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": "bbbd68780f20b0315d7a36f5a0356a461835a08a2fef3351d5866ce71a89c492",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_pow] at hxy ⊢
rw [hxy] | true | {
"chosen_tokens": 23,
"rejected_tokens": 8,
"token_jaccard": 0.043478,
"token_length_ratio": 0.347826
} | unsolved_goals | unsolved_goals | 5c7529cda976d85a9a9adf1b9b0f3b53232f647a423d16d2aec41c4810f0c835 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Module.Submodule.Map
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.RingTheory.Ideal.Quotient.Defs
public import Mathlib.Alge... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "504115d569001f0a67a9ce455f0e9cd144d5f834fa1d24a2642020756e258b38",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/SModEq",
"family_id": "pow",
"file_id": "mathlib/Mathlib/LinearAlgebra/SModEq/Basic.lean",
"sample_id": "550eea517b6f57ead3df265df6cf73d2563e67f3ae71b6c49159805a9ce8003d"
} | train |
{
"chosen_proof_sha256": "7333d1a239a893fe44382cfe648dc46c728566269ad1ec5413c976d55e86e914",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b445b02c8a845e7a62745b861f317ebf3f... | by
simpa only [Ordinal.liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, Order.succ_le_iff] using!
le_of_lt (Ordinal.liftPrincipalSeg.{u, u + 1}.lt_top (Order.succ c).ord) | false | {
"chosen_tokens": 44,
"rejected_tokens": 48,
"token_jaccard": 0.90625,
"token_length_ratio": 1.090909
} | missing_wrong_import | import | 5cda1d4857e72035b57376b1472fbb443a60d895d5995a95110903e3f63a40b8 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.SetTheory.Ordinal.Basic
Namespace:
Cardinal
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as de... | import Mathlib.WrongImport
by
simpa only [Ordinal.liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, Order.succ_le_iff] using!
le_of_lt (Ordinal.liftPrincipalSeg.{u, u + 1}.lt_top (Order.succ c).ord) | {
"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": "SetTheory/Ordinal",
"family_id": "lift_lt_univ",
"file_id": "mathlib/Mathlib/SetTheory/Ordinal/Univ.lean",
"sample_id": "dabe8f52856a803fb511e285a4cf7be82ea7a0120495bcf033465a0c9f492e10"
} | train |
{
"chosen_proof_sha256": "bbb068c1b254f9a62b1fe49e6c1cd52f49bd7cbde7d3a811059b2617b8439517",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
have h0 : v πᵥ ≠ 0 := by simpa using uniformizingPolynomial_ne_zero hle
induction f using RatFunc.induction_on with
| f p q hq =>
use (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {p})).factors -
(Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {q})).factors
simp only... | false | {
"chosen_tokens": 116,
"rejected_tokens": 2,
"token_jaccard": 0.019231,
"token_length_ratio": 0.017241
} | sorry_admit_sorryAx_proof_hole | sorry | 5d25718484196511abe16bf9004a3b4d745cd58cff3c2facf14b202951e3da9d | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.Finite.Valuation
public import Mathlib.NumberTheory.FunctionField
public import Mathlib.RingTheory.Valuation.Discrete.Basic
Namespace:
RatFunc
Lo... | by
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/RatFunc",
"family_id": "exists_zpow_uniformizingpolynomial",
"file_id": "mathlib/Mathlib/NumberTheory/RatFunc/Ostrowski.lean",
"sample_id": "c87ad3ae24029059963e03348c139139f0d135dcb45808f3d2427d4e578980f8"
} | train |
{
"chosen_proof_sha256": "4525ed279745c1a825a2d966ade6667dde0e9859d5d810c453aa56b8f6012282",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7b0349090d26641392842231a42304f290... | by
simpa only using! congr_fun (f.isCompat' φ) x | false | {
"chosen_tokens": 13,
"rejected_tokens": 17,
"token_jaccard": 0.8125,
"token_length_ratio": 1.307692
} | missing_wrong_import | import | 5d38c44ba3feb80d6feedf76a67ab657983c710f80fe522050ca6aea967f4929 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorProduct.RightExactness
public import Mathlib.RingTheory.Congruence.Hom
public import Mathlib.RingTheory.FiniteType
public import Mathlib.Ri... | import Mathlib.WrongImport
by
simpa only using! congr_fun (f.isCompat' φ) x | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/PolynomialLaw",
"family_id": "polynomiallaw",
"file_id": "mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean",
"sample_id": "06f153f627e2ad13220f5061c7800c30d1bdd6c1da9c7ac7f9963af2c8d4a748"
} | train |
{
"chosen_proof_sha256": "00de7867ae9086f2cf9c529fdfa5d87816a8332564eca883373884a7662e8cc7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "85d7ced1570d9b1f378e8a36e8670d4ffe... | by
rw [rowLen, Nat.lt_find_iff]
push Not
exact ⟨fun h _ hmj => μ.up_left_mem (by rfl) hmj h, fun h => h _ (by rfl)⟩ | false | {
"chosen_tokens": 38,
"rejected_tokens": 45,
"token_jaccard": 0.923077,
"token_length_ratio": 1.184211
} | markdown_preamble_trailing | fence | 5d7e4ddf47490630e40bfbde73f1f4c6c8a607b0e9ea032ed0b24da7a730e6d7 | 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.Preimage
public import Mathlib.Data.Finset.Prod
public import Mathlib.Data.SetLike.Basic
public import Mathlib.Order.UpperLower.Basic
Namespace:
Y... | ```lean
by
rw [rowLen, Nat.lt_find_iff]
push Not
exact ⟨fun h _ hmj => μ.up_left_mem (by rfl) hmj h, fun h => h _ (by rfl)⟩
``` | {
"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/Young",
"family_id": "mem_iff_lt_rowlen",
"file_id": "mathlib/Mathlib/Combinatorics/Young/YoungDiagram.lean",
"sample_id": "7d66ee4f35ab6035f8dacc7a5cf8575522e26c3c67988759a6c4ebffd4318014"
} | train |
{
"chosen_proof_sha256": "839d48065eb916980f8e3d750584330734f1d7501b43b9c8e7cc3615d69e6548",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "18639f8ebcd25d6e01b9f3933b9d096b5b... | by
classical
exact if a0 : a = 0 then
a0.symm ▸ H0 b
else
have _ := mod_lt b a0
H1 _ _ a0 (GCD.induction (b % a) a H0 H1)
termination_by a | true | {
"chosen_tokens": 42,
"rejected_tokens": 5,
"token_jaccard": 0.185185,
"token_length_ratio": 0.119048
} | circular_dependency | circular_dependency | 5ddc9cd3ae4188262e96bfd70d39d88718fe5cbeceab145bf34e53bdfa885299 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.Order.RelClasses
Namespace:
EuclideanDomain
Local context:
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
R... | by
exact GCD.induction | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "044f9cb2078be90a5358d28fe7970f2f36fc21a24ff61936a30a0975b9ddb53d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/EuclideanDomain",
"family_id": "gcd",
"file_id": "mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean",
"sample_id": "04cd638bd1240f2fe7fe9709e6cdc0eae054c396fecbe87a5f13fcb8dac94627"
} | train |
{
"chosen_proof_sha256": "3e9d48f580176762fa7a5e0ebfd29db0eb21f956c84ea97eca970101095c5878",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "352387881c46e504868b78447c6e9de91b... | by
letI : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
letI : MeasurableSpace Ω := 𝓕 n
let X m := (Function.curry (h𝓕.approx m) (n + 1))
refine ⟨(fun m ↦ SimpleFunc.mk (X m) ?_ ?_), (fun ω ↦ h𝓕.tendsto_approx ⟨(n + 1), ω⟩)⟩
· intro s
rw [(by aesop : X m ⁻¹' {s} = {ω | (n + 1, ω) ∈ h𝓕.approx m ⁻¹' {s}})]
... | true | {
"chosen_tokens": 154,
"rejected_tokens": 5,
"token_jaccard": 0.033333,
"token_length_ratio": 0.032468
} | wrong_namespace_qualification | wrong_namespace_qualification | 5e01c0d2e10bc2323650bf0dd3b56888dba0a4aa7ac264fafd110ff226bee327 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Probability.Process.Adapted
Namespace:
MeasureTheory.IsStronglyPredictable
Local context:
/-
Copyright (c) 2025 Kexing Ying. All rights reserved.
Released un... | by
exact OProverMissing.missing_57ea28a7b08d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c306a7148b10c6b61881053d17fe2a9398891b75e0c836e1ab98c3e7021ce396",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability/Process",
"family_id": "measurable_add_one",
"file_id": "mathlib/Mathlib/Probability/Process/Predictable.lean",
"sample_id": "57ea28a7b08d8f885b958267bc30048df940f453f6ddd0117c4636da80751a9c"
} | train |
{
"chosen_proof_sha256": "da2d9310157ed9ceff8c58c2de745bed1ed2329391fedb4f8010b3557e4b56ed",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
have h : ∃ n, x ≤ t n := by
simp only [tendsto_atTop_atTop_iff_of_monotone ht_mono.monotone] at ht_tendsto
exact ht_tendsto x
have h' m := Nat.find_min h (m := m)
simp only [not_le] at h'
exact ⟨Nat.find h - 1, h' _ (by simp [hx]), by simp [Nat.find_spec h, hx]⟩ | false | {
"chosen_tokens": 76,
"rejected_tokens": 2,
"token_jaccard": 0.026316,
"token_length_ratio": 0.026316
} | by_exact_placeholder | by? | 5ecfac8ac7314e2dfdcfa6a3c4aa619ffe9a03c12334bbcf7ce588d77263e0b7 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecificLimits.ArithmeticGeometric
public import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
public import Mathlib.MeasureTheor... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Probability/Distributions",
"family_id": "strictmono",
"file_id": "mathlib/Mathlib/Probability/Distributions/Fernique.lean",
"sample_id": "9b4e3e0d6a7653b54818bb9ece9503e507bf5b7abbd8d2d4b99515af39aba54b"
} | train |
{
"chosen_proof_sha256": "bc3578532551ba1c8a4369ed77aebd3d2f1775c8fadb93080f656df6af2cd8fe",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b97fb9e2cc68aedd758bfae5333c8c15c2... | by
have ⟨φ⟩ := ne
refine h.toIsAlgebraic.of_injective φ φ.injective
|>.normalClosure_eq_iSup_adjoin_of_splits fun x ↦ ?_
rw [← minpoly.algHom_eq _ φ.injective]
apply h.splits | true | {
"chosen_tokens": 41,
"rejected_tokens": 2,
"token_jaccard": 0.033333,
"token_length_ratio": 0.04878
} | invalid_tactic | invalid_tactic | 5ed4bb05bdf22b4c2a32b645b078308803bebee357a3cb730607d3493fcc0bfe | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.SimpleRing.Basic
public import Mathlib.FieldTheory.Normal.Basic
public import Mathlib.Order.Closure
public import Mathlib.LinearAlgebra.FreeModule.F... | by
__oprover_invalid_tactic_f6fe57d0752e | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "7b1518385a5ba9d0a83bd1d8e58195e5c9ddaf1c9c9b2760a13f5c857f9ce18a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/Normal",
"family_id": "normalclosure_eq_isup_adjoin",
"file_id": "mathlib/Mathlib/FieldTheory/Normal/Closure.lean",
"sample_id": "f6fe57d0752e0cd229648ee764e154901c2c974592574e9f5ce3b35211409c9e"
} | train |
{
"chosen_proof_sha256": "1f1ea0e13f9e5b99d77e0d90716326ac0d10ebd31e81aeaa4cd98f72f2f5c63c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
induction n, hn using Nat.le_induction with
| base => simp
| succ k hk ih =>
have hck : 0 ≤ c k := hc k hk
have heq : c k * ∑ j ∈ Ico n₀ k, b j * ∏ i ∈ Ico (j + 1) k, c i + b k =
∑ j ∈ Ico n₀ (k + 1), b j * ∏ i ∈ Ico (j + 1) (k + 1), c i := by
rw [sum_Ico_succ_top hk, mul_sum, Ico_self, p... | true | {
"chosen_tokens": 286,
"rejected_tokens": 8,
"token_jaccard": 0.054054,
"token_length_ratio": 0.027972
} | unsolved_goals | unsolved_goals | 5ed7f629890456a76dc7af2a5eab7208c0fe428066ab25de49288d6fe67797ca | 1 | 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.ExpDeriv
Namespace:
(root)
Local context:
/-
Copyright (c) 2026 Dennj Osele. All rights reserved.
