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": "603b10c8e8d8264cdfe3ea1be9c171071aba17b8e3411dbc942a8512b4e99c56",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
haveI := Fintype.ofFinite ι
haveI := fun i => Fintype.ofFinite (κ i)
classical
refine (Basis.piTensorProduct b).dualBasis.ext (fun p ↦ ?_)
refine (Basis.piTensorProduct b).ext (fun q ↦ ?_)
simp [Finsupp.single_apply, Fintype.prod_ite_zero, ← funext_iff] | true | {
"chosen_tokens": 67,
"rejected_tokens": 8,
"token_jaccard": 0.051282,
"token_length_ratio": 0.119403
} | unsolved_goals | unsolved_goals | 926dcba658fcf33b5e50cacd2845a67e2362fd2216d9bc0917af1c5bd2dca98d | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Dual.Basis
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
public import Mathlib.LinearAlgebra.PiTensorProduct.Basis
Namespace:
PiTe... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "cb05b30a78cf980bf36ff77c6e80f66db4663fa5980f198787430dfdc4567007",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/PiTensorProduct",
"family_id": "dualdistrib_dualdistribinvofbasis_left_inverse",
"file_id": "mathlib/Mathlib/LinearAlgebra/PiTensorProduct/Dual.lean",
"sample_id": "57919ee9fe831200d4f5697405ab93efe8a6295ef8776bc2c7eff0fee53e76ad"
} | train |
{
"chosen_proof_sha256": "91037b5a5d4107cfc6e47498527b6da8e785a774a3e33005feebe3831fc4a2a4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [← tendsto_norm_atTop_iff_cobounded]
simpa [norm_intCast_eq_abs_mul_norm_one] using tendsto_intCast_atTop_atTop
|>.comp (tendsto_abs_atBot_atTop.sup tendsto_abs_atTop_atTop)
|>.atTop_mul_const (norm_pos_iff.mpr one_ne_zero) | false | {
"chosen_tokens": 32,
"rejected_tokens": 2,
"token_jaccard": 0.041667,
"token_length_ratio": 0.0625
} | sorry_admit_sorryAx_proof_hole | sorry | 92a944633a45622cf5f4fd7400117cfbba626757da487dd20f23729f6cc67f22 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Module
public import Mathlib.Algebra.Order.Field.Power
public import Mathlib.Algebra.Polynomial.Monic
public import Mathlib.Analysis.Asymp... | 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": "Analysis/SpecificLimits",
"family_id": "tendsto_intcast_atbot_sup_attop_cobounded",
"file_id": "mathlib/Mathlib/Analysis/SpecificLimits/Normed.lean",
"sample_id": "2031f82d6bd5b2086402ba4895d772d5452bfbf03175b7504ea5ff92425b4486"
} | train |
{
"chosen_proof_sha256": "d39e8a584cb9cadcbcf3f438cba249cdb73b9b66b3d2c3c40bd6185edf5ea6c9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
have h : f = 𝟙 j := by cat_disch
rw [h]
simp | false | {
"chosen_tokens": 16,
"rejected_tokens": 3,
"token_jaccard": 0.0625,
"token_length_ratio": 0.1875
} | by_exact_placeholder | placeholder | 92dc07b66192b973ad262a1c76c1392e51822ffbe94f4efe4c817f85ad7f35b5 | 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.Pi.Basic
Namespace:
CategoryTheory.Discrete
Local context:
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 ... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Discrete",
"family_id": "functor_map_id",
"file_id": "mathlib/Mathlib/CategoryTheory/Discrete/Basic.lean",
"sample_id": "75d34468da2034a80fca64c4e5869a12c798eeacfe03a8618a187bb63ae913dc"
} | train |
{
"chosen_proof_sha256": "3939293b987d00efdc8989f961d8fe481cabdbec2fac5708f8344374642f9236",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [TopologicalSpace.nhds_generateFrom]
simp only [comap_iInf, comap_principal]
intro s hs
rw [← le_principal_iff]
refine iInf_le_of_le { u | s ∈ u } ?_
refine iInf_le_of_le ⟨hs, ⟨s, rfl⟩⟩ ?_
exact principal_mono.2 fun _ ↦ id | false | {
"chosen_tokens": 54,
"rejected_tokens": 2,
"token_jaccard": 0.027778,
"token_length_ratio": 0.037037
} | sorry_admit_sorryAx_proof_hole | sorry | 933a2df7f7791c3680d346b31c5f9372773340ca48a4ddd8f3ec567f2a0d3ce8 | 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.Bases
public import Mathlib.Topology.DenseEmbedding
public import Mathlib.Topology.Connected.TotallyDisconnected
Namespace:
(root)
Local context:
/-... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/Compactification",
"family_id": "ultrafilter_comap_pure_nhds",
"file_id": "mathlib/Mathlib/Topology/Compactification/StoneCech.lean",
"sample_id": "dd448d7bddd39539db63b6005ceef026bb709e5272ecd6e5386e656fa08db32f"
} | train |
{
"chosen_proof_sha256": "ab4cb4aa5cd70f70198092337097cb05b18354e67218420118c17e78a73513e0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "88d7c6c796cca4e10e9b92b371cecd5010... | by
refine ⟨fun h ↦ ?_, fun ⟨h₁, h₂⟩ ↦ ?_⟩
· refine evalInduction h fun a h IH ↦ ?_
rcases e : f a with - | a'
· rw [Part.mem_unique h
(PFun.mem_fix_iff.2 <| Or.inl <| Part.mem_some_iff.2 <| by rw [e]; rfl)]
exact ⟨ReflTransGen.refl, e⟩
· rcases PFun.mem_fix_iff.1 h with (h | ⟨_, h, _⟩) <... | true | {
"chosen_tokens": 215,
"rejected_tokens": 3,
"token_jaccard": 0.035714,
"token_length_ratio": 0.013953
} | unknown_identifier | unknown_identifier | 935af83f304e69d2db817e7d37adbf8b99b6ef96754aba393851550da6f32d0f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Data.PFun
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Fintype.Defs
Nam... | by
exact __oprover_missing_9fe3719b8a58 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "7680089adf39523b071822545c9422f78acfbc1c1267b530ac9d11d5bb1ed0bb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Computability",
"family_id": "mem_eval",
"file_id": "mathlib/Mathlib/Computability/StateTransition.lean",
"sample_id": "9fe3719b8a587cc320eadf69f2619355af22157136ae9400bafc484518741ba5"
} | train |
{
"chosen_proof_sha256": "a923c894044509b83369abcd9431b69d970849ba97f75437eb89f230df9823e5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [IsHausdorff.eq_iff_smodEq (I := 𝔭)]
simp only [smul_eq_mul, mul_top]
intro n
cases n
· simp
· simp [SModEq, mk_pow_fontaineTheta] | false | {
"chosen_tokens": 32,
"rejected_tokens": 2,
"token_jaccard": 0.041667,
"token_length_ratio": 0.0625
} | by_exact_placeholder | by? | 93941d41ab48a4f62798eb27e56dce2fd3fdad510c71bc294c6f872a87fe2998 | 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.AdicCompletion.Functoriality
public import Mathlib.RingTheory.AdicCompletion.RingHom
public import Mathlib.RingTheory.Perfectoid.Untilt
public impor... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Perfectoid",
"family_id": "fontainetheta_teichmuller",
"file_id": "mathlib/Mathlib/RingTheory/Perfectoid/FontaineTheta.lean",
"sample_id": "0884596d77461ad97de009c6f8b8d539833c319659cef7be0f8610e17da4299a"
} | train |
{
"chosen_proof_sha256": "8c34499c0ebf931c33761750b78ccc8342159b860f73a3673d688956dbdeb67f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8c140a2ef00922a349e0f673ab6bb93e99... | by
constructor <;> rintro ⟨u, h₂⟩
· refine ⟨f <$$> u, ?_⟩
apply (h₁ u).mp h₂
· refine ⟨g <$$> u, ?_⟩
rw [h₁]
simp only [MvFunctor.map_map, h₀, LawfulMvFunctor.id_map, h₂] | true | {
"chosen_tokens": 70,
"rejected_tokens": 3,
"token_jaccard": 0.027027,
"token_length_ratio": 0.042857
} | circular_dependency | circular_dependency | 93e4531309605192a328a98a9a127a64aa25eb2271a781e647aae687eb70835b | 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 exists_iff_exists_of_mono | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "cb185bb97261df45a50d1ec28e331d7692cc2a91b9d22399172cd8a6b49d2cc3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Control/Functor",
"family_id": "exists_iff_exists_of_mono",
"file_id": "mathlib/Mathlib/Control/Functor/Multivariate.lean",
"sample_id": "267c884cb305e890be2d32bd1dd81644507811de08af6ce289cc15754404e6fb"
} | train |
{
"chosen_proof_sha256": "36e2743b3c488bb6e8cf2bf83c88a0f4aa1e0278b4cda351f399f8fa4b568278",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
refine OpenPartialHomeomorph.ext _ _ (fun x => rfl) (fun x => rfl) ?_
simp [e.open_source.interior_eq, ← inter_assoc] | false | {
"chosen_tokens": 32,
"rejected_tokens": 3,
"token_jaccard": 0.041667,
"token_length_ratio": 0.09375
} | by_exact_placeholder | placeholder | 93f777d45c6e104ae37303811b60c8847d91e243e0c791cb8b986b21c96b9170 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.OpenPartialHomeomorph.Continuity
Namespace:
OpenPartialHomeomorph
Local context:
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Relea... | 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": "Topology/OpenPartialHomeomorph",
"family_id": "restr_source_inter",
"file_id": "mathlib/Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean",
"sample_id": "a3683d3fecf5a78a2e088915121e34b1d7028592a27520975260840e3084875a"
} | train |
{
"chosen_proof_sha256": "230ad1a22636c034f298a25415dfe821b738da09682e6993796fa2ce84eb0a0a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "977e118700d760b10bf27f87641e8ca5e4... | by
rw [← nsmul_one, D.map_smul_of_tower n, map_one_eq_zero, smul_zero] | true | {
"chosen_tokens": 15,
"rejected_tokens": 5,
"token_jaccard": 0.125,
"token_length_ratio": 0.333333
} | wrong_namespace_qualification | wrong_namespace_qualification | 942452b9c7e5dc65573672e93beca690224414f6371b92fa26fcf0a06c49e9c9 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Derivative
Namespace:
Derivation
Local context:
/-
Copyright (c) 2020 Nicolò Cavalleri... | by
exact OProverMissing.missing_16bdb0569806 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1406c1d6d59b2bd47e9ab3099314aab3ff0677f666c8ff57863ae067cb244c1a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Derivation",
"family_id": "map_natcast",
"file_id": "mathlib/Mathlib/RingTheory/Derivation/Basic.lean",
"sample_id": "16bdb0569806200171e13b3e0d9b0d935a38c6f276941a0946af8ff38e8cee10"
} | train |
{
"chosen_proof_sha256": "96180be036a5aa5c1b27a5d6a0c89b53acf0cf1ee820da9c2ba106ab55ce193e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "39bb38d8237a1e91907a351e01371e5a01... | by
unfold traverse
simp only [Functor.map_map, fmap_def, map_comp_coe_apply, Multiset.fmap_def, ←
Multiset.map_traverse] | true | {
"chosen_tokens": 23,
"rejected_tokens": 3,
"token_jaccard": 0.117647,
"token_length_ratio": 0.130435
} | circular_dependency | circular_dependency | 942d06e75eed46f68f0e61a9e78f8d790bd50ef4f174e4a05e2f26dd3cdc5c89 | 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.Control.AlternativeMonad
public import Mathlib.Data.Finset.Lattice.Union
public import Mathlib.Data.Finset.NAry
public import Mathlib.Data.Multiset.Functor
... | by
exact map_traverse | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "c44e8da96e746b4df7a03133b15fb3addc95b1b7f2d5d5908c42221cd369ccfe",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Finset",
"family_id": "map_traverse",
"file_id": "mathlib/Mathlib/Data/Finset/Functor.lean",
"sample_id": "1daa9f465ccaa4038e45b7b45c48af664510fd77f6ef72028805211ad957a3be"
} | train |
{
"chosen_proof_sha256": "69cb17d2f43d6924d494a8be87b9a52df302f4a4d1c3a416993a289830e67b47",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c2d4f1022a58d69b30e8702a765bce86c3... | by
by_cases hR : R = 0
· simp_all
have H : ContinuousOn (circleAverage f c) (Set.Ioc 0 |R|) := by
refine (h₁f.2.mono ?_).circleAverage (fun z hz ↦ hz.1.le)
intro x hx
rw [closure_ball _ (by aesop), mem_closedBall_iff_norm]
exact hx.2
rw [← circleAverage_abs_radius]
apply H.eq_of_eqOn_Ioo (by a... | true | {
"chosen_tokens": 114,
"rejected_tokens": 5,
"token_jaccard": 0.051724,
"token_length_ratio": 0.04386
} | wrong_namespace_qualification | wrong_namespace_qualification | 944cdb2952f47c03bb39cb7e8557f652cd05e2c3393ec3cac075c2a707bc59f8 | 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.Complex.Harmonic.Analytic
public import Mathlib.Analysis.Complex.MeanValue
public import Mathlib.Analysis.InnerProductSpace.Harmonic.HarmonicContOnCl
... | by
exact OProverMissing.missing_2c9653db60a3 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "3d104c18db9b972e50396b9a392cda3ff9b2c9dded66cd0da7d76145b7678814",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Complex",
"family_id": "harmoniccontoncl",
"file_id": "mathlib/Mathlib/Analysis/Complex/Harmonic/MeanValue.lean",
"sample_id": "2c9653db60a3ddae63cedc1cc9e2ba6e8d6ce8800e7284483f64df67132c5de6"
} | train |
{
"chosen_proof_sha256": "536871cc5d7d8fd19b1680026dd5fcc3bae6cf4953adabbd17e9ab3ecd407260",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
refine ⟨fun h ↦ ⟨?_, ?_, ?_⟩, fun h ↦ ?_⟩
any_goals rintro a b rfl rfl; exact map_eq_empty.1 h
· rintro a b rfl rfl; exact disjSum_eq_empty.1 h
cases a <;> cases b
· exact map_eq_empty.2 (h.1 _ _ rfl rfl)
· simp [h.2.1 _ _ rfl rfl]
· rfl
· exact map_eq_empty.2 (h.2.2 _ _ rfl rfl) | true | {
"chosen_tokens": 99,
"rejected_tokens": 8,
"token_jaccard": 0.027027,
"token_length_ratio": 0.080808
} | unsolved_goals | unsolved_goals | 946357e3723c2286b6e2e31701f41d9cf43873c3591fae492ea919389abe64bc | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finset.Sum
public import Mathlib.Data.Sum.Order
public import Mathlib.Order.Interval.Finset.Defs
Namespace:
Finset
Local context:
/-
Copyright (c) 2022 ... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "53d569c0052ce94b94afc3253716bce29a821b82b471b97a92dcf3254037f6c2",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Sum",
"family_id": "sumlexlift_eq_empty",
"file_id": "mathlib/Mathlib/Data/Sum/Interval.lean",
"sample_id": "7fa1587a449af9bd025ff7fcf8e9e6160d9d026e5e59b7be7c90ab324dc63b3b"
} | train |
{
"chosen_proof_sha256": "c7cf3138eb7d775d3fb111ddb9cf2f6f27656c166d40a0c93cd557614d2fd3af",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c2d1819f1557f27ef928e9cc08a0a1fce5... | by
cases xs
simp only [reverse, snoc, cons, toList_mk]
congr
simp [toList, append_def] | false | {
"chosen_tokens": 21,
"rejected_tokens": 28,
"token_jaccard": 0.882353,
"token_length_ratio": 1.333333
} | markdown_preamble_trailing | fence | 94c14bb6dbd5e734777f02dad5535744faa179a9efc6544bc526098d0b9ba230 | 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.Vector.Basic
Namespace:
List.Vector
Local context:
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as describe... | ```lean
by
cases xs
simp only [reverse, snoc, cons, toList_mk]
congr
simp [toList, append_def]
``` | {
"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/Vector",
"family_id": "reverse_snoc",
"file_id": "mathlib/Mathlib/Data/Vector/Snoc.lean",
"sample_id": "2b323eed78bdd6ac7a8d6ff6b4eecb4cf02033f76acf0f63ce6e3ba7ef4dfad0"
} | train |
{
"chosen_proof_sha256": "3bdeb5d69c9d68024e63a03b42978b4ed5db36efc7c46e5dd1c026762b0bff70",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "97fbd19fb2bbaefce301241f4a558e6928... | by
ext
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· obtain @⟨n, i, h0, hn⟩ := h
simp only [iSup_iff, ofHoms_iff, Subtype.exists, exists_prop]
use n, i
· simp only [iSup_iff, ofHoms_iff] at h
obtain ⟨n, ⟨i, h0, hn⟩, _, _⟩ := h
exact horn_ι_mem_innerHornInclusions h0 hn | true | {
"chosen_tokens": 81,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.037037
} | unknown_identifier | unknown_identifier | 94c6e1e545462c81213d784e2df1d795227dc44992a5722b28bc9ccfcd084744 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicTopology.Quasicategory.Basic
Namespace:
SSet
Local context:
/-
Copyright (c) 2026 Jack McKoen. All rights reserved.
