathlib/Data/Finite/Prod.lean",
"sample_id": "335fd55413b4f96955767217c73d17f78f377c305db1438e043702f98d245c46"
} | train |
{
"chosen_proof_sha256": "fcb2f3f3485ce6a70b3afb69269f67e2beb686789a095f74d88d7d21aa424579",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c5ea797717dbad7344a1184169096320fb... | by
ext S; simp only [smul_def, _root_.zero_smul, zero_def, Pi.zero_apply] | false | {
"chosen_tokens": 19,
"rejected_tokens": 24,
"token_jaccard": 0.8,
"token_length_ratio": 1.263158
} | markdown_preamble_trailing | trailing | bde33bd6c990ddf351ecc0a02c5bb46922d5374460e512cf6e8bf84b6059d2d8 | 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.TensorProduct.RightExactness
public import Mathlib.RingTheory.Congruence.Hom
public import Mathlib.RingTheory.FiniteType
public import Mathlib.Ri... | by
ext S; simp only [smul_def, _root_.zero_smul, zero_def, Pi.zero_apply]
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/PolynomialLaw",
"family_id": "zero_smul",
"file_id": "mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean",
"sample_id": "502371bcc584859610c49145091eb701ac3e0cd90f5beb5f633781b964813dd7"
} | train |
{
"chosen_proof_sha256": "10363aa97aac5a8e4a6ed13d0ebf67594ed188c89c81a730878f4693d1b889fa",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cd6445d62ea0160fdadc6b7b9535f61ac5... | by
ext x
simp only [LinearMap.comp_apply, apply, map_add, map_smul, add_assoc]
simp only [smul_add, ← mul_smul, ← add_smul, ← mul_add (f x), mul_assoc] | true | {
"chosen_tokens": 38,
"rejected_tokens": 2,
"token_jaccard": 0.04,
"token_length_ratio": 0.052632
} | invalid_tactic | invalid_tactic | be0aef3bab710dee073eaf0a6e8f15f08539a0c444548aa1adb21017809dd5c4 | 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.GroupAction.SubMulAction.OfFixingSubgroup
public import Mathlib.LinearAlgebra.Charpoly.BaseChange
public import Mathlib.LinearAlgebra.Dual.BaseChan... | by
__oprover_invalid_tactic_19c9230dd953 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "385719a1adfad89cc12f79a73ff05551050d39afbfee5702224580fb5780ec35",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Transvection",
"family_id": "comp_smul_smul",
"file_id": "mathlib/Mathlib/LinearAlgebra/Transvection/Basic.lean",
"sample_id": "19c9230dd9532a6df572a3d4cf4ed8cfd2fc18850e231e933699018c9fe60632"
} | train |
{
"chosen_proof_sha256": "b1b2588f46af06b1f0c76c618d2cd12402ece06fbea01ebed4a3cb2260d14513",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e38015f0095dc11e18da07e2408293619a... | by
rw [mapRange_def]
refine (Finset.prod_subset support_mk_subset ?_).trans ?_
· intro i h1 h2
simp only [mem_support_toFun, ne_eq] at h1
simp only [Finset.coe_sort_coe, mem_support_toFun, mk_apply, ne_eq, h1, not_false_iff,
dite_eq_ite, ite_true, not_not] at h2
simp [h2, h0]
· refine Finset.p... | true | {
"chosen_tokens": 87,
"rejected_tokens": 3,
"token_jaccard": 0.027027,
"token_length_ratio": 0.034483
} | circular_dependency | circular_dependency | be160afeb02fcd319494f44f81b0452ef860b378300a5795751baaa26516d585 | 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.GroupWithZero.Action
public import Mathlib.Data.DFinsupp.Ext
public import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
Namespace:
DFi... | by
exact prod_mapRange_index | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "a2f14ecb13921862f3b89a5fe7809bb51283b0f5e72623843fbb2784b0a193fb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Data/DFinsupp",
"family_id": "prod_maprange_index",
"file_id": "mathlib/Mathlib/Data/DFinsupp/BigOperators.lean",
"sample_id": "345ae0fc64c4223b685a8043b5b363bbd84f8c383d47f5ccfa20bde13d5436d1"
} | train |
{
"chosen_proof_sha256": "313c9cf4c1b5eea4156e212ff571b19fbbfa824c1fd6f777992a3ed4cc19b0b1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "eec7b09ae0be76f3b02bd63e3447ae8dcf... | by
by_cases hq : 1 < q
· rw [← pow_zero q, ofPowerSeries_apply_pow hq, PowerSeries.coeff_zero_eq_constantCoeff]
· simp [ofPowerSeries, dif_neg hq] | true | {
"chosen_tokens": 29,
"rejected_tokens": 3,
"token_jaccard": 0.043478,
"token_length_ratio": 0.103448
} | unknown_identifier | unknown_identifier | be28e710c68e328f83374b91e821216ae8bc4f3204687b111a044ca5e583f76e | 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.ArithmeticFunction.Defs
public import Mathlib.Order.Northcott
public import Mathlib.RingTheory.PowerSeries.Basic
public import Mathlib.RingTheory.... | by
exact __oprover_missing_dd0a6c86eaf7 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "8967cd195d564f7c2cd89655430989b937bc29d2583c3008fd2c583048117b91",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/ArithmeticFunction",
"family_id": "ofpowerseries_apply_one",
"file_id": "mathlib/Mathlib/NumberTheory/ArithmeticFunction/LFunction.lean",
"sample_id": "dd0a6c86eaf74b4453f65b352e9a02cbc6a9bc8e7367f1c0ef0f634a91f60ff2"
} | train |
{
"chosen_proof_sha256": "7d031507a20582e22ca4729c3a20ae04f876f5f252c02c99897df94087378755",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e4a8552d1a248e6c893e3831be7b2e444a... | by
ext j
rw [assoc, assoc, enrichedComp_π, id_comp, tensorHom_def', assoc,
← MonoidalCategory.whiskerLeft_comp_assoc, enrichedId_π,
whisker_exchange_assoc, MonoidalCategory.whiskerRight_id, assoc, assoc,
Iso.inv_hom_id_assoc]
dsimp
rw [e_comp_id, comp_id] | false | {
"chosen_tokens": 47,
"rejected_tokens": 52,
"token_jaccard": 0.857143,
"token_length_ratio": 1.106383
} | markdown_preamble_trailing | trailing | becaac8453286340912807970a3cd8b7a6c70279ed811cfba300a0128763f6a5 | 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.Monoidal.FunctorCategory
public import Mathlib.CategoryTheory.Enriched.Ordinary.Basic
public import Mathlib.CategoryTheory.Functor.Category
publ... | by
ext j
rw [assoc, assoc, enrichedComp_π, id_comp, tensorHom_def', assoc,
← MonoidalCategory.whiskerLeft_comp_assoc, enrichedId_π,
whisker_exchange_assoc, MonoidalCategory.whiskerRight_id, assoc, assoc,
Iso.inv_hom_id_assoc]
dsimp
rw [e_comp_id, comp_id]
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/Enriched",
"family_id": "enriched_comp_id",
"file_id": "mathlib/Mathlib/CategoryTheory/Enriched/FunctorCategory.lean",
"sample_id": "623cf51d346b229ca13fed1da60cd6df5ba3c3809b92103ef6f67007cec38024"
} | train |
{
"chosen_proof_sha256": "12e284d615394feaae0bf91b6d2469b44129b24d28aa683dddb8dd6ad90c1ad2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "733568c1cfdc13c4513e891277fa1d4790... | by
grw [hv.apply_sum_le, ← cbiSup_eq_of_forall (by grind)]
simp | true | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.0625,
"token_length_ratio": 0.2
} | circular_dependency | circular_dependency | beddc8e185731c21f3582ee3d00edaeef1a441f87c87ebe50cf5d59df05aa720 | 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.Homogenize
public import Mathlib.NumberTheory.Height.Basic
import Mathlib.Algebra.Order.Ring.IsNonarchimedean
import Mathlib.Data.Fintype.Or... | by
exact apply_sum_univ_le | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "9cafc93385394096ca1d871f33c8bf36ee8a696d3a6c946fbc1d2e1eb529be0b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Height",
"family_id": "apply_sum_univ_le",
"file_id": "mathlib/Mathlib/NumberTheory/Height/MvPolynomial.lean",
"sample_id": "141f039ee25caf569389190297e8e170fe2b5f957ffbed666e13fe2928910cf7"
} | train |
{
"chosen_proof_sha256": "9863212e81e0182629bd15d927cf3f594339d8438f634ee6f333a33c3de5d6eb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0b9613c0f8f4ed0424d4b806c7d2c4c3dd... | by
conv_rhs => simp [← h.commute_iff, Commute.symm_iff (a := e), commute_iff_eq, ← mem_commutant_iff] | true | {
"chosen_tokens": 24,
"rejected_tokens": 3,
"token_jaccard": 0.045455,
"token_length_ratio": 0.125
} | unknown_identifier | unknown_identifier | bf69b8068a4f044aeb110178a284673006e47a724a6526e5270a56d6407d30d0 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.CStarAlgebra.Classes
public import Mathlib.Analysis.InnerProductSpace.Adjoint
Namespace:
VonNeumannAlgebra
Local context:
/-
Copyright (c) 2022 Kim ... | by
exact __oprover_missing_b3ad2032874b | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "55112e9037835c620d6ba0fb1a4edc0edec6cac84a7f722d5b0e6f70ca22063c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/VonNeumannAlgebra",
"family_id": "isidempotentelem",
"file_id": "mathlib/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean",
"sample_id": "b3ad2032874ba6a0b6f50397e2b035a5757e486348575d6dc25170f7fefc69ec"
} | train |
{
"chosen_proof_sha256": "63ce66eb5d2d85e6c3aa93e37342e57afc36c526d7b717542f12e151e6f9e034",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "947f4998f81845289fe2e622281075006b... | by
let := Localization.AtPrime.algebraOfLiesOver (p.under R) p
rw [isUnramifiedAt_iff_map_eq R (p.under R) p, and_iff_right,
Ideal.IsDedekindDomain.ramificationIdx_eq_one_iff hp Ideal.map_comap_le]
have : Finite (R ⧸ p.under R) :=
Ideal.finiteQuotientOfFreeOfNeBot _ (mt Ideal.eq_bot_of_comap_eq_bot hp)
... | false | {