Released under Apache 2.0 license... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "af10428edcc082dd3f679ee77f769d82028cd6e459370567dcf0dc593de70cc1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/ODE",
"family_id": "discrete_gronwall_prod_general",
"file_id": "mathlib/Mathlib/Analysis/ODE/DiscreteGronwall.lean",
"sample_id": "19249b12421b93ef9aca4c5c6690238801a0a01bd844a6075bfc5c80d4237b2d"
} | train |
{
"chosen_proof_sha256": "fd406c6900b20879ac4c2cbd6762020da2098ae8a94437f8bd328ff516fb751d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
have := Adjunction.rightAdjointUniq_hom_counit (mapPullbackAdj (𝟙 X)) (id X).mapPullbackAdj
rw [pullbackId, Adjunction.rightAdjointUniq_hom_counit] | false | {
"chosen_tokens": 27,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.074074
} | sorry_admit_sorryAx_proof_hole | sorry | 5ef28815007bfd414a8fb7a9b009a5a3650e905cdf4e5575bf130ad6334448c3 | 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.Comma.Over.Pullback
public import Mathlib.CategoryTheory.Monoidal.Cartesian.Basic
public import Mathlib.CategoryTheory.Adjunction.Unique
public ... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/LocallyCartesianClosed",
"family_id": "pullbackid_hom_counit",
"file_id": "mathlib/Mathlib/CategoryTheory/LocallyCartesianClosed/ChosenPullbacksAlong.lean",
"sample_id": "9b1bbd8b0b0a1747cccd2009098a56d3ba588503f823228d885fa9ccd0f30c49"
} | train |
{
"chosen_proof_sha256": "0c64ae53f4b1abd57d7d7d427b1d8d5ae6a8abade34b2f455335c0dfb08d0184",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [subsingleton_iff_support_subset, PrimeSpectrum.basicOpen_eq_zeroLocus_compl,
disjoint_compl_left_iff, Set.le_iff_subset] | true | {
"chosen_tokens": 15,
"rejected_tokens": 8,
"token_jaccard": 0.055556,
"token_length_ratio": 0.533333
} | unsolved_goals | unsolved_goals | 5f349b6c05d09baa879acf42cc9ecb0a4f46f00d4dba0361dcd14f99d23de449 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Spectrum.Prime.Topology
public import Mathlib.RingTheory.Support
Namespace:
(root)
Local context:
/-
Copyright (c) 2024 Andrew Yang. All rights re... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "37d4ff0ae14f24830f528636dabca2fbba5b391e7c706d600a69dbb19bd9f263",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Spectrum",
"family_id": "localizedmodule",
"file_id": "mathlib/Mathlib/RingTheory/Spectrum/Prime/Module.lean",
"sample_id": "f05312ee1b649b954d0ad3d3cedaebdac4a9c0766690dacc1ff7776f9463beba"
} | train |
{
"chosen_proof_sha256": "9427a4321d73c521b7283942cf3246fe882d0cae4ae971903987493e7ed9579b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [isUnit_iff_exists]
exact ⟨n, inv_mul_of_unit _ h, mul_inv_of_unit _ h⟩ | false | {
"chosen_tokens": 17,
"rejected_tokens": 3,
"token_jaccard": 0.133333,
"token_length_ratio": 0.176471
} | by_exact_placeholder | placeholder | 5f4f14ece8555daf0ac0a747863fb91a74a3c7bc21a64b2c7748898e4495f8e1 | 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.Associated
public import Mathlib.Data.ZMod.Basic
public import Mathlib.RingTheory.Coprime.Lemmas
Namespace:
ZMod
Local context:
/-
Copyr... | 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": "Data/ZMod",
"family_id": "isunit_inv",
"file_id": "mathlib/Mathlib/Data/ZMod/Units.lean",
"sample_id": "7c3c6fabe63da232039bfcfad352f1cc6f001b2ce6fe12a8d6099283072984c1"
} | train |
{
"chosen_proof_sha256": "3b83d6405db8d7b67a22120881e91f0e3ce367fb9b3ca08e1bae760a47811d62",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "02dd01e0fea4843490de53f3241bde2b1e... | by
cases Nat.eq_zero_or_pos n with
| inl hn =>
rw [if_pos hn, hn, dp_zero]
| inr hn =>
rw [if_neg (ne_of_gt hn), ← zero_smul R (0 : M), dp_smul]
rw [zero_pow (Nat.pos_iff_ne_zero.mp hn), zero_smul] | false | {
"chosen_tokens": 57,
"rejected_tokens": 62,
"token_jaccard": 0.888889,
"token_length_ratio": 1.087719
} | markdown_preamble_trailing | trailing | 5f7f5e7698bac394bffbae83e9ad9f779c3209c99213a89ea051e4792f65683f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MvPolynomial.Eval
public import Mathlib.Algebra.RingQuot
public import Mathlib.RingTheory.DividedPowers.Basic
Namespace:
DividedPowerAlgebra
Local co... | by
cases Nat.eq_zero_or_pos n with
| inl hn =>
rw [if_pos hn, hn, dp_zero]
| inr hn =>
rw [if_neg (ne_of_gt hn), ← zero_smul R (0 : M), dp_smul]
rw [zero_pow (Nat.pos_iff_ne_zero.mp hn), zero_smul]
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/DividedPowerAlgebra",
"family_id": "dp_null",
"file_id": "mathlib/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean",
"sample_id": "ae60c1f3772d8f03a4244e5c6a05fff7f8704657d4e3b6abd61ebd01d5c23d05"
} | train |
{
"chosen_proof_sha256": "b654cbf08d5ade13b5e6184e32b41bf84d2fa139990565902f32727bb1db98ee",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "de40c21181d595bbe1edc7f9ec432ad2cd... | by
refine ⟨fun ⟨g, hg⟩ ↦ ?_, fun h ↦ (isCuspForm_iff f).mpr (isZeroAt_of_coeffZero_eq_zero f h)⟩
rw [← hg, qExpansion_coeff_zero one_pos
(ModularFormClass.analyticAt_cuspFunction_zero _ one_pos one_mem_strictPeriods_SL)
(periodic_comp_ofComplex _ one_mem_strictPeriods_SL)]
exact (CuspFormClass.zero_at_inf... | true | {
"chosen_tokens": 58,
"rejected_tokens": 3,
"token_jaccard": 0.058824,
"token_length_ratio": 0.051724
} | circular_dependency | circular_dependency | 5fe429e9e2acd7222f74c4432aa1608622c4b2dd167763db7f56386493186b9e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.NumberTheory.ModularForms.QExpansion
public import Mathlib.NumberTheory.ModularForms.LevelOne.Basic
public import Mathlib.NumberTheory.ModularForms.EisensteinS... | by
exact isCuspForm_iff_coeffZero_eq_zero | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "93960059c6b35838146de0a4e3d0d098404924f5c50df6b54a77f288340c28b7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/ModularForms",
"family_id": "iscuspform_iff_coeffzero_eq_zero",
"file_id": "mathlib/Mathlib/NumberTheory/ModularForms/CuspFormSubmodule.lean",
"sample_id": "0eb67f824175dac229b7646063b2648b75a17b30e9821e1a51528362c79353d3"
} | train |
{
"chosen_proof_sha256": "17f240f800994d7074febbde9e3f6dc2b5012103ed8b58582f7bf051f16e73b0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c5a217f06acc8ac4d7b7d814d63ee49880... | by
intro b₁ b₂ h
simpa only [and_true, eq_self_iff_true, mk_inj] using h | false | {
"chosen_tokens": 18,
"rejected_tokens": 25,
"token_jaccard": 0.882353,
"token_length_ratio": 1.388889
} | markdown_preamble_trailing | fence | 604faa073f61840109fecb5a03b4046738a28b6a8f2702eb202cee2a295c533d | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Lean.PrettyPrinter.Delaborator.Builtins
public import Mathlib.Logic.Function.Defs
public import Mathlib.Logic.Function.Iterate
public import Mathlib.Tactic.Inhabit
pub... | ```lean
by
intro b₁ b₂ h
simpa only [and_true, eq_self_iff_true, mk_inj] 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": "Data/Prod",
"family_id": "mk_left_injective",
"file_id": "mathlib/Mathlib/Data/Prod/Basic.lean",
"sample_id": "dd825554f724a499bc7ed68c1db6d9b3cab6e04c69fa70bc2acb5b885250233e"
} | train |
{
"chosen_proof_sha256": "d4554f84e387bbee7a732a6c3f4886d4878a68fc65f7eb3a496b2c61095b621d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
simp [-objD, -obj_X] | false | {
"chosen_tokens": 9,
"rejected_tokens": 2,
"token_jaccard": 0.111111,
"token_length_ratio": 0.222222
} | sorry_admit_sorryAx_proof_hole | admit | 608812857d9c8de1bda119fba6700eebdc92519d6b923357e61649b0da01a71e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Homology.HomologicalComplex
public import Mathlib.AlgebraicTopology.SimplicialObject.Basic
public import Mathlib.CategoryTheory.Abelian.Basic
Namespac... | 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": "AlgebraicTopology",
"family_id": "normalizedmoorecomplex_objd",
"file_id": "mathlib/Mathlib/AlgebraicTopology/MooreComplex.lean",
"sample_id": "95cc52a04da20a5f5c49fe48e2d7c757cb686f4efaddac32474d2f88abd5ceb5"
} | train |
{
"chosen_proof_sha256": "280119e14881b51170fc3d0d4d95c5bf43762c9b1e1b31c95c9da1e53082f7ea",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "68756b8c7ed9cf05ae44a5c6d82ef89bec... | 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 | true | {
"chosen_tokens": 35,
"rejected_tokens": 3,
"token_jaccard": 0.064516,
"token_length_ratio": 0.085714
} | unknown_identifier | unknown_identifier | 61186375c9c82fd6e361fb61ba2dc721e531b03f1918cda05e874ccdd64363d6 | 1 | 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 __oprover_missing_b84cec6b1d75 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "0a26ae39026f0e5b2ca8202bfab8de17cf723df1a9e3af431f76b965b8751387",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "5bd7c7167db7f215623bed49eb0f0085e5528c13934e7559ad6540e992b06a8a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
letI := h𝕜.rclike
exact hs.mono_field | false | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.181818
} | by_exact_placeholder | by? | 6180baa8390b4f66256715cc7521fc44755d22acbdf801a1932536dcc6cf7e7a | 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.RCLike.Basic
public import Mathlib.Topology.Instances.RealVectorSpace
public import Mathlib.Analysis.Calculus.TangentCone.Real
Namespace:
(root)
Loc... | 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": "Analysis/RCLike",
"family_id": "uniquediffwithinat",
"file_id": "mathlib/Mathlib/Analysis/RCLike/TangentCone.lean",
"sample_id": "2e71596933c22369bea1ce7d19358deda1acb57a1b511e60283fae72b0ed1051"
} | train |
{
"chosen_proof_sha256": "ad3461d90bc1a2d6d0744349016f752c25c2128b46b806f78f2e03cb5879efbf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "958dff2c8ca01f64b80069695c6cce9df9... | by
have uc : (iUnion fun i => Option.elim' sᶜ u i) = univ := by
apply Subset.antisymm (subset_univ _)
· simp_rw [← compl_union_self s, Option.elim', iUnion_option]
apply union_subset_union_right sᶜ us
rcases precise_refinement (Option.elim' sᶜ u) (Option.forall.2 ⟨isOpen_compl_iff.2 hs, uo⟩)
uc ... | true | {
"chosen_tokens": 143,
"rejected_tokens": 3,
"token_jaccard": 0.029412,
"token_length_ratio": 0.020979
} | circular_dependency | circular_dependency | 61836ccf1928cb838a95af86894df1d7fafc1594cdc566807366fbaa41adfec0 | 1 | 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.Option.Basic
public import Mathlib.Topology.Separation.Regular
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserv... | by
exact precise_refinement_set | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "66ebdfc1bef0defc4f2675a58a412d95d58da93a2301f1ea001a9135307ef2d1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Compactness",
"family_id": "precise_refinement_set",
"file_id": "mathlib/Mathlib/Topology/Compactness/Paracompact.lean",
"sample_id": "45c618e63e4bfb258f2eb5bccb2da6bd795c57ebdbec6a654acc6ef23d1bc7ee"
} | train |
{
"chosen_proof_sha256": "85d4d418bf899b74281aa7eeb44015fb0b192ec7ec0ce946edf50b0cafb9b526",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
ext
simp [map, hw] | false | {
"chosen_tokens": 8,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.375
} | by_exact_placeholder | placeholder | 618debe6a91ef617a7cddbc17196549300383a5942679bf6a64d65d14f661198 | 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.LocallyFinsupp
public import Mathlib.Topology.Spectral.Basic
Namespace:
Function.locallyFinsupp
Local context:
/-
Copyright (c) 2026 Raphael Douglas... | 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/LocallyFinsupp",
"family_id": "map_id",
"file_id": "mathlib/Mathlib/Topology/LocallyFinsupp/Pushforward.lean",
"sample_id": "ed1374ebe28e9b99f5e81874724199c0edfb80ce0f4bc23069e0f6178e331a7a"
} | train |
{
"chosen_proof_sha256": "0678ccbc2752fb8380a44655810f825f8a9182914dd6e363b01acde3b055d878",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "501f3f54663e4d1e0765f46e1474477e03... | by
induction y generalizing x with
| nil => simpa using h
| cons y ys ih => exact T.prop (ih (by simpa)) | true | {
"chosen_tokens": 29,
"rejected_tokens": 2,
"token_jaccard": 0.045455,
"token_length_ratio": 0.068966
} | invalid_tactic | invalid_tactic | 61b3372f1461b554ffd6e07a6d5ce191b26e2b99d5b44bb6f80de9a043ea9c3b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.CompleteLattice.SetLike
Namespace:
Descriptive.Tree
Local context:
/-
Copyright (c) 2024 Sven Manthe. All rights reserved.