Released under Apache 2.0 licens... | by
exact __oprover_missing_0499f9ddadea | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "eb85c3ff5246f5f01dca8603d7f065d6a0e0f60b154ec42e4fdd05a4649cdb27",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/Quasicategory",
"family_id": "innerhorninclusions_eq_isup",
"file_id": "mathlib/Mathlib/AlgebraicTopology/Quasicategory/InnerFibration.lean",
"sample_id": "0499f9ddadeadf4353cd9529cac7c5ef8cb81587f40f09d0d81cd202b497e41b"
} | train |
{
"chosen_proof_sha256": "feace96534351d67a98372e9c910b5669a41f603a705ba51814a94d00a6c6ea8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
intro _ ha _ hb hne
have := hrc ha hb (fun h ↦ hne (hf h))
grind | false | {
"chosen_tokens": 23,
"rejected_tokens": 2,
"token_jaccard": 0.058824,
"token_length_ratio": 0.086957
} | sorry_admit_sorryAx_proof_hole | sorry | 95bc59b4c6500a353a17c85a8ea26f8f60ee1d9dfa1ab3fe427f5eeba5722740 | 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.Notation
public import Mathlib.Data.Set.Pairwise.Basic
public import Mathlib.Data.SetLike.Basic
public import Mathlib.Order.Directed
public import Mat... | 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": "Order/Preorder",
"family_id": "ischain",
"file_id": "mathlib/Mathlib/Order/Preorder/Chain.lean",
"sample_id": "5bb17713869ded8211fecb07cd342899842c304348ff9eddfbbe96f8cb169aa9"
} | train |
{
"chosen_proof_sha256": "4bc1dfb496047d2c2f53f0ca55577975a296dcb639bf7ace11b6e910a7a97f44",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [← SetLike.mem_coe, coe_commutant]
rfl | false | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.272727
} | by_exact_placeholder | exact? | 95c0e2a1d3caa42aa4f85980b24053ba69d137c4abd8c07579f401368524423c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.CStarAlgebra.Classes
public import Mathlib.Analysis.InnerProductSpace.Adjoint
Namespace:
VonNeumannAlgebra
Local context:
/-
Copyright (c) 2022 Kim ... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/VonNeumannAlgebra",
"family_id": "mem_commutant_iff",
"file_id": "mathlib/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean",
"sample_id": "452212369db5a32584e53a4244413eb62217af8a259514599237694d17941ea9"
} | train |
{
"chosen_proof_sha256": "0220babf450eb445fb686e51233d9dc7bd43814cdb2cf6da4315eb591a53bef9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b57d7d0c5ab9e4702f316c1b1e6199c14d... | by
rw [← natCast_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (Int.natCast_ne_zero.mpr hn),
Int.cast_natCast]
rfl | true | {
"chosen_tokens": 22,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.136364
} | circular_dependency | circular_dependency | 95ccc628dd08a7e7fde8db89401fa53c9cd03294a350cd6a2688966422b3a367 | 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.Field.Basic
public import Mathlib.Algebra.Order.Group.Unbundled.Int
public import Mathlib.Algebra.Module.NatInt
public import Mathlib.GroupTheory.Quoti... | by
exact nsmul_mem_zmultiples_iff_exists_sub_div | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "6ef9a5a4ed12c70089f813f6df4ce39ed255fbb003ad6d1e279c9986230c6369",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/CharZero",
"family_id": "nsmul_mem_zmultiples_iff_exists_sub_div",
"file_id": "mathlib/Mathlib/Algebra/CharZero/Quotient.lean",
"sample_id": "a86ece7891869444431a13676c31d6de05fdf4898f0e667ba12ab2a36fc2b894"
} | train |
{
"chosen_proof_sha256": "f55e39083a696313fcfdd6bb61c3e62ddbca985973b59bf86a8381597fddf755",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d380a35b9ecdb938e51a37459a7daa5151... | by
have : Nontrivial (Subobject X) := nontrivial_of_not_isZero h
obtain ⟨Y, s⟩ := (IsAtomic.eq_bot_or_exists_atom_le (⊤ : Subobject X)).resolve_left top_ne_bot
exact ⟨Y, (subobject_simple_iff_isAtom _).mpr s.1⟩
/-- Choose an arbitrary simple subobject of a non-zero Artinian object. -/
noncomputable def simpleSub... | true | {
"chosen_tokens": 145,
"rejected_tokens": 3,
"token_jaccard": 0.04918,
"token_length_ratio": 0.02069
} | circular_dependency | circular_dependency | 95d1bab8f3e93b2e8c13e1dbf50c71a773e9322fb2cfc579c802c5a4d7823e21 | 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.ObjectProperty.ContainsZero
public import Mathlib.CategoryTheory.ObjectProperty.EpiMono
p... | by
exact exists_simple_subobject | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "bdc71e8c5e534085dbdb9990398b70f7d2e10e12b69d4c15900767c955adb26e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Subobject",
"family_id": "exists_simple_subobject",
"file_id": "mathlib/Mathlib/CategoryTheory/Subobject/ArtinianObject.lean",
"sample_id": "9044b1a9ec5361ec717fcb988a173f7c747b987f4a363348924f9ee3f41d531b"
} | train |
{
"chosen_proof_sha256": "212d1f84cff2b596f7c1f93804fd9c282408c326066650fb7feba4fd785cf41f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp_rw [max_def, val_le_val, ← apply_ite]
rfl | true | {
"chosen_tokens": 11,
"rejected_tokens": 8,
"token_jaccard": 0.0625,
"token_length_ratio": 0.727273
} | unsolved_goals | unsolved_goals | 963b2498c5963f3336d9f0e776e6de80aa4f4585fc818971222980e418db7e85 | 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.Hom.Basic
public import Mathlib.Algebra.Group.Units.Defs
Namespace:
Units
Local context:
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Rele... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "ded86b16da47f9d82c38ee6c592a843cf0becb292209ed58b815e439fe400d26",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Algebra/Order",
"family_id": "max_val",
"file_id": "mathlib/Mathlib/Algebra/Order/Monoid/Units.lean",
"sample_id": "a53b7c1c6dc7380167b7eef2678e6391bfc432a0b3abcb53f2f3e41f881be729"
} | train |
{
"chosen_proof_sha256": "252205b7b358cac3e7fbb348dc9a6e22e848772b9ebec8a5f89dd1c88ca1cccb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9293c8a0161f13a7f11277c44e7e1055a3... | by
induction t with
| nil => rw [traverse, map]
| node v l r hl hr =>
rw [traverse, map, hl, hr, Function.comp_apply, map_pure, pure_seq, map_pure, pure_seq,
map_pure] | false | {
"chosen_tokens": 45,
"rejected_tokens": 50,
"token_jaccard": 0.857143,
"token_length_ratio": 1.111111
} | markdown_preamble_trailing | trailing | 967ecca4df948b08929d730358f8d6375d9a3974b635dff02b90e02298f1504f | 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.Tree.Basic
public import Mathlib.Control.Applicative
public import Mathlib.Control.Traversable.Basic
Namespace:
BinaryTree
Local context:
/-
Copyright (... | by
induction t with
| nil => rw [traverse, map]
| node v l r hl hr =>
rw [traverse, map, hl, hr, Function.comp_apply, map_pure, pure_seq, map_pure, pure_seq,
map_pure]
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Tree",
"family_id": "traverse_eq_map_id",
"file_id": "mathlib/Mathlib/Data/Tree/Traversable.lean",
"sample_id": "932a2516a9f3261c5acb0efee2adbd6bd7291142ae5d6a59e51bba4d80664223"
} | train |
{
"chosen_proof_sha256": "8690ef82dc2fef54f22a4e762da7bbf0fe4e6499a977029b3fad0c51781aa987",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "61af39f4fdc1dd36cdc67664e32ec30006... | by
classical
rw [ofDirectSumEquiv]
convert! ofDFinsuppEquiv_symm_single_tprod p x | true | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.181818
} | invalid_tactic | invalid_tactic | 968d0e3de119c581ca76b720d9e63432bacf43339c4bfa462602dc861b4d7f7e | 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.PiTensorProduct.Basic
public import Mathlib.LinearAlgebra.PiTensorProduct.DFinsupp
public import Mathlib.Algebra.DirectSum.Module
Namespace:
PiT... | by
__oprover_invalid_tactic_29db24e141a0 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "60e98fc682da6c40c20f1888a7dac37a54624c1127574650db49dc7a58e7dc6b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/PiTensorProduct",
"family_id": "ofdirectsumequiv_symm_lof_tprod",
"file_id": "mathlib/Mathlib/LinearAlgebra/PiTensorProduct/DirectSum.lean",
"sample_id": "29db24e141a0e3e320d003b92675922c58df588b3780d126668224314302b43a"
} | train |
{
"chosen_proof_sha256": "9a21e4324d10fe830df7d8256fcb74edf6972d935383079b80c071c0dccb073e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "bbdda3acdd91e6499b48bf858ce581113a... | by
simpa using toInt_le_toInt val j i | false | {
"chosen_tokens": 7,
"rejected_tokens": 12,
"token_jaccard": 0.583333,
"token_length_ratio": 1.714286
} | markdown_preamble_trailing | trailing | 96c0a6bb3fccd66d890e9aa1139345b35d4bb3b840cbec98a20bfa55e0e25d81 | 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... | by
simpa using toInt_le_toInt val j i
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": "Tactic/Order",
"family_id": "toint_nlt_toint",
"file_id": "mathlib/Mathlib/Tactic/Order/ToInt.lean",
"sample_id": "c7eeb279213fe096be67e549c7a7f69ed83061d62764c53bfce17ab42221096a"
} | train |
{
"chosen_proof_sha256": "084e758470934f76ebf17be847788f3818be1118ff79efcd27a52969227aebb9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
have h₁ : x⁻¹ ≠ 1 := by rwa [inv_eq_one_div, Ne, div_eq_iff_mul_eq hx0, one_mul]
have h₂ : x⁻¹ - 1 ≠ 0 := mt sub_eq_zero.1 h₁
have h₃ : x - 1 ≠ 0 := mt sub_eq_zero.1 hx1
have h₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x :=
Nat.recOn n (by simp) fun n h => by
rw [pow_succ', mul_inv_rev, ← mul_assoc, h, mul_as... | true | {
"chosen_tokens": 194,
"rejected_tokens": 8,
"token_jaccard": 0.059701,
"token_length_ratio": 0.041237
} | unsolved_goals | unsolved_goals | 970e9b469f8efd896abe579511e6d0d37fec5a2ec1805815c11c52293eaf140a | 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.Field.Basic
public import Mathlib.Algebra.Ring.GeomSum
Namespace:
(root)
Local context:
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
R... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "f96373c1c75b4007a6925536ea2b0e6fdb6d46243df83d9435f508ad9378de14",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Field",
"family_id": "geom_sum_inv",
"file_id": "mathlib/Mathlib/Algebra/Field/GeomSum.lean",
"sample_id": "2357e78341a07f48e92532f87e0bfabcf4fba0a7567069cbcd02ca8502dd6972"
} | train |
{
"chosen_proof_sha256": "24551563a4e7d2141b3625f37d2a2f5a066ec7e2be8f2231f8bc3aa32f433342",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [isUnit_iff_and, and_iff_right ha] | false | {
"chosen_tokens": 8,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.375
} | by_exact_placeholder | placeholder | 97b39bb023980107fa94eec4a97fcacbae69f242147a5828b14a1b8d4e680fe7 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Ideal.Maximal
Namespace:
Submonoid
Local context:
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as descr... | 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/Ideal",
"family_id": "isunit_iff_of_ne_zero",
"file_id": "mathlib/Mathlib/RingTheory/Ideal/Nonunits.lean",
"sample_id": "8d47a007615a250bf24cd20685fe57ad901c62e3e8a8706c9506725d94de99c7"
} | train |
{
"chosen_proof_sha256": "4ba70e5cd3f178f8ef12d7930e49952b41e1f76063cf50cc900cfa723b52ba14",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [Iff.comm]
obtain ⟨s₁, rfl⟩ := s₁.exists_eq_ofList
obtain ⟨s₂, rfl⟩ := s₂.exists_eq_ofList
simp only [lt_iff_toList_lt, String.Legacy.iter, String.Legacy.mkIterator, String.toList_ofList]
induction s₁ generalizing s₂ <;> cases s₂
· unfold ltb; decide
· rename_i c₂ cs₂; apply iff_of_true
· unfold... | true | {
"chosen_tokens": 329,
"rejected_tokens": 8,
"token_jaccard": 0.030303,
"token_length_ratio": 0.024316
} | unsolved_goals | unsolved_goals | 97b860f5cab14977a1271c9c93d88b68bae13b8792ccc8e0b9508ca3957652c8 | 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
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "048a0061ab271ef253655400b9b43ee00fad1618e82571b7a796b34b47ff1cd6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/String",
"family_id": "lt_iff_ltb",
"file_id": "mathlib/Mathlib/Data/String/Basic.lean",
"sample_id": "e0f31a2fa4b167750af0775ad92818537ced61f211e8620b229e5bbd135fd43d"
} | train |
{
"chosen_proof_sha256": "5416e2b8428ca9ff8cc9626edea3896864ca77c250a86dc31c56a1ba22b8daf5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0a9c671be9f78b6b39ea625e04972d8197... | by
unfold Measure.mconv
rw [integral_map (by fun_prop) hf.1, integral_prod]
exact (integrable_map_measure hf.1 (by fun_prop)).mp hf | false | {
"chosen_tokens": 32,
"rejected_tokens": 39,
"token_jaccard": 0.904762,
"token_length_ratio": 1.21875
} | markdown_preamble_trailing | fence | 97d4c78baa86f065ced236a308ed68b6dc4da72b4f37d9840a8b30e6cc788a08 | 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.Convolution
public import Mathlib.MeasureTheory.Integral.Prod
Namespace:
MeasureTheory
Local context:
/-
Copyright (c) 2025 Rémy Degenne.... | ```lean
by
unfold Measure.mconv
rw [integral_map (by fun_prop) hf.1, integral_prod]
exact (integrable_map_measure hf.1 (by fun_prop)).mp hf
``` | {
"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/Group",
"family_id": "integral_mconv",
"file_id": "mathlib/Mathlib/MeasureTheory/Group/IntegralConvolution.lean",
"sample_id": "91fa9e01efb72d4bb46c66a3c3321fad75748aa4aba9170a103de81feb5b067d"
} | train |
{
"chosen_proof_sha256": "c337966156f7fa0b44c661383f2baec35f6094ce1c36181a113cc67c17fd2ac2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "27381825cd323e32e19bcfb43a0af2db00... | by
by_cases c <;> simp [*] | true | {
"chosen_tokens": 10,
"rejected_tokens": 5,
"token_jaccard": 0.071429,
"token_length_ratio": 0.5
} | wrong_namespace_qualification | wrong_namespace_qualification | 9820b6c1733f1ef51279c5a3fea94bec792e240dfc7e1191548ee1d3eec3bb2c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Ordering.Basic
public import Mathlib.Order.Defs.Unbundled
Namespace:
Ordering
Local context:
/-
Copyright (c) 2017 Microsoft Corporation. All rights res... | by
exact OProverMissing.missing_4771507a2eda | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "7978311a381272ea846814facc81a9663890bface912b0c8eb134e18a5bfb5cc",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Ordering",