"chosen_tokens": 92,
"rejected_tokens": 97,
"token_jaccard": 0.9,
"token_length_ratio": 1.054348
} | markdown_preamble_trailing | preamble | bf89e9579f627d977ad17dcb48950dbbcc62b5d75150dfcd9ccff5f24f810488 | 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.RamificationInertia.Basic
public import Mathlib.RingTheory.LocalRing.ResidueField.Instances
public import Mathlib.RingTheory.Unramified.LocalRing
... | Here is the proof:
by
let := Localization.AtPrime.algebraOfLiesOver (p.under R) p
rw [isUnramifiedAt_iff_map_eq R (p.under R) p, and_iff_right,
Ideal.IsDedekindDomain.ramificationIdx_eq_one_iff hp Ideal.map_comap_le]
have : Finite (R ⧸ p.under R) :=
Ideal.finiteQuotientOfFreeOfNeBot _ (mt Ideal.eq_bot_of_... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/RamificationInertia",
"family_id": "algebra",
"file_id": "mathlib/Mathlib/NumberTheory/RamificationInertia/Unramified.lean",
"sample_id": "ed0bc1a8156a91f3c77b4c93e0e33c2312cab82d14a2bd0d77876ca7788349af"
} | train |
{
"chosen_proof_sha256": "99f33d97a3ff079eb01e172cf32d5125fc28f1ebfb2b8c076d51536e340d9ee6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "afc67d04fe81026f8c10733aa377845ea3... | by
apply tendsto_integral_filter_of_dominated_convergence _ _ _ hf.norm
· filter_upwards with v
nth_rewrite 2 [show f v = cexp (- (0 : ℝ) * ‖v‖ ^ 2) • f v by simp]
apply (Tendsto.cexp _).smul_const
exact tendsto_inv_atTop_zero.ofReal.neg.mul_const _
· filter_upwards with c using
AEStronglyMeasur... | true | {
"chosen_tokens": 144,
"rejected_tokens": 3,
"token_jaccard": 0.028571,
"token_length_ratio": 0.020833
} | unknown_identifier | unknown_identifier | bfa5b8eace368b97ef5d76e061c3331c0c68cac32282eb03ba5245a9ae1574e7 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Integral.PeakFunction
public import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
Namespace:
Real
Local context:
/-
Copyright (c)... | by
exact __oprover_missing_dce31b4f3667 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "510d9da003c113862690226ac0151db9fe7da08152bd3adf43d17651e352bfc3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Fourier",
"family_id": "tendsto_integral_cexp_sq_smul",
"file_id": "mathlib/Mathlib/Analysis/Fourier/Inversion.lean",
"sample_id": "dce31b4f3667c3830b421cb76a3fefb37b0de149d8e70ed049af7ee6287ae930"
} | train |
{
"chosen_proof_sha256": "63cd494fa84f771bdfe24b254aba32aae9ae2b28ee1e64d13bc4951087609f7b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "64b0828b72ceff37d3d9baf47b05b8e5a0... | by
induction γ using Quotient.ind with | mk γ =>
simpa [← mk_trans, ← mk_symm, ← mk_refl, eq] using Homotopic.trans_symm γ | true | {
"chosen_tokens": 30,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.066667
} | invalid_tactic | invalid_tactic | bfad231e82430113335d411ff8a08f5d5bc98e82e3dc570bc59196f27c7a72a1 | 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.Grpd.Basic
public import Mathlib.Topology.Category.TopCat.Basic
public import Mathlib.Topology.Homotopy.Path
public import Mathlib.Data... | by
__oprover_invalid_tactic_3d37610ea90f | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e91437c0d2c6b48ba87952c8f66e370e39310c3c2fec5f9fa0210d7b9d00a0f3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/FundamentalGroupoid",
"family_id": "trans_symm",
"file_id": "mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean",
"sample_id": "3d37610ea90fd4d8210d4ae3653f0e831fc85b4617585bafc5fe04742a37ccee"
} | train |
{
"chosen_proof_sha256": "105decf31a73d8f03c28fa8bbcf536ea59b795c7d4d6c1c8dd7fade7fd71c4b3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "69ddf01226811bd3e646477572e8227879... | by
refine ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => ?_⟩
have a0 := K0.trans_le ha
have b0 := K0.trans_le hb
rw [inv_sub_inv' ((abv_pos abv).1 a0) ((abv_pos abv).1 b0), abv_mul abv, abv_mul abv, abv_inv abv,
abv_inv abv, abv_sub abv]
refine lt_of_mul_lt_mul_left (lt_of_mul_lt_mul_right ?... | true | {
"chosen_tokens": 131,
"rejected_tokens": 3,
"token_jaccard": 0.037037,
"token_length_ratio": 0.022901
} | unknown_identifier | unknown_identifier | bfae18879023840b877ad962e6b5d163f3e718e1dc726152329984b6b3768b47 | 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.Pi
public import Mathlib.Algebra.Order.AbsoluteValue.Basic
public import Mathlib.Algebra.Order.Field.Basic
public import Mathlib.Algebra.O... | by
exact __oprover_missing_2ecf221e60a1 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "0e6829ec1178c92b549141908bb745c977605ae9db5344e4cb50a2da19ec4586",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Order",
"family_id": "rat_inv_continuous_lemma",
"file_id": "mathlib/Mathlib/Algebra/Order/CauSeq/Basic.lean",
"sample_id": "2ecf221e60a1de5319a947fdeadd18115cd6ed88d23420eaeb85a40fe0314ea2"
} | train |
{
"chosen_proof_sha256": "4a41798036ff76446dc750a37ef284895b5d38963c6bb7b381d67845e6108343",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
have : JacobsonSpace ↥X := LocallyOfFiniteType.jacobsonSpace (f ≫ i)
refine ext_of_fromSpecResidueField_eq f g i (S ∩ closedPoints X) ?_ ?_ H'
· rwa [dense_iff_closure_eq, JacobsonSpace.closure_inter_closedPoints_eq_closure hS,
← dense_iff_closure_eq]
· intro x ⟨hxS, hx⟩
rw [← cancel_epi (Spec.map ... | false | {
"chosen_tokens": 126,
"rejected_tokens": 2,
"token_jaccard": 0.016129,
"token_length_ratio": 0.015873
} | sorry_admit_sorryAx_proof_hole | sorry | bfd210fe30b2ff65a38babfcb4d8ba3ea9daf294d32826be8d14276131d922b7 | 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.AlgebraicGeometry.Morphisms.Finite
public import Mathlib.FieldTheory.IsAlgClosed.Basic
Namespace:
AlgebraicGeometry
Local context:
/-
Copyright (c) 2026 Andr... | 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": "AlgebraicGeometry/AlgClosed",
"family_id": "ext_of_apply_eq",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/AlgClosed/Basic.lean",
"sample_id": "da422d22caf26f3e5568440ff4f4ce39cf647f59520cb7416b45b8017ec51f47"
} | train |
{
"chosen_proof_sha256": "06f47103349d9ae4b5fa9c701936477e3308e59432a0c09cdb18eb8a9e2de388",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6cb7857e8c992cbf8974094de8e796c632... | by
contrapose!; exact parts_eq_empty_iff | true | {
"chosen_tokens": 6,
"rejected_tokens": 2,
"token_jaccard": 0.142857,
"token_length_ratio": 0.333333
} | invalid_tactic | invalid_tactic | bfe88c55f540db35856d776df6cb318428f25f275c95f3e3cfea5e9dcf913a0f | 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.Data.Finset.Pairwise
public import Mathlib.Data.Finset.Preimage
public import Mathlib.Data.Fintyp... | by
__oprover_invalid_tactic_ea19c96b1833 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "1f1f15d8bb54b35d8fbae03733e059798ff7fab941495223e03a5e90c270d8a2",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Partition",
"family_id": "parts_nonempty_iff",
"file_id": "mathlib/Mathlib/Order/Partition/Finpartition.lean",
"sample_id": "ea19c96b1833a290f3ad4a2c8868771d2cf2aee1459ba1b113ef3ef0cf9c4a4c"
} | train |
{
"chosen_proof_sha256": "db758460f3feb84a60fe6730f2e7a0ca8f4614d60686027e4a4aa0bdb93eeac5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
by_cases hμ : SFinite μ
swap; · rw [compProd_of_not_sfinite _ _ hμ]; simp
by_cases hν : SFinite ν
swap; · rw [compProd_of_not_sfinite _ _ hν]; simp
let s := κ.mutuallySingularSet η
have hs : MeasurableSet s := Kernel.measurableSet_mutuallySingularSet κ η
symm
refine ⟨s, hs, ?_⟩
rw [compProd_apply h... | false | {
"chosen_tokens": 184,
"rejected_tokens": 3,
"token_jaccard": 0.030769,
"token_length_ratio": 0.016304
} | by_exact_placeholder | exact? | c0233fccdadf7eb2d3c3b685c0af797c8080e40510af741c1b6a8535e507692d | 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.Probability.Kernel.Composition.MeasureCompProd
public import Mathlib.Probability.Kernel.RadonNikodym
Namespace:
MeasureTheory.Measure
Local context:
/-
Copyr... | 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": "Probability/Kernel",
"family_id": "mutuallysingular",
"file_id": "mathlib/Mathlib/Probability/Kernel/Composition/AbsolutelyContinuous.lean",
"sample_id": "56824650124ef37b3559baee31b1277720d3a9f018ce7db07f8712e4b6c90072"
} | train |
{
"chosen_proof_sha256": "1917f8da7209b00ed86e3ac8e7571325a8372a2d3f50420181edba9e230a5abb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
simp_all [SProd.sprod, product, mem_flatMap, mem_map, Prod.ext_iff, and_left_comm] | false | {
"chosen_tokens": 19,
"rejected_tokens": 2,
"token_jaccard": 0.066667,
"token_length_ratio": 0.105263
} | sorry_admit_sorryAx_proof_hole | sorry | c03c7b3cc128fe980190764f2d5ba4b5183c6575284ad99c6faa8c4f22a08d19 | 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.List.Defs
public import Mathlib.Tactic.Attr.Core
public import Mathlib.Tactic.Common
Namespace:
List
Local context:
/-
Copyright (c) 2015 Leonardo de Mo... | 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": "Data/List",
"family_id": "mem_product",
"file_id": "mathlib/Mathlib/Data/List/ProdSigma.lean",