Released under Apache 2.0 li... | by
__oprover_invalid_tactic_2d5cb89a9cfd | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "0f354678c0f83971db7925ae6659b7f54eeeac4a8991428d1c82ccdc51e8ad88",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "SetTheory/Descriptive",
"family_id": "mem_of_append",
"file_id": "mathlib/Mathlib/SetTheory/Descriptive/Tree.lean",
"sample_id": "2d5cb89a9cfda0560e3764a0fa8315a7c93635c11c81d041a427e94157d40461"
} | train |
{
"chosen_proof_sha256": "05b7a960b09bb29c3bdb02159f0748a158edd33fe39806e604a0595973e7a4e9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [isSemilinearSet_iff] at hs
rcases hs with ⟨S, hS, rfl⟩
choose φ hφ using fun s : S => (hS s.1 s.2).definable
refine ⟨Formula.iSup φ, ?_⟩
ext x
have := fun s hs x => Set.ext_iff.1 (hφ ⟨s, hs⟩).symm x
simp only [mem_setOf_eq] at this
simp [this] | false | {
"chosen_tokens": 85,
"rejected_tokens": 3,
"token_jaccard": 0.021277,
"token_length_ratio": 0.035294
} | by_exact_placeholder | placeholder | 61be77ea0fb5c21e05ae0939392bda1474e0bdfd53521e34f8bf2a08d19e3453 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.ModelTheory.Arithmetic.Presburger.Basic
public import Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic
public import Mathlib.ModelTheory.Definability... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "ModelTheory/Arithmetic",
"family_id": "issemilinearset",
"file_id": "mathlib/Mathlib/ModelTheory/Arithmetic/Presburger/Definability.lean",
"sample_id": "861b7d8a1fe1c1ac90ef7cb56d31d4772a1a7fac4746b22305f0316ae345cc5c"
} | train |
{
"chosen_proof_sha256": "66e860b3b879ffe015e76af344d6b2852af86ab51c1939b65912c04b3badbf97",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8bccf914e3afe1f74007d8940af4723ed3... | by
refine ⟨fun h => ?_, fun h => h.symm ▸ mulSingle_one i⟩
rw [← mulSingle_eq_same i x, h, one_apply] | true | {
"chosen_tokens": 30,
"rejected_tokens": 3,
"token_jaccard": 0.041667,
"token_length_ratio": 0.1
} | unknown_identifier | unknown_identifier | 620ec99f2740f4221cdd0448a9e955acc76d3351042523d9f4916d9731993ca4 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Notation.Pi.Defs
public import Mathlib.Logic.Function.Basic
Namespace:
Pi
Local context:
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Rele... | by
exact __oprover_missing_6c05ada6fb86 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "28b1d82e3402bc5d5fd255754a4dbc2634054d3077ac89f8a5f7b48a8ebf634d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Algebra/Notation",
"family_id": "mulsingle_eq_one_iff",
"file_id": "mathlib/Mathlib/Algebra/Notation/Pi/Basic.lean",
"sample_id": "6c05ada6fb867e067b57945bee363d3116da5202db3c69d06f6b508b9b741204"
} | train |
{
"chosen_proof_sha256": "db5676c2a4baaaa8601683c727e173d0b3b0fe87aa92feb60828f3a50296fa53",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
have hp : ((f.support : Set G).smulAntidiagonal (Function.support x) p).Finite :=
Set.SMulAntidiagonal.finite_of_finite_fst f.support.finite_toSet x.support p
set s : Set (G × P) := ↑(Finset.SMulAntidiagonal p hp)
have h₁ : s.InjOn Prod.fst := fun _ h₁ _ h₂ h ↦ by
rw [Finset.mem_coe, mem_smulAntidiagon... | false | {
"chosen_tokens": 224,
"rejected_tokens": 2,
"token_jaccard": 0.014706,
"token_length_ratio": 0.008929
} | sorry_admit_sorryAx_proof_hole | sorry | 62368d7d15cb1037595ffb41f0c7fe4914ae356d9bb9dff027d838a28737f86a | 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.MonoidAlgebra.Defs
public import Mathlib.Data.Finset.SMulAntidiagonal
Namespace:
MonoidAlgebra
Local context:
/-
Copyright (c) 2025 Scott Carnahan. A... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/MonoidAlgebra",
"family_id": "smul_apply_mulaction",
"file_id": "mathlib/Mathlib/Algebra/MonoidAlgebra/PointwiseSMul.lean",
"sample_id": "ba002fff735a1eceaabf1a8e0dbcc41b8470b3783bbd4a79e63a536436ef24f3"
} | train |
{
"chosen_proof_sha256": "e99a691cbba97dd50b462f822d8ea54c6c328e4bf1d6437c6b809f75afb37a33",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "02f4eef5948f491574f8f5278e997fcef4... | by
have key : ∀ x ∈ hs.toFinset, s.indicator (fun _ ↦ a) x = a := by
intro x hx
rw [indicator_of_mem (hs.mem_toFinset.1 hx) (fun _ ↦ a)]
rw [Nat.card_eq_card_finite_toFinset hs, ← sum_eq_card_nsmul key, eventually_atTop]
obtain ⟨m, hm⟩ := hs.bddAbove
refine ⟨m + 1, fun n n_m ↦ (sum_subset ?_ ?_).symm⟩ <... | false | {
"chosen_tokens": 146,
"rejected_tokens": 153,
"token_jaccard": 0.967213,
"token_length_ratio": 1.047945
} | markdown_preamble_trailing | fence | 62759a6568879ae87178790e10e987d6c4740171c5c1c3c29c7171506bd2f3b4 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
public import Mathlib.Algebra.Order.Archimedean.Basic
public import Mathlib.Algebra.Order.BigOperators.Group.Finset... | ```lean
by
have key : ∀ x ∈ hs.toFinset, s.indicator (fun _ ↦ a) x = a := by
intro x hx
rw [indicator_of_mem (hs.mem_toFinset.1 hx) (fun _ ↦ a)]
rw [Nat.card_eq_card_finite_toFinset hs, ← sum_eq_card_nsmul key, eventually_atTop]
obtain ⟨m, hm⟩ := hs.bddAbove
refine ⟨m + 1, fun n n_m ↦ (sum_subset ?_ ?_)... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Order",
"family_id": "sum_indicator_eventually_eq_card",
"file_id": "mathlib/Mathlib/Algebra/Order/Archimedean/IndicatorCard.lean",
"sample_id": "71930e9f307d51dc22d21feef3aebed1568f0f6f3ee70012583f25208fbd5c02"
} | train |
{
"chosen_proof_sha256": "d9b98cd109749b4d841c4e5ec47310732f89bc8df99f9f6e87e3aa11beb61e37",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c35d695e227f0e1f879e46ee2607f0e367... | by
have ⟨hμν_ac, hκη_ac⟩ := Measure.absolutelyContinuous_compProd_iff.mp h_ac
rw [← integrable_rnDeriv_mul_log_iff h_ac] at h_int
replace h_int := convexOn_mul_log.integrable_apply_rnDeriv_of_integrable_compProd
continuous_mul_log.stronglyMeasurable continuous_mul_log.continuousWithinAt h_int hκη_ac
exact (... | true | {
"chosen_tokens": 56,
"rejected_tokens": 5,
"token_jaccard": 0.085714,
"token_length_ratio": 0.089286
} | wrong_namespace_qualification | wrong_namespace_qualification | 62856dbab2d46edb96ea8c66e80ad1efd3877284e4b9f9d7c9dca173d75617a3 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.InformationTheory.KullbackLeibler.Basic
public import Mathlib.Probability.Kernel.Composition.MeasureCompProd
public import Mathlib.Probability.Notation
import ... | by
exact OProverMissing.missing_33493801268b | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a4ba8a8fa0074b81cb1ecf70db7e8657c8312096326e034e9548a6cea40d85f5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "InformationTheory/KullbackLeibler",
"family_id": "integrable_llr_of_integrable_llr_compprod",
"file_id": "mathlib/Mathlib/InformationTheory/KullbackLeibler/ChainRule.lean",
"sample_id": "33493801268bbdfd455e186fa50a17ff9c38e950d8c9e9bcab0dd5bfd9827ec7"
} | train |
{
"chosen_proof_sha256": "35928b4ec6c4dd8e48530f64dacc44b471422b62b6f0a8ae08f8ed59d2ec96f4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ca8114b0fec5bc25a1e786573058e15d24... | by
calc
_ = 𝟙 _ ⊗≫ (F.map₂ β ▷ η.app b ≫ η.naturality g) ▷ θ.app b ⊗≫
η.app a ◁ θ.naturality g ⊗≫ 𝟙 _ := by
bicategory
_ = 𝟙 _ ⊗≫ η.naturality f ▷ θ.app b ⊗≫
η.app a ◁ (G.map₂ β ▷ θ.app b ≫ θ.naturality g) ⊗≫ 𝟙 _ := by
rw [η.naturality_naturality]
bicategory
_ = _... | true | {
"chosen_tokens": 112,
"rejected_tokens": 3,
"token_jaccard": 0.030303,
"token_length_ratio": 0.026786
} | unknown_identifier | unknown_identifier | 632510a50b9f9f135c5f684847fb97e145382bb48e573e530ea5d49d88e71174 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Bicategory.Functor.Lax
public import Mathlib.Tactic.CategoryTheory.Bicategory.Basic
Namespace:
CategoryTheory.Lax.OplaxTrans
Local context:
/-... | by
exact __oprover_missing_325825a927e5 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a78cef6cfb9e129191d6991a931fff261f59f3f6a2a11e298e4eae50d348cd40",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Bicategory",
"family_id": "vcomp_naturality_naturality",
"file_id": "mathlib/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Lax.lean",
"sample_id": "325825a927e5735338dd52a28d2dff4f12d1ad248dc0bc8ca3daba7d87f856c6"
} | train |
{
"chosen_proof_sha256": "1927463d606bf10bea7c2ddb5c31b00ff037d766b129f9620262d1b12ca44ebf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [← tsum_dirichletSummand χ hs]
apply eulerProduct_completely_multiplicative <| summable_dirichletSummand χ hs | false | {
"chosen_tokens": 15,
"rejected_tokens": 2,
"token_jaccard": 0.071429,
"token_length_ratio": 0.133333
} | sorry_admit_sorryAx_proof_hole | admit | 634cfe713e0c566ce7a19af7bdb61ece92a7405e61e54ec080facf6a6dfc2c3e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.NumberTheory.EulerProduct.ExpLog
public import Mathlib.NumberTheory.LSeries.Dirichlet
Namespace:
(root)
Local context:
/-
Copyright (c) 2023 Michael Stoll. A... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/EulerProduct",
"family_id": "dirichletcharacter",
"file_id": "mathlib/Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean",
"sample_id": "f7a04a0bdfb9132901b240788f15f017596751bca9984db5e9bf9d3c080b9f4c"
} | train |
{
"chosen_proof_sha256": "a329996be1763b313924738bf153d1074388b636f52e211888a4d4f3a26dea07",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e335eb5a3b25d52b6c9d5323a6623c91e5... | by
simpa using w
-- It would be nice to mark this as `aesop cases` if
-- https://github.com/leanprover-community/aesop/issues/59
-- is resolved. | false | {
"chosen_tokens": 44,
"rejected_tokens": 51,
"token_jaccard": 0.967742,
"token_length_ratio": 1.159091
} | markdown_preamble_trailing | fence | 634e54a5b06cd9594b6ca2b20da84e8e030c881b55ab4b0e475264e8137f277e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Logic.Equiv.Defs
public import Mathlib.Tactic.MkIffOfInductiveProp
public import Mathlib.Tactic.PPWithUniv
Namespace:
(root)
Local context:
/-
Copyright (c) ... | ```lean
by
simpa using w
-- It would be nice to mark this as `aesop cases` if
-- https://github.com/leanprover-community/aesop/issues/59
-- is resolved.
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Logic/Small",
"family_id": "shrink",
"file_id": "mathlib/Mathlib/Logic/Small/Defs.lean",
"sample_id": "1f97d0501e044f00fde760711297ef657cb86ce42d441db370fdf596ff19fb0c"
} | train |
{
"chosen_proof_sha256": "ea338c8c6deda19356f39b2f9dc6bfce64075bf47da045c1a31eb2b19f1fe070",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "67ec5d4508cf381b8aa1ef1018c768d831... | by
refine ⟨fun h => ?_, fun _ => inferInstance⟩
contrapose! h
infer_instance | true | {
"chosen_tokens": 18,
"rejected_tokens": 2,
"token_jaccard": 0.066667,
"token_length_ratio": 0.111111
} | invalid_tactic | invalid_tactic | 63d2385d914bc74f975fe25fe9cc5282f49069d52970ace258073ad6dc4990b3 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.Sets.Closeds
public import Mathlib.Topology.QuasiSeparated
Namespace:
TopologicalSpace.Compacts
Local context:
/-
Copyright (c) 2020 Floris van Door... | by
__oprover_invalid_tactic_7c30455a13ea | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "a6c490ca941c9b42e2ab8981efb38dcc907ea993d97bb32302ae1a52b003eb25",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Sets",
"family_id": "subsingleton_iff",
"file_id": "mathlib/Mathlib/Topology/Sets/Compacts.lean",
"sample_id": "7c30455a13ea31c0fdc5ee48caccc8329851df034a9e411c0a2bb1d61bbf2a76"
} | train |
{
"chosen_proof_sha256": "0678ccbc2752fb8380a44655810f825f8a9182914dd6e363b01acde3b055d878",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
induction y generalizing x with
| nil => simpa using h
| cons y ys ih => exact T.prop (ih (by simpa)) | false | {
"chosen_tokens": 29,
"rejected_tokens": 2,
"token_jaccard": 0.045455,
"token_length_ratio": 0.068966
} | sorry_admit_sorryAx_proof_hole | admit | 640dcd7df4f7de5a236b33951f7fefab47a70e5015704c0c3904c6f7e803618e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.CompleteLattice.SetLike
Namespace:
Descriptive.Tree
Local context:
/-
Copyright (c) 2024 Sven Manthe. All rights reserved.
Released under Apache 2.0 li... | by
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": "SetTheory/Descriptive",
"family_id": "mem_of_append",
"file_id": "mathlib/Mathlib/SetTheory/Descriptive/Tree.lean",
"sample_id": "2d5cb89a9cfda0560e3764a0fa8315a7c93635c11c81d041a427e94157d40461"
} | train |
{
"chosen_proof_sha256": "de8a32b5f73d36b3c5675db1bfb4e3c6a3d479eee57e39397db99ac101c60702",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
refine ⟨fun H ↦ ?_, isTopologicallyNilpotent_of_constantCoeff⟩
replace H : Tendsto (fun n ↦ constantCoeff (f ^ n)) atTop (nhds 0) :=
continuous_constantCoeff R |>.tendsto' 0 0 constantCoeff_zero |>.comp H
simpa only [map_pow] using! H | false | {
"chosen_tokens": 54,
"rejected_tokens": 2,
"token_jaccard": 0.025,
"token_length_ratio": 0.037037
} | sorry_admit_sorryAx_proof_hole | admit | 64420a671444cbc235088b592bf837243bbb5a41c7867e300934638a683548c3 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finsupp.Interval
public import Mathlib.RingTheory.Ideal.Quotient.Defs
public import Mathlib.RingTheory.MvPowerSeries.PiTopology
public import Mathlib.Topo... | 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/MvPowerSeries",
"family_id": "istopologicallynilpotent_iff_constantcoeff",
"file_id": "mathlib/Mathlib/RingTheory/MvPowerSeries/LinearTopology.lean",
"sample_id": "460fd408bbfa2c9d3696e93eaf82711398320704b30a3dd64ec1d4fbb298afc9"
} | train |
{
"chosen_proof_sha256": "078c3ba86f93f2f5c0b7638f81f2db59b5ef54f0de3d73360e15d4f2f0ce975a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7825f2858d4abf33c4a06f455ce84c7b78... | by
simp only [normThreshold_zero]
calc a / (1 - √2)
_ ≤ 0 := div_nonpos_of_nonneg_of_nonpos ha_pos.le (by simp)
_ < a := ha_pos | true | {
"chosen_tokens": 32,
"rejected_tokens": 3,
"token_jaccard": 0.038462,
"token_length_ratio": 0.09375
} | unknown_identifier | unknown_identifier | 6488fb7e4416697b83503b794308f7c1c74f5ec15d365ad6bf10641665f194c6 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecificLimits.ArithmeticGeometric
public import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
public import Mathlib.MeasureTheor... | by
exact __oprover_missing_5f55f4262f31 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f01ee2a99ebe44248b4fedba58b2999d353f5813f2b0d44d5b296ed14e75fadb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability/Distributions",
"family_id": "lt_normthreshold_zero",
"file_id": "mathlib/Mathlib/Probability/Distributions/Fernique.lean",
"sample_id": "5f55f4262f31682a3e12d8625b11abf778bb7c39ed9ebef5aef18cf9e8a8d94d"
} | train |
{
"chosen_proof_sha256": "c165c49ef010a17e120cfdcd1a6cc6afaacc95eed29ab44ee041da167fa437e8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "882301a1333b1d1e59e6d22aa8bb0161b0... | by
-- this follows from a covering argument using the sets satisfying `ρ a ≤ ν a`.
apply ENNReal.le_of_forall_pos_le_add fun ε εpos _ => ?_
obtain ⟨U, sU, U_open, νU⟩ : ∃ (U : Set α), s ⊆ U ∧ IsOpen U ∧ ν U ≤ ν s + ε :=
exists_isOpen_le_add s ν (ENNReal.coe_pos.2 εpos).ne'
let f : α → Set (Set α) := fun _ =... | false | {
"chosen_tokens": 324,
"rejected_tokens": 328,
"token_jaccard": 0.969697,
"token_length_ratio": 1.012346
} | missing_wrong_import | import | 64add9dafbb27d4d82980b7a0a2da1e071227280adb535bec9613782bda38ca7 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Covering.VitaliFamily
public import Mathlib.MeasureTheory.Function.AEMeasurableOrder
public import Mathlib.MeasureTheory.Integral.Average
public ... | import Mathlib.WrongImport
by
-- this follows from a covering argument using the sets satisfying `ρ a ≤ ν a`.