"family_id": "ite_eq_gt_distrib",
"file_id": "mathlib/Mathlib/Data/Ordering/Lemmas.lean",
"sample_id": "4771507a2edaa490165ad6dbbb8bbee190b6099c2dfd923f1e5c8b47e63ba2b7"
} | train |
{
"chosen_proof_sha256": "d6d98f4113b14782373b73a8bb3f9c78ad64b883beec8e3deb78731e5a49b554",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9864f227c35a74c0fbf4907c25450f6f1a... | by
simp [← toList_inj] | true | {
"chosen_tokens": 6,
"rejected_tokens": 2,
"token_jaccard": 0.142857,
"token_length_ratio": 0.333333
} | invalid_tactic | invalid_tactic | 9877080e2db5e3aec6f116c9b7bde8bbb00c6997655155cbf7a88118471a81ef | 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
__oprover_invalid_tactic_7e250c2a0a36 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "93521dac595be4f3b0345f5f9145f1b7c69e5813d522c0e2b542bab2a0a04911",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/String",
"family_id": "oflist_eq",
"file_id": "mathlib/Mathlib/Data/String/Basic.lean",
"sample_id": "7e250c2a0a366c6dc511bcc2bdae4c79d5eb964d8cab919edc4393813525710c"
} | train |
{
"chosen_proof_sha256": "e1c4c37ea1905b854b1db5a6620e0ed4d40197fe584c2e9cb453bf8d89ed1d01",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
simpa using ideal_biInf I Set.finite_univ | false | {
"chosen_tokens": 8,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.375
} | by_exact_placeholder | exact? | 987729c212f0231cb03a5f250fbf2feef9c28d2b82c41baad07d44f7da76c554 | 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.QuasiCompact
public import Mathlib.AlgebraicGeometry.Properties
public import Mathlib.Tactic.DepRewrite
Namespace:
AlgebraicGeomet... | 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": "AlgebraicGeometry/IdealSheaf",
"family_id": "ideal_iinf",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/Basic.lean",
"sample_id": "322ab09370ffc7ba02e5b0d458715ab5e05e1b425cf1b69dc3d33d89b75fd5ba"
} | train |
{
"chosen_proof_sha256": "5a23d360fd69064fcfa72d36f2b8acb7fa78696675236423d19ef91035657ba8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
have _ := t.adj₂.isLeftAdjoint
refine ⟨fun h ↦ by rw [← map_rightToLeft_app]; exact G.map_epi _, fun h ↦ ?_⟩
rw [epi_rightToLeft_app_iff_epi_map_adj₁_unit_app]
simpa using epi_comp (t.adj₂.unit.app (H.obj X)) (H.map (t.adj₂.counit.app X ≫ t.adj₁.unit.app X)) | false | {
"chosen_tokens": 85,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.035294
} | by_exact_placeholder | exact? | 9880c5cb4932518794dcfdd1f10f8bb74767dbc571e68a39f0c79916c117e868 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Adjunction.Opposites
public import Mathlib.CategoryTheory.Adjunction.Unique
public import Mathlib.CategoryTheory.Monad.Adjunction
Namespace:
Ca... | 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/Adjunction",
"family_id": "epi_righttoleft_app_iff",
"file_id": "mathlib/Mathlib/CategoryTheory/Adjunction/Triple.lean",
"sample_id": "45bdf565a874c8590b8b9e375ecaaa19d5c42795fb168fac5a8abcf519632618"
} | train |
{
"chosen_proof_sha256": "f258ff7f229ca2a8c793329e07d0fcda1df2d4bda138c89c5b29fe00a0404c2f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "76dc56f0319fe7f1f9c6eb862aa810c366... | by
refine summable_geometric_iff_norm_lt_one.mp ?_
simp_rw [← map_pow]
exact hsum.comp_injective <| Nat.pow_right_injective hp | true | {
"chosen_tokens": 22,
"rejected_tokens": 3,
"token_jaccard": 0.095238,
"token_length_ratio": 0.136364
} | unknown_identifier | unknown_identifier | 98a1a82616377f6b815d79483c3738fb673bd9d4081509e8ad626a1dd8987e4e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.Normed.Ring.InfiniteSum
public import Mathlib.Analysis.SpecificLimits.Normed
public import Mathlib.NumberTheory.ArithmeticFunction.Defs
public import ... | by
exact __oprover_missing_1caba13c5bfb | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1fc684596d6aa267652ee86ed26f91783864f05a42f9eefb5edf36459b4cf348",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/EulerProduct",
"family_id": "summable",
"file_id": "mathlib/Mathlib/NumberTheory/EulerProduct/Basic.lean",
"sample_id": "1caba13c5bfb2e48e2aa824af4b9e227bed2165274d63dd634f59f34ffe47e2e"
} | train |
{
"chosen_proof_sha256": "62c33a874b3140b3e0d71815632c5a3a67b39c9eeeb9eb02b51c7aa26e877fd4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4291a8efb1c861482a7cda69054b40fffe... | by
constructor <;> grind [IsZero] | true | {
"chosen_tokens": 9,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.555556
} | wrong_namespace_qualification | wrong_namespace_qualification | 998ecf59e80ec1dbfc22b47e8643ad91a2cf909c15aedfa44adaef524b0f8d5b | 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.Tactic.ComputeAsymptotics.Multiseries.Defs
Namespace:
Tactic.ComputeAsymptotics.MultiseriesExpansion.IsZero
Local context:
/-
Copyright (c) 2026 Vasilii Nest... | by
exact OProverMissing.missing_b7d8bbdbc675 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "05ad2133f9a9623f30c52faa0929a33e147e84c8e5ec6777c2d4a145bb09ce7d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/ComputeAsymptotics",
"family_id": "const_iff",
"file_id": "mathlib/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Trimming.lean",
"sample_id": "b7d8bbdbc675ed1b504fd154e9b2c440cdd8d44f35c4f66773b08147245061fc"
} | train |
{
"chosen_proof_sha256": "d02b7bd3abbc9b6e64a37811efb4eb652c5a1b34ba47dca636dd232d861f2cb3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3a981c0403dd9005b3944835a67adab490... | by
ext
simpa [Units.ext_iff] using pow_card_sub_one_eq_one (Units.ne_zero _) | true | {
"chosen_tokens": 16,
"rejected_tokens": 5,
"token_jaccard": 0.117647,
"token_length_ratio": 0.3125
} | wrong_namespace_qualification | wrong_namespace_qualification | 99909038c68bf16bd11cd2b9484f8b25af96fc969863a9b506d026c5356590c8 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity
public import Mathlib.FieldTheory.Finite.Basic
Namespace:
ZMod
Local context:
/-
Copyright (c) 2025 Yakov Pechersk... | by
exact OProverMissing.missing_f90ebae3e238 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1304ea7bce7ecaab3e83cf83d81bb69172bcf1cd1f54b6dc0b909bd563eeebff",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/ZMod",
"family_id": "rootsofunity_eq_top",
"file_id": "mathlib/Mathlib/RingTheory/ZMod/Torsion.lean",
"sample_id": "f90ebae3e238223bb942229ba9072bef620bae81190f54e359d35b11e6164e6b"
} | train |
{
"chosen_proof_sha256": "988485a8c756571de67c1e7554cc8678471929e0b79e0159e23042969b87deb3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f392082ed9ba8b8ad90648b283f41e7a07... | by
induction p with
| nil => rfl
| cons _ _ ih => dsimp [pathToList] at *; rw [ih] | false | {
"chosen_tokens": 25,
"rejected_tokens": 32,
"token_jaccard": 0.904762,
"token_length_ratio": 1.28
} | markdown_preamble_trailing | fence | 9a3408f22216413ea087729ec80ea551a048c0779415db79701e88bbb9f99dc5 | 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.Quiver.Cast
public import Mathlib.Combinatorics.Quiver.Symmetric
Namespace:
Quiver.SingleObj
Local context:
/-
Copyright (c) 2023 Antoine Label... | ```lean
by
induction p with
| nil => rfl
| cons _ _ ih => dsimp [pathToList] at *; rw [ih]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/Quiver",
"family_id": "listtopath_pathtolist",
"file_id": "mathlib/Mathlib/Combinatorics/Quiver/SingleObj.lean",
"sample_id": "5dc82f08b0f93626082032ca9aa64a8e948617dc0aa5acf4d820aa7f216565eb"
} | train |
{
"chosen_proof_sha256": "107f7ce7855f94c8cc4f2d6c868596a255522f4e82b0b2ecfe00fd289a468b21",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "00be9ebd31a702d21957e4d4af4dc0b354... | by
classical exact continuous_piecewise hs hf continuousOn_const | true | {
"chosen_tokens": 7,
"rejected_tokens": 3,
"token_jaccard": 0.25,
"token_length_ratio": 0.428571
} | circular_dependency | circular_dependency | 9a344deac6ec155c9d19bc7b6e53f219b5374acbcdf18793b511c44adeb62500 | 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.Notation.Indicator
public import Mathlib.Topology.Piecewise
public import Mathlib.Topology.Clopen
Namespace:
(root)
Local context:
/-
Copyright (c) 2... | by
exact continuous_mulIndicator | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "48f7006cee49963e7e0e0aed59a97bd4dc5ac7b2640ac2c1dfb5bb709993f14c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Topology/Algebra",
"family_id": "continuous_mulindicator",
"file_id": "mathlib/Mathlib/Topology/Algebra/Indicator.lean",
"sample_id": "a1e841521763274b65199f7d26a8e6592c777a58ce061b317bd17dcc7a139bd0"
} | train |
{
"chosen_proof_sha256": "4db2e60816a954672941e1fdfd4788e81ed7fe826b34170d024d9dedb9483621",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
have := W.isClosedUnderColimitsOfShape_isLocal J κ hW
infer_instance | false | {
"chosen_tokens": 10,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.2
} | sorry_admit_sorryAx_proof_hole | admit | 9a4347faff1eeaa1541ca5f18cd5b8947856b7717766b1d9c045d2429329ceb2 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Adjunction.PartialAdjoint
public import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
public import Mathlib.CategoryTheory.Localization.Bo... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Presentable",
"family_id": "morphismproperty",
"file_id": "mathlib/Mathlib/CategoryTheory/Presentable/OrthogonalReflection.lean",
"sample_id": "00a4fa01aa3988f187453211eea6fee689951bcc17ccb3f8913ec66689b3d85f"
} | train |
{
"chosen_proof_sha256": "3431d3593784b843a6d2cba92338ea3a61fb0adeecb9190810dad7f00a815cd9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "88c32e9180e9f0dad07f72ecceffc95651... | by
induction N with
| zero => simp
| succ N ih => calc
_ ⊆ Ico (a 0) (a N) ∪ Ico (a N) (a (N + 1)) := Ico_subset_Ico_union_Ico
_ ⊆ _ := by simpa [Finset.range_add_one] using
union_subset_union_right (Ico (a N) (a (N + 1))) ih | false | {
"chosen_tokens": 70,
"rejected_tokens": 75,
"token_jaccard": 0.861111,
"token_length_ratio": 1.071429
} | markdown_preamble_trailing | preamble | 9a44dc810c68526fc6f62790d7b53e23564780145a5afd1607383043dfa93d04 | 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.Range
public import Mathlib.Data.Set.Lattice
public import Mathlib.Order.Interval.Set.LinearOrder
Namespace:
(root)
Local context:
/-
Copyright (... | Here is the proof:
by
induction N with
| zero => simp
| succ N ih => calc
_ ⊆ Ico (a 0) (a N) ∪ Ico (a N) (a (N + 1)) := Ico_subset_Ico_union_Ico
_ ⊆ _ := by simpa [Finset.range_add_one] using
union_subset_union_right (Ico (a N) (a (N + 1))) ih | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/Interval",
"family_id": "ico_subset_biunion_ico",
"file_id": "mathlib/Mathlib/Order/Interval/Set/Union.lean",
"sample_id": "80e66e4358b5a047cdbe0254803df37a3bda8284661c9db0a5c755ad758dbabf"
} | train |
{
"chosen_proof_sha256": "e65faed5f7af3fe66504654d3728889128f1a2b5c77ec046ad6fc37f70906b6d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by simp [left_distrib, *, sub_eq_add_neg] | true | {
"chosen_tokens": 9,
"rejected_tokens": 8,
"token_jaccard": 0.071429,
"token_length_ratio": 0.888889
} | unsolved_goals | unsolved_goals | 9a5da8a7ec02dedf77a68b414659a7ee5fb69a03b55acb1a6894df268ee9d1d5 | 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.Field.Basic
public import Mathlib.Algebra.Order.Ring.Defs
public import Mathlib.Data.Tree.Basic
public import Mathlib.Tactic.NormNum.Core
public import... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "3d6e41fedb96281cd87efe84955437aef603bb2faead83f7993cb089c0299ee9",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/CancelDenoms",
"family_id": "sub_subst",
"file_id": "mathlib/Mathlib/Tactic/CancelDenoms/Core.lean",
"sample_id": "104ce9cbaebdaa87f4904559e1ae0c57adfb3fa19b70fd7be28fa5278b9ac29a"
} | train |
{
"chosen_proof_sha256": "fb278dc04d580276be7c164bc304823032754bdaf50110401d324e4d50eff42b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp [mem_def, or_and_right, exists_or] | true | {
"chosen_tokens": 9,
"rejected_tokens": 8,
"token_jaccard": 0.071429,
"token_length_ratio": 0.888889
} | unsolved_goals | unsolved_goals | 9a62d60bcdf5a9db49bb7e3c6fc6d9845aae008fcbb95fa76f607bc15afe57d0 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.Monoid.NatCast
public import Mathlib.Algebra.Ring.Nat
public import Mathlib.Data.Sigma.Basic
public... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "7e75af5eef05ad6aa7c56ce700d8affc4ba550a5dbebbb3e54c1afd93be140f0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "SetTheory",
"family_id": "mem_cons",
"file_id": "mathlib/Mathlib/SetTheory/Lists.lean",
"sample_id": "96dbde4f4151db83b5ae77d34dc896b57800591a2988fbcdab30f0f92312396b"
} | train |
{
"chosen_proof_sha256": "9c5374a4a76445b21ceb5288576943f3dfabe49605e7c1eedae05176a4a561c7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "dd7e7c7724f251b0bbfef38f1495fa3c1b... | by
rw [← isLocallySurjective_iff_epi']
exact LightCondensed.isLocallySurjective_iff_locallySurjective_on_lightProfinite _ f | true | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.153846,
"token_length_ratio": 0.25
} | unknown_identifier | unknown_identifier | 9a995ba4b83be33a1b021bf1f80920c591fd17b22f89ee0a6785e57b243a9eff | 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.Shapes.SequentialProduct
public import Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit
public import Mathlib.Condensed.Light.Functo... | by
exact __oprover_missing_cb787666449d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d29eb7e8dd70c805f3e054806d51e06406fa46814cc85869619e939ff0b78244",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Condensed/Light",
"family_id": "epi_iff_locallysurjective_on_lightprofinite",
"file_id": "mathlib/Mathlib/Condensed/Light/Epi.lean",
"sample_id": "cb787666449dd4a52c1a3d0c14107bfed4df4b3d94d031fd1a98bc977a411bf3"
} | train |
{