"sample_id": "c126f860a1889f0eed701887869d359f44f2980da9414820754ce91fa0c2db26"
} | train |
{
"chosen_proof_sha256": "4de448a7d71906907793d1dc7af096bf74aa03a276d6cf774b8ce324536dc08e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ad39024358756a3c192f53ac5f381a4640... | by
refine tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => ?_
rw [Set.indicator_of_notMem]
simp only [not_le, Set.mem_setOf_eq]
refine lt_of_le_of_lt (Nat.le_ceil _) ?_
refine lt_of_lt_of_le (lt_add_one _) ?_
norm_cast
variable {p : ℝ≥0∞} | true | {
"chosen_tokens": 71,
"rejected_tokens": 2,
"token_jaccard": 0.020833,
"token_length_ratio": 0.028169
} | invalid_tactic | invalid_tactic | c074818763fcc463b6a6dd997c6e3044fec551c379f014baca056b3128895e14 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
public import Mathlib.MeasureTheory.Function.L1Space.Integrable
Namespace:
MeasureTheory
Local context:
/-
Copyri... | by
__oprover_invalid_tactic_e762d0b76da2 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "1b6a170d9c37790b9de218cacd60bf88b8968bde11daeaf89f7fb76c8b183eaf",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Function",
"family_id": "tendsto_indicator_ge",
"file_id": "mathlib/Mathlib/MeasureTheory/Function/UniformIntegrable.lean",
"sample_id": "e762d0b76da2ca7a3ba42e4fe5df7c426f0bb2153a670147e34342a186a3e955"
} | train |
{
"chosen_proof_sha256": "3dbc0fabaa57897ff1e49ce5f7faadddfa1be34ceeabe2d7c9fe2b87f820ba31",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "30a1d731d06c0889b5d0dec83922f82193... | by
refine .mk _ Etale.respectsIso fun R S T _ _ _ _ _ h h' ↦ ?_
rw [etale_algebraMap] at h' ⊢
rw [faithfullyFlat_algebraMap_iff] at h
exact .of_etale_tensorProduct_of_faithfullyFlat S | true | {
"chosen_tokens": 39,
"rejected_tokens": 5,
"token_jaccard": 0.115385,
"token_length_ratio": 0.128205
} | wrong_namespace_qualification | wrong_namespace_qualification | c0902f6804ee78ea9803d579c38617cb2162f0303a43f92782a6dd9d88e69ee6 | 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.Etale
public import Mathlib.RingTheory.Finiteness.Descent
public import Mathlib.RingTheory.Extension.Cotangent.BaseChange
Namespace:
RingHo... | by
exact OProverMissing.missing_3087fe77077a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "569be107d7730fe97c5e0e6ca16e9b4cdf8170260586047481181b658fe1b9a8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Etale",
"family_id": "etale",
"file_id": "mathlib/Mathlib/RingTheory/Etale/Descent.lean",
"sample_id": "3087fe77077a4804fd4adc39f3b2f67cf5b7a70da11ffe0c839d1c6a058f86e4"
} | train |
{
"chosen_proof_sha256": "d74b30d157e4d96b787ba5699f29a97d3d77e2405ded9e8ea9925e08645a5660",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "597f39dd347d0f59b6dc44976880e8d552... | by
rw [coeff_eq_ncoeff, ncoeff_eq_zero_of_lt_order A (-n - 1) x]
lia
/-- Given an endomorphism-valued function on integers satisfying a pointwise bounded-pole condition,
we produce a vertex operator. -/
noncomputable def of_coeff (f : ℤ → Module.End R V)
(hf : ∀ x, BddBelow (Function.support fun y ↦ f y x)) : ... | true | {
"chosen_tokens": 99,
"rejected_tokens": 3,
"token_jaccard": 0.016667,
"token_length_ratio": 0.030303
} | circular_dependency | circular_dependency | c0e9428ec1ddde99a716b52c49659a919237fed57fb053a67457e8cd0c023804 | 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.Vertex.HVertexOperator
public import Mathlib.Data.Int.Interval
Namespace:
VertexOperator
Local context:
/-
Copyright (c) 2024 Scott Carnahan. All rig... | by
exact coeff_eq_zero_of_lt_order | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "7b18f3f54dd5bfb9411f5e773f1653d7dd05981d964f1ec06c8c8d595b85ff58",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Vertex",
"family_id": "coeff_eq_zero_of_lt_order",
"file_id": "mathlib/Mathlib/Algebra/Vertex/VertexOperator.lean",
"sample_id": "aad755a0feeacee18eda311e51c7bdd7c4de3325eadaa0977eddb936e76e9ea5"
} | train |
{
"chosen_proof_sha256": "909daa16cd181b78a203d073bc4331c1f30425ac8b8431620e4cd7f332d4a635",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
constructor
· intro H
induction H using Submodule.span_induction with
| mem a ha =>
obtain ⟨a, ha, rfl⟩ := ha
exact ⟨C a, fun i ↦ by rw [coeff_C]; aesop, aeval_C _ _⟩
| zero => exact ⟨0, by simp, aeval_zero _⟩
| add a b ha hb ha' hb' =>
obtain ⟨a, ha, ha'⟩ := ha'
obtain ⟨b... | false | {
"chosen_tokens": 312,
"rejected_tokens": 2,
"token_jaccard": 0.010638,
"token_length_ratio": 0.00641
} | sorry_admit_sorryAx_proof_hole | admit | c15a49c61392328c75b7c9291a6394da75cffb394894910940bcbc2164652b44 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.RingDivision
public import Mathlib.RingTheory.Adjoin.Polynomial.Basic
public import Mathlib.RingTheory.Ideal.Maps
Namespace:
Algebra
Local... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Polynomial",
"family_id": "mem_ideal_map_adjoin",
"file_id": "mathlib/Mathlib/RingTheory/Polynomial/Ideal.lean",
"sample_id": "a13d8df10b3157064561e6f5629998f83e9b759c9ef6010480db3810fec50f71"
} | train |
{
"chosen_proof_sha256": "ebb5a2a7a52df61da4119d7165c562d64bfda64efb43081a7067ed8d5938555e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "108b8bdc62a57e8b5f470c5fc576e999d2... | by
rw [toComplex_def₂, toComplex_def₂]
exact congr_arg₂ _ rfl (Int.cast_neg _) | true | {
"chosen_tokens": 20,
"rejected_tokens": 2,
"token_jaccard": 0.058824,
"token_length_ratio": 0.1
} | invalid_tactic | invalid_tactic | c1679ab85d22b5c2037b5a20a6175948b8b39f116f32c6d6f9dea6e8b729454f | 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.Archimedean.Real.Basic
public import Mathlib.Data.Complex.Basic
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.NumberTheory.Zsq... | by
__oprover_invalid_tactic_40b2cc766330 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "0649046269b815a6c7b72f826b9aa9322615538f6cc6b95b9f60978f422ca0c0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Zsqrtd",
"family_id": "tocomplex_star",
"file_id": "mathlib/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean",
"sample_id": "40b2cc76633014d1e5dccc9d2cd9aa43dc95f8c9291dfcfb08d96f928e0a655e"
} | train |
{
"chosen_proof_sha256": "cb2db471013d02747d266722ac3b318ad74596861a15ff8911bcdfb53d010405",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5dcc120b10457a5e5ed4d89893c537fd3c... | by
rw [GradedAlgebra.proj_apply, decompose_symm_of, Equiv.apply_symm_apply] | true | {
"chosen_tokens": 13,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.153846
} | invalid_tactic | invalid_tactic | c16a22330d974c47587ef3e38f585c505ad72e1987e9c353922f65a170f94f23 | 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.DirectSum.Algebra
public import Mathlib.Algebra.DirectSum.Decomposition
public import Mathlib.Algebra.DirectSum.Internal
public import Mathlib.Algebra.... | by
__oprover_invalid_tactic_7730bf9984af | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "a4389f0ef08c32b3f06bc1d4370eed33ac3ff832db59e2ecf1897fb9550abcd1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/GradedAlgebra",
"family_id": "gradedalgebra",
"file_id": "mathlib/Mathlib/RingTheory/GradedAlgebra/Basic.lean",
"sample_id": "7730bf9984af8e1d55b7e5d3174ec963362e837effdd837bd031f3763c30060e"
} | train |
{
"chosen_proof_sha256": "98bbccba21050a97745f14c454bcd7300ae6caac0c9cdef2d7b7857a2d6f0324",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "44e7cadea26a9e80398866fe2e0034ed41... | by
rw [← totallyDisconnectedSpace_subtype_iff]
have : Countable s := hs
infer_instance | true | {
"chosen_tokens": 13,
"rejected_tokens": 7,
"token_jaccard": 0.117647,
"token_length_ratio": 0.538462
} | circular_dependency | circular_dependency | c2535c5a88682902dce7ca90b630ba7885842731491d5583b8873a630a719c79 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.GDelta.MetrizableSpace
public import Mathlib.Topology.Separation.CompletelyRegular
public import Mathlib.Topology.Separation.Profinite
Namespace:
(ro... | by
exact Set.Countable.isTotallyDisconnected | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "10e73213d1dd8472f9126b7f7950ef68dd1835640f7557eca13341f677964830",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Separation",
"family_id": "set",
"file_id": "mathlib/Mathlib/Topology/Separation/Lemmas.lean",
"sample_id": "28dab043652a1525adfd5ddc52e6243324bae2d9c4d94e5b8f18a41fa2a7d11f"
} | train |
{
"chosen_proof_sha256": "3f5dfa9a05050a7e8f6e2944c4416e1a85109aff6c3b31b2844a70ac199a91e0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "48fc0507f8a383ca9960ad8374aaee847a... | by
obtain ⟨C, D, rfl⟩ := hNM
exact hI.of_delete.of_contract | false | {
"chosen_tokens": 17,
"rejected_tokens": 22,
"token_jaccard": 0.75,
"token_length_ratio": 1.294118
} | markdown_preamble_trailing | preamble | c293062afcc42ecc7b5bd7d2cb04e3f1a23c877abe8cb170f292472cbef648f4 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Combinatorics.Matroid.Minor.Contract
Namespace:
Matroid
Local context:
/-
Copyright (c) 2025 Peter Nelson. All rights reserved.