apply ENNReal.le_of_forall_pos_le_add fun ε εpos _ => ?_
obtain ⟨U, sU, U_open, νU⟩ : ∃ (U : Set α), s ⊆ U ∧ IsOpen U ∧ ν U ≤ ν s + ε :=
exists_isOpen_le_add s ν (ENNReal.coe_pos.2 εpos).ne'
let f :... | {
"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": "MeasureTheory/Covering",
"family_id": "measure_le_of_frequently_le",
"file_id": "mathlib/Mathlib/MeasureTheory/Covering/Differentiation.lean",
"sample_id": "c0fbb14cf7d5c8d77a74ea9851f8a6d43a29d0444f221fd7a52f0350a1b23692"
} | train |
{
"chosen_proof_sha256": "4acb529aee5f1ece8076fbbdc844587e9d93366665ac8d5f6045ad6127fa4525",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1cd3353ed9a27852a0187d966636342809... | by
have := lift_lt.{_, max (u + 1) v}.2 (lift_lt_univ c)
rw [lift_lift, lift_univ, univ_umax.{u, v}] at this
exact this | false | {
"chosen_tokens": 40,
"rejected_tokens": 45,
"token_jaccard": 0.848485,
"token_length_ratio": 1.125
} | markdown_preamble_trailing | preamble | 64dd6e8952d65c5a85ce27a1fbaa734169e4af4263fffde0142f4ee593a2dac2 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.SetTheory.Ordinal.Basic
Namespace:
Cardinal
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as de... | Here is the proof:
by
have := lift_lt.{_, max (u + 1) v}.2 (lift_lt_univ c)
rw [lift_lift, lift_univ, univ_umax.{u, v}] at this
exact this | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "SetTheory/Ordinal",
"family_id": "lift_lt_univ",
"file_id": "mathlib/Mathlib/SetTheory/Ordinal/Univ.lean",
"sample_id": "beebaf6401da7ba6ae02cab75a18db90132912fa7666bc24027b84d9741587a1"
} | train |
{
"chosen_proof_sha256": "29e959508ac284606b1ea79d1f9c44fa587eb78cdc7fa0d4ba1b2b14a0490f11",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "955ffaab0169b8f61dba8b3842d08064f9... | by
simp only [Shrink.rec, eqRec_eq_cast, cast_eq_iff_heq]
rw [Equiv.symm_apply_apply] | false | {
"chosen_tokens": 18,
"rejected_tokens": 23,
"token_jaccard": 0.777778,
"token_length_ratio": 1.277778
} | markdown_preamble_trailing | trailing | 64f2397ee56431cda2008184d76ea5b1ed7d3227a0e48385730de97374df7702 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Logic.Equiv.Defs
public import Mathlib.Tactic.MkIffOfInductiveProp
public import Mathlib.Tactic.PPWithUniv
Namespace:
(root)
Local context:
/-
Copyright (c) ... | by
simp only [Shrink.rec, eqRec_eq_cast, cast_eq_iff_heq]
rw [Equiv.symm_apply_apply]
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Logic/Small",
"family_id": "shrink",
"file_id": "mathlib/Mathlib/Logic/Small/Defs.lean",
"sample_id": "e5f7ea238c907be9131815ee500db761a2036d4f0c717eb8bb8747df40a1767e"
} | train |
{
"chosen_proof_sha256": "1c6acec1b0c2300f662b80f0c922eb6275df7e3daf8325d6423e0848393437e0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1aa534d9cd1750e6e0c840502eacb09027... | by simpa using! I.ringCon.add hx hy | true | {
"chosen_tokens": 11,
"rejected_tokens": 5,
"token_jaccard": 0.153846,
"token_length_ratio": 0.454545
} | wrong_namespace_qualification | wrong_namespace_qualification | 654c971ae9ea183b29debfc713366287ddebc4ed82e7d009f11bde879cc48575 | 1 | 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.Abel
public import Mathlib.Algebra.Ring.Opposite
public import Mathlib.GroupTheory.GroupAction.SubMulAction
public import Mathlib.RingTheory.Congruence.... | by
exact OProverMissing.missing_76c80b8cd380 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f3eba94b9e0af848eb1dedaf22b82d19e96f5adcf1b3be938ef972ac295c4bac",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/TwoSidedIdeal",
"family_id": "add_mem",
"file_id": "mathlib/Mathlib/RingTheory/TwoSidedIdeal/Basic.lean",
"sample_id": "76c80b8cd380741542148048f4f9d53c485cf357afdef4b9fbe25f15cafe33a7"
} | train |
{
"chosen_proof_sha256": "cfefd25aa03c8c5b75a26d1fb3bac847d72b8789c719fd94bddb3d949b0e73ab",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "deb583f09156ce6e73c3a31850ee7ecaa7... | by
refine ⟨fun hn x ↦ lt_of_le_of_lt (norm_coe_le_norm f x) hn, ?_⟩
· obtain (he | he) := isEmpty_or_nonempty α
· simpa
· obtain ⟨x, hx⟩ := exist_norm_eq h
exact fun h ↦ hx ▸ h x | false | {
"chosen_tokens": 48,
"rejected_tokens": 55,
"token_jaccard": 0.935484,
"token_length_ratio": 1.145833
} | markdown_preamble_trailing | fence | 65505ed8673c7d32d973ff8a63b30de2f5c4fb9bd575f47eddf9cdc43a009af4 | 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.ContinuousMap.Bounded.Normed
Namespace:
(root)
Local context:
/-
Copyright (c) 2024 Yoh Tanimoto. All rights reserved.
Released under Apache 2.0 lic... | ```lean
by
refine ⟨fun hn x ↦ lt_of_le_of_lt (norm_coe_le_norm f x) hn, ?_⟩
· obtain (he | he) := isEmpty_or_nonempty α
· simpa
· obtain ⟨x, hx⟩ := exist_norm_eq h
exact fun h ↦ hx ▸ h x
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/ContinuousMap",
"family_id": "norm_lt_iff_of_compactlysupported",
"file_id": "mathlib/Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean",
"sample_id": "3872ad2e2e46d2587f8e296dc30252bf359d8b645c5d7c7851970aa025ddb54e"
} | train |
{
"chosen_proof_sha256": "b656fc39071ddeaac02c29e4861f9ce7d1311a7b76fad806354a82acd6bcd29b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "719c1265d6da97aed3f9b0cf9aa1b7a5fc... | by
cases isEmpty_or_nonempty ι
· rw [iSup_of_empty', iSup_of_empty']
· exact hdf.ciSup_le fun x ↦ hdg.le_ciSup_of_le B x (H x) | false | {
"chosen_tokens": 28,
"rejected_tokens": 32,
"token_jaccard": 0.884615,
"token_length_ratio": 1.142857
} | missing_wrong_import | import | 6556d98de994045a73f4ab436191c1831078216e81b8597188c61414471682cd | 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.ConditionallyCompletePartialOrder.Basic
public import Mathlib.Order.GaloisConnection.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2026 Jireh... | import Mathlib.WrongImport
by
cases isEmpty_or_nonempty ι
· rw [iSup_of_empty', iSup_of_empty']
· exact hdf.ciSup_le fun x ↦ hdg.le_ciSup_of_le B x (H x) | {
"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": "Order/ConditionallyCompletePartialOrder",
"family_id": "directed",
"file_id": "mathlib/Mathlib/Order/ConditionallyCompletePartialOrder/Indexed.lean",
"sample_id": "66a9d446584881e6ae181bf60c62b6c4435815bff91f04bcf763c4007fc9aaae"
} | train |
{
"chosen_proof_sha256": "7308aa6374fea597c48250db2901872d2efda7ec2b018c03749527c542943e3a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "581df8840e83eb5edacbd18f72ceb46915... | by
obtain ⟨n, hn⟩ := exists_rat_eq_convergent h
exact ⟨n, hn.symm ▸ convs_eq_convergent ξ n⟩ | true | {
"chosen_tokens": 22,
"rejected_tokens": 3,
"token_jaccard": 0.117647,
"token_length_ratio": 0.136364
} | unknown_identifier | unknown_identifier | 655ec6c808d7df615d43df0d5ba7f03a31b4cf4bd6f5727cb54f8ecb46d1ec1b | 1 | 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.ContinuedFractions.Computation.ApproximationCorollaries
public import Mathlib.Algebra.ContinuedFractions.Computation.Translations
public import Mathlib... | by
exact __oprover_missing_88dc792ee34d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f97653a8c17f20bb89110d4a80754c612af3085d28b5120373ead1492c0ba914",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/DiophantineApproximation",
"family_id": "exists_convs_eq_rat",
"file_id": "mathlib/Mathlib/NumberTheory/DiophantineApproximation/ContinuedFractions.lean",
"sample_id": "88dc792ee34d82bdbe14dbea6a9bf9efe530e0a2ea1640a11d02fb687a45c35f"
} | train |
{
"chosen_proof_sha256": "2b5a7d59d6ab27179c1c1bc56a6c116c83ca6fccc9d560e5875bd86a9aa70480",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8d0fad0c6786d754062706cdf367d3b518... | by
simp [List.mem_iff_get, Fin.exists_iff, eq_comm (b := b)] | true | {
"chosen_tokens": 18,
"rejected_tokens": 2,
"token_jaccard": 0.0625,
"token_length_ratio": 0.111111
} | invalid_tactic | invalid_tactic | 657e2a6bb5fc7af0634486c4df803989c7fa4355e29e4e8a00c7c1af0632b500 | 1 | 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.List.Defs
Namespace:
List
Local context:
/-
Copyright (c) 2024 Miyahara Kō. All rights reserved.
Released ... | by
__oprover_invalid_tactic_32ce66990d1f | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "df3c87cd4b27ed940158d498e269e644378bdf3d8c1ddad7a4c60c248c2c2b46",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/List",
"family_id": "mem_iterate",
"file_id": "mathlib/Mathlib/Data/List/Iterate.lean",
"sample_id": "32ce66990d1ffba436ebb7dc106eea97303764f97b8b35040bc36958f838afdb"
} | train |
{
"chosen_proof_sha256": "9c6b8baf38f61be33d84aa02acad26a64e1f3aa8d253f05349f099a61ea5244e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f176d26ff830ef796b098e912be2e03d67... | by
refine ⟨fun h p hp hp' ↦ hp.not_isUnit (h p hp'), fun h d hd ↦ by_contra fun hdu ↦ ?_⟩
have hd₀ : d ≠ 0 := ne_zero_of_dvd_ne_zero (ne_zero_of_dvd_ne_zero hx₀ hd) (dvd_mul_left d d)
obtain ⟨p, irr, dvd⟩ := WfDvdMonoid.exists_irreducible_factor hdu hd₀
exact h p irr ((mul_dvd_mul dvd dvd).trans hd) | true | {
"chosen_tokens": 79,
"rejected_tokens": 5,
"token_jaccard": 0.075,
"token_length_ratio": 0.063291
} | wrong_namespace_qualification | wrong_namespace_qualification | 65d3fff3408719a2ede93cd67e1dabf020bb63d559df78942f238c48f1eaa51e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Coprime.Lemmas
public import Mathlib.RingTheory.Nilpotent.Basic
public import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid
public import M... | by
exact OProverMissing.missing_ff5f243c3159 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "3715e83bf682fb1105ead0708dd43592344fab389870a44de509aa50284f0d6e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Squarefree",
"family_id": "squarefree_iff_no_irreducibles",
"file_id": "mathlib/Mathlib/Algebra/Squarefree/Basic.lean",
"sample_id": "ff5f243c31598fd2aadb8dc36b39e21f486eb66be282ba3b720cbff07478671d"
} | train |
{
"chosen_proof_sha256": "1c6acec1b0c2300f662b80f0c922eb6275df7e3daf8325d6423e0848393437e0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2a6176a9db1730286a0ff010222bf3e543... | by simpa using! I.ringCon.add hx hy | false | {
"chosen_tokens": 11,
"rejected_tokens": 16,
"token_jaccard": 0.714286,
"token_length_ratio": 1.454545
} | markdown_preamble_trailing | trailing | 65db2051249a7f09929bde4a94b45756f68d8ad980f49fc516e07d8f55dc140b | 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.Tactic.Abel
public import Mathlib.Algebra.Ring.Opposite
public import Mathlib.GroupTheory.GroupAction.SubMulAction
public import Mathlib.RingTheory.Congruence.... | by simpa using! I.ringCon.add hx hy
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/TwoSidedIdeal",
"family_id": "add_mem",
"file_id": "mathlib/Mathlib/RingTheory/TwoSidedIdeal/Basic.lean",
"sample_id": "76c80b8cd380741542148048f4f9d53c485cf357afdef4b9fbe25f15cafe33a7"
} | train |
{
"chosen_proof_sha256": "a6a0730acc445e18a71efd0b82b6e6d57aa672e42e9dfc4ec8e49f142e23c559",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "afb40734c45bc026b3d01a0cc527b84f04... | by
rw [irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree, ← not_exists]
simp only [hr, not_true, false_or, and_false] | false | {
"chosen_tokens": 19,
"rejected_tokens": 24,
"token_jaccard": 0.736842,
"token_length_ratio": 1.263158
} | markdown_preamble_trailing | preamble | 65dc9b76c0a5b94548647162ba8b5957d753078f0ca43cc7f27727c95a425ece | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Coprime.Lemmas
public import Mathlib.RingTheory.Nilpotent.Basic
public import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid
public import M... | Here is the proof:
by
rw [irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree, ← not_exists]
simp only [hr, not_true, false_or, and_false] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Squarefree",
"family_id": "squarefree_iff_irreducible_sq_not_dvd_of_exists_irreducible",
"file_id": "mathlib/Mathlib/Algebra/Squarefree/Basic.lean",