"chosen_proof_sha256": "f3a8fc7a4995882f5a643c316d6ec43e868e48edcba54745dfaa0649321d1a67",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cf1a2f0ff6036835a4e6c170684ceaf7b1... | by
have hb' : b.Dom := Part.dom_iff_mem.mpr ⟨a.get ha, hb⟩
rwa [← eq_get_iff_mem hb', eq_iff_of_dom ha hb'] at hb | true | {
"chosen_tokens": 33,
"rejected_tokens": 5,
"token_jaccard": 0.071429,
"token_length_ratio": 0.151515
} | wrong_namespace_qualification | wrong_namespace_qualification | 9ad0e24a5382afc758cd382915951af34eabab401d09f09c74dda861a4700552 | 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.Notation.Defs
public import Mathlib.Data.Set.Subsingleton
public import Mathlib.Logic.Equiv.Defs
Namespace:
Part
Local context:
/-
Copyright (c) 2017... | by
exact OProverMissing.missing_28eb45ede628 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "420c3f3ce3306fb489c0309a0d061c8cb39ccfa13f433fd6b2b617199a97f64a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data",
"family_id": "eq_of_mem",
"file_id": "mathlib/Mathlib/Data/Part.lean",
"sample_id": "28eb45ede6284ff2c969d0ba34870b558934c40303d056e5740787b98d7dcd28"
} | train |
{
"chosen_proof_sha256": "9d69af709cb1a99e4edeafcd19d4f6a19e8a85efb124f30d847acee7dc29b2e3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e01fdda376cb29cd87e36b0bb679b1859d... | by
suffices ∀ x : FreeMonoid α, (xs.map FreeMonoid.of).foldl (· * ·) x = x * ofList xs by
simpa [← List.foldl_map, lift_mk_eq_foldl, toFreeMonoid, lift] using this (FreeMonoid.of x)
induction xs with grind [ofList_nil, ofList_cons] | true | {
"chosen_tokens": 60,
"rejected_tokens": 3,
"token_jaccard": 0.027027,
"token_length_ratio": 0.05
} | unknown_identifier | unknown_identifier | 9b1dcfe8fff3a797fc57e600936a39c4bee7b6a9202f893c0c9945adae545eeb | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.FreeMonoid.Basic
public import Mathlib.Algebra.Free
public import Mathlib.Algebra.Group.WithOne.Basic
public import Mathlib.Algebra.Group.Units.Basic
p... | by
exact __oprover_missing_b0369a5a54c5 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "9f44ebac9cf93fac99ed2d7a2b320900c5324a3d8680b6c03c27d0a55dac15ed",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Algebra/FreeMonoid",
"family_id": "tofreemonoid_mk_eq_cons",
"file_id": "mathlib/Mathlib/Algebra/FreeMonoid/FreeSemigroup.lean",
"sample_id": "b0369a5a54c56e81b58aafefa0901fd2dbc75fdc809ee1103b64d4a18103c12d"
} | train |
{
"chosen_proof_sha256": "280119e14881b51170fc3d0d4d95c5bf43762c9b1e1b31c95c9da1e53082f7ea",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rcases h with ⟨u, hu⟩
rw [← Units.inv_mul_eq_one, mul_add] at hu
apply Or.imp _ _ (isUnit_or_isUnit_of_add_one hu) <;> exact isUnit_of_mul_isUnit_right | false | {
"chosen_tokens": 35,
"rejected_tokens": 3,
"token_jaccard": 0.064516,
"token_length_ratio": 0.085714
} | by_exact_placeholder | placeholder | 9b25e5872f7c33656ac93d706c40fdfee0148e4fcfd27967933aaa24ab5e8a9d | 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.LocalRing.Defs
public import Mathlib.RingTheory.Ideal.Nonunits
Namespace:
IsLocalRing
Local context:
/-
Copyright (c) 2018 Kenny Lau. All rights r... | 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/LocalRing",
"family_id": "isunit_or_isunit_of_isunit_add",
"file_id": "mathlib/Mathlib/RingTheory/LocalRing/Basic.lean",
"sample_id": "b84cec6b1d751705167f6f103cf7edb4e8a75633ff7d571654ff5ea97e2ce6d3"
} | train |
{
"chosen_proof_sha256": "4ba70e5cd3f178f8ef12d7930e49952b41e1f76063cf50cc900cfa723b52ba14",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [Iff.comm]
obtain ⟨s₁, rfl⟩ := s₁.exists_eq_ofList
obtain ⟨s₂, rfl⟩ := s₂.exists_eq_ofList
simp only [lt_iff_toList_lt, String.Legacy.iter, String.Legacy.mkIterator, String.toList_ofList]
induction s₁ generalizing s₂ <;> cases s₂
· unfold ltb; decide
· rename_i c₂ cs₂; apply iff_of_true
· unfold... | false | {
"chosen_tokens": 329,
"rejected_tokens": 5,
"token_jaccard": 0.030928,
"token_length_ratio": 0.015198
} | sorry_admit_sorryAx_proof_hole | sorryAx | 9b3dd045183297fbcc2ed9b1d6e7a18363637b6fa592ed2885577c023f9e4c39 | 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.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 sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/String",
"family_id": "lt_iff_ltb",
"file_id": "mathlib/Mathlib/Data/String/Basic.lean",
"sample_id": "e0f31a2fa4b167750af0775ad92818537ced61f211e8620b229e5bbd135fd43d"
} | train |
{
"chosen_proof_sha256": "5e0e8b084461b68f53dd226528eec6d4daa60789f672fc8f9edb2d39518f89fb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
apply c.isPushout.hom_ext h₀
apply Cofan.IsColimit.hom_ext c.isColimit₂
simpa [cell_def] using h | false | {
"chosen_tokens": 25,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.08
} | by_exact_placeholder | by? | 9b42f9cc2e63d691c00b76901b42ccb95064875fe4a90268959961fadd1cf126 | 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.MorphismProperty.Limits
Namespace:
HomotopicalAlgebra.AttachCells
Local context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Release... | 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": "AlgebraicTopology/RelativeCellComplex",
"family_id": "hom_ext",
"file_id": "mathlib/Mathlib/AlgebraicTopology/RelativeCellComplex/AttachCells.lean",
"sample_id": "f5cf3adf5b2f007282e6afc982afeb8adfc7d1ca08c6a7becb4328354365d479"
} | train |
{
"chosen_proof_sha256": "95f0a046edcc88c2f0897d1f675b7af88bcf9e64df19c8ace7d838acef7a93e4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
have ⟨n, S, d, _, hd, ⟨e⟩⟩ := exists_end_algEquiv_pi_matrix_end R₀ R M
classical exact ⟨n, _, d, inferInstance, inferInstance, hd, ⟨e⟩⟩ | false | {
"chosen_tokens": 42,
"rejected_tokens": 2,
"token_jaccard": 0.05,
"token_length_ratio": 0.047619
} | sorry_admit_sorryAx_proof_hole | admit | 9bb33355c2a4fd8075be3c51c0cb1c6be0fa271188e64ea622f8051e40cd631e | 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.FreeModule.Finite.Basic
public import Mathlib.RingTheory.FiniteLength
public import Mathlib.RingTheory.SimpleModule.Isotypic
public import Mathli... | 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/SimpleModule",
"family_id": "exists_end_algequiv_pi_matrix_divisionring",
"file_id": "mathlib/Mathlib/RingTheory/SimpleModule/WedderburnArtin.lean",
"sample_id": "929feb6854c6bd754df0ddca63934028a11faf17c7ad84065207bb38e2efb282"
} | train |
{
"chosen_proof_sha256": "b9e374e8c54a58f587ed34013c46e429e34757a52725bc1a7e7b25ce0f218592",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "adc475a92322c09535483773caf67bedfb... | by
apply injective_of_mono (X.mapIso (sigmaIso f).op).hom
apply injective_of_mono (sigmaComparison X (fun a ↦ (fiber f a).1))
ext a
specialize h a
rw [← sigmaComparison_comp_sigmaIso] at h
exact h | true | {
"chosen_tokens": 48,
"rejected_tokens": 5,
"token_jaccard": 0.1,
"token_length_ratio": 0.104167
} | wrong_namespace_qualification | wrong_namespace_qualification | 9bdb705348dc55b3a87d665eeb42b1227cd3508e37f3fbe270ddcd680196e5a6 | 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.Condensed.Discrete.Basic
public import Mathlib.Condensed.TopComparison
public import Mathlib.Topology.Category.CompHausLike.SigmaComparison
public import Mathl... | by
exact OProverMissing.missing_5b180b286925 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "fbb4490cc5cc1702b1ab3cd7930c7b7c59d7bdf7f526f7b9bc324b8689c3989f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Condensed/Discrete",
"family_id": "presheaf_ext",
"file_id": "mathlib/Mathlib/Condensed/Discrete/LocallyConstant.lean",
"sample_id": "5b180b286925047b6265b1d6b6387babba0a99b025076644a75ae77bbbe0b4d6"
} | train |
{
"chosen_proof_sha256": "886ef528de3066c5206bdbcb8b42b2d7a0a7e9340445f6b246cca5e2a1bdcafc",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [smul_Spec_def, smul_Spec_def]
simp_rw [smul_restrictAppIso_hom_apply, ← ConcreteCategory.comp_apply, Category.assoc]
have :
f ≫ (ΓSpecIso S).inv ≫ (Spec S).presheaf.map U.leTop.op ≫ (Hom.appIso (Spec.map f) U).inv =
(ΓSpecIso R).inv ≫ (Spec R).presheaf.map (Spec.map f ''ᵁ U).leTop.op := by
... | true | {
"chosen_tokens": 108,
"rejected_tokens": 8,
"token_jaccard": 0.086957,
"token_length_ratio": 0.074074
} | unsolved_goals | unsolved_goals | 9c24c402ff87d2ee49592979f5cfce060dc04de150731a97f57382f97e3ebbfe | 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.Category.ModuleCat.Localization
public import Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent
public import Mathlib.AlgebraicGeometry.AffineSche... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "3cc5681fb2aadb1c7d4fe839814f0ad6ebf6c93803fcff4920efd664a92338a8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Modules",
"family_id": "restrictappiso_smul_spec",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Modules/Tilde.lean",
"sample_id": "cfdd53edfe1acc3d2b51c43a1c120daa3d309e493c2f938aa9d41f5faeec220c"
} | train |
{
"chosen_proof_sha256": "ca01e87830c1462b512c1bad10dd02834fa50907bb2a15cdffba13757dee4a07",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "85cb31cb85bde4e36fe5f40e3de45ee208... | by
refine mt ((injective_iff_map_eq_zero _).mp bS.algebraMap_injective _) ?_
simp only [Finset.prod_eq_zero_iff, not_exists]
rintro x ⟨hx, rfl⟩
exact finsetApprox.zero_notMem bS adm hx | true | {
"chosen_tokens": 40,
"rejected_tokens": 5,
"token_jaccard": 0.09375,
"token_length_ratio": 0.125
} | wrong_namespace_qualification | wrong_namespace_qualification | 9c59fe52dcae363d069c8b6dde13987209b6e7c96dc6c7418b033677f2cdace0 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecialFunctions.Pow.Real
public import Mathlib.LinearAlgebra.FreeModule.PID
public import Mathlib.LinearAlgebra.Matrix.AbsoluteValue
public import Ma... | by
exact OProverMissing.missing_32a4aa28ca53 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "78f27bb7efc29cdeff6f19b6f4a5035916aa4fb1a7484ad25cdf828673e36092",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/ClassNumber",
"family_id": "prod_finsetapprox_ne_zero",
"file_id": "mathlib/Mathlib/NumberTheory/ClassNumber/Finite.lean",
"sample_id": "32a4aa28ca53ec5f375880fb329a6c92a05a17c1c9d5e8c664a77a4bde982211"
} | train |
{
"chosen_proof_sha256": "c849c5df0799eac828ab19895ffb8d4649eaf0e6c519838aa60e6b56e2b819cc",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
have a := hP.card_parts_eq_average ht
lia | false | {
"chosen_tokens": 9,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.555556
} | sorry_admit_sorryAx_proof_hole | sorryAx | 9c63ab419bdd2c0d33b5d4adb5f70ce08314517b4fdfe1928c31d828b40958b0 | 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.Nat
public import Mathlib.Data.Set.Equitable
public import Mathlib.Logic.Equiv.Fin.Basic
public import Mathlib.Order.Partition.Finpartition
... | 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/Partition",
"family_id": "isequipartition",
"file_id": "mathlib/Mathlib/Order/Partition/Equipartition.lean",
"sample_id": "a6cc210366796b93c3087a5747f28bf3396e3f7d848298d6e55848e06cc71e8d"
} | train |
{
"chosen_proof_sha256": "08f7647cfd9ab05023d94bc2fcca4631870d5f71d7f24de348f6c58aa5102a1d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [isPosSemidef_def, Matrix.posSemidef_iff_dotProduct_mulVec]
apply and_congr (B.isSymm_iff_isHermitian_toMatrix b)
rw [isNonneg_def]
refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
· rw [star_dotProduct_toMatrix₂_mulVec]
exact h _
· rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]
exact h _ | true | {
"chosen_tokens": 58,
"rejected_tokens": 8,
"token_jaccard": 0.026316,
"token_length_ratio": 0.137931
} | unsolved_goals | unsolved_goals | 9c63c5ecad1eedb815533e5209838e77eb95557b83334c565ffb3c1f7431d13a | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Matrix.PosDef
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as ... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "a9386e192ec9ab948f6d25238318c978127a3e7797faf0be0ba97a5e688dcb74",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/SesquilinearForm",
"family_id": "linearmap",
"file_id": "mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Star.lean",
"sample_id": "9b67a823dcea17b7fc9288545c47193c34fe1b90d1bd9cf3ef9be1964ef22f5d"
} | train |
{
"chosen_proof_sha256": "3486bfbcb341bbc6602e3e4c5334d8af2187c1a15d03540967ecccbb4983efe2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [← cancel_mono (mapBifunctorLeftUnitor F X e p hp Y').inv, assoc, assoc, Iso.hom_inv_id,
comp_id, mapBifunctorLeftUnitor_inv_naturality, Iso.hom_inv_id_assoc] | false | {
"chosen_tokens": 33,
"rejected_tokens": 3,
"token_jaccard": 0.038462,
"token_length_ratio": 0.090909
} | by_exact_placeholder | exact? | 9c7dd68cebbe856a6934a266678842835bccedaf321335716d34c5692ddbd0f8 | 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.GradedObject.Associator
public import Mathlib.CategoryTheory.GradedObject.Single
Namespace:
CategoryTheory.GradedObject
Local context:
/-
Copy... | 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/GradedObject",
"family_id": "mapbifunctorleftunitor_naturality",
"file_id": "mathlib/Mathlib/CategoryTheory/GradedObject/Unitor.lean",
"sample_id": "9547247b6d603996c4b3fb961825dcb69ba77ace42e69a17a558d84380f3ec34"
} | train |
{
"chosen_proof_sha256": "65460242af6486d44c1e55ce08a5b548dbca87ef42697a0dbcd7738054ac5b9c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f2228c66df4392ccdb499fe036ba44562c... | by
convert_to Tendsto (f ∘ (fun x ↦ c + x) ∘ Neg.neg ∘ Inv.inv) atTop l at h
· ext
simp [AddGroupWithOne.sub_eq_add_neg]
simpa [Tendsto, ← Filter.map_map] using h | false | {
"chosen_tokens": 46,
"rejected_tokens": 53,
"token_jaccard": 0.944444,
"token_length_ratio": 1.152174
} | markdown_preamble_trailing | fence | 9c8ea8ce79ce4a6881c98d6e3a7624e0b7ae160840d324b2bf5f82001a404ac1 | 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.Algebra.Order.Field
public import Mathlib.Topology.Maps.Basic
public import Mathlib.Analysis.Asymptotics.Lemmas
Namespace:
Tactic.ComputeAsymptotics