Released under Apache 2.0 lic... | Here is the proof:
by
obtain ⟨C, D, rfl⟩ := hNM
exact hI.of_delete.of_contract | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/Matroid",
"family_id": "indep",
"file_id": "mathlib/Mathlib/Combinatorics/Matroid/Minor/Order.lean",
"sample_id": "44d2da225f20cd3b6e6276fb821a5a22fe9ec474c9711a39a7cbc160460c406f"
} | train |
{
"chosen_proof_sha256": "e849d95b5db02f9c868ba8b0e417961eeea08fc774f074fe05a938dadf630d6e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
by_contra!
exact hA (hP.subobject_eq_top this) | false | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.181818
} | sorry_admit_sorryAx_proof_hole | sorry | c29a91f5e5643b2586363f3479a54d015712abed8aa03ed3cb577f2b5415a34f | 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.ExtremalEpi
public import Mathlib.CategoryTheory.Generator.Basic
public import Mathlib.CategoryTheory.Limits.Presentation
Namespace:
CategoryTh... | 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/Generator",
"family_id": "exists_of_subobject_ne_top",
"file_id": "mathlib/Mathlib/CategoryTheory/Generator/StrongGenerator.lean",
"sample_id": "bf6d1403eb4460ab7069e5f743251c32cc0683a5c043812b0396ebf8e760f9d6"
} | train |
{
"chosen_proof_sha256": "1c8bfaba84928466d869f23a6dc4de48708f2d6673fcc79573e01b4de03f6a29",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c4f806797736ab64809b45b40862e20cdb... | by
rw [normal_iff] at h ⊢
intro x; specialize h (f.symm x)
rw [← f.apply_symm_apply x, minpoly.algEquiv_eq, ← f.toAlgHom.comp_algebraMap, ← map_map]
exact ⟨h.1.map f, h.2.map _⟩ | true | {
"chosen_tokens": 57,
"rejected_tokens": 5,
"token_jaccard": 0.088235,
"token_length_ratio": 0.087719
} | circular_dependency | circular_dependency | c29aaf23e73dbf05cb8a8d4bc4e81042e68e1ade4ebefe830adc5e8de3197e15 | 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.Splits
public import Mathlib.FieldTheory.Galois.Notation
public import Mathlib.FieldTheory.IntermediateField.Basic
public import Mathlib.Fie... | by
exact Normal.of_algEquiv | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "cbdd080a81701a668239d4e5a7d1758d4c6b0bd6ffaa7b91509a9b2085371949",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/Normal",
"family_id": "normal",
"file_id": "mathlib/Mathlib/FieldTheory/Normal/Defs.lean",
"sample_id": "30e2dcca73b76516afafa57bf58fa1f528e0a923278fa6441a9972d75d07dd33"
} | train |
{
"chosen_proof_sha256": "56b4e98d034de1994cb22dfefff34d5e35c25783ae7eef0d3fe3f45cbf3ff033",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8b929f3d1885371d6b55ef0cdc5e96bf8f... | by
simpa using mul_vle_mul_left ((vle_total 0 1).resolve_right not_vle_one_zero) x | false | {
"chosen_tokens": 15,
"rejected_tokens": 22,
"token_jaccard": 0.866667,
"token_length_ratio": 1.466667
} | markdown_preamble_trailing | fence | c2a8a70eccaabbb4f95b3eff60005b1aeef1f60fff0eabebbb61a5b4498607fb | 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.Valuation.Basic
public import Mathlib.Data.NNReal.Defs
public import Mathlib.Topology.Defs.Filter
public import Mathlib.Order.Filter.Bases.Basic
Na... | ```lean
by
simpa using mul_vle_mul_left ((vle_total 0 1).resolve_right not_vle_one_zero) x
``` | {
"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/Valuation",
"family_id": "zero_vle",
"file_id": "mathlib/Mathlib/RingTheory/Valuation/ValuativeRel/Basic.lean",
"sample_id": "27226b4cef484b3c92a34e5c76b5413440a1f9efdabd65d7d5bb3fbb80618898"
} | train |
{
"chosen_proof_sha256": "335297b55a581bb6fe6dcb38f704182637c77f8f85b5c081f025c29aaa5ebb61",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [← Option.some_inj]
simp [finSuccAboveOrderIso, finSuccAboveEquiv, OrderIso.symm] | false | {
"chosen_tokens": 18,
"rejected_tokens": 5,
"token_jaccard": 0.055556,
"token_length_ratio": 0.277778
} | sorry_admit_sorryAx_proof_hole | sorryAx | c2eb304b244fc4f70332e2a4d38b94ad25cbfa3bf2eec28009a1011b9eff7bb1 | 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.Fin.VecNotation
public import Mathlib.Logic.Equiv.Fin.Basic
public import Mathlib.Order.Fin.Basic
public import Mathlib.Order.PiLex
public import Mathlib.... | 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/Fin",
"family_id": "finsuccaboveorderiso_symm_apply_last",
"file_id": "mathlib/Mathlib/Order/Fin/Tuple.lean",
"sample_id": "c092795e5fd0e22d540c502fd1b4fd017f05beade7817be0597a69f2c004e7d6"
} | train |
{
"chosen_proof_sha256": "e483053df1cef9a9899f381185253ce0593370fe3c8c56da6f87d63a4270c28a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e538998df3d05f93bc2e2817a045d944fa... | by
dsimp [localization]
simp only [Localization.liftNatTrans_app, Functor.id_obj, Functor.whiskerRight_app,
NatTrans.naturality, Functor.comp_map, Functor.id_map, Iso.hom_inv_id_app_assoc]
include W | false | {
"chosen_tokens": 38,
"rejected_tokens": 45,
"token_jaccard": 0.916667,
"token_length_ratio": 1.184211
} | markdown_preamble_trailing | fence | c300e8616e64cb5278f68df964d7d2bd474d20aa3d8f5f2b4a17d9c29e3a4a54 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Center.Preadditive
public import Mathlib.CategoryTheory.Localization.Predicate
public import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
... | ```lean
by
dsimp [localization]
simp only [Localization.liftNatTrans_app, Functor.id_obj, Functor.whiskerRight_app,
NatTrans.naturality, Functor.comp_map, Functor.id_map, Iso.hom_inv_id_app_assoc]
include W
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Center",
"family_id": "localization_app",
"file_id": "mathlib/Mathlib/CategoryTheory/Center/Localization.lean",
"sample_id": "81608c9debe142f31aa7efa37f974504e0e8f8ceff0f0397335ad80e4df5060e"
} | train |
{
"chosen_proof_sha256": "65fb0bf51735f2fd54b4e2c9d7d6b383f1f337560f0bc27999124bd7b667a991",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ec42e30620eff4d5664c510c36fb816ac3... | by
have hA_T_nonneg : ∀ i j, 0 ≤ Aᵀ i j := fun i j => by
simpa [Matrix.transpose_apply] using hA.nonneg j i
refine ⟨hA_T_nonneg, ?_⟩
intro i j
letI : Quiver n := toQuiver A
obtain ⟨p, hp_pos⟩ := hA.connected j i
cases p with
| nil =>
simp at hp_pos
| @cons b _ q e =>
let qT := transposePath ... | true | {
"chosen_tokens": 121,
"rejected_tokens": 2,
"token_jaccard": 0.016949,
"token_length_ratio": 0.016529
} | invalid_tactic | invalid_tactic | c383ae8416b8f5f30fd609b398bcfaa20389d9329f1d39efd5609f11ada07fd4 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Combinatorics.Quiver.ConnectedComponent
public import Mathlib.Combinatorics.Quiver.Path.Vertices
public import Mathlib.Data.Matrix.Mul
Namespace:
Matrix
Loca... | by
__oprover_invalid_tactic_99fd3b1c7dbb | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b88570995ea7a47ee5c63db757dd22f09fa8725a6f4313168317e911e33af73a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Matrix",
"family_id": "isirreducible",
"file_id": "mathlib/Mathlib/LinearAlgebra/Matrix/Irreducible/Defs.lean",
"sample_id": "99fd3b1c7dbbd31133ecfe7e89f6d8fdbe8581f23efe8e63f4413b7eb18ed016"
} | train |
{
"chosen_proof_sha256": "01a9a5bebe8404a63c3e4dd6c8ed6a63ca7ab9f3512e93006d4e0432218d9a02",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "301632548b2cf18455f3303dcaa3cf3638... | by
let ⟨⟨κ, b⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [(equivFreeAlgebra b).toLinearEquiv.rank_eq, FreeAlgebra.rank_eq, mk_list_eq_sum_pow,
Basis.mk_eq_rank'' b] | true | {
"chosen_tokens": 47,
"rejected_tokens": 3,
"token_jaccard": 0.074074,
"token_length_ratio": 0.06383
} | circular_dependency | circular_dependency | c3d97e0e62d3c6d29f11d3ab8b0293bdcd548cacc679d04ca3bd997fd9321abf | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorAlgebra.Basic
public import Mathlib.LinearAlgebra.FreeAlgebra
Namespace:
TensorAlgebra
Local context:
/-
Copyright (c) 2023 Eric Wieser. ... | by
exact rank_eq | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "ee50573d078d5041c122f54fe08b9d8601c9649d459f6a3a579cf71cc6658164",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/TensorAlgebra",
"family_id": "rank_eq",
"file_id": "mathlib/Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean",
"sample_id": "d99fd6fb9456101e1d28de19e801f40e15b75e0d01ea3c483f56e794b32e60bf"
} | train |
{
"chosen_proof_sha256": "f6142b8f4680c491cc9a84d46312116399269b0b324f299edcee5c0f95b61ddc",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
intro x y hxy
have := M.isUnit_algebraMap_end_of_le_basicOpen _ hle
rw [Module.End.isUnit_iff] at this
exact this.injective hxy | false | {
"chosen_tokens": 27,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.074074
} | sorry_admit_sorryAx_proof_hole | admit | c3e8ce175e8d37cbe89b0260163757984727c7119f61167ae676d43220174807 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Category.ModuleCat.Localization