"sample_id": "510546c198df0e51ff7856781031b1d5536b836338687232036e8eccc1052100"
} | train |
{
"chosen_proof_sha256": "8390e404c259cefd040f4b8484fc4ca8346f13b8985655df805e82d4eca600cf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2b3879b0cad92f3b44a2e7ca566d11d48d... | by
ext x
induction x <;> rfl | false | {
"chosen_tokens": 9,
"rejected_tokens": 14,
"token_jaccard": 0.615385,
"token_length_ratio": 1.555556
} | markdown_preamble_trailing | trailing | 65de2a2715f4002c0b1034c38c8b15ff718a1b421e05bc56f46bd6f3e61dbb02 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Basic
public import Mathlib.Algebra.Group.Equiv.Defs
public import Mathlib.Algebra.Group.WithOne.Defs
Namespace:
WithOne
Local context:
/-
Copy... | by
ext x
induction x <;> rfl
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Group",
"family_id": "mapmulhom_id",
"file_id": "mathlib/Mathlib/Algebra/Group/WithOne/Basic.lean",
"sample_id": "923ffa343160dc678018bf65a4585befc8d6632387419cd5ee47dffbd11b76e4"
} | train |
{
"chosen_proof_sha256": "d99c571d1e3b54f4a7c947a3bff9621121704efdf01dd61bab5cf6b4a711ab1c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
simp [Functor.IsDense.leftKanExtensionIso] | false | {
"chosen_tokens": 9,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.333333
} | by_exact_placeholder | placeholder | 65df635bab0e9d6233cbf5181ff045a971a88c78a56ba8f7d34bdadfdcf0a4a6 | 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.KanExtension.DenseAt
public import Mathlib.CategoryTheory.Limits.Presheaf
public import Mathlib.CategoryTheory.Generator.StrongGenerator... | 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/Functor",
"family_id": "isdense",
"file_id": "mathlib/Mathlib/CategoryTheory/Functor/KanExtension/Dense.lean",
"sample_id": "052befed77426c858368b90b76b9540e8aaef3d27651275eef8d687539481881"
} | train |
{
"chosen_proof_sha256": "16b3be2598502346de2754703a4649aa758c506146fa34fc091635ed0bf9b555",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rcases isEmpty_or_nonempty X with h | h
· exact ⟨Classical.arbitrary Y, funext <| h.elim⟩
· exact ⟨f (Classical.arbitrary X), hf.eq_const _⟩ | false | {
"chosen_tokens": 39,
"rejected_tokens": 5,
"token_jaccard": 0.111111,
"token_length_ratio": 0.128205
} | sorry_admit_sorryAx_proof_hole | sorryAx | 65e3d6b9cf5695a9237acfca994779d50791b309e06b70d11686ba8dd8ed00fc | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Notation.Indicator
public import Mathlib.Topology.Connected.LocallyConnected
public import Mathlib.Topology.Sets.Closeds
Namespace:
IsLocallyConstant
... | 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": "Topology/LocallyConstant",
"family_id": "exists_eq_const",
"file_id": "mathlib/Mathlib/Topology/LocallyConstant/Basic.lean",
"sample_id": "41091d79c58a55895294fdd103ee3f3bbfd8236b3858691c14a8a3dfe9a1b2a2"
} | train |
{
"chosen_proof_sha256": "4c0ce5d128f69a2b99881bee610f584c1a319bb4f885c9faba74ae83efc61828",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | 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 | false | {
"chosen_tokens": 42,
"rejected_tokens": 5,
"token_jaccard": 0.035714,
"token_length_ratio": 0.119048
} | sorry_admit_sorryAx_proof_hole | sorryAx | 666d03667d0bdc320217bf01ffe15b5005697344c8cf473a49e27d9fd9bddec0 | 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.Subobject.Lattice
public import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
Namespace:
CategoryTheory
Local context:
/-
Copyright (c) 2025 ... | 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",
"family_id": "extremalepi",
"file_id": "mathlib/Mathlib/CategoryTheory/ExtremalEpi.lean",
"sample_id": "c8cd16ca7c7e5a007fb195fa59aed53c5734a477ccb7bfa5ed293e06fc1f996c"
} | train |
{
"chosen_proof_sha256": "d7f9b791b4990ce10c90fe8f0ae29086af43a18750773d3fc0cc6aa2bef28c2c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
refine hs.induction_on _ (by simp) fun {i s} his hs e ↦ ?_
simp only [iInf_insert, e, ideal_inf] | false | {
"chosen_tokens": 30,
"rejected_tokens": 5,
"token_jaccard": 0.074074,
"token_length_ratio": 0.166667
} | sorry_admit_sorryAx_proof_hole | sorryAx | 6684815c752a340c53cb511cd0fe2d62e976fa1128da9d4d0a30df235753ff55 | 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 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": "AlgebraicGeometry/IdealSheaf",
"family_id": "ideal_biinf",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/Basic.lean",
"sample_id": "fd1fd9a53ab94e93ee71cb526c00c66f283a440ab2cd926e2cae6762d9e62179"
} | train |
{
"chosen_proof_sha256": "76fa0ee83574746f588fc7899220fecd9e66129f439898c7d8ec00fbf1b68c95",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
obtain ⟨i', hi'_lub⟩ : ∃ i', IsLUB (Set.Iio i) i' := exists_lub_Iio i
rcases lub_Iio_eq_self_or_Iio_eq_Iic i hi'_lub with hi'_eq_i | h_Iio_eq_Iic
· rw [← hi'_eq_i] at hi'_lub ⊢
exact hτ.measurableSet_lt_of_isLUB i' hi'_lub
· have h_lt_eq_preimage : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iio i := rfl
have h_Ii... | false | {
"chosen_tokens": 128,
"rejected_tokens": 3,
"token_jaccard": 0.033898,
"token_length_ratio": 0.023438
} | by_exact_placeholder | placeholder | 66a36a29a184c930f6350812b46806ac0e2e0deecc516ee3520dceb931d92c66 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Probability.Process.Adapted
public import Mathlib.MeasureTheory.Constructions.BorelSpace.WithTop
public import Mathlib.Data.ENat.Lattice
Namespace:
MeasureThe... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Probability/Process",
"family_id": "isstoppingtime",
"file_id": "mathlib/Mathlib/Probability/Process/Stopping.lean",
"sample_id": "bfdf8a5735fb83291498154fe55688e7c5eabb3111989c0465481333ca9088fd"
} | train |
{
"chosen_proof_sha256": "cff2f50ea6340f1c0d647e96a5ab7bcbcc74041793ed6f6cea3ea0d685f9013d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
intro s hs
induction hs with
| base t t_mem => exact self_subset_generateSetAlgebra (h t_mem)
| empty => exact isSetAlgebra_generateSetAlgebra.empty_mem
| compl t _ t_mem => exact isSetAlgebra_generateSetAlgebra.compl_mem t_mem
| union t u _ _ t_mem u_mem => exact isSetAlgebra_generateSetAlgebra.union_me... | false | {
"chosen_tokens": 51,
"rejected_tokens": 2,
"token_jaccard": 0.035714,
"token_length_ratio": 0.039216
} | by_exact_placeholder | by? | 6710b8a328a307d8c286174af117e894bec061d556e9dec71cc8840dbeb34e45 | 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.Finite.Prod
public import Mathlib.MeasureTheory.SetSemiring
Namespace:
MeasureTheory
Local context:
/-
Copyright (c) 2024 Etienne Marion. All rights res... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "MeasureTheory",
"family_id": "generatesetalgebra_mono",
"file_id": "mathlib/Mathlib/MeasureTheory/SetAlgebra.lean",
"sample_id": "865430ae7ad7a3af9b967fd8cc81590df7b116ddc79d82d6edeca41aba827e49"
} | train |
{
"chosen_proof_sha256": "e5473258af32213de8f9e9baf793bc2d791875a7c5122f73158f25e7055ff270",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simpa only [sub_eq_add_neg] using!
(f.commute_add_nat n).inverses_right (Equiv.addRight _).right_inv (Equiv.addRight _).left_inv | true | {
"chosen_tokens": 32,
"rejected_tokens": 8,
"token_jaccard": 0.038462,
"token_length_ratio": 0.25
} | unsolved_goals | unsolved_goals | 6713b2fe40bc8084465e68eb63f7b2f9048095eb37d0ab888c597a2d504c31e1 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Order.SemiconjSup
Namespace:
CircleDeg1Lift
Local context:
/-
Copyright (c) 2020 Yury Kudryashov. All rig... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "de5527eb48fab754a24ba3ead9d2db7dd1db73cca77b65990218b8090075d3ca",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Dynamics/Circle",
"family_id": "commute_sub_nat",
"file_id": "mathlib/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean",
"sample_id": "bff817808a5cc69dcd74325b621cd3b889cc91e9bdb2f5fd6f80c7083b9ec6d9"
} | train |
{
"chosen_proof_sha256": "81f6811c61bda2ac7e16fe95d39238e8cafe9933dcb6310481ee06482505a998",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simp [fiber, fiberι, Scheme.Pullback.range_fst, Scheme.range_fromSpecResidueField] | false | {
"chosen_tokens": 18,
"rejected_tokens": 2,
"token_jaccard": 0.076923,
"token_length_ratio": 0.111111
} | by_exact_placeholder | by? | 67273ce46ef66fa6a713d881ae64136ead2fdd59cfbf30133608db85640e3fc4 | 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.PullbackCarrier
public import Mathlib.RingTheory.LocalRing.ResidueField.Fiber
public import Mathlib.RingTheory.Spectrum.Prime.Jacobson
public... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry",
"family_id": "scheme",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Fiber.lean",
"sample_id": "9ea116b7daa0ff9aa81b54eb159a02ebd9756dce658d8964cedd444ceb6ca675"
} | train |
{
"chosen_proof_sha256": "baf74685d38b5f76627c288eda6c3033c2f20e6a31a28995ec3f9cc5ab112e2a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
haveI := Fintype.ofFinite ι
classical
refine (Basis.piTensorProduct b).ext_elem (fun q ↦ ?_)
simp [Finsupp.single_apply, Fintype.prod_ite_zero, ← funext_iff] | false | {
"chosen_tokens": 37,
"rejected_tokens": 3,
"token_jaccard": 0.032258,
"token_length_ratio": 0.081081
} | by_exact_placeholder | placeholder | 673f03b25af2f45630eedfcfc586275d00ab166858ba76142234df68f06fc64c | 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.Finsupp.VectorSpace
public import Mathlib.LinearAlgebra.PiTensorProduct.Finsupp
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Daniel M... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/PiTensorProduct",
"family_id": "basis",
"file_id": "mathlib/Mathlib/LinearAlgebra/PiTensorProduct/Basis.lean",
"sample_id": "317b87a5beef988d47043cbd23eccd8456a804914b949c3de8ba8b8d221e791d"
} | train |
{
"chosen_proof_sha256": "32e319c4110fd1b6339d57505a0712e65f79dbff575ad63d342ca445e0185468",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simp [← HomologicalComplex.comp_f]
/-- The homotopy equivalence from `X.chainComplex R` to `X.normalizedChainComplex R`. -/
noncomputable def homotopyEquivNormalizedChainComplex :
HomotopyEquiv (X.chainComplex R) (X.normalizedChainComplex R) :=
SimplicialObject.Splitting.homotopyEquivNondegComplex _ | false | {
"chosen_tokens": 55,
"rejected_tokens": 2,
"token_jaccard": 0.030303,
"token_length_ratio": 0.036364
} | by_exact_placeholder | by? | 6742fd3b9761a4ba69c34fcd1c66f483db8e770c7eac0dc54e04656af0f16f92 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Homology.QuasiIso
public import Mathlib.AlgebraicTopology.SimplicialSet.Homology.Basic
public import Mathlib.AlgebraicTopology.SimplicialSet.Splitting
... | 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/SimplicialSet",
"family_id": "tonormalizedchaincomplex_f_fromnormalizedchaincomplex_f",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplicialSet/Homology/Nondegenerate.lean",
"sample_id": "5f583e779a0f586168eb801c8f41ed46b5920677578a5ac3171bab04bd27faca"
} | train |
{
"chosen_proof_sha256": "eeea18d1418cb641b169bfeb8bce930a2742ccdb2c216d63523e2c130f059c37",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "26ccc8781af1080819c179c8d94b6ddbd4... | by
obtain ⟨χ, hχ⟩ := Finset.rado_selection_subtype (β := β) (fun s ↦ g s s.finite_toSet)
refine ⟨χ, fun s hs ↦ ?_⟩
obtain ⟨t, ht, hst⟩ := hχ hs.toFinset
simp only [Set.Finite.toFinset_subset] at ht
exact ⟨t, by simp_all⟩ | true | {
"chosen_tokens": 70,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.071429
} | wrong_namespace_qualification | wrong_namespace_qualification | 677ae943ce0f5d5b5a0122fe65d9c3bab5252e5ca6d009c13043ec4e3daf4701 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
import Mathlib.Topology.Compactness.Compact
public import Mathlib.Data.Set.Finite.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Bhavik Mehta. All rights reserved.