... | ```lean
by
convert_to Tendsto (f ∘ (fun x ↦ c + x) ∘ Neg.neg ∘ Inv.inv) atTop l at h
· ext
simp [AddGroupWithOne.sub_eq_add_neg]
simpa [Tendsto, ← Filter.map_map] using 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": "Tactic/ComputeAsymptotics",
"family_id": "tendsto_nhdslt_of_tendsto_attop",
"file_id": "mathlib/Mathlib/Tactic/ComputeAsymptotics/Lemmas.lean",
"sample_id": "8415a8306797307a1399dfa46e7bc4bffb6c158dd15399817b536ab5989fba40"
} | train |
{
"chosen_proof_sha256": "4db2e60816a954672941e1fdfd4788e81ed7fe826b34170d024d9dedb9483621",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "102237e83b0e666517b452145f83069b8f... | by
have := W.isClosedUnderColimitsOfShape_isLocal J κ hW
infer_instance | true | {
"chosen_tokens": 10,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.2
} | invalid_tactic | invalid_tactic | 9cae314c6bf42d5a2de6529c1d251aa190ee4f8c2b1772263f47f694889796b7 | 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.Adjunction.PartialAdjoint
public import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
public import Mathlib.CategoryTheory.Localization.Bo... | by
__oprover_invalid_tactic_00a4fa01aa39 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "fe7ecb9d9e91afc254d08c07d93f9e830ea33bbc5a327e56e3ff714888f4f813",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Presentable",
"family_id": "morphismproperty",
"file_id": "mathlib/Mathlib/CategoryTheory/Presentable/OrthogonalReflection.lean",
"sample_id": "00a4fa01aa3988f187453211eea6fee689951bcc17ccb3f8913ec66689b3d85f"
} | train |
{
"chosen_proof_sha256": "2b60f1251ce380c3c88892d3140d40e19399072a863e960ad4ea10cd3795106f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "031a4f415c5dae30902ba08057eae47c3a... | by
rcases exists_pair_ne α with ⟨x, y, hxy⟩
cases lt_or_gt_of_ne hxy <;> exact ⟨_, _, ‹_›⟩ | false | {
"chosen_tokens": 28,
"rejected_tokens": 33,
"token_jaccard": 0.8,
"token_length_ratio": 1.178571
} | markdown_preamble_trailing | preamble | 9cc69174c315e968d49d9d961dd9e0821a1e4c73f6999e8ca4a73f061acd75a7 | 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.Prod.Basic
public import Mathlib.Logic.Function.Basic
public import Mathlib.Logic.Nontrivial.Defs
public import Mathlib.Logic.Unique
public import Mathlib... | Here is the proof:
by
rcases exists_pair_ne α with ⟨x, y, hxy⟩
cases lt_or_gt_of_ne hxy <;> exact ⟨_, _, ‹_›⟩ | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Logic/Nontrivial",
"family_id": "exists_pair_lt",
"file_id": "mathlib/Mathlib/Logic/Nontrivial/Basic.lean",
"sample_id": "9a258eabbca29a5c3771ee588ea1b0dec5082cf5ef32ba82797313921b1c69f8"
} | train |
{
"chosen_proof_sha256": "7f6de06b7c519dc56f72cfe0e561560f9ccb6c57f8383a711dc7fbaa5be562ae",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [absNorm_eq' 1 I₀ (by rw [one_smul]; rfl), OneMemClass.coe_one, map_one, abs_one,
Int.cast_one, _root_.div_one] | false | {
"chosen_tokens": 33,
"rejected_tokens": 2,
"token_jaccard": 0.041667,
"token_length_ratio": 0.060606
} | sorry_admit_sorryAx_proof_hole | sorry | 9cf7c55109a65d05cbfd1d40e54ae8152ae76b531282ce05b2e7897fcc6a9fda | 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.FractionalIdeal.Basic
public import Mathlib.RingTheory.Ideal.Norm.AbsNorm
public import Mathlib.RingTheory.Localization.NormTrace
Namespace:
Fracti... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/FractionalIdeal",
"family_id": "coeideal_absnorm",
"file_id": "mathlib/Mathlib/RingTheory/FractionalIdeal/Norm.lean",
"sample_id": "04175e8f69832579c88e53e7defc7abb906d1749bebae6932b35a3cc3ac66b47"
} | train |
{
"chosen_proof_sha256": "08d1285c21c0227003f0de3a7a5739a4341408268171b9be42c7528b7aade63d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3a1459c8dec68bfc213585aa1c5fccaa2c... | by
rw [Function.isPeriodicPt_iff_minimalPeriod_dvd, Nat.dvd_prime hp.out,
← minimalPeriod_eq_one_iff_isFixedPt.not, or_and_right, and_not_self_iff, false_or,
iff_self_and]
exact fun h ↦ ne_of_eq_of_ne h hp.out.ne_one | true | {
"chosen_tokens": 38,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.078947
} | circular_dependency | circular_dependency | 9cf89843f5e5126abe2fefb4179117a4e62fe8d9c418dca1e3425a2243117877 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GCDMonoid.Finset
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.Data.PNat.Basic
public import Mathlib.Dynamics.PeriodicPts.Defs
publi... | by
exact minimalPeriod_eq_prime_iff | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "63614e997eb4c88fc6d4055bd7b38fb697773987f80bfa9be0ccf703b218e0b2",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Dynamics/PeriodicPts",
"family_id": "minimalperiod_eq_prime_iff",
"file_id": "mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean",
"sample_id": "f7b620c51cfaab544fc583af36da411df3fc24767a9c25ebd60f0ee43a35db07"
} | train |
{
"chosen_proof_sha256": "8af251c6a8eb77af2dfa75888d7c394a97af0ecb8aa758dfff389aa06e6d13fe",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7ef421e6d8431cec7f0feeba135adb69e4... | by
rw [← LinearMap.comp_assoc, map_map_comp_assoc_eq]
ext
rfl | true | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.25
} | unknown_identifier | unknown_identifier | 9d2b2d5072a78df278d479cb392b8e2cc67921b37ff2797076f094cedfb4bf3d | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorProduct.Tower
public import Mathlib.RingTheory.Coalgebra.Basic
import Mathlib.Tactic.Attr.Register
Namespace:
CoassocSimps
Local context:... | by
exact __oprover_missing_6a8e8150ef6c | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c503fdb4c044847a6cd79274f5ada48e49be7c7ef35657c384aa33c4ebf099e7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Coalgebra",
"family_id": "assoc_comp_map_map_comp",
"file_id": "mathlib/Mathlib/RingTheory/Coalgebra/CoassocSimps.lean",
"sample_id": "6a8e8150ef6c1755c0f36af56608a5c55a4714f68f327a0666a7f1b00c8049b8"
} | train |
{
"chosen_proof_sha256": "ec2ccc87cb1b0580e84707f89c57da330dcbae57aa0cfba902f545e3a992cb11",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c2db1f63c6a5a140290dd7cda06b1d0b04... | by
by_contra h
exact absurd Valuation.IsEquiv.refl (h ▸ adicValuation_not_isEquiv_infty_valuation p) | false | {
"chosen_tokens": 16,
"rejected_tokens": 20,
"token_jaccard": 0.823529,
"token_length_ratio": 1.25
} | missing_wrong_import | import | 9d58290ca2321094e9261d6f984f309bc96734b8326058e91ea9a6767e0db4e3 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.Finite.Valuation
public import Mathlib.NumberTheory.FunctionField
public import Mathlib.RingTheory.Valuation.Discrete.Basic
Namespace:
RatFunc
Lo... | import Mathlib.WrongImport
by
by_contra h
exact absurd Valuation.IsEquiv.refl (h ▸ adicValuation_not_isEquiv_infty_valuation p) | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/RatFunc",
"family_id": "adicvaluation_ne_inftyvaluation",
"file_id": "mathlib/Mathlib/NumberTheory/RatFunc/Ostrowski.lean",
"sample_id": "d885a32647abb9166c82b5317470ebda2b4ddd48a31b71c34892848d64e4f116"
} | train |
{
"chosen_proof_sha256": "cc860a95c409479a8d192fe8d26a650d014f625436d7b17247ecda1e72c0ab68",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "548db6b888ddc843c2de1b43678e5e79b3... | by
induction ψ using Quotient.inductionOn
induction θ using Quotient.inductionOn
simp_rw [← QuotientAddGroup.mk_zsmul, ← QuotientAddGroup.mk_add,
QuotientAddGroup.eq_iff_sub_mem, ← smul_sub, ← sub_sub]
exact AddSubgroup.zsmul_mem_zmultiples_iff_exists_sub_div hz | true | {
"chosen_tokens": 40,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.075
} | unknown_identifier | unknown_identifier | 9d850d5b8966c198b519969a9841abbbba8cebfc8d449c1ef4e88859573c8cd5 | 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.Field.Basic
public import Mathlib.Algebra.Order.Group.Unbundled.Int
public import Mathlib.Algebra.Module.NatInt
public import Mathlib.GroupTheory.Quoti... | by
exact __oprover_missing_c6688be02d1d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "3723345e40798c4ab175004908b05fa9883a3ae63476a1a7621409b451e089fc",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/CharZero",
"family_id": "zmultiples_zsmul_eq_zsmul_iff",
"file_id": "mathlib/Mathlib/Algebra/CharZero/Quotient.lean",
"sample_id": "c6688be02d1db8888f8f73bfb3dc49d8f7954b075b4191fe5618031c2b9977e0"
} | train |
{
"chosen_proof_sha256": "b03c7a526376dec68a09b72eb773aacc63927789fd0d788813f105124df25955",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
have : Algebra.IsSeparable (FractionRing A) (FractionRing R₂) := by
refine Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv A K).symm.toRingEquiv
(FractionRing.algEquiv R₂ F₂).symm.toRingEquiv ?_
ext x
exact IsFractionRing.algEquiv_commutes (FractionRing.algEquiv A K).symm
(Fra... | true | {
"chosen_tokens": 260,
"rejected_tokens": 8,
"token_jaccard": 0.047059,
"token_length_ratio": 0.030769
} | unsolved_goals | unsolved_goals | 9e20db1b0e31edaf703b947788146350fa89589f3a5654f489ea19e2111c48ee | 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.LinearDisjoint
public import Mathlib.RingTheory.DedekindDomain.Different
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Xavier Roblot. Al... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "a4b71ad099ac2394ce9495b0a3866c1ec8daf5ad15e3394048e1ec89dd27e8b1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/DedekindDomain",
"family_id": "submodule",
"file_id": "mathlib/Mathlib/RingTheory/DedekindDomain/LinearDisjoint.lean",
"sample_id": "a86a08c70d72b421064dab529cf5e5d4ac90c62963744456fb1461e7c0e42c96"
} | train |
{
"chosen_proof_sha256": "6dbb4280aff812e88179a089ca7e30391af8146bdc02e8489a28db8f88a2918d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5879b2913a081f78e72683e0e490903029... | by
ext a
apply Subtype.ext
ext ⟨i, hi⟩
obtain ⟨⟨n, a, ⟨b, hb'⟩, i, rfl : _ = b⟩, rfl⟩ := mk_surjective a
simp only [homOfLE_leOfHom, CommRingCat.hom_comp, RingHom.coe_comp, Function.comp_apply]
erw [ProjectiveSpectrum.Proj.awayToSection_apply]
rw [CommRingCat.hom_ofHom, val_awayMap_mk, Localization.mk_eq_... | false | {
"chosen_tokens": 107,
"rejected_tokens": 112,
"token_jaccard": 0.929825,
"token_length_ratio": 1.046729
} | markdown_preamble_trailing | preamble | 9e3b71242247ee3589ac6c094170efd21dc949539acd28cc1c8b1266a300a89a | 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.ProjectiveSpectrum.Scheme
public import Mathlib.AlgebraicGeometry.AffineScheme
public import Mathlib.AlgebraicGeometry.Gluing
Namespace:
Alg... | Here is the proof:
by
ext a
apply Subtype.ext
ext ⟨i, hi⟩
obtain ⟨⟨n, a, ⟨b, hb'⟩, i, rfl : _ = b⟩, rfl⟩ := mk_surjective a
simp only [homOfLE_leOfHom, CommRingCat.hom_comp, RingHom.coe_comp, Function.comp_apply]
erw [ProjectiveSpectrum.Proj.awayToSection_apply]
rw [CommRingCat.hom_ofHom, val_awayMap_mk, ... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/ProjectiveSpectrum",
"family_id": "awaymap_awaytosection",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean",
"sample_id": "f56b4dc966c61de3f4b4118e398dfcc1e44b0ddfc9b9e3ab8b9799975c8612d7"
} | train |
{
"chosen_proof_sha256": "76978dc753664885955d4019ab183b19aab997594734243a4e6b99373e308b2b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simp_rw [isCompactElement_iff_exists_le_sSup_of_le_sSup]
refine ⟨fun h k s hs => ?_, fun h s => ?_⟩
· obtain ⟨t, ⟨hts, htsup⟩⟩ := h s
use t, hts
rwa [← htsup]
· obtain ⟨t, ⟨hts, htsup⟩⟩ := h (sSup s) s (by rfl)
have : sSup s = t.sup id := by
suffices t.sup id ≤ sSup s by apply le_antisymm <... | false | {
"chosen_tokens": 120,
"rejected_tokens": 2,
"token_jaccard": 0.037037,
"token_length_ratio": 0.016667
} | by_exact_placeholder | by? | 9e449cadd4c0806f41fbc179f26ca137d3d5ce5dafdae5489cac128ea8e573dd | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.Atoms
public import Mathlib.Order.OrderIsoNat
public import Mathlib.Order.RelIso.Set
public import Mathlib.Order.SupClosed
public import Mathlib.Order.Su... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/CompactlyGenerated",
"family_id": "issupfinitecompact_iff_all_elements_compact",
"file_id": "mathlib/Mathlib/Order/CompactlyGenerated/Basic.lean",
"sample_id": "4ea77ade5bcaf82f263f63d2e35e31706e2c9a6c7ae719c3b1558eff0efaaa66"
} | train |
{
"chosen_proof_sha256": "4d69ce6ddd118e205dc838354a84f5c72260ded77b2d93b7242fcdda08dda204",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3e1c2bd9695ae8682b25bbfbf9b301964e... | by
rwa [← hasPointwiseRightDerivedFunctorAt_iff F L W] | false | {
"chosen_tokens": 9,
"rejected_tokens": 16,
"token_jaccard": 0.818182,
"token_length_ratio": 1.777778
} | markdown_preamble_trailing | fence | 9e8cfabca8d02169b5e0ea7b5a3ea70d33985d84b8fa47d0b0a74c5a09906a59 | 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.Functor.Derived.RightDerived
public import Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
public import Mathlib.CategoryTheory.Localizati... | ```lean
by
rwa [← hasPointwiseRightDerivedFunctorAt_iff F L 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/Functor",
"family_id": "haspointwiserightderivedfunctorat",
"file_id": "mathlib/Mathlib/CategoryTheory/Functor/Derived/PointwiseRightDerived.lean",
"sample_id": "039e1da8660af549f0b84de8a8d3a1a21828f8928f6164e5d53b92bab06d1d90"
} | train |
{
"chosen_proof_sha256": "0dd5faeb64813fa427f25f46617ddcf71a676309c87e033eb80368aa47d425e1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "734c8700a702ad20f6bf7de6d00c2776a2... | by
simp only [← C_eq_intCast, coeff_C, Int.cast_ite, Int.cast_zero] | true | {
"chosen_tokens": 17,
"rejected_tokens": 5,
"token_jaccard": 0.125,
"token_length_ratio": 0.294118
} | wrong_namespace_qualification | wrong_namespace_qualification | 9eb7357adebead7df1c91c91320390c086c901b96a074d91332458347d46991c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.Degree.Lemmas
Namespace:
Mathlib.Tactic.ComputeDegree
Local context:
/-
Copyright (c) 2023 Damiano Testa. All rights reserved.