public import Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent
public import Mathlib.AlgebraicGeometry.AffineSche... | 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": "AlgebraicGeometry/Modules",
"family_id": "issmulregular_of_le_basicopen",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Modules/Tilde.lean",
"sample_id": "ebebed96f1f59661bd31143f834b43b5b9941049affa74d994da26eb1385a91c"
} | train |
{
"chosen_proof_sha256": "a14f9a077bdd5b6f8dbacb2f648d28154c7ace639c1cada2afcd1dc3728bb20d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
constructor
intro Z ⟨g, gc⟩ ⟨h, hc⟩ e
apply_fun InducedCategory.Hom.hom at e
obtain rfl : g = h := ConcreteCategory.hom_ext _ _ fun y ↦ by
rw [← (h₁ y).choose_spec]
simpa using congr(($e).base.hom (h₁ y).choose)
refine SheafedSpace.hom_stalk_ext ⟨g, gc⟩ ⟨g, hc⟩ rfl fun y ↦ ?_
rw [← (h₁ y).choose_... | false | {
"chosen_tokens": 158,
"rejected_tokens": 2,
"token_jaccard": 0.017241,
"token_length_ratio": 0.012658
} | sorry_admit_sorryAx_proof_hole | admit | c435151482dde8e322d7017ef7aefc216329eb87cb05d3f63dfd75eb7ede5df5 | 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.Geometry.RingedSpace.PresheafedSpace.HasColimits
public import Mathlib.Geometry.RingedSpace.Stalks
public import Mathlib.Topology.Sheaves.Functors
Namespace:
... | 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": "Geometry/RingedSpace",
"family_id": "epi_of_base_surjective_of_stalk_mono",
"file_id": "mathlib/Mathlib/Geometry/RingedSpace/SheafedSpace.lean",
"sample_id": "36f6292b2bea01ba2949cab99df0942afa3d01db01b2c1e30655e18e62dad252"
} | train |
{
"chosen_proof_sha256": "c1390084cb2580392cbfe88281f49b6df9fd83f38d46b0fc57029acaa73ff3f3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a8f650db8d1b0bf326b19579d79bab9b82... | by
rw [← span_span_of_tower R, finrank_span_eq_finrank] | false | {
"chosen_tokens": 9,
"rejected_tokens": 14,
"token_jaccard": 0.642857,
"token_length_ratio": 1.555556
} | markdown_preamble_trailing | preamble | c4561dcb2df0eedef243beba495a3027f05dd4a823dd427707c6293ffbc8c6fe | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Epi
public import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
public import Mathlib.LinearAlgebra.Finsupp.LinearCombination
public impo... | Here is the proof:
by
rw [← span_span_of_tower R, finrank_span_eq_finrank] | {
"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/Span",
"family_id": "finrank_span_eq_finrank_span",
"file_id": "mathlib/Mathlib/LinearAlgebra/Span/TensorProduct.lean",
"sample_id": "f378af96f4b655711415908ce319014735dc5bb779d91dee4115105bae787859"
} | train |
{
"chosen_proof_sha256": "99a954ce5f0b66b4d53e9b41552229a51be60d1da33548a2621f7dcf8fed671d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "bb92462adba4b538731ab9f0c55d878b2d... | by
use 1, S.one_mem
intro b hb
rw [one_smul]
obtain ⟨b', b'_mem, rfl⟩ := mem_one.mp (h hb)
exact Set.mem_range_self b' | false | {
"chosen_tokens": 35,
"rejected_tokens": 39,
"token_jaccard": 0.90625,
"token_length_ratio": 1.114286
} | missing_wrong_import | import | c4c6088d2958298470367aebf187ef566d21f267e6f55bcc5ba761f960a9f79e | 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.Localization.Integer
public import Mathlib.RingTheory.Localization.Submodule
Namespace:
FractionalIdeal
Local context:
/-
Copyright (c) 2020 Anne ... | import Mathlib.WrongImport
by
use 1, S.one_mem
intro b hb
rw [one_smul]
obtain ⟨b', b'_mem, rfl⟩ := mem_one.mp (h hb)
exact Set.mem_range_self b' | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/FractionalIdeal",
"family_id": "isfractional_of_le_one",
"file_id": "mathlib/Mathlib/RingTheory/FractionalIdeal/Basic.lean",
"sample_id": "4fa9f7ffaee479f67dcc3f60683ec20977b8a4ed853209875ba4ba0d88e8525f"
} | train |
{
"chosen_proof_sha256": "50b6a5e16d3844921370e0938982952c322ae0fcb0f107c81a94eedb5977aaa4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b6c7165c0bd788e389a9bc6e6e492d5bb7... | by
rw [← lintegral_sum_measure, h.sum_restrict_of_ac hν] | false | {
"chosen_tokens": 12,
"rejected_tokens": 17,
"token_jaccard": 0.6875,
"token_length_ratio": 1.416667
} | markdown_preamble_trailing | preamble | c544df3040aa49a9fe64a495ebb56bf8731013c66f929a22697d190bca352a6f | 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.Group.Action
public import Mathlib.MeasureTheory.Group.Pointwise
public import Mathlib.MeasureTheory.Integral.Lebesgue.Map
public import Mathlib.... | Here is the proof:
by
rw [← lintegral_sum_measure, h.sum_restrict_of_ac hν] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Group",
"family_id": "lintegral_eq_tsum_of_ac",
"file_id": "mathlib/Mathlib/MeasureTheory/Group/FundamentalDomain.lean",
"sample_id": "70a4170ae953a085b2ee61b70a1cefea9bde7f6d81dae6658409edfa420b8c2e"
} | train |
{
"chosen_proof_sha256": "3816114db1c077f621bfeb2d3b1b98b7b671ab19074e0d158d715632b3696809",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
intro x y hxy
let xy : G.obj (pullback π π) := (PreservesPullback.iso G π π).inv <|
(TopCat.pullbackIsoProdSubtype (G.map π) (G.map π)).inv ⟨(x, y), hxy⟩
have ha' := congr_fun ha xy
dsimp at ha'
have h₁ : ∀ y, G.map (pullback.fst _ _) ((PreservesPullback.iso G π π).inv y) =
pullback.fst (G.map π)... | true | {
"chosen_tokens": 211,
"rejected_tokens": 8,
"token_jaccard": 0.068966,
"token_length_ratio": 0.037915
} | unsolved_goals | unsolved_goals | c55a0667ce6320a7d8c219565f1f502e4fb0077c0ef37bea82fdc312238df32d | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Preserves.Opposites
public import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
public import Mathlib.Condensed.Basic
public impo... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "0b618490eacd78b02138e83eb99423e432e36ab7122ae0c28c40876a46f14a1a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Condensed",
"family_id": "factorsthrough_of_pullbackcondition",
"file_id": "mathlib/Mathlib/Condensed/TopComparison.lean",
"sample_id": "9584eeedec12f1270f773e97e07803ac6bace4ed2fabdce1f1ae60147a31ccc0"
} | train |
{
"chosen_proof_sha256": "be2b6932dda4e0b3c56c94860ddf8d756e2f7715633872f9e4a77499f7decf69",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
simp only [cmpUsing, Ordering.ite_eq_gt_distrib, if_false_right, and_true, if_false_left,
and_iff_right_iff_imp, reduceCtorEq]
exact fun hba hab ↦ (irrefl a) (_root_.trans hab hba) | false | {
"chosen_tokens": 36,
"rejected_tokens": 3,
"token_jaccard": 0.074074,
"token_length_ratio": 0.083333
} | by_exact_placeholder | exact? | c5a023fbc19f95476990a981d665d46620209eb34c022a1e6a0f22a28eb17604 | 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:
(root)
Local context:
/-
Copyright (c) 2017 Microsoft Corporation. All rights reser... | 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/Ordering",
"family_id": "cmpusing_eq_gt",
"file_id": "mathlib/Mathlib/Data/Ordering/Lemmas.lean",
"sample_id": "0700ede7788b015baae01a8e4aa3cd674212af76354c06e79abb4f1dca42ce86"
} | train |
{
"chosen_proof_sha256": "197871194b11ccd2f7b68c20098508f4fb8d58251817640d3a8e7c9a33767c9c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "56c83df4fcbbc0fb6cb913621257dbef79... | by
refine ⟨fun h _ _ ↦ by rw [h], fun h ↦ ?_⟩
rw [DFunLike.ext'_iff]
apply Function.Surjective.injective_comp_right mkAlgHom_surjective
simpa [← AlgHom.coe_comp] using MvPolynomial.algHom_ext fun ⟨n, m⟩ ↦ h n m | false | {
"chosen_tokens": 54,
"rejected_tokens": 58,
"token_jaccard": 0.909091,
"token_length_ratio": 1.074074
} | missing_wrong_import | import | c5cee8610374bccd198e38f2c84561f0425d3c89b6e6c0dbec3968cf6c6cff7f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MvPolynomial.Eval
public import Mathlib.Algebra.RingQuot
public import Mathlib.RingTheory.DividedPowers.Basic
Namespace:
DividedPowerAlgebra
Local co... | import Mathlib.WrongImport
by
refine ⟨fun h _ _ ↦ by rw [h], fun h ↦ ?_⟩
rw [DFunLike.ext'_iff]
apply Function.Surjective.injective_comp_right mkAlgHom_surjective
simpa [← AlgHom.coe_comp] using MvPolynomial.algHom_ext fun ⟨n, m⟩ ↦ h n m | {
"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/DividedPowerAlgebra",
"family_id": "alghom_ext_iff",
"file_id": "mathlib/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean",
"sample_id": "41bafb9f17936ff03795282fab01c1edc2ec5df90a2dc068b3d884811a5b8b02"
} | train |
{
"chosen_proof_sha256": "7333d1a239a893fe44382cfe648dc46c728566269ad1ec5413c976d55e86e914",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9fe351be3c3d262f5458bdcaa60348aa5e... | by
simpa only [Ordinal.liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, Order.succ_le_iff] using!