R... | by
exact OProverMissing.missing_88630b1e90f4 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "5e9c1d83ea08a6ddfbb8aa6b76237d5783cdc9f356579e982c7c54e27365a53a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics",
"family_id": "set",
"file_id": "mathlib/Mathlib/Combinatorics/Compactness.lean",
"sample_id": "88630b1e90f48b536f58e1419d7c32cb382c083b0e23e75073e0b1e0fdf80749"
} | train |
{
"chosen_proof_sha256": "d7f9b791b4990ce10c90fe8f0ae29086af43a18750773d3fc0cc6aa2bef28c2c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
refine hs.induction_on _ (by simp) fun {i s} his hs e ↦ ?_
simp only [iInf_insert, e, ideal_inf] | true | {
"chosen_tokens": 30,
"rejected_tokens": 8,
"token_jaccard": 0.033333,
"token_length_ratio": 0.266667
} | unsolved_goals | unsolved_goals | 6791e7fedcd4c4f203b942538cc6f8d58f31db99d551ae4e9413ef79853b3fdd | 1 | 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
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "9526e73d9c63205db28236294bbda493881a30958fda9ee27c078ae44ee7adc5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/IdealSheaf",
"family_id": "ideal_biinf",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/Basic.lean",
"sample_id": "fd1fd9a53ab94e93ee71cb526c00c66f283a440ab2cd926e2cae6762d9e62179"
} | train |
{
"chosen_proof_sha256": "be5d4d87008e815276be22f67ac5016b50a8ac860e36c4ba227914947bb79f69",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4c373cfd20145617555f6ba51d8c75290a... | by
ext <;> simp; ext <;> simp <;> (rw [← Category.assoc]; congr 1; simp) | false | {
"chosen_tokens": 29,
"rejected_tokens": 36,
"token_jaccard": 0.894737,
"token_length_ratio": 1.241379
} | markdown_preamble_trailing | fence | 679a872412b6fc7dda7f6ef14e44c7ba3509764dfbca7d5b142ec1cff274c983 | 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.Subobject.Lattice
public import Mathlib.CategoryTheory.Monoidal.Braided.Basic
public import Mathlib.CategoryTheory.Dialectica.Basic
Namespace:
... | ```lean
by
ext <;> simp; ext <;> simp <;> (rw [← Category.assoc]; congr 1; simp)
``` | {
"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/Dialectica",
"family_id": "tensorhom_comp_tensorhom",
"file_id": "mathlib/Mathlib/CategoryTheory/Dialectica/Monoidal.lean",
"sample_id": "9e0d38f023447d9d9affb59a11c3c96b5f0f51e0a6c3e310742ed531f794b9ce"
} | train |
{
"chosen_proof_sha256": "02be6bd02ac451c737251c72bcfefcd2bda23393af373ca35cd3c4b0bb93ffd2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [coe_inv, inv_def, normSq_coe, inv_one, ofReal_one, mul_one] | false | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.2
} | by_exact_placeholder | placeholder | 67f27c6a5ce84c4753b9a6f0ca89084d8df644348fd59d40bf89c421ccd9947a | 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.Trigonometric.Basic
public import Mathlib.Analysis.Normed.Field.UnitBall
Namespace:
Circle
Local context:
/-
Copyright (c) 2021 Hea... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/Complex",
"family_id": "coe_inv_eq_conj",
"file_id": "mathlib/Mathlib/Analysis/Complex/Circle.lean",
"sample_id": "4dda6d8eed8aa9990b9ce99d74443691206268bc0aae2f322f2853e45e3b954a"
} | train |
{
"chosen_proof_sha256": "f5148887f1a7dfc8799d4c5143dcee9d63a0a776aa7b9be04cad5dfcce2d500a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2200214482fedc100475ea58f565cc5394... | by
refine ⟨fun {U} hU ↦ ?_⟩
exact .of_comp (a := f.s₀) (𝒱.f ·) (f.h₀ ·)
(fun _ ↦ Scheme.Hom.continuous _) (fun i ↦ funext <| by simp [← Scheme.Hom.comp_apply])
(fun _ ↦ Scheme.Hom.continuous _) U.2 (hU.isCompactOpenCovered 𝒱) | true | {
"chosen_tokens": 85,
"rejected_tokens": 2,
"token_jaccard": 0.025,
"token_length_ratio": 0.023529
} | invalid_tactic | invalid_tactic | 67ff61369669a0c8a6eeb5d825966cbfee8361b9d31c7841367860f815fa7a7e | 1 | 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.Affine
public import Mathlib.AlgebraicGeometry.Properties
public import Mathlib.AlgebraicGeometry.PullbackCarrier
public import Mat... | by
__oprover_invalid_tactic_b53835cfef00 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "2822a633e8c431be1dd8a28f37dd57884322e6dc3c13429d6036183992312270",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Cover",
"family_id": "of_hom",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Cover/QuasiCompact.lean",
"sample_id": "b53835cfef00f711bf8e92a6529c08ab65e2ebdbf08f272d243eb3a0265fab0b"
} | train |
{
"chosen_proof_sha256": "9f76daa2d17794583c7f31dfaabd84b49dfce6e55e1f6c203ac803f9bd29220e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by rw [← subAt_pullSub S x]; gcongr
mpr _ := le_trans (by gcongr) (pullSub_subAt T x) | false | {
"chosen_tokens": 23,
"rejected_tokens": 2,
"token_jaccard": 0.052632,
"token_length_ratio": 0.086957
} | sorry_admit_sorryAx_proof_hole | sorry | 6845695eec8ae84259197debf5701f9513a2145885e44fbc1921cf37f216c477 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.CompleteLattice.SetLike
Namespace:
Descriptive.Tree
Local context:
/-
Copyright (c) 2024 Sven Manthe. All rights reserved.
Released under Apache 2.0 li... | by
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": "SetTheory/Descriptive",
"family_id": "pullsub_adjunction",
"file_id": "mathlib/Mathlib/SetTheory/Descriptive/Tree.lean",
"sample_id": "1e270d1bd1eeb14c5bf62361efc62c628e247ff0c8b0d9ddc26f734fc3b353a7"
} | train |
{
"chosen_proof_sha256": "82d206dfe82328407b70971415868cac0e9eb31b2da76433329c285b45739547",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0cdce9457755c2ffaac6240c2fe685201f... | by
apply Linear.comp_smul | false | {
"chosen_tokens": 5,
"rejected_tokens": 10,
"token_jaccard": 0.5,
"token_length_ratio": 2
} | markdown_preamble_trailing | preamble | 684c0855b5d0b353cd3f9c82d2ff4a4d2717ca322fe95383dc9c51c0bcdc46f1 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Defs
public import Mathlib.Algebra.Module.TransferInstance
public import Mathlib.Algebra.Group.Invertible.Defs
public import Mathlib.Algebra.Mo... | Here is the proof:
by
apply Linear.comp_smul | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Linear",
"family_id": "comp_units_smul",
"file_id": "mathlib/Mathlib/CategoryTheory/Linear/Basic.lean",
"sample_id": "f2d217efd8dc67e5b8a5a62c60104ed7db1fe692e856a2299afd32cbd082efda"
} | train |
{
"chosen_proof_sha256": "ae21492afcd16aacb24a3ebd74e7f36439c097260b4eee6f2b4d2294937fc244",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
cases F; rfl | false | {
"chosen_tokens": 5,
"rejected_tokens": 5,
"token_jaccard": 0.111111,
"token_length_ratio": 1
} | sorry_admit_sorryAx_proof_hole | sorryAx | 68791f88bfdeff9ffcdd53a1dcffe6b6ae0b0a311b306542362c396e1289518d | 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.Analysis.Filter
public import Mathlib.Topology.Bases
public import Mathlib.Topology.LocallyFinite
Namespace:
Ctop
Local context:
/-
Copyright (c) 2017 M... | 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/Analysis",
"family_id": "ofequiv_val",
"file_id": "mathlib/Mathlib/Data/Analysis/Topology.lean",
"sample_id": "0aeef97f4cfea55c0640efb59e6d6792baab329447e04559e5b0bd1794a70844"
} | train |
{
"chosen_proof_sha256": "08d1285c21c0227003f0de3a7a5739a4341408268171b9be42c7528b7aade63d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | 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 | false | {
"chosen_tokens": 38,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.078947
} | by_exact_placeholder | exact? | 68a40bff9b7512c92c56067e25eccc951172a6683864aee191761f917c945c8f | 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.GCDMonoid.Finset
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.Data.PNat.Basic
public import Mathlib.Dynamics.PeriodicPts.Defs
publi... | 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": "Dynamics/PeriodicPts",
"family_id": "minimalperiod_eq_prime_iff",
"file_id": "mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean",
"sample_id": "f7b620c51cfaab544fc583af36da411df3fc24767a9c25ebd60f0ee43a35db07"
} | train |
{
"chosen_proof_sha256": "d216e83441f49f61f8d591a6008f2029d6a48f6f63ee5482956becb053185e63",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [commute_iff_eq, hp.le_iff_mul_eq_right hq |>.mp h, hp.le_iff_mul_eq_left hq |>.mp h] | true | {
"chosen_tokens": 25,
"rejected_tokens": 8,
"token_jaccard": 0.047619,
"token_length_ratio": 0.32
} | unsolved_goals | unsolved_goals | 68ad3d6f87bb505ac46bb83ed5986e0a31cf1eb79e259cde78fa937d1879e154 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order
public import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric
import Mat... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "d86ce97fb9dbeb6d52941debacee15a7914437ecfd602b701b6d053181cf4eca",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/CStarAlgebra",
"family_id": "commute_of_le",
"file_id": "mathlib/Mathlib/Analysis/CStarAlgebra/Projection.lean",
"sample_id": "9c77b43c796b9a66a9282957c2b67d46b27f3beda4c44634fa70becc056cd2ca"
} | train |
{
"chosen_proof_sha256": "2f2eac7accbcc17b05097bd1c9316c1f6c3d52ca67232fbfd3c0c187a0e1b260",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b373b3ff8a6dcb9dbdb8440546b2bd1bad... | by
rw [← le_div_iff₀ pi_div_two_pos, div_eq_inv_mul (W k) _]
rw [W_eq_integral_sin_pow_div_integral_sin_pow, le_div_iff₀ (integral_sin_pow_pos _)]
convert! integral_sin_pow_succ_le (2 * k + 1)
rw [integral_sin_pow (2 * k)]
simp | true | {
"chosen_tokens": 46,
"rejected_tokens": 3,
"token_jaccard": 0.035714,
"token_length_ratio": 0.065217
} | unknown_identifier | unknown_identifier | 68f4916afcb2098ea0252d2851954499d6aa9a027e4102ae335bfc3e7e7eddae | 1 | 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.Integrals.Basic
Namespace:
Real.Wallis
Local context:
/-
Copyright (c) 2021 Hanting Zhang. All rights reserved.
Released under Apac... | by
exact __oprover_missing_01224b14a9dc | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d07ddd786dcbe9c2f4a38ffa0a10f91217cb68af13e15bff0af81a4e5bd42ca4",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Real",
"family_id": "le_w",
"file_id": "mathlib/Mathlib/Analysis/Real/Pi/Wallis.lean",
"sample_id": "01224b14a9dc704c2fd71fd678b713d03a65ff9d6ee15bc743185909d5552045"
} | train |
{
"chosen_proof_sha256": "de4e82cdf62623b5f6eb51d83e7f17e091632bf91d46d2109ff7b01e77e7f85f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8ba782304e1ea50b35011d546ef6e91a73... | by
have hinj : Function.Injective (algebraMap F[X] K) := by
rw [IsScalarTower.algebraMap_eq F[X] F⟮X⟯ K]
exact (algebraMap F⟮X⟯ K).injective.comp (IsFractionRing.injective F[X] F⟮X⟯)
rw [injective_iff_map_eq_zero (algebraMap F[X] (↥(ringOfIntegers F K)))]
intro p hp
rw [← Subtype.coe_inj, Subalgebra.coe... | true | {
"chosen_tokens": 111,
"rejected_tokens": 3,
"token_jaccard": 0.051282,
"token_length_ratio": 0.027027
} | unknown_identifier | unknown_identifier | 68fd92fdd2ad301daa925c724aa59b9c9e4419553a2e801718d641cdc8d877a7 | 1 | 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.RatFunc.Degree
public import Mathlib.RingTheory.DedekindDomain.IntegralClosure
public import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
pu... | by
exact __oprover_missing_143ab2863e5e | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "292c27b988f5649ff312b116f57c1b3ef448808a632e250b862f34bf016acb19",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory",
"family_id": "algebramap_injective",
"file_id": "mathlib/Mathlib/NumberTheory/FunctionField.lean",
"sample_id": "143ab2863e5ef8203ad9cdb078d9a96faf3dccd3f496872d4f958e2b6081b962"
} | train |
{
"chosen_proof_sha256": "d93c674782daaf325f075697a98c3704ebb23b2b31510b6d8f8e1890f07f7bba",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b824477d2857f6ad9c20787126a537e758... | by
simp_rw [noZeroSMulDivisors_iff, or_iff_not_imp_left]
exact ⟨fun h r hr m eq ↦ h eq hr, fun h r m eq hr ↦ h r hr m eq⟩ | true | {
"chosen_tokens": 33,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.060606
} | invalid_tactic | invalid_tactic | 690f14e8598a642816c7abeba4d718f382096d1cc402705a725b04455c72108a | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Module.Torsion.Free
public import Mathlib.Tactic.Contrapose
Namespace:
(root)
Local context:
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reser... | by
__oprover_invalid_tactic_dac74140b502 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e16d571c851e82e00549e48f917c69df3f57df4d1743b52c94e5dcf916423f57",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/NoZeroSMulDivisors",
"family_id": "nozerosmuldivisors_iff_right_eq_zero_of_smul",
"file_id": "mathlib/Mathlib/Algebra/NoZeroSMulDivisors/Defs.lean",
"sample_id": "dac74140b5028292313ba8ba24d22052ad637c133b6e08b981669e5ff7752981"
} | train |
{
"chosen_proof_sha256": "471c71a8aa6e81210c7b7ea449c4bb32d4af4dfb79ecb5a4425fcd21336ff9fe",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simp only [ncoeff, HVertexOperator.coeff, LinearMap.coe_mk, AddHom.coe_mk]
exact HahnSeries.coeff_eq_zero_of_lt_order h | false | {
"chosen_tokens": 23,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.086957
} | by_exact_placeholder | by? | 6922f341ebd1f0184ca530051e47869bb1a13748a9acc47b35d422683bc70f33 | 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.Vertex.HVertexOperator
public import Mathlib.Data.Int.Interval
Namespace:
VertexOperator
Local context:
/-
Copyright (c) 2024 Scott Carnahan. All rig... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Vertex",
"family_id": "ncoeff_eq_zero_of_lt_order",
"file_id": "mathlib/Mathlib/Algebra/Vertex/VertexOperator.lean",
"sample_id": "a0a40b53d14535de5a165c9e14107454e6cc6b4afceef9775bdaaf0c31600cd0"
} | train |
{
"chosen_proof_sha256": "46daf5b2e7bf4d90e3013b1a193808b8c73aa5defa9de5956e0736e8e4908309",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
apply of_span_eq_top_target _ hs
rintro - ⟨i, rfl⟩
exact .equiv <| (IsLocalization.algEquiv (.powers <| s i) _ (T i)).symm |>.restrictScalars R | true | {
"chosen_tokens": 42,
"rejected_tokens": 8,
"token_jaccard": 0.029412,
"token_length_ratio": 0.190476