Released un... | by
exact OProverMissing.missing_fc41a1bfba60 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "86215d490ba04ebddcafbf9574b975189850bfea0add5aea2dd7ee4e9a9da4fe",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic",
"family_id": "coeff_intcast_ite",
"file_id": "mathlib/Mathlib/Tactic/ComputeDegree.lean",
"sample_id": "fc41a1bfba603ad2786ab361571962d38f904641b8aa32668047f0fd3d10b946"
} | train |
{
"chosen_proof_sha256": "94092e3c9ae7aafc8bfd4b060b03f1f7bf7ca38bd35810157e3b4c1cdb58864b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
ext1
apply map_C
-- mixing the two monad structures | false | {
"chosen_tokens": 11,
"rejected_tokens": 5,
"token_jaccard": 0.071429,
"token_length_ratio": 0.454545
} | sorry_admit_sorryAx_proof_hole | sorryAx | 9eea0249dbc312bea1db7020140451fc160cb511f9cb6fa65fecb5201089dfa4 | 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.Rename
public import Mathlib.Algebra.MvPolynomial.Variables
Namespace:
MvPolynomial
Local context:
/-
Copyright (c) 2020 Johan Commelin,... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/MvPolynomial",
"family_id": "map_comp_c",
"file_id": "mathlib/Mathlib/Algebra/MvPolynomial/Monad.lean",
"sample_id": "bd9359a574dcaca31acf15cd4ed4c49c4962b2ea1c2de798364f401e03500718"
} | train |
{
"chosen_proof_sha256": "dae1c5b0ed2fe337b0ef71d64dc3ef20222db1f81ac15117533aa09e94ca0f93",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1f8f5e1811f5aaefcb9cbe3919d7531d35... | by
obtain ⟨u, u_cov, u_props⟩ := sc_st
exact
⟨u,
s_sub.trans u_cov,
fun n ↦
⟨(u_props n).1,
disjoint_of_subset (fun ⦃_⦄ a ↦ a) t_sub (u_props n).2⟩⟩ | true | {
"chosen_tokens": 50,
"rejected_tokens": 5,
"token_jaccard": 0.103448,
"token_length_ratio": 0.1
} | circular_dependency | circular_dependency | 9f0bf0cf60f05c9f74d2a6e900a1cd4125d598ef8d66ef3c79cdb9d2d3e3f032 | 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.Continuous
public import Mathlib.Topology.NhdsSet
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Releas... | by
exact HasSeparatingCover.mono | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "ac9c6e1008ff2b0129ae0cfabce8d936833e1cbb736dda7e778c65da8daa1a91",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Separation",
"family_id": "hasseparatingcover",
"file_id": "mathlib/Mathlib/Topology/Separation/SeparatedNhds.lean",
"sample_id": "f8bf5611d48800247cbe0df58105517cb729c0618719c6fcdd68a62dbe1a0152"
} | train |
{
"chosen_proof_sha256": "3628ed809b6ca5e99b1c81051989cfa58d947fe9238f4a04564fff2b92ccbda9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9fd7dac785c481bc273cd04cdc4c367188... | 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] | false | {
"chosen_tokens": 58,
"rejected_tokens": 63,
"token_jaccard": 0.848485,
"token_length_ratio": 1.086207
} | markdown_preamble_trailing | preamble | 9f215292416da8302890ca8acb37a62ec213ead2ba931f96fe1f5c01d4c962b7 | 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.Birational.RationalMap
Namespace:
AlgebraicGeometry.Scheme.PartialMap
Local context:
/-
Copyright (c) 2026 Justus Springer. All rights rese... | Here is the proof:
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] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | 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": "4c0ce5d128f69a2b99881bee610f584c1a319bb4f885c9faba74ae83efc61828",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
have := hf (equalizer.lift f h) (equalizer.ι p q) (by simp)
rw [← cancel_epi (equalizer.ι p q), equalizer.condition]
isIso := by tauto | true | {
"chosen_tokens": 42,
"rejected_tokens": 8,
"token_jaccard": 0.107143,
"token_length_ratio": 0.190476
} | unsolved_goals | unsolved_goals | 9f574bf755d6e38061b37b87010e639fa37a64a0ad4f390f7daa4ffd4534305f | 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.Limits.Shapes.StrongEpi
Namespace:
CategoryTheory
Local context:
/-
Copyright (c) 2025 ... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "c48e85f3a0dd1885e32e041cdf5815ff823cca7a66bae88376559c0383284a5b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory",
"family_id": "extremalepi",
"file_id": "mathlib/Mathlib/CategoryTheory/ExtremalEpi.lean",
"sample_id": "c8cd16ca7c7e5a007fb195fa59aed53c5734a477ccb7bfa5ed293e06fc1f996c"
} | train |
{
"chosen_proof_sha256": "4a140838d46443a223247967ef703140b4be6ea2808c8d4f93f75968349bd809",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8490e25c33757674cf12d7337a39354059... | by
have := h.out
simp [this]
/- evalCast -/ | true | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.2
} | circular_dependency | circular_dependency | 9f8ef36fc6f1a9b3e76c8106fd82701a6fbc31c08206d24aa3e8542d43bbafb5 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Algebra.Algebra.Basic
public import Mathlib.Tactic.Ring.RingNF
Namespace:
Mathlib.Tactic.Algebra
Local context:
/-... | by
exact isNat_zero_eq | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "707b7c8a83869e6b8cd1e78df8a7605e31b2246bee96abc87d99aa16187fd170",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/Algebra",
"family_id": "isnat_zero_eq",
"file_id": "mathlib/Mathlib/Tactic/Algebra/Lemmas.lean",
"sample_id": "b579425a25a44dbf03d7aec5fcfc9f633b6db243ba8c3f42fbd8c2796e7b54f2"
} | train |
{
"chosen_proof_sha256": "252205b7b358cac3e7fbb348dc9a6e22e848772b9ebec8a5f89dd1c88ca1cccb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6654eaee1efe2f7a4c8a60caecda062301... | by
induction t with
| nil => rw [traverse, map]
| node v l r hl hr =>
rw [traverse, map, hl, hr, Function.comp_apply, map_pure, pure_seq, map_pure, pure_seq,
map_pure] | true | {
"chosen_tokens": 45,
"rejected_tokens": 5,
"token_jaccard": 0.074074,
"token_length_ratio": 0.111111
} | wrong_namespace_qualification | wrong_namespace_qualification | a019863c0476ef5f5db56248c4c9c6dc75a2a33fed0bc89af19fde77466f6454 | 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.Tree.Basic
public import Mathlib.Control.Applicative
public import Mathlib.Control.Traversable.Basic
Namespace:
BinaryTree
Local context:
/-
Copyright (... | by
exact OProverMissing.missing_932a2516a9f3 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "6f6643d507164b618da6939ed9b473049553adc6abcf0577ddcbfafbfd3bf48c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Tree",
"family_id": "traverse_eq_map_id",
"file_id": "mathlib/Mathlib/Data/Tree/Traversable.lean",
"sample_id": "932a2516a9f3261c5acb0efee2adbd6bd7291142ae5d6a59e51bba4d80664223"
} | train |
{
"chosen_proof_sha256": "f1c76c9c21cec19c82ec1c29aff36596a09a140cc79532bb60d64518174b61b0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
convert! LinearMap.iSupIndep_map _ hv (iSupIndep_range_lsingle ι R R)
ext; simp [mem_span_singleton] | false | {
"chosen_tokens": 20,
"rejected_tokens": 3,
"token_jaccard": 0.047619,
"token_length_ratio": 0.15
} | by_exact_placeholder | exact? | a14e2bf7cbbdc701323bb0ea40b270c73feea423df4f617c9e24a091b21b60b2 | 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.Tuple.Reflection
public import Mathlib.LinearAlgebra.Dual.Defs
public import Mathlib.LinearAlgebra.Finsupp.SumProd
public import Mathlib.LinearAlgebra... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/LinearIndependent",
"family_id": "linearindependent",
"file_id": "mathlib/Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean",
"sample_id": "2f46fd8127f7a4707cce4dc2e25df1a6780c736d06d1699ff4e93a369a16eb74"
} | train |
{
"chosen_proof_sha256": "871db309f2a7e188f5170224e4eada7ffe3aa87ef44b3d70e3a18c5c524571b5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a04ab4cdf2e6dd7b879e0c3709f89419da... | by
refine le_antisymm (map_continuous <| toWeakSpaceCLM 𝕜 E).continuousOn.image_closure
(Set.compl_subset_compl.mp fun x hx ↦ ?_)
obtain ⟨x, -, rfl⟩ := (toWeakSpace 𝕜 E).toEquiv.image_compl (closure s) |>.symm.subset hx
have : ContinuousSMul ℝ E := IsScalarTower.continuousSMul 𝕜
obtain ⟨f, u, hus, hux⟩ :... | true | {
"chosen_tokens": 266,
"rejected_tokens": 3,
"token_jaccard": 0.022222,
"token_length_ratio": 0.011278
} | unknown_identifier | unknown_identifier | a154114063fd163044a9c149f00709d301e20f7afab29154863c14412d91aaf1 | 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.LocallyConvex.Separation
public import Mathlib.Analysis.LocallyConvex.SeparatingDual
public import Mathlib.LinearAlgebra.Dual.Defs
public import Mathl... | by
exact __oprover_missing_d3baf770853c | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "89ba6f2bcfc1b178d9d73811da45d89a3dd56b21b3beea3228eb1bd32e984d45",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/LocallyConvex",
"family_id": "convex",
"file_id": "mathlib/Mathlib/Analysis/LocallyConvex/WeakSpace.lean",
"sample_id": "d3baf770853c9de1742f6952ed6591411750a3d98ef5c34871eada480093b869"
} | train |
{
"chosen_proof_sha256": "843286c15dd0e5c3687da59e28ddbf32542077397522d384d533e4f852e034a9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [map_comp]
exact hg.trans <| map_mono _ hf | false | {
"chosen_tokens": 14,
"rejected_tokens": 3,
"token_jaccard": 0.133333,
"token_length_ratio": 0.214286
} | by_exact_placeholder | exact? | a1b4e9eb194cfcda8927f2db8dbd9e46ef20345db6b77b3d60529558c528fdd1 | 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.GradedAlgebra.Homogeneous.Ideal
public import Mathlib.RingTheory.GradedAlgebra.RingHom
Namespace:
HomogeneousIdeal
Local context:
/-
Copyright (c)... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/GradedAlgebra",
"family_id": "irrelevant_le_map_comp",
"file_id": "mathlib/Mathlib/RingTheory/GradedAlgebra/Homogeneous/Maps.lean",
"sample_id": "2c90f2e32be3da6fb260df3bb94bf0c15d7de35d2f1513e19789e1136caf26c6"
} | train |
{
"chosen_proof_sha256": "281c14df73dbd3d58dd1e8053eb70fe2ed484fdf0d2d7eb3b862de0b5a2b7009",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d3834ad3829318d82a78287b79886e2fbb... | by
have : Nonempty J := IsFiltered.nonempty
let hc' := Under.isColimitLiftCocone D s c (p ≫ f) h hc
obtain ⟨j, q, hq⟩ := exists_hom_of_isColimit (X := Under.mk p) hc' (Under.homMk f rfl)
use j, q.right, Under.w q, congr($(hq).right) | true | {
"chosen_tokens": 72,
"rejected_tokens": 5,
"token_jaccard": 0.046512,
"token_length_ratio": 0.069444
} | circular_dependency | circular_dependency | a1bb0b6fdb4bc45df4dad7ea100332fb12284ef9e5bf54a30e74158a222c342e | 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.Filtered
public import Mathlib.CategoryTheory.Limits.Preserves.Filtered
public import Mathlib.CategoryTheory.Limits.Types.Filtered
public... | by
exact IsFinitelyPresentable.exists_hom_of_isColimit_under | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "af1ddfa5d3d5dd1978afb2e241b6e0883a3df9c61d341d229e9dcf838133591f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Presentable",
"family_id": "isfinitelypresentable",
"file_id": "mathlib/Mathlib/CategoryTheory/Presentable/Finite.lean",
"sample_id": "e9c86bc71f86c9474c5b0e70638d93a26490f015f3c3db0caf5be6d93809c4ae"
} | train |
{
"chosen_proof_sha256": "90ae162d9072def41fe3c1f6b71c6fb09d8794f02fb213178dc3f4878196efbb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [← cancel_epi (prodComparisonIso (Over.pullback f) _ _).hom.left,
Over.hom_left_inv_left_assoc]
simp [CartesianMonoidalCategory.prodComparison, snd] | true | {
"chosen_tokens": 33,
"rejected_tokens": 8,
"token_jaccard": 0.035714,
"token_length_ratio": 0.242424
} | unsolved_goals | unsolved_goals | a1fe908d8c94804196228f945e342fa8fc94d2cdd114882879712b5808532b04 | 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.Adjunction.Limits
public import Mathlib.CategoryTheory.Comma.Over.Pullback
public import Mathlib.CategoryTheory.Limits.Constructions.Over.Produc... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "fa711ec91a0c6b5ff1ef81a1aec0a6032c4b166288b7aff2c36b94b7e6f99218",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Monoidal",
"family_id": "prodcomparisoniso_pullback_inv_left_fst_snd",
"file_id": "mathlib/Mathlib/CategoryTheory/Monoidal/Cartesian/Over.lean",
"sample_id": "3129c4684252f138a34ea19d6ab7a0e7965bec43e42276b25a947e2819c0b66a"
} | train |
{
"chosen_proof_sha256": "dbcf29961f4c4a120ac8530a8dcc28a30bfa6b6db329e1836f9e00211fe2e46d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f7bf80e04d0a2e1788d695cec2bec9e4d9... | by
simpa [CochainComplex.of.d] using (inhomogeneousCochains A).d_comp_d n (n + 1) (n + 2) | true | {
"chosen_tokens": 27,
"rejected_tokens": 2,
"token_jaccard": 0.052632,
"token_length_ratio": 0.074074
} | invalid_tactic | invalid_tactic | a2067ac654d697a1f792b395ccc80fc1673291eda74fdcb42b0f59422849c8c3 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Homology.Opposite
public import Mathlib.Algebra.Homology.ConcreteCategory
public import Mathlib.RepresentationTheory.Homological.Resolution
public impo... | by
__oprover_invalid_tactic_405cc27ec849 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "13168a73d4b76a5139b9aa1f337c85935681cfb54b706059f94f1f39e7156fe5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RepresentationTheory/Homological",
"family_id": "inhomogeneouscochains",
"file_id": "mathlib/Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean",