le_of_lt (Ordinal.liftPrincipalSeg.{u, u + 1}.lt_top (Order.succ c).ord) | true | {
"chosen_tokens": 44,
"rejected_tokens": 3,
"token_jaccard": 0.032258,
"token_length_ratio": 0.068182
} | circular_dependency | circular_dependency | c5fde899450b8eb48b93cd0927762d7b2ee700ea0a4c3dcba306ea1eac1e48a0 | 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.SetTheory.Ordinal.Basic
Namespace:
Cardinal
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as de... | by
exact lift_lt_univ | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "dd1078bf964280d36c3d51095d2a3f4371dcbe8fe09875bd684346014d99989b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "SetTheory/Ordinal",
"family_id": "lift_lt_univ",
"file_id": "mathlib/Mathlib/SetTheory/Ordinal/Univ.lean",
"sample_id": "dabe8f52856a803fb511e285a4cf7be82ea7a0120495bcf033465a0c9f492e10"
} | train |
{
"chosen_proof_sha256": "768478b669f1b0c48360ff5cb94ef6db06e9b1b6432c6287e4928bb1cb2132ad",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b506c66ce42848c4a4ad0e476f485d2b07... | by
rw [eq_comm, ← ltb_cons_addChar' c]
simp | true | {
"chosen_tokens": 10,
"rejected_tokens": 5,
"token_jaccard": 0.071429,
"token_length_ratio": 0.5
} | wrong_namespace_qualification | wrong_namespace_qualification | c6187b5e167e246c8b350825a72b760f2cbbe887323447151567e861c05e2737 | 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.String.Lemmas
public import Mathlib.Data.List.Lex
public import Mathlib.Data.Char
public import Mathlib.Algebra.Order.Group.Nat
import all Init.Data.Str... | by
exact OProverMissing.missing_2ad48a41750f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c112f948655427f7e212f06151c0a653e2a7a34517ec7d039ee6fde2a4b16326",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/String",
"family_id": "ltb_cons_addchar",
"file_id": "mathlib/Mathlib/Data/String/Basic.lean",
"sample_id": "2ad48a41750f9ee7416f3ed47f9d4bdc606c25a12a288f7161359dab4138e50a"
} | train |
{
"chosen_proof_sha256": "18e5d95a99a28ddbd089bbdf5652dbad7467c92bf9a278d6d749571d6703ac97",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6019d53cd390889e024e24ee9eef9da9a2... | by
rw [descFactorial_eq_prod_range, factorial_eq_prod_range_add_one]
simp only [cast_prod]
nth_rw 2 [← card_range n]
rw [pow_card_mul_prod]
refine prod_congr rfl ?_
intro x hx
rw [← tsub_add_eq_tsub_tsub_swap,
Nat.cast_sub <| Nat.le_trans (Nat.add_one_le_iff.mpr (List.mem_range.mp hx)) h,
CharP.ca... | false | {
"chosen_tokens": 81,
"rejected_tokens": 88,
"token_jaccard": 0.960784,
"token_length_ratio": 1.08642
} | markdown_preamble_trailing | fence | c6528826cec183f7207e67d577463e9fb38e134b4532f1ff420bc68845aedfe8 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.Factorial.BigOperators
public import Mathlib.Data.ZMod.Basic
Namespace:
ZMod
Local context:
/-
Copyright (c) 2023 Moritz Firsching. All rights reser... | ```lean
by
rw [descFactorial_eq_prod_range, factorial_eq_prod_range_add_one]
simp only [cast_prod]
nth_rw 2 [← card_range n]
rw [pow_card_mul_prod]
refine prod_congr rfl ?_
intro x hx
rw [← tsub_add_eq_tsub_tsub_swap,
Nat.cast_sub <| Nat.le_trans (Nat.add_one_le_iff.mpr (List.mem_range.mp hx)) h,
... | {
"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/ZMod",
"family_id": "cast_descfactorial",
"file_id": "mathlib/Mathlib/Data/ZMod/Factorial.lean",
"sample_id": "8c92966f8a3dcfb46f7baa7761df8474960f52637455e799bcbf00ab2a68364d"
} | train |
{
"chosen_proof_sha256": "3628ed809b6ca5e99b1c81051989cfa58d947fe9238f4a04564fff2b92ccbda9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
obtain ⟨W, hW, hWl, hWr, h⟩ := h
have e₁ := isDominant_hom_iff_isDominant_restrict_hom f W hW hWl
have e₂ := isDominant_hom_iff_isDominant_restrict_hom g W hW hWr
dsimp only [restrict_domain, restrict_hom] at ⊢ e₁ e₂ h
rw [e₁, h, ← e₂] | true | {
"chosen_tokens": 58,
"rejected_tokens": 8,
"token_jaccard": 0.09375,
"token_length_ratio": 0.137931
} | unsolved_goals | unsolved_goals | c6cf8abfbcde9d0c3809197297df66581ef505eaf88921dcf253c59169be4837 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Birational.RationalMap
Namespace:
AlgebraicGeometry.Scheme.PartialMap
Local context:
/-
Copyright (c) 2026 Justus Springer. All rights rese... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "08fe55745aee9ead66f239113c1f502f1051027dadf5aeb75b805ea75370f88f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Birational",
"family_id": "isdominant_hom_iff_of_equiv",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Birational/Dominant.lean",
"sample_id": "0b655641f0fbcc76ef7401981695bc0221f56804bae473d2205adf3e718b621a"
} | train |
{
"chosen_proof_sha256": "aedbd85f784d006a3b8e0a1c284ce353d4c9554e1c007c5246c151cde67cde07",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b463641116e31fe41b30a3ddc8948cc471... | by
exact ⟨fun h ↦ ⟨⟨@effectiveEpiFamilyStructOfEffectiveEpiDesc _ _ _ _ X π _ h _ _ (fun g ↦
(FinitaryPreExtensive.isIso_sigmaDesc_fst (fun a ↦ Sigma.ι X a) g inferInstance).epi_of_iso)⟩⟩,
fun _ ↦ inferInstance⟩
variable {D : Type*} [Category* D] [FinitaryPreExtensive D]
variable (F : C ⥤ D) [PreservesFinite... | false | {
"chosen_tokens": 80,
"rejected_tokens": 85,
"token_jaccard": 0.904762,
"token_length_ratio": 1.0625
} | markdown_preamble_trailing | trailing | c6d45450d1ab0aaf2f040125f71b0f33e9dba281c490657d38512ae8c6546d2c | 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.EffectiveEpi.Preserves
public import Mathlib.CategoryTheory.EffectiveEpi.Coproduct
public import Mathlib.CategoryTheory.Extensive
public import ... | by
exact ⟨fun h ↦ ⟨⟨@effectiveEpiFamilyStructOfEffectiveEpiDesc _ _ _ _ X π _ h _ _ (fun g ↦
(FinitaryPreExtensive.isIso_sigmaDesc_fst (fun a ↦ Sigma.ι X a) g inferInstance).epi_of_iso)⟩⟩,
fun _ ↦ inferInstance⟩
variable {D : Type*} [Category* D] [FinitaryPreExtensive D]
variable (F : C ⥤ D) [PreservesFinite... | {
"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/EffectiveEpi",
"family_id": "effectiveepi_desc_iff_effectiveepifamily",
"file_id": "mathlib/Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean",
"sample_id": "a3caeca3aca5af45210b187259a35456fb858a420f6704c94186f1b14ec00a25"
} | train |
{
"chosen_proof_sha256": "83fc6e2c4e62e7f25a2d9e2ffddabb7d5875dc78574f118790a50383327ea178",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | 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... | true | {
"chosen_tokens": 142,
"rejected_tokens": 8,
"token_jaccard": 0.05,
"token_length_ratio": 0.056338
} | unsolved_goals | unsolved_goals | c702dddbd722a5f62b3a894af81033a47a71d2864dcd1dcd0fecb0ed476ae025 | 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.Filter.AtTopBot.Basic
public import Mathlib.RingTheory.Finiteness.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2018 Mario Carneiro, Kevin Bu... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "f558a8bc29b2dda1c8db968c2e595d63e502ecf0d40a46e52a1cc5d336b9c083",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "fc005df75a037e9e437ea57ce191d036c65586027b4016497e6ed6b62c40e9f3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "eb5890534d5da8e62ba89ed4014fd0be99... | by
dsimp only [LiftP, LiftP']
apply exists_iff_exists_of_mono F (f _ n α) (g _ n α)
· ext i ⟨x, _⟩
cases i <;> rfl
· intros
rw [MvFunctor.map_map]
dsimp +unfoldPartialApp [(· ⊚ ·)]
suffices (fun i => Subtype.val) = (fun i x => (MvFunctor.f P n α i x).val) by rw [this]
ext i ⟨x, _⟩
cases ... | true | {
"chosen_tokens": 111,
"rejected_tokens": 5,
"token_jaccard": 0.037037,
"token_length_ratio": 0.045045
} | wrong_namespace_qualification | wrong_namespace_qualification | c73ca49f94f9c68cc0d5ff46f43be466703c44aad78feba9649fbedafefb41e9 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fin.Fin2
public import Mathlib.Data.TypeVec
public import Mathlib.Logic.Equiv.Defs
Namespace:
MvFunctor
Local context:
/-
Copyright (c) 2018 Jeremy Avig... | by
exact OProverMissing.missing_9f7fcfe93c14 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "30dd9fdd086dd9549026b8bf8a622901404ac2baa283626e4fafd465303a6cef",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Control/Functor",
"family_id": "liftp_predlast_iff",
"file_id": "mathlib/Mathlib/Control/Functor/Multivariate.lean",
"sample_id": "9f7fcfe93c14fb87931e79553027be989fd1cf1216f6e4592f76a5cfd75311b1"
} | train |
{