} | unsolved_goals | unsolved_goals | 698779a8993fc8c1ef6b8bc616dd712300b2818803d8d409e07e425853219289 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Finiteness.FiniteTypeLocal
public import Mathlib.RingTheory.Localization.Away.AdjoinRoot
Namespace:
Algebra.FinitePresentation
Local context:
/-
C... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "bfd92c5e7fda4a0dd3ca7960f5ef17fe540591adb6ec4018c9b213f68f9f8b99",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Finiteness",
"family_id": "of_span_eq_top_target_of_islocalizationaway",
"file_id": "mathlib/Mathlib/RingTheory/Finiteness/FinitePresentationLocal.lean",
"sample_id": "9439016a6f4e6a57c85536640298da69f3a589358dd7d886fc77db2d3daea612"
} | train |
{
"chosen_proof_sha256": "c66eb3df5899011ef7909b5b28a8dbc7ead7f1cd7984dec7b597a92d174e6203",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [SModEq.sub_mem] at hxy ⊢
rw [← smul_sub]
exact smul_mem_smul hc hxy | false | {
"chosen_tokens": 19,
"rejected_tokens": 3,
"token_jaccard": 0.125,
"token_length_ratio": 0.157895
} | by_exact_placeholder | placeholder | 6995f95510fe39e14f87ea7517134b55811a4d8808f651a17caf3d4a0fc12265 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Operations
public import Mathlib.LinearAlgebra.SModEq.Basic
Namespace:
SModEq
Local context:
/-
Copyright (c) 2025 Jiedong Jiang. All rights ... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/SModEq",
"family_id": "smul",
"file_id": "mathlib/Mathlib/LinearAlgebra/SModEq/Pointwise.lean",
"sample_id": "bf0246d237d6d9933a85d317c9082d92e62a7779e9ca287e29ec9e8f9ef938b0"
} | train |
{
"chosen_proof_sha256": "4854fca9d1b2dda857b810abc21527817558defad53eaaee4f2088331645faca",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "506ccf9dfd99be292019dce873814f6f38... | by cases x; rfl
nonrec def ReaderT.callCC {ε} [MonadCont m] {α β : Type _}
(f : Label α (ReaderT ε m) β → ReaderT ε m α) : ReaderT ε m α :=
ReaderT.mk (fun r => callCC fun f' => (f <| ReaderT.mkLabel _ f').run r) | true | {
"chosen_tokens": 72,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.069444
} | circular_dependency | circular_dependency | 69b0652e12395bee9c58c776dd46abe21cc386066abad279daba36ae579c072b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Control.Monad.Basic
public import Mathlib.Control.Monad.Writer
public import Mathlib.Control.Lawful
public import Batteries.Tactic.Congr
public import Batterie... | by
exact ReaderT.goto_mkLabel | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "d489f6e26f60be43fa848a04ea14587c7a614fd6bbff11c171842c866d6688c4",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Control/Monad",
"family_id": "readert",
"file_id": "mathlib/Mathlib/Control/Monad/Cont.lean",
"sample_id": "063710952a3a4d125df5a3e180296e39817ba1f9ff1c0061323053fb46b49fc4"
} | train |
{
"chosen_proof_sha256": "2f0e35fce05624abb1b88e07b05ba59ec3da85bd198bf90cf2c93b8503cfca8a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4f3514ad322903372735ab2658d2f37307... | by
let g : X ⟶ (⨆ i, U i : Y.Opens) := IsOpenImmersion.lift (Scheme.Opens.ι _) f (by simpa using H)
rw [← IsOpenImmersion.lift_fac (⨆ i, U i).ι f (by simpa using H)]
apply MorphismProperty.RespectsRight.postcomp (Q := @IsOpenImmersion) _ inferInstance
rw [iff_of_iSup_eq_top (P := P) (U := fun i : ι ↦ (⨆ i, U i)... | true | {
"chosen_tokens": 294,
"rejected_tokens": 3,
"token_jaccard": 0.026316,
"token_length_ratio": 0.010204
} | unknown_identifier | unknown_identifier | 69cf1ac5948ba337fa46dd69b6ae7453902383f185eae090a740b7c77981300d | 1 | 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.Limits
public import Mathlib.CategoryTheory.MorphismProperty.Local
public import Mathlib.Data.List.TFAE
Namespace:
AlgebraicGeometry.IsZaris... | by
exact __oprover_missing_93a0ad9cdbe5 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "10e8a49c43a1fabc5110ea9d976f96c17de8cda685b1c84f147a87943e7d6267",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Morphisms",
"family_id": "of_range_subset_isup",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Morphisms/Basic.lean",
"sample_id": "93a0ad9cdbe52633a2a6476e78bca93ac766df3668d5e36441bd4a7ce8340870"
} | train |
{
"chosen_proof_sha256": "de0e9cf3147279392578300fd27bc7777015480616482bce0bced9bd64ff6f36",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [NatTrans.ext_iff]
simp [le_def, Set.subset_def, ConcreteCategory.hom_ext_iff, funext_iff] | true | {
"chosen_tokens": 21,
"rejected_tokens": 8,
"token_jaccard": 0.047619,
"token_length_ratio": 0.380952
} | unsolved_goals | unsolved_goals | 69e5edc15184e562bee70be0f06e43f072f345b376d1fe3b9e9009a209aaf5a7 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
public import Mathlib.CategoryTheory.Subfunctor.Image
Namespace:
CategoryTheory.Subfunctor
Local context:
/-
Copyrigh... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "6558a8ef63f8f3040d5ff8daa4b0aaa1b1e75f13d587e3f6e90d13e3f6a26edb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Subfunctor",
"family_id": "range_le_equalizer_iff",
"file_id": "mathlib/Mathlib/CategoryTheory/Subfunctor/Equalizer.lean",
"sample_id": "7c2cc263cb08cd0ae7b3b6f1e5d4cbec38b9a07797c8b210ba27cbac169d95dc"
} | train |
{
"chosen_proof_sha256": "a40d38439b063352a0a57202db0bbc83f3e167b29ffee4ace96a68c95b1c7096",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
apply Subtype.ext; funext n
rcases h : s.val n; · rfl
refine absurd ?_ H; exact ⟨⟨_, _, h.symm⟩⟩ | false | {
"chosen_tokens": 36,
"rejected_tokens": 2,
"token_jaccard": 0.038462,
"token_length_ratio": 0.055556
} | sorry_admit_sorryAx_proof_hole | sorry | 69ebd5249d8f6940aa394412a311a854931b8d15cb413b67a43fef099ed79df0 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.Find
public import Mathlib.Data.Stream.Init
public import Mathlib.Logic.Relator
public import Mathlib.Tactic.Common
public import Batteries.Tactic.Lin... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Seq",
"family_id": "eq_empty_of_not_terminates",
"file_id": "mathlib/Mathlib/Data/Seq/Computation.lean",
"sample_id": "74e41a4b5bb950621cded82f802859d90e618c5ffbeaff846b1d0a22fbcf641d"
} | train |
{
"chosen_proof_sha256": "f95ac83c92042b8eca33c773ee62f7012472f34ee0a3ebc546b077931f47c21d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "10a43da42697c190a54beae2ffcca52222... | by
induction hs using Finset.Nonempty.cons_induction with
| singleton i => exact ⟨i, by simp⟩
| @cons i s hi hs ih =>
obtain ⟨j, hj⟩ := ih
by_cases hji : f j ≤ f i
· refine ⟨i, mem_cons_self .., ?_⟩
simp only [mem_cons, forall_eq_or_imp, imp_self, true_and]
exact fun k hk hik ↦ _root_.tran... | true | {
"chosen_tokens": 112,
"rejected_tokens": 3,
"token_jaccard": 0.033898,
"token_length_ratio": 0.026786
} | unknown_identifier | unknown_identifier | 69fe836f2fc4a509a1d269b96ad6f05d2ab30fa4e01aedd3d31c0cce6f45be40 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Set.Finite.Basic
public import Mathlib.Order.Minimal
Namespace:
Finset
Local context:
/-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released ... | by
exact __oprover_missing_83bbcd0a12ed | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ad6db25547f15ab0b280e78eba944e995a578aa92c68be7e3422f6192f6573b0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Preorder",
"family_id": "exists_maximalfor",
"file_id": "mathlib/Mathlib/Order/Preorder/Finite.lean",
"sample_id": "83bbcd0a12ed90d64932bbbf23b8435e8984a966305ad698b755df700c2893b8"
} | train |
{
"chosen_proof_sha256": "de9ab835bffa0c1a69cfad4d6364d7e5552c21f89d63ae535eda5295f7d4c8b1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7f4f13cee3f4a102def98fffe2db00bfee... | by
refine ⟨reg.1.of_flat_of_isBaseChange hf, ?_⟩
rw [← Ideal.map_ofList]
exact ((hf.map_smul_top_ne_top_iff_of_faithfullyFlat R M _).mpr reg.2.symm).symm | true | {
"chosen_tokens": 40,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.075
} | unknown_identifier | unknown_identifier | 6a1fde328b4eae961e1fdcc614348279a1bf08527d0da39c1eb08c85c84bf412 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
public import Mathlib.RingTheory.Flat.Localization
public import Mathlib.RingTheory.Regular.RegularSequence
Namespace:
Ri... | by
exact __oprover_missing_3a6d8b994748 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "88bdca814c3785c377d41f98c9f068f19c050a7f914ac811ab3a3d626c141925",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Regular",
"family_id": "isregular",
"file_id": "mathlib/Mathlib/RingTheory/Regular/Flat.lean",
"sample_id": "3a6d8b994748d4e07066dd4b842c5e18a4a267de9cc4efed1e2d52e8ad6a24cb"
} | train |
{
"chosen_proof_sha256": "0726e99880ae39e0c58cb38c38f992cfa6af39d47cf8ffe5ea36831ca442aef8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9a39f51210f01f82a58423ea9a2a56c213... | by
obtain ⟨Us, hUs, heq⟩ := TopologicalSpace.Opens.isBasis_iff_cover.mp
(PrespectralSpace.isBasis_opens X) ⟨f ⁻¹' U, hU.preimage hfc⟩
obtain ⟨t, ht⟩ := by
refine hc.elim_finite_subcover (fun s : Us ↦ f '' s.1) (fun s ↦ h _ s.1.2) (fun x hx ↦ ?_)
obtain ⟨x, rfl⟩ := hs hx
obtain ⟨i, hi, hx⟩ := mem_sSu... | false | {
"chosen_tokens": 252,
"rejected_tokens": 257,
"token_jaccard": 0.951807,
"token_length_ratio": 1.019841
} | markdown_preamble_trailing | trailing | 6a3acf6f1286ac55078124496c97862927be9b3a20f7c2b8ebf051369a1f63fe | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.Ideal
public import Mathlib.Topology.Sets.Compacts
public import Mathlib.Topology.Sets.OpenCover
public import Mathlib.Topology.Spectral.Hom
Namespace:
... | by
obtain ⟨Us, hUs, heq⟩ := TopologicalSpace.Opens.isBasis_iff_cover.mp
(PrespectralSpace.isBasis_opens X) ⟨f ⁻¹' U, hU.preimage hfc⟩
obtain ⟨t, ht⟩ := by
refine hc.elim_finite_subcover (fun s : Us ↦ f '' s.1) (fun s ↦ h _ s.1.2) (fun x hx ↦ ?_)
obtain ⟨x, rfl⟩ := hs hx
obtain ⟨i, hi, hx⟩ := mem_sSu... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/Spectral",
"family_id": "isopenmap",
"file_id": "mathlib/Mathlib/Topology/Spectral/Prespectral.lean",
"sample_id": "06e54fa94e0402146a1056a8e38c4177d4d7f6c2eb72f119267197e2349465f8"
} | train |
{
"chosen_proof_sha256": "74e6e717b4029a5bb695646d264a47c3c431da50efbba97f42340e634e0683b0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
induction hs using Nonempty.cons_induction with
| singleton => simp [Hlt _ (mem_singleton_self _)]
| cons _ _ _ _ ih =>
simp only [sum_cons, forall_mem_cons] at Hlt ⊢
exact ENat.add_lt_add Hlt.1 (ih Hlt.2) | false | {
"chosen_tokens": 51,
"rejected_tokens": 2,
"token_jaccard": 0.030303,
"token_length_ratio": 0.039216
} | sorry_admit_sorryAx_proof_hole | admit | 6a3ad530e9773fd254b1cd97a1d286b8209c699291516714cc2bedb7f6f8190e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Ring.Finset
public import Mathlib.Algebra.BigOperators.WithTop
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import... | by
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": "Data/ENat",
"family_id": "sum_lt_sum_of_nonempty",
"file_id": "mathlib/Mathlib/Data/ENat/BigOperators.lean",
"sample_id": "7736e8daaa8f61cdcc7d061cb79e4a68846e752fbc7a81c150325816b7000c0c"
} | train |
{
"chosen_proof_sha256": "defdc0751f53e30ab813268ab9f2c357764562c8bfc11a4ae2b13dad9c027777",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq] | false | {
"chosen_tokens": 15,
"rejected_tokens": 5,
"token_jaccard": 0.066667,
"token_length_ratio": 0.333333
} | sorry_admit_sorryAx_proof_hole | sorryAx | 6a7f576cef4e95140a4e5d45e5d10c5272cc752092d430b16d0c8655a04c9687 | 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.GaloisConnection.Basic
Namespace:
Heyting
Local context:
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as ... | 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/Heyting",
"family_id": "isregular",
"file_id": "mathlib/Mathlib/Order/Heyting/Regular.lean",
"sample_id": "9a7f488b9072e1c3cd3e743c9cf1c9c590011f4701530db321fda7898e28d6e1"
} | train |
{
"chosen_proof_sha256": "2f6287791c3459636988da88fa63c514ca770a07a4d1537e256b0a039a8246d2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "04c6531705ec9500b581b2d2a796d16ba0... | by
rcases l.exists_antitone_basis with ⟨u, hu⟩
rcases (Filter.hasBasis_self.mpr hl'.exists_measurable_subset).exists_antitone_subbasis with
⟨v, v_meas, hv⟩
simp only [hu.tendsto_iff hv.toHasBasis, true_imp_iff, true_and, setOf_forall, setOf_exists]
exact .iInter fun n ↦ .iUnion fun _ ↦ .biInter (to_countabl... | true | {
"chosen_tokens": 83,
"rejected_tokens": 5,
"token_jaccard": 0.06383,
"token_length_ratio": 0.060241
} | wrong_namespace_qualification | wrong_namespace_qualification | 6ae91d2e8a07a5f7f30f318fd79ffb0596aba5a13b1aecd7fa41f4135057a965 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.MeasurableSpace.Constructions
public import Mathlib.Order.Filter.AtTopBot.CompleteLattice
public import Mathlib.Order.Filter.AtTopBot.CountablyGe... | by
exact OProverMissing.missing_f6e07b955781 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "49f342f124d6d08c8514ce9671c7728ffe347c2e31f59b41de24d33bf7352c27",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/MeasurableSpace",
"family_id": "measurableset_tendsto",
"file_id": "mathlib/Mathlib/MeasureTheory/MeasurableSpace/MeasurablyGenerated.lean",
"sample_id": "f6e07b9557819a58269647c6b5189639d71a69d808a07410892e1fb416291719"
} | train |
{
"chosen_proof_sha256": "991657dbd4edd454359c31dd8bdf594c016e0afa8edd3fccac619f800a61e43a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ffd1e1b04297070915458a9e6cd813666c... | by
simp [pointOfClosedPoint, ← SpecMap_residueFieldIsoBase_inv, ← Spec.map_comp] | false | {
"chosen_tokens": 13,
"rejected_tokens": 17,
"token_jaccard": 0.785714,
"token_length_ratio": 1.307692
} | missing_wrong_import | import | 6b0c7332ef0556832036e8c1e4b88e9452d121e887677bd78489e2b34c4c3b0f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.Finite
public import Mathlib.FieldTheory.IsAlgClosed.Basic
Namespace:
AlgebraicGeometry
Local context:
/-
Copyright (c) 2026 Andr... | import Mathlib.WrongImport
by
simp [pointOfClosedPoint, ← SpecMap_residueFieldIsoBase_inv, ← Spec.map_comp] | {