"sample_id": "405cc27ec849e33c860448f43291f011dd4708a3dd857ace59279e3e185b0743"
} | train |
{
"chosen_proof_sha256": "9740f164ce7086b555af4b7554cbcc58f11573780fea0b6812b9fc64a3703b2b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "35a2e7b8e31ff1274ea9232d6e5039032b... | by
apply MonoidHom.eq_of_eqOn_denseM (closure_range_of _)
grind [Set.eqOn_range] | true | {
"chosen_tokens": 15,
"rejected_tokens": 2,
"token_jaccard": 0.066667,
"token_length_ratio": 0.133333
} | invalid_tactic | invalid_tactic | a220e6a934593c52d646236b01d1ff4279126d1687666f9757cbf19059dc9eee | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.FreeMonoid.Basic
public import Mathlib.Algebra.Group.Submonoid.Operations
public import Mathlib.GroupTheory.Congruence.Hom
Namespace:
PresentedMonoid
... | by
__oprover_invalid_tactic_27c7b061eb2d | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "f81701716254788fe52c04e43bcebb225d8becb879d53705942d7de3afa7dc28",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/PresentedMonoid",
"family_id": "ext",
"file_id": "mathlib/Mathlib/Algebra/PresentedMonoid/Basic.lean",
"sample_id": "27c7b061eb2df69ed4438c60ac705043bf7b1c99567810c2e2cbfe95dec0ba49"
} | train |
{
"chosen_proof_sha256": "d7c3832f7227fb403f537f073b6c62458e059e39149445706fd5cbe0c34cde70",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
simpa only [tendsto_iff_comap] using le_iff_ultrafilter | false | {
"chosen_tokens": 8,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.625
} | sorry_admit_sorryAx_proof_hole | sorryAx | a27a528238bbf853f57a70a677fecc3574a3c31b16b7ceb272a0a8bb9c189594 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.Filter.Ultrafilter.Defs
public import Mathlib.Order.Filter.Cofinite
public import Mathlib.Order.ZornAtoms
Namespace:
Filter
Local context:
/-
Copyright... | 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/Filter",
"family_id": "tendsto_iff_ultrafilter",
"file_id": "mathlib/Mathlib/Order/Filter/Ultrafilter/Basic.lean",
"sample_id": "4a94ca7d62a9fac81c4789f5e6cb34fc62a67f792fb4fbe276c40abbf6855644"
} | train |
{
"chosen_proof_sha256": "543d94c4b8beec8d8219800c1a531bf78be218b6d235d8e934ed1722e2e75902",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
refine measure_mono_null (fun x hx ↦ ?_)
(Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet _
(isClosed_closure (s := s)).measurableSet)
by_cases h : x ∈ closure s <;>
simp only [mem_compl_iff, mem_setOf, h, not_false_eq_true, indicator_of_notMem,
indicator_of_mem, Pi.one_apply]
· r... | false | {
"chosen_tokens": 91,
"rejected_tokens": 5,
"token_jaccard": 0.057692,
"token_length_ratio": 0.054945
} | sorry_admit_sorryAx_proof_hole | sorryAx | a2df7fbcc1d1983151a230f218fab8275b725132466812adedbabdbf1f5fee25 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.Normed.Order.UpperLower
public import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
public import Mathlib.Topology.Order.DenselyOrdered
Names... | 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": "MeasureTheory/Order",
"family_id": "isupperset",
"file_id": "mathlib/Mathlib/MeasureTheory/Order/UpperLower.lean",
"sample_id": "4561b215385a78d721dfe36030198c0459e29f4db8031b7c27e613ae90c4a078"
} | train |
{
"chosen_proof_sha256": "f258ff7f229ca2a8c793329e07d0fcda1df2d4bda138c89c5b29fe00a0404c2f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
refine summable_geometric_iff_norm_lt_one.mp ?_
simp_rw [← map_pow]
exact hsum.comp_injective <| Nat.pow_right_injective hp | false | {
"chosen_tokens": 22,
"rejected_tokens": 2,
"token_jaccard": 0.047619,
"token_length_ratio": 0.090909
} | sorry_admit_sorryAx_proof_hole | sorry | a2ec5e38c441b6f51bdf897b02b35374df0d2b6d2b9d901769846c28559ab449 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.Normed.Ring.InfiniteSum
public import Mathlib.Analysis.SpecificLimits.Normed
public import Mathlib.NumberTheory.ArithmeticFunction.Defs
public import ... | 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": "NumberTheory/EulerProduct",
"family_id": "summable",
"file_id": "mathlib/Mathlib/NumberTheory/EulerProduct/Basic.lean",
"sample_id": "1caba13c5bfb2e48e2aa824af4b9e227bed2165274d63dd634f59f34ffe47e2e"
} | train |
{
"chosen_proof_sha256": "0dd5faeb64813fa427f25f46617ddcf71a676309c87e033eb80368aa47d425e1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1f592e66efeee00906ccbae7089ea605c8... | by
simp only [← C_eq_intCast, coeff_C, Int.cast_ite, Int.cast_zero] | false | {
"chosen_tokens": 17,
"rejected_tokens": 22,
"token_jaccard": 0.764706,
"token_length_ratio": 1.294118
} | markdown_preamble_trailing | trailing | a331ea818521809f6ab59e5a011aa6065e295974751e1f77b5649b65f721a37f | 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.Degree.Lemmas
Namespace:
Mathlib.Tactic.ComputeDegree
Local context:
/-
Copyright (c) 2023 Damiano Testa. All rights reserved.
Released un... | by
simp only [← C_eq_intCast, coeff_C, Int.cast_ite, Int.cast_zero]
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Tactic",
"family_id": "coeff_intcast_ite",
"file_id": "mathlib/Mathlib/Tactic/ComputeDegree.lean",
"sample_id": "fc41a1bfba603ad2786ab361571962d38f904641b8aa32668047f0fd3d10b946"
} | train |
{
"chosen_proof_sha256": "5710d934b0159d30d30972bfc5b7ddfd3eb8cd27b8c73a16ac7a346cdcff98c4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "df1deb7781ab53caab0682db75c3859714... | by
rfl
variable (φ : A →ₐc[R] B) | true | {
"chosen_tokens": 15,
"rejected_tokens": 5,
"token_jaccard": 0.052632,
"token_length_ratio": 0.333333
} | wrong_namespace_qualification | wrong_namespace_qualification | a340e681742a6ee9487c30fbfe743b01a3b74e26509fa694a1669b0fbf5bf999 | 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.Coalgebra.Hom
public import Mathlib.RingTheory.Bialgebra.Basic
Namespace:
BialgHom
Local context:
/-
Copyright (c) 2024 Amelia Livingston. All rig... | by
exact OProverMissing.missing_519e1718b1b4 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "bff67f76e82abb9c386c8872c74302d76034ec0f62fa7fe37fc18c8690890f71",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Bialgebra",
"family_id": "toalghom_tolinearmap",
"file_id": "mathlib/Mathlib/RingTheory/Bialgebra/Hom.lean",
"sample_id": "519e1718b1b4803bcd063f65500a31317c48e550e0d65b418a9de8968712bef5"
} | train |
{
"chosen_proof_sha256": "679574657f773b5361b65d8bc217854b5dee221b7dde4fd6f07b322f738d2333",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f352361c545b1f8b216789e3adf4cb0dda... | by
rwa [← map_natCast' _ he, ← map_natCast' _ he, he'.eq_iff, Nat.cast_inj] at h | true | {
"chosen_tokens": 23,
"rejected_tokens": 2,
"token_jaccard": 0.058824,
"token_length_ratio": 0.086957
} | invalid_tactic | invalid_tactic | a3a33a4e720737d60648e64d87942495f4ceeeee8903db1b00e6a784e4be22ac | 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.CharZero.Defs
public import Mathlib.Algebra.Group.Equiv.Defs
public import Mathlib.Data.Nat.Cast.Basic
Namespace:
(root)
Local context:
/-
Copyright ... | by
__oprover_invalid_tactic_bf23546e5597 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b48a9cf12347d77462a3bba311a51487fe50c86a6c82fefce933044f59edb2bb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/CharZero",
"family_id": "charzero",
"file_id": "mathlib/Mathlib/Algebra/CharZero/AddMonoidHom.lean",
"sample_id": "bf23546e5597548a8a703fba0f21b06b38585a3dc1e1cc164336dbb8b96373a4"
} | train |
{
"chosen_proof_sha256": "63a4845b3fb713f383d6aefe7ca814d2f2062db02f00a1f2f03bbbb49cf53c38",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
replace h (a m) : Q₁ (a ⊗ₜ m) = Q₂ (a ⊗ₜ m) := by
rw [← mul_one a, ← smul_eq_mul, ← smul_tmul', QuadraticMap.map_smul, QuadraticMap.map_smul, h]
ext x
induction x with
| tmul => simp [h]
| zero => simp
| add x y hx hy =>
have : Q₁.polarBilin = Q₂.polarBilin := by
ext
dsimp [polar]
... | false | {
"chosen_tokens": 129,
"rejected_tokens": 5,
"token_jaccard": 0.017544,
"token_length_ratio": 0.03876
} | sorry_admit_sorryAx_proof_hole | sorryAx | a3d9c38dfc97419de8d1b30a70a23a68125200039ca1ea1a7676abafdb9aabb0 | 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.BilinearForm.TensorProduct
public import Mathlib.LinearAlgebra.QuadraticForm.Basic
public import Mathlib.Tactic.LinearCombination
Namespace:
(ro... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/QuadraticForm",
"family_id": "basechange_ext",
"file_id": "mathlib/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean",
"sample_id": "2b78fa8807ec50bfc8ab5c93ca7eeaa86e8f4d73fc76cde03273374bc7683948"
} | train |
{
"chosen_proof_sha256": "1a3ab110d23e9c5a2fcf5c8f847d9eb4eca9aceac610399fa60f5ab4f28d4c1a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "59745f19014a0741162cbebbbb6846c12b... | by
obtain ⟨c₁, hc₁, c₂, _, h⟩ := hf b hb
refine ⟨c₁, hc₁, ?_⟩
filter_upwards [h]
exact fun _ H u hu => (H u hu).1 | false | {
"chosen_tokens": 49,
"rejected_tokens": 54,
"token_jaccard": 0.882353,
"token_length_ratio": 1.102041
} | markdown_preamble_trailing | trailing | a3ff053968b3576f47dda96b8c08f173f4de721127156565c9f02189d990f8be | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecialFunctions.Pow.Real
public import Mathlib.Algebra.Order.ToIntervalMod
public import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Al... | by
obtain ⟨c₁, hc₁, c₂, _, h⟩ := hf b hb
refine ⟨c₁, hc₁, ?_⟩
filter_upwards [h]
exact fun _ H u hu => (H u hu).1
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Computability/AkraBazzi",
"family_id": "eventually_attop_ge",
"file_id": "mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean",
"sample_id": "a2b4875ff5d193c274945a397a372732cc39a1c971c7cc22bbea2e4368fe786d"
} | train |
{
"chosen_proof_sha256": "0f3c5c923c48eece1af3a7c60f1fa94f43ad2e6d58c10a4a8ef6046b414d8933",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "738f9ce2b5be5545426b3823582006b3d8... | by
obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sSup_empty, Real.sInf_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csSup_singleton 0
by_cases h : BddBelow s
· exact ((OrderIso.smulRightDual ℝ ha').map_csInf' hs h).symm
· rw [Real.sS... | true | {
"chosen_tokens": 89,
"rejected_tokens": 5,
"token_jaccard": 0.090909,
"token_length_ratio": 0.05618
} | circular_dependency | circular_dependency | a42a69cd28452e347ff46141da72ccf84bc27094ef054c76ac0253cd455de258 | 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.Pointwise.Set
public import Mathlib.Algebra.Order.Archimedean.Real.Basic
public import Mathlib.Algebra.Order.Module.Pointwise
publ... | by
exact Real.sSup_smul_of_nonpos | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "06cfa16a2f30bef1f3fa2b63ba647fcb6efa532049c95193258886fc0080f3ad",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Real",
"family_id": "real",
"file_id": "mathlib/Mathlib/Data/Real/Pointwise.lean",
"sample_id": "d0df3140da7fa364792a7ce4f3fa0fb068f28c3068cdfb76bb78b78e0d7a5c60"
} | train |
{
"chosen_proof_sha256": "871db309f2a7e188f5170224e4eada7ffe3aa87ef44b3d70e3a18c5c524571b5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
refine le_antisymm (map_continuous <| toWeakSpaceCLM 𝕜 E).continuousOn.image_closure
(Set.compl_subset_compl.mp fun x hx ↦ ?_)
obtain ⟨x, -, rfl⟩ := (toWeakSpace 𝕜 E).toEquiv.image_compl (closure s) |>.symm.subset hx
have : ContinuousSMul ℝ E := IsScalarTower.continuousSMul 𝕜
obtain ⟨f, u, hus, hux⟩ :... | false | {
"chosen_tokens": 266,
"rejected_tokens": 2,
"token_jaccard": 0.011111,
"token_length_ratio": 0.007519
} | sorry_admit_sorryAx_proof_hole | sorry | a44b982519654340f86b214831d62a92748b3435479141cb5dcdcf19fb02f84e | 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.LocallyConvex.Separation
public import Mathlib.Analysis.LocallyConvex.SeparatingDual
public import Mathlib.LinearAlgebra.Dual.Defs
public import Mathl... | 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": "Analysis/LocallyConvex",
"family_id": "convex",
"file_id": "mathlib/Mathlib/Analysis/LocallyConvex/WeakSpace.lean",
"sample_id": "d3baf770853c9de1742f6952ed6591411750a3d98ef5c34871eada480093b869"
} | train |
{
"chosen_proof_sha256": "5110e356531b5aca6a5cc8c637eddfe770b9e7e43c44858448f93b443e747ce9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "35aaf71a5609ad5967e4c59b74a8bb922b... | by
refine ⟨fun ⟨H⟩ ↦ ⟨?_⟩, fun ⟨H⟩ ↦ ⟨?_⟩⟩ <;>
simp only [isCardinalFiltered_aleph0_iff] at * <;>
exact H | true | {
"chosen_tokens": 38,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.131579
} | wrong_namespace_qualification | wrong_namespace_qualification | a46ac2d5f7bced63d57a0fa800eb58638242929bffa0cff5ac5b2529b1564b30 | 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.Filtered
public import Mathlib.CategoryTheory.Limits.Preserves.Filtered
public import Mathlib.CategoryTheory.Limits.Types.Filtered
public... | by
exact OProverMissing.missing_948388cf0ece | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "0157dc8d36be3b8cc1ad5eeb90d9ff4eeae1dce4cd0baf91ac2ecc4674ffadf3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Presentable",
"family_id": "hascardinalfilteredcolimits_iff_hasfilteredcolimitsofsize",
"file_id": "mathlib/Mathlib/CategoryTheory/Presentable/Finite.lean",