"chosen_proof_sha256": "a956e0393455f70ac3f0634d07a69fa1a653d99212eafe809fbf92b858839bc8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a95656ffd38538e1dc9aa485eecac2357b... | by
constructor
· induction fs generalizing t with
| nil =>
simp only [sequence, mem_pure, imp_self, forall₂_nil_left_iff, exists_eq_left, Set.pure_def,
singleton_subset_iff, traverse_nil]
| cons b fs ih =>
intro ht
rcases mem_seq_iff.1 ht with ⟨u, hu, v, hv, ht⟩
rcases mem_ma... | true | {
"chosen_tokens": 127,
"rejected_tokens": 2,
"token_jaccard": 0.016393,
"token_length_ratio": 0.015748
} | invalid_tactic | invalid_tactic | c741a92a9b54e4bf90f0ef20c42e6e02337cbeca59644b8e832659f0ab5427d5 | 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.Control.Traversable.Instances
public import Mathlib.Order.Filter.Map
Namespace:
Filter
Local context:
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserv... | by
__oprover_invalid_tactic_a1347b35a1da | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "047acb989ff296fd86a71da279282c412282b3ce5deba21d79152cb3ac5ba2e4",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Filter",
"family_id": "mem_traverse",
"file_id": "mathlib/Mathlib/Order/Filter/ListTraverse.lean",
"sample_id": "a1347b35a1da940b0cf7bb9fa19ec8fb67a45c4aec52bd9f946f2c2f04d77b83"
} | train |
{
"chosen_proof_sha256": "665f9b24ed35095c9258c237fa3353b33a4af46379e1ca75f0d890346defc187",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
apply EqualizerCondition.mk
intro Z B π _ _
refine ⟨fun a b h ↦ ?_, fun ⟨a, ha⟩ ↦ ?_⟩
· simp only [yonedaPresheaf, comp, Quiver.Hom.unop_op, TypeCat.Fun.coe_mk,
Set.coe_setOf, mapToEqualizer, Set.mem_setOf_eq, ConcreteCategory.hom_ofHom, Subtype.mk.injEq,
mk.injEq] at h
simp only [yonedaPresh... | false | {
"chosen_tokens": 325,
"rejected_tokens": 2,
"token_jaccard": 0.009615,
"token_length_ratio": 0.006154
} | sorry_admit_sorryAx_proof_hole | admit | c778555600e29517ab9267992523e6f71e122438ad968160ffaecc3bc669b9cf | 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.Preserves.Opposites
public import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
public import Mathlib.Condensed.Basic
public impo... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Condensed",
"family_id": "equalizercondition_yonedapresheaf",
"file_id": "mathlib/Mathlib/Condensed/TopComparison.lean",
"sample_id": "df00ef8b543a3838a8dcfd15337dcf5374c33e276a9693b751546ead28ce4bfb"
} | train |
{
"chosen_proof_sha256": "81f6811c61bda2ac7e16fe95d39238e8cafe9933dcb6310481ee06482505a998",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "936168df007e15fb85b6f7d1e2a4ec6ccd... | by
simp [fiber, fiberι, Scheme.Pullback.range_fst, Scheme.range_fromSpecResidueField] | true | {
"chosen_tokens": 18,
"rejected_tokens": 5,
"token_jaccard": 0.133333,
"token_length_ratio": 0.277778
} | wrong_namespace_qualification | wrong_namespace_qualification | c7c27c1930ea9b6da3dad4d064a11eefa4ea90d775f39ed99eb36b0ed22c4f84 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.PullbackCarrier
public import Mathlib.RingTheory.LocalRing.ResidueField.Fiber
public import Mathlib.RingTheory.Spectrum.Prime.Jacobson
public... | by
exact OProverMissing.missing_9ea116b7daa0 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a98d6c6c6a65f5bb99e7c6e1ecfc81c0f48acb301906f82b1bb3c855493b0266",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry",
"family_id": "scheme",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Fiber.lean",
"sample_id": "9ea116b7daa0ff9aa81b54eb159a02ebd9756dce658d8964cedd444ceb6ca675"
} | train |
{
"chosen_proof_sha256": "06d455a8b5ee3bf3c5458bfc466011b102068e1311619ad2681ed0ee2fe0c4c3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "73e2230058ad9af7f90bd906974327f3df... | by
by_cases hβ : Nonempty β
· refine small_of_surjective (f := Function.invFun g ∘ f) (fun b => ?_)
obtain ⟨a, ha⟩ := h b
exact ⟨a, by rw [Function.comp_apply, ha, Function.leftInverse_invFun hg]⟩
· simp only [not_nonempty_iff] at hβ
infer_instance
/-!
We don't define `Countable.toSmall` in this file... | false | {
"chosen_tokens": 93,
"rejected_tokens": 98,
"token_jaccard": 0.935484,
"token_length_ratio": 1.053763
} | markdown_preamble_trailing | preamble | c7de665b3ae280a1442fefd5b27719ebc4fdc4395a5fad5a6901eaaad899e7ec | 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.Small.Defs
public import Mathlib.Logic.Equiv.Set
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released und... | Here is the proof:
by
by_cases hβ : Nonempty β
· refine small_of_surjective (f := Function.invFun g ∘ f) (fun b => ?_)
obtain ⟨a, ha⟩ := h b
exact ⟨a, by rw [Function.comp_apply, ha, Function.leftInverse_invFun hg]⟩
· simp only [not_nonempty_iff] at hβ
infer_instance
/-!
We don't define `Countable.to... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Logic/Small",
"family_id": "small_of_injective_of_exists",
"file_id": "mathlib/Mathlib/Logic/Small/Basic.lean",
"sample_id": "900df9f57575ae9ae5d97a81a010ab1e20e766cd1a88e2d731f34af9d7eca3f8"
} | train |
{
"chosen_proof_sha256": "153be3b3963b7b4e295fef95384346f85509c50a1d50e691a207cccee178ce99",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "826b1e14ead13499112deffa7b08d1cf24... | by
constructor
· rw [Presheaf.isLocallySurjective_iff_whisker_forget (J := extensiveTopology C)]
exact fun h _ ↦
surjective_of_isLocallySurjective_sheaf_of_types (Functor.whiskerRight f (forget D)) h
· exact fun a ↦
Presheaf.isLocallySurjective_of_surjective _ _ (fun _ ↦ a _) | true | {
"chosen_tokens": 49,
"rejected_tokens": 2,
"token_jaccard": 0.034483,
"token_length_ratio": 0.040816
} | invalid_tactic | invalid_tactic | c82929fdeb2f3bcfb24c97fd8bb674e0b7d958c7475d18f7738af5e1e78ae8f2 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
public import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
public import Mathlib.CategoryTheory.Sites.... | by
__oprover_invalid_tactic_fffbd570a9bc | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "35d8298951273c49ae6c129d7bea7a08db796bef45304d744145b2f187cd8783",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Sites",
"family_id": "extensivetopology",
"file_id": "mathlib/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean",
"sample_id": "fffbd570a9bc968d3091c7ef16d1e7aecf0edf42a3c5fce61027b019b7b6b538"
} | train |
{
"chosen_proof_sha256": "0e10a34c52b1aa6caddc8087552eba2fdbebe27ef42d67264d88c7c0b6ced53f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b4314e3aa68947f9c2ef21b50c145bfc86... | by
induction H with
| basic h =>
cases h
· exact Or.inr ⟨Set.univ, cond.univ_mem, by simpa⟩
· exact Or.inl (by assumption)
| univ => exact Or.inl cond.univ_mem
| @inter T1 T2 _ _ h1 h2 =>
rcases h1 with (h | ⟨Q, hQ, rfl⟩) <;> rcases h2 with (i | ⟨R, hR, rfl⟩)
· exact Or.inl (cond.inter_mem h... | false | {
"chosen_tokens": 183,
"rejected_tokens": 188,
"token_jaccard": 0.920635,
"token_length_ratio": 1.027322
} | markdown_preamble_trailing | preamble | c879fef9c8a937258494a90e703ff9764d8ec8964ab99417bba3d74fe9175804 | 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.Insert
public import Mathlib.Data.Set.Lattice
Namespace:
FiniteInter
Local context:
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Releas... | Here is the proof:
by
induction H with
| basic h =>
cases h
· exact Or.inr ⟨Set.univ, cond.univ_mem, by simpa⟩
· exact Or.inl (by assumption)
| univ => exact Or.inl cond.univ_mem
| @inter T1 T2 _ _ h1 h2 =>
rcases h1 with (h | ⟨Q, hQ, rfl⟩) <;> rcases h2 with (i | ⟨R, hR, rfl⟩)
· exact Or.in... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Set",
"family_id": "finiteinterclosure_insert",
"file_id": "mathlib/Mathlib/Data/Set/Constructions.lean",
"sample_id": "b69b8cf11d81b81760b7e2d01790b8cb08e792859b8026925d68c9b27e00c6b1"
} | train |
{
"chosen_proof_sha256": "c975c088fa8efb7c0cc15ea499939f263cd071ef49bfcc3a4f706c833af7cab0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "77ab52f257e20e46da01266890c31363fd... | by
tfae_have 1 → 2 := fun _ ↦ inferInstance
tfae_have 2 → 3 := fun _ ↦ IsAtomic.exists_atom _
tfae_have 3 → 1 := fun ⟨I, hI⟩ ↦ by
have ⟨_, h⟩ := isSimpleRing_iff_isTwoSided_imp.mp ‹IsSimpleRing R›
simp_rw [← isFullyInvariant_iff_isTwoSided] at h
have := isSimpleModule_iff_isAtom.mpr hI
obtain eq |... | true | {
"chosen_tokens": 127,
"rejected_tokens": 3,
"token_jaccard": 0.034483,
"token_length_ratio": 0.023622