"sample_id": "948388cf0ecec93fe348cbe9fadcd98478b5894a8ffcc1cd67d7ba5e880077f1"
} | train |
{
"chosen_proof_sha256": "90ae162d9072def41fe3c1f6b71c6fb09d8794f02fb213178dc3f4878196efbb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [← cancel_epi (prodComparisonIso (Over.pullback f) _ _).hom.left,
Over.hom_left_inv_left_assoc]
simp [CartesianMonoidalCategory.prodComparison, snd] | false | {
"chosen_tokens": 33,
"rejected_tokens": 5,
"token_jaccard": 0.08,
"token_length_ratio": 0.151515
} | sorry_admit_sorryAx_proof_hole | sorryAx | a4aa8834606200485227db4654d309cea0b3c7d938e82f7e9186ec5227c6f509 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Adjunction.Limits
public import Mathlib.CategoryTheory.Comma.Over.Pullback
public import Mathlib.CategoryTheory.Limits.Constructions.Over.Produc... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Monoidal",
"family_id": "prodcomparisoniso_pullback_inv_left_fst_snd",
"file_id": "mathlib/Mathlib/CategoryTheory/Monoidal/Cartesian/Over.lean",
"sample_id": "3129c4684252f138a34ea19d6ab7a0e7965bec43e42276b25a947e2819c0b66a"
} | train |
{
"chosen_proof_sha256": "baf74685d38b5f76627c288eda6c3033c2f20e6a31a28995ec3f9cc5ab112e2a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c41500350895bfb16aa2300c4a21bbaf66... | by
haveI := Fintype.ofFinite ι
classical
refine (Basis.piTensorProduct b).ext_elem (fun q ↦ ?_)
simp [Finsupp.single_apply, Fintype.prod_ite_zero, ← funext_iff] | true | {
"chosen_tokens": 37,
"rejected_tokens": 3,
"token_jaccard": 0.032258,
"token_length_ratio": 0.081081
} | unknown_identifier | unknown_identifier | a50eb09c5dc13b9e7baffe35b1cbd2d4f9fa916c62c67d91bc4b8c94100a413c | 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.Finsupp.VectorSpace
public import Mathlib.LinearAlgebra.PiTensorProduct.Finsupp
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Daniel M... | by
exact __oprover_missing_317b87a5beef | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "51b630ec23df456c39a3f66e21e6af079c4c2cd5981346cc82443591834c6712",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/PiTensorProduct",
"family_id": "basis",
"file_id": "mathlib/Mathlib/LinearAlgebra/PiTensorProduct/Basis.lean",
"sample_id": "317b87a5beef988d47043cbd23eccd8456a804914b949c3de8ba8b8d221e791d"
} | train |
{
"chosen_proof_sha256": "54c464d6ce26bdef8908aa10edaaa4627d275c464a5704aa3ef40ec7a4c12203",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "39b50e23b82ce5537ed61a7d157a011c0e... | by
rintro ⟨m, hm⟩
have hm' : ∀ a, f a ≤ m := fun a ↦ hm <| Set.mem_range_self _
obtain ⟨a₀⟩ := ‹Nonempty α›
suffices ∀ b, f a₀ ≤ b → ∃ a, b < f a by
obtain ⟨a, ha⟩ : ∃ a, m < f a := this m (hm' a₀)
exact ha.not_ge (hm' a)
have h : ∀ a, ∃ a', f a < f a' := fun a ↦ (exists_gt a).imp (fun a' h ↦ hf h)
... | true | {
"chosen_tokens": 153,
"rejected_tokens": 5,
"token_jaccard": 0.056604,
"token_length_ratio": 0.03268
} | circular_dependency | circular_dependency | a51bdaecde2fd85d06fd381bca6bbf98e63583d37902c33a2a7cbe7040ee0fd5 | 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.SuccPred.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described... | by
exact StrictMono.not_bddAbove_range_of_isSuccArchimedean | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "d695199e13f0b965550c9f5560525db137aecb9838811b90f0a2ba7be0022612",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Order/SuccPred",
"family_id": "strictmono",
"file_id": "mathlib/Mathlib/Order/SuccPred/Archimedean.lean",
"sample_id": "d0ac2882ec427829126b8598a0819730b07105bce20b1228031035f70c0b76c9"
} | train |
{
"chosen_proof_sha256": "fa7ef57b668670c0961ba6cf31ace864279e791239ac9d2e2fff89a3656780bf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b98968742f45cb65d12f6f4c121c37ed0e... | by
rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id] | true | {
"chosen_tokens": 18,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.166667
} | unknown_identifier | unknown_identifier | a52ea9dda4cf5510a1acdc902343ddd6470392bcaef241db5416810ca1f94da9 | 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.Products.Basic
Namespace:
CategoryTheory.Bifunctor
Local context:
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apac... | by
exact __oprover_missing_f4231e8dda39 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "aa8c8af28b716053b8fa5ffe35ff9f072b8b4fbdeeceed70db27f2990957b21f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Products",
"family_id": "diagonal",
"file_id": "mathlib/Mathlib/CategoryTheory/Products/Bifunctor.lean",
"sample_id": "f4231e8dda39dee1bbddb1e4ffb21d16d698adcaa0cf76876d45ddbd2a042b9f"
} | train |
{
"chosen_proof_sha256": "9347b24919ebf6db54344437810a6552ffb014d358370aa936d28051c4570548",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d8c92fb88ac14308e19c63829929f1118e... | by
rw [eq_comm, ← Ultrafilter.coe_le_coe]
change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α | s ∈ v } ∈ u
simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff,
mem_setOf_eq]
constructor
· intro h a ha
exact h _ ⟨ha, a, rfl⟩
· rintro h a ⟨xi, a, rfl⟩
exact ... | false | {
"chosen_tokens": 80,
"rejected_tokens": 84,
"token_jaccard": 0.94,
"token_length_ratio": 1.05
} | missing_wrong_import | import | a553b6a93311ea378b1a9d2cb2c686abfa56cdf93685b941e0d54af26521c693 | 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.Bases
public import Mathlib.Topology.DenseEmbedding
public import Mathlib.Topology.Connected.TotallyDisconnected
Namespace:
(root)
Local context:
/-... | import Mathlib.WrongImport
by
rw [eq_comm, ← Ultrafilter.coe_le_coe]
change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α | s ∈ v } ∈ u
simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff,
mem_setOf_eq]
constructor
· intro h a ha
exact h _ ⟨ha, a, rfl⟩
· rintro ... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/Compactification",
"family_id": "ultrafilter_converges_iff",
"file_id": "mathlib/Mathlib/Topology/Compactification/StoneCech.lean",
"sample_id": "02e40ac0605619b9d35a8cf4395e31d8750e7cb631f2db38ba43f21adcd702b7"
} | train |
{
"chosen_proof_sha256": "d23fb76aeeed9353afe60037f9d9b0091fa23e08b64b45bbb0d2d34201990525",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [loopOfHom, ← Category.assoc, IsIso.comp_inv_eq, Category.id_comp]
rcases H with H | H
· rw [treeHom_eq T (Path.cons default ⟨Sum.inl e, H⟩), homOfPath]
rfl
· rw [treeHom_eq T (Path.cons default ⟨Sum.inr e, H⟩), homOfPath]
simp only [IsIso.inv_hom_id, Category.comp_id, Category.assoc, treeHom] | false | {
"chosen_tokens": 86,
"rejected_tokens": 2,
"token_jaccard": 0.025641,
"token_length_ratio": 0.023256
} | sorry_admit_sorryAx_proof_hole | admit | a571e3c35aa7440511981858dcd61caa6cc8c057635cfa26894f9ad9199d865f | 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.Action
public import Mathlib.Combinatorics.Quiver.Arborescence
public import Mathlib.Combinatorics.Quiver.ConnectedComponent
public import Mathl... | 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": "GroupTheory/FreeGroup",
"family_id": "loopofhom_eq_id",
"file_id": "mathlib/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean",
"sample_id": "0e496e19d8169f840fb42721b4ba0ad691cd3e68e48df661f9f6fc16f635b298"
} | train |
{
"chosen_proof_sha256": "b2fc7f29216c9249b22ab675eb45a0b56030097a0051e019b9740cd584e92cb3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "90a524481a706873c8f8be57d178063511... | by
simp only [liftFun_iff_succ r, forall_iff_succ, cons_val_succ, cons_val_zero, ← succ_castSucc,
castSucc_zero]
variable [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α} | true | {
"chosen_tokens": 45,
"rejected_tokens": 3,
"token_jaccard": 0.03125,
"token_length_ratio": 0.066667
} | unknown_identifier | unknown_identifier | a5a98b2b82056923ba1ae5cc9b220874a47e971c32b4e864ca7ab0b417211dcc | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Logic.Equiv.Fin.Basic
public import Mathlib.Order.Fin.Basic
public import Mathlib.Order.PiLex
public import Mathlib.... | by
exact __oprover_missing_34e8994f5ec2 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "4545c51f78b10c4746e0a81afaa7ab82f17a56df42cf05e6f1c5d0c4f8059431",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Fin",
"family_id": "liftfun_veccons",
"file_id": "mathlib/Mathlib/Order/Fin/Tuple.lean",
"sample_id": "34e8994f5ec220f3f31f1cf7f6b1b16bc75863bde20089fcb254304fbec368fa"
} | train |
{
"chosen_proof_sha256": "991657dbd4edd454359c31dd8bdf594c016e0afa8edd3fccac619f800a61e43a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8b50f5b0dc46f4363a12bc6de0e2753f76... | by
simp [pointOfClosedPoint, ← SpecMap_residueFieldIsoBase_inv, ← Spec.map_comp] | true | {
"chosen_tokens": 13,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.230769
} | circular_dependency | circular_dependency | a5c81dc0255fab260b6e440824d90dcb075b869f129ad952c0695073e0c0a6ec | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.Finite
public import Mathlib.FieldTheory.IsAlgClosed.Basic
Namespace:
AlgebraicGeometry
Local context:
/-
Copyright (c) 2026 Andr... | by
exact pointOfClosedPoint_comp | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "48fe44c62faf4a71a9dae781c9c0b690b8e9bf64729deabd5648596f0f4ecb2b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/AlgClosed",
"family_id": "pointofclosedpoint_comp",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/AlgClosed/Basic.lean",
"sample_id": "2adf0561a9917b239f775543158667a626bcb1a48a3b86163579bd9e8eb95116"
} | train |
{
"chosen_proof_sha256": "0bd39f4463d7de80fb25819949e6b30d4389ae1eaeb6979f0ff98507bcbb9bbf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b95a85964efe4bfd41af2d45319f1c087d... | by
cat_disch
variable {D : Type u'} [𝒟 : Category.{u'} D] (I J : D ⥤ C) (ρ : I ⟶ J) {W : D} | false | {
"chosen_tokens": 39,
"rejected_tokens": 43,
"token_jaccard": 0.884615,
"token_length_ratio": 1.102564
} | missing_wrong_import | import | a630f0cff0b0ca9510bc167e5a818c3f816f7577dfbc55013e371950544f4953 | 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.Elementwise
public import Mathlib.CategoryTheory.EpiMono
public import Mathlib.Tactic.PPWithUniv
public import Mathlib.Tactic.ToAdditive
Namesp... | import Mathlib.WrongImport
by
cat_disch
variable {D : Type u'} [𝒟 : Category.{u'} D] (I J : D ⥤ C) (ρ : I ⟶ J) {W : D} | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Types",
"family_id": "eqtohom_map_comp_apply",
"file_id": "mathlib/Mathlib/CategoryTheory/Types/Basic.lean",
"sample_id": "f97af12d0eda400dcad66bb72ae73c904310f1b01ffca3c206a7820ced091c83"
} | train |
{
"chosen_proof_sha256": "61dd4a4b69745b7e0fa68c4103cf026764280161c8ce4459f05b5e5caa88e47f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "113a4b35611419334ebc80fdab8a9b41f3... | by
simp_rw [univLE_iff_cardinal_le]; apply le_total | true | {
"chosen_tokens": 8,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.625
} | wrong_namespace_qualification | wrong_namespace_qualification | a64d1f7253bedebff92cced156c06005fa6b375b79f2836e32b8d55d6759bf9e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Logic.UnivLE
public import Mathlib.SetTheory.Ordinal.Univ
Namespace:
(root)
Local context:
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released und... | by
exact OProverMissing.missing_7de96c1ef780 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a7cef40ca6bea23817455a518594ece5bb094d201f96f748c81aa510036c97df",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "SetTheory/Cardinal",
"family_id": "univle_total",
"file_id": "mathlib/Mathlib/SetTheory/Cardinal/UnivLE.lean",
"sample_id": "7de96c1ef7802d716a78d8f34f82e51e820ae7e56b3d2af666ad2f6428d02fa0"
} | train |
{
"chosen_proof_sha256": "41ba997d60350b8410456fab7fd86644a15e80166bdc8dbee3ebfacac2920661",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "869b7b26a3015184cd23c518f62b205765... | by
induction l with
| nil => simp only [List.not_mem_nil, false_and, exists_false] at hl
| cons x l ih =>
simp only [List.mem_cons, exists_eq_or_imp] at hl
rcases hl with h | hal
· simpa only [List.map_cons, List.prod_cons] using I.mul_mem_right _ _ h
· simpa only [List.map_cons, List.prod_cons] u... | false | {
"chosen_tokens": 84,
"rejected_tokens": 89,
"token_jaccard": 0.880952,
"token_length_ratio": 1.059524
} | markdown_preamble_trailing | preamble | a68115044813e79127693707677f47018f1c1270241d0e99b4d86216d7770ffb | 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.Congruence.BigOperators
public import Mathlib.RingTheory.TwoSidedIdeal.Basic
Namespace:
TwoSidedIdeal
Local context:
/-
Copyright (c) 2024 Jujian ... | Here is the proof:
by
induction l with
| nil => simp only [List.not_mem_nil, false_and, exists_false] at hl
| cons x l ih =>
simp only [List.mem_cons, exists_eq_or_imp] at hl
rcases hl with h | hal
· simpa only [List.map_cons, List.prod_cons] using I.mul_mem_right _ _ h
· simpa only [List.map_cons... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/TwoSidedIdeal",
"family_id": "listprod_mem",
"file_id": "mathlib/Mathlib/RingTheory/TwoSidedIdeal/BigOperators.lean",
"sample_id": "ab793cebc21f139d2eab524640ed552866c6f6bd28b0f8c110edadda18cfb9c9"
} | train |
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