} | unknown_identifier | unknown_identifier | c88115c5b63f5725066c994334b90b1701b1f854fee64cc0c2a074f154903d5f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
public import Mathlib.RingTheory.FiniteLength
public import Mathlib.RingTheory.SimpleModule.Isotypic
public import Mathli... | by
exact __oprover_missing_e7de025d50f8 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "3ca9173408122d90bbd2d1cc256b0eecafe21120d00538b390272d6224118338",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/SimpleModule",
"family_id": "issimplering",
"file_id": "mathlib/Mathlib/RingTheory/SimpleModule/WedderburnArtin.lean",
"sample_id": "e7de025d50f83819819386b2f880ed49a8eebee597e932160cded70716ac05cb"
} | train |
{
"chosen_proof_sha256": "475307141c3a1cbf5e416ce7878a4f3fd1a2e5cb9833e295c816f2bfea6746a8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "607d86a3d00d2a167cfec5e1d61d8780b4... | by
simp [Fintype.subtype_card, Finset.card_univ] | true | {
"chosen_tokens": 11,
"rejected_tokens": 5,
"token_jaccard": 0.153846,
"token_length_ratio": 0.454545
} | wrong_namespace_qualification | wrong_namespace_qualification | c8c487a8893b0f0f627c1f9bcf93a2d0484260b7b2481fcc661667fd70d28c92 | 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.Powerset
public import Mathlib.Data.Fintype.EquivFin
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
... | by
exact OProverMissing.missing_8718a7560479 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d550a9dd5c7d512b8a62f6a897276d41ffa03f3b78bea00e7968cf43d1720249",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "340281fbf0a45d34c3a34d2f4a61a24c355dc2f03e29d91afa10e3fb584bf2fd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | 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'] | false | {
"chosen_tokens": 61,
"rejected_tokens": 2,
"token_jaccard": 0.033333,
"token_length_ratio": 0.032787
} | sorry_admit_sorryAx_proof_hole | sorry | c8de96fd1d3b2bea39328b9aef7fa962b0ff7d4e775ef9ca767eda0f64f1d7e2 | 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.Finsupp.Order
public import Mathlib.Data.DFinsupp.Lex
public import Mathlib.Data.Finsupp.ToDFinsupp
Namespace:
Finsupp
Local context:
/-
Copyright (c) 2... | 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": "Data/Finsupp",
"family_id": "lex",
"file_id": "mathlib/Mathlib/Data/Finsupp/Lex.lean",
"sample_id": "3b594b8efd7138c1edf9cf4beb8848fba944bf82946b75971c431aea36b7d8df"
} | train |
{
"chosen_proof_sha256": "36b9bfe8ecf433634e45203e8b895f570c57f0adf9b70b7ca8ace15a79e21d3e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2a9d151a1126ceeaa73613f006206f0692... | by
ext
simp [single, ← Finsupp.smul_single] | true | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.166667,
"token_length_ratio": 0.272727
} | circular_dependency | circular_dependency | c9850ca9de24f6253853a2069672e47cf4845bc7220e42116361236e439fcd50 | 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.Action.TransferInstance
public import Mathlib.Algebra.Module.Defs
public import Mathlib.Data.Finsupp.Basic
public import Mathlib.Data.Fin... | by
exact smul_single | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "cc05767c2f159dbd97091dc9144cf3af5d7c6e0540f926bdf1aa9cee1bf9a0e1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Algebra/MonoidAlgebra",
"family_id": "smul_single",
"file_id": "mathlib/Mathlib/Algebra/MonoidAlgebra/Defs.lean",
"sample_id": "bcfecfd0c0cb2071cab40553cc6bdd48d75c67302392e5f99abe6ae29aa62d18"
} | train |
{
"chosen_proof_sha256": "63ac599b07e7436f21a63f1b73d4a7f54c73a40c1b4a50f48b5455d9ae2693f2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a4a38b92b5df1cd680b4f1767b59c251ab... | by
ext j
dsimp
rw [mapBifunctorLeftUnitor_inv_apply, mapBifunctorLeftUnitor_inv_apply, assoc, assoc,
ι_mapBifunctorMapMap]
dsimp
rw [Functor.map_id, NatTrans.id_app, id_comp, ← NatTrans.naturality_assoc,
← NatTrans.naturality_assoc]
rfl | true | {
"chosen_tokens": 41,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.121951
} | wrong_namespace_qualification | wrong_namespace_qualification | c9bb09ec8112b748dc3c91abcccff58534697f8e7fe93d1c5e04e506aea549b5 | 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.GradedObject.Associator
public import Mathlib.CategoryTheory.GradedObject.Single
Namespace:
CategoryTheory.GradedObject
Local context:
/-
Copy... | by
exact OProverMissing.missing_9614d072a910 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "182eea63fb00055802188d40d115deecf013ebd54a204df7d6d56a61235cdbdf",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/GradedObject",
"family_id": "mapbifunctorleftunitor_inv_naturality",
"file_id": "mathlib/Mathlib/CategoryTheory/GradedObject/Unitor.lean",
"sample_id": "9614d072a9105eea2ab6f57e5011e3855902130db93743053ebaa70a564b484b"
} | train |
{
"chosen_proof_sha256": "0e11e10bbaaaed2dc873b84e165eba0bc5466c4e31acb825aeb49b92a5fe94d8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0da4ea72451dc03589d0ca83202fa7e907... | by
-- Let `Z` be a minimal set of prime ideals such that their product is contained in `J`.
obtain ⟨Z₀, hZ₀⟩ := exists_primeSpectrum_prod_le_and_ne_bot_of_domain hNF hI0
obtain ⟨Z, ⟨hZI, hprodZ⟩, h_eraseZ⟩ :=
wellFounded_lt.has_min
{Z | (Z.map asIdeal).prod ≤ I ∧ (Z.map asIdeal).prod ≠ ⊥}
⟨Z₀, hZ₀... | true | {
"chosen_tokens": 346,
"rejected_tokens": 2,
"token_jaccard": 0.008197,
"token_length_ratio": 0.00578
} | invalid_tactic | invalid_tactic | c9f23e7ff9469079c9e017a90c3dff1e9d3c1b111f58a2032fe9bb800321eb1f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
public import Mathlib.RingTheory.DedekindDomain.Basic
public import Mathlib.RingTheory.FractionalIdeal.Inverse
public impo... | by
__oprover_invalid_tactic_0996acbca200 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "04744208de7d7bdddb9dac48ae8cf6a515aca5ab180f1139e812531dac867fc4",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/DedekindDomain",
"family_id": "primespectrum",
"file_id": "mathlib/Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean",
"sample_id": "0996acbca200c0dccec01a46d4e81205ed46c98fd950cb9f91af86182dcfbe7d"
} | train |
{
"chosen_proof_sha256": "1bc941a2b3a47d131821876175410f2047d53e35df24a12c023e660701f0a1b1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [← isLowerSet_compl, ← coe_compl] at h𝒜
have := h𝒜.card_inter_le_finset hℬ
rwa [card_compl, Fintype.card_finset, tsub_mul, le_tsub_iff_le_tsub, ← mul_tsub, ←
card_sdiff_of_subset inter_subset_right, sdiff_inter_self_right, sdiff_compl,
_root_.inf_comm] at this
· grw [inter_subset_right]
· grw ... | false | {
"chosen_tokens": 64,
"rejected_tokens": 3,
"token_jaccard": 0.028571,
"token_length_ratio": 0.046875
} | by_exact_placeholder | placeholder | ca2f03f376599b0ab7731ddc7d924f56fca241f65fcc7aaf9ec8ee85d6d874c6 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Ring.Canonical
public import Mathlib.Algebra.Order.Ring.Nat
public import Mathlib.Combinatorics.SetFamily.Compression.Down
public import Mathlib.... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/SetFamily",
"family_id": "isupperset",
"file_id": "mathlib/Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean",
"sample_id": "f74f3c30220b04da1fe4f1b1519530cecb6ea35343f0fc2bc9e9549863b219a6"
} | train |
{
"chosen_proof_sha256": "7f54226990013dd5c5400459b377a357ecf590ee9b93a656a04179c0ea205d15",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [isUnit_iff_coprime]
exact (Prime.dvd_iff_not_coprime <| prime_of_mem_primeFactors h).mp <| dvd_of_mem_primeFactors h | true | {
"chosen_tokens": 21,
"rejected_tokens": 8,
"token_jaccard": 0.043478,
"token_length_ratio": 0.380952
} | unsolved_goals | unsolved_goals | ca60b76884b829a0c1f9e3277150bb48b5858066fdc3a96eca24ec9598f1ff64 | 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.Associated
public import Mathlib.Data.ZMod.Basic
public import Mathlib.RingTheory.Coprime.Lemmas
Namespace:
ZMod
Local context:
/-
Copyr... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "d795702a186429c0c2a7edc8a9baa750dc174c3b70c4d0d1eecd3d5c261fa5c5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/ZMod",
"family_id": "not_isunit_of_mem_primefactors",
"file_id": "mathlib/Mathlib/Data/ZMod/Units.lean",
"sample_id": "4e7c68a0ae1706e7fbfa84a0b8e103da95b624a1edb0fcbc393efbb9763f79d8"
} | train |
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