"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": "AlgebraicGeometry/AlgClosed",
"family_id": "pointofclosedpoint_comp",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/AlgClosed/Basic.lean",
"sample_id": "2adf0561a9917b239f775543158667a626bcb1a48a3b86163579bd9e8eb95116"
} | train |
{
"chosen_proof_sha256": "7749888414d06cd3f5ca86c4419d34b5759e4af27af9cf5a4232c6e45cfd9c6a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
simpa only [add_comm _ (n : ℝ)] using f.commute_add_int n | false | {
"chosen_tokens": 17,
"rejected_tokens": 2,
"token_jaccard": 0.058824,
"token_length_ratio": 0.117647
} | sorry_admit_sorryAx_proof_hole | admit | 6b10aa030836b7e419d3385406285d77f7ef146ba48538967b81391736625f8f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Order.SemiconjSup
Namespace:
CircleDeg1Lift
Local context:
/-
Copyright (c) 2020 Yury Kudryashov. All rig... | by
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": "Dynamics/Circle",
"family_id": "commute_add_int",
"file_id": "mathlib/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean",
"sample_id": "099bd67010bd3bdf0391a625cbe08e92d2569c8b6c447860d4abbc5d4f43fc92"
} | train |
{
"chosen_proof_sha256": "8e1e8082a0a84f2600d4003971ff82d46d02db0f4b2396697813b11973e2972c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rintro ⟨p⟩
rw [← @WeaklyConnectedComponent.eq (Generators G), eq_comm, ← FreeGroup.of_injective.eq_iff, ←
mul_inv_eq_one]
let X := FreeGroup (WeaklyConnectedComponent <| Generators G)
let f : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)
let F : G ⥤ CategoryTheory.SingleObj.{u} (X : Ty... | false | {
"chosen_tokens": 168,
"rejected_tokens": 2,
"token_jaccard": 0.014493,
"token_length_ratio": 0.011905
} | sorry_admit_sorryAx_proof_hole | sorry | 6b2c493e064f4ac62cadbade8561dcd815bd28b9332d632c5a3b79c3d2373a24 | 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
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": "GroupTheory/FreeGroup",
"family_id": "path_nonempty_of_hom",
"file_id": "mathlib/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean",
"sample_id": "660e9e4b92da9e1464e29ce063157b6ff472ec59e69123eed2b8146bcadb1cb1"
} | train |
{
"chosen_proof_sha256": "12eef625fe3216831384e04422da6efdbadfd13d594c33e0c21289712e29f94e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "792bc3b530c92020c59f266814860e6bb8... | by
classical
rw [monomial_def]
exact Pi.single_eq_same _ _ | true | {
"chosen_tokens": 12,
"rejected_tokens": 5,
"token_jaccard": 0.230769,
"token_length_ratio": 0.416667
} | wrong_namespace_qualification | wrong_namespace_qualification | 6b57fc5b26c9de708910d4bdfe077ff55f862c59480538f36ce537c67415ec35 | 1 | 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.Antidiag.Finsupp
public import Mathlib.Data.Finsupp.Weight
public import Mathlib.LinearAlgebra.Pi
public import Mathlib.Algebra.MvPolynomial.Basi... | by
exact OProverMissing.missing_3a3c329ac7ef | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "80fc3728752a516898d07abf38ea6c636e6b9d30e9a52332e0117e665d609f1a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/MvPowerSeries",
"family_id": "coeff_monomial_same",
"file_id": "mathlib/Mathlib/RingTheory/MvPowerSeries/Basic.lean",
"sample_id": "3a3c329ac7ef7bd97b6db71b43a7efe1eaaa001c53c4b487b813a42f94dd4a0c"
} | train |
{
"chosen_proof_sha256": "e99a691cbba97dd50b462f822d8ea54c6c328e4bf1d6437c6b809f75afb37a33",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "29d94a2fea1431fff864f71743a56f3cc6... | by
have key : ∀ x ∈ hs.toFinset, s.indicator (fun _ ↦ a) x = a := by
intro x hx
rw [indicator_of_mem (hs.mem_toFinset.1 hx) (fun _ ↦ a)]
rw [Nat.card_eq_card_finite_toFinset hs, ← sum_eq_card_nsmul key, eventually_atTop]
obtain ⟨m, hm⟩ := hs.bddAbove
refine ⟨m + 1, fun n n_m ↦ (sum_subset ?_ ?_).symm⟩ <... | true | {
"chosen_tokens": 146,
"rejected_tokens": 2,
"token_jaccard": 0.016667,
"token_length_ratio": 0.013699
} | invalid_tactic | invalid_tactic | 6b71a8d59f9fd0d3e53fa9335ecda90cd757a2ec34da85e4bb882746c0e3fc6c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
public import Mathlib.Algebra.Order.Archimedean.Basic
public import Mathlib.Algebra.Order.BigOperators.Group.Finset... | by
__oprover_invalid_tactic_71930e9f307d | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "3c908ace2349f6a695ba3590395460653d9e87667f67513c1b144d979648c6db",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Order",
"family_id": "sum_indicator_eventually_eq_card",
"file_id": "mathlib/Mathlib/Algebra/Order/Archimedean/IndicatorCard.lean",
"sample_id": "71930e9f307d51dc22d21feef3aebed1568f0f6f3ee70012583f25208fbd5c02"
} | train |
{
"chosen_proof_sha256": "8af251c6a8eb77af2dfa75888d7c394a97af0ecb8aa758dfff389aa06e6d13fe",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [← LinearMap.comp_assoc, map_map_comp_assoc_eq]
ext
rfl | false | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.25
} | by_exact_placeholder | placeholder | 6b7ee87143d14f8efe72be43fea5fb1a15f3de815c197c659eb56bed7306df3f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorProduct.Tower
public import Mathlib.RingTheory.Coalgebra.Basic
import Mathlib.Tactic.Attr.Register
Namespace:
CoassocSimps
Local context:... | 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/Coalgebra",
"family_id": "assoc_comp_map_map_comp",
"file_id": "mathlib/Mathlib/RingTheory/Coalgebra/CoassocSimps.lean",
"sample_id": "6a8e8150ef6c1755c0f36af56608a5c55a4714f68f327a0666a7f1b00c8049b8"
} | train |
{
"chosen_proof_sha256": "68b7446878a3dfdb301fca861ee4b86aba230785df09d24821618a061945374b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
classical
let ⟨f⟩ := quotient_prod_linearEquiv p
exact rank_prod'.symm.trans f.rank_eq | false | {
"chosen_tokens": 18,
"rejected_tokens": 2,
"token_jaccard": 0.0625,
"token_length_ratio": 0.111111
} | by_exact_placeholder | by? | 6b8596949063684467dad9ad77e3c53c4ef4e4f3815f36075a288e4e57c3ff10 | 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.Basis.VectorSpace
public import Mathlib.LinearAlgebra.Dimension.Finite
public import Mathlib.LinearAlgebra.Dimension.RankNullity
Namespace:
(roo... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Dimension",
"family_id": "rank_quotient_add_rank_of_divisionring",
"file_id": "mathlib/Mathlib/LinearAlgebra/Dimension/DivisionRing.lean",
"sample_id": "a8f96640d4ba327fc7e9e6bd4ae65eab83d12cf3037d5abef2ad29e199cb314c"
} | train |
{
"chosen_proof_sha256": "a8a91e5033815085839eaad1f6a9d8721d67997ed1799815646887dd0bdc0a37",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
simp [AlgEquiv.ext_iff, Subtype.ext_iff, AlgEquiv.restrictNormal_apply] | false | {
"chosen_tokens": 15,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.133333
} | sorry_admit_sorryAx_proof_hole | sorry | 6b859716db3520013cc49d56dd93f6f55b040b688487c35ee5125302f7cbad45 | 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.Splits
public import Mathlib.FieldTheory.Galois.Notation
public import Mathlib.FieldTheory.IntermediateField.Basic
public import Mathlib.Fie... | 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": "FieldTheory/Normal",
"family_id": "algequiv",
"file_id": "mathlib/Mathlib/FieldTheory/Normal/Defs.lean",
"sample_id": "e7bd69cfd0dadb9fdaab62111c02ae20ae7bf73cdd0cc212f534d85bcf679dc1"
} | train |
{
"chosen_proof_sha256": "355d8928efd1f0f2dd702b5a22182d531523711ce4f995520e27017b7c2e8b46",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c7578c9e54d861a424808b1e99539d9386... | by
refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
· filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy
· intro ε εpos
filter_upwards [Icc_mem_nhdsLT <| show x - ε < x by linarith] with y hy
rw [closedBall_eq_Icc]
exact Icc_subset_Icc hy.1 (by linarith) | true | {
"chosen_tokens": 63,
"rejected_tokens": 3,
"token_jaccard": 0.051282,
"token_length_ratio": 0.047619
} | unknown_identifier | unknown_identifier | 6b9085aede07a4702af03a394daff29550439a0c92aaaa34e2bc8decf849b527 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Covering.DensityTheorem
public import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
Namespace:
Real
Local context:
/-
Copyright (c) 2022 Sébast... | by
exact __oprover_missing_8ad925533b8c | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f3aa7baa2d27b96138f6a76ee851cf06d9c56d057604b3c2c8bc5cf2fee34068",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Covering",
"family_id": "tendsto_icc_vitalifamily_left",
"file_id": "mathlib/Mathlib/MeasureTheory/Covering/OneDim.lean",
"sample_id": "8ad925533b8c56254faf2ab0de8790476e52ea1f36cb7dddd76cb2f24ea5a395"
} | train |
{
"chosen_proof_sha256": "1e710a9d7e1e6f7c130e5c280474bf460f6d3246c3fb6e99473017f60576e601",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by cases x; rfl
nonrec def StateT.callCC {σ} [MonadCont m] {α β : Type _}
(f : Label α (StateT σ m) β → StateT σ m α) : StateT σ m α :=
StateT.mk (fun r => callCC fun f' => (f <| StateT.mkLabel f').run r) | true | {
"chosen_tokens": 71,
"rejected_tokens": 8,
"token_jaccard": 0.073171,
"token_length_ratio": 0.112676
} | unsolved_goals | unsolved_goals | 6b93d0267d91c871c2a0115ea1384e4a3c0eccaa373d44bc6b8389a7225158f9 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Control.Monad.Basic
public import Mathlib.Control.Monad.Writer
public import Mathlib.Control.Lawful
public import Batteries.Tactic.Congr
public import Batterie... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "575ec23ce4a082f312691784f5746a7ae8e8f898dcf4e503b927436679856375",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Control/Monad",
"family_id": "statet",
"file_id": "mathlib/Mathlib/Control/Monad/Cont.lean",
"sample_id": "8451c1001183081b0718334cf931fa3560f5cf1ca44e66d4b164528ee6a8ec57"
} | train |
{
"chosen_proof_sha256": "0c7a4b44bba5c20da7810399c16de1e098a5c39bdee940dd765e6735e6e9d672",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4b02dfc32f7fdee24d8ec38c3b482d9b41... | by
simp [IsSelfAdjoint.star_eq, ← sub_eq_add_neg, (CFC.sqrt_nonneg (1 - a ^ 2 : A)).isSelfAdjoint] | false | {
"chosen_tokens": 27,
"rejected_tokens": 32,
"token_jaccard": 0.846154,
"token_length_ratio": 1.185185
} | markdown_preamble_trailing | trailing | 6bb66eaf4323e4c7deebd13d2072ca1ce8ebb1ce92f587cdd4841fa51a18ce32 | 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.ContinuousFunctionalCalculus.Order
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary
public import Mathlib... | by
simp [IsSelfAdjoint.star_eq, ← sub_eq_add_neg, (CFC.sqrt_nonneg (1 - a ^ 2 : A)).isSelfAdjoint]
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": "Analysis/CStarAlgebra",
"family_id": "selfadjoint",
"file_id": "mathlib/Mathlib/Analysis/CStarAlgebra/Unitary/Span.lean",
"sample_id": "af27a105e43a072189a2b49287038416eed37dc52c7226788734b79f2baf138e"
} | train |
{
"chosen_proof_sha256": "b0a03b5526b62c3517381ba5457b6ebaafee0e4da110c931ab1a6ee0269bdd8b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3a5c41e6c5f8d96bb9ba0e28b19215e242... | by
cases p
rw [← ofFractionRing_smul] | false | {
"chosen_tokens": 8,
"rejected_tokens": 13,
"token_jaccard": 0.615385,
"token_length_ratio": 1.625
} | markdown_preamble_trailing | preamble | 6c370c96e94f198256381a159b24f1cac3a35319a9f4b580c77ad8170c7ebf72 | 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.CharP.Algebra
public import Mathlib.FieldTheory.RatFunc.Defs
public import Mathlib.RingTheory.Polynomial.Content
public import Mathlib.RingTheory.Algeb... | Here is the proof:
by
cases p
rw [← ofFractionRing_smul] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "FieldTheory/RatFunc",
"family_id": "tofractionring_smul",
"file_id": "mathlib/Mathlib/FieldTheory/RatFunc/Basic.lean",
"sample_id": "5b0c69fc729fb329df53e747032d70f91d7a865fc26c9ee5566a2b68840871a7"
} | train |
{
"chosen_proof_sha256": "e1c4c37ea1905b854b1db5a6620e0ed4d40197fe584c2e9cb453bf8d89ed1d01",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c2825dbc1dbcd1603529867e278ac4ffcf... | by
simpa using ideal_biInf I Set.finite_univ | true | {
"chosen_tokens": 8,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.375
} | circular_dependency | circular_dependency | 6c451ec27cae6a09e2ca9ff7543b25c26bb8324dcb0f6e5a6e000449522fef58 | 1 | 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 ideal_iInf | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "1fd8ad3507d7671b7fe89ee7943e17682e5178be8bbde12ede02413b3aa492a8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "30f83bf0d483bdb24895df71acc1365551186d762bcd20d14e7f1ac4b162ed7c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
obtain ⟨s, -, hscard, hs⟩ := addRothNumber_spec (univ : Finset α)
haveI := noAccidental hs
rw [← hscard, ← card_triangleIndices, ← card_triangles]
exact (locallyLinear hs).le_ruzsaSzemerediNumber | false | {
"chosen_tokens": 41,
"rejected_tokens": 5,
"token_jaccard": 0.0625,
"token_length_ratio": 0.121951
} | sorry_admit_sorryAx_proof_hole | sorryAx | 6c6bec73a3eab774723079f876f3c0074c3a0fd987e9a27c689bc1097add8305 | 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.Additive.AP.Three.Behrend
public import Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite
public import Mathlib.Tactic.Rify
public import Mat... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/Extremal",
"family_id": "addrothnumber_le_ruzsaszemeredinumber",
"file_id": "mathlib/Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean",
"sample_id": "862a0c729b5cd3749b546c796ee66cf2c56a927378f8d912626bfe6707f1fa0a"
} | train |
{
"chosen_proof_sha256": "6ef68f198f9c3ddb8b2e403677d5bebc09f01883ade66189972cc6ad03aa6c3a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e52881820a620469d9332d0452bd209c49... | by
let f := (lmul R S).toLinearMap.prodMap (lmul R T).toLinearMap
have : (lmul R (S × T)).toLinearMap = (prodMapLinear R S T S T R).comp f :=
LinearMap.ext₂ Prod.mul_def
simp_rw [trace, this]
exact trace_prodMap' _ _ | false | {
"chosen_tokens": 64,
"rejected_tokens": 69,
"token_jaccard": 0.891892,
"token_length_ratio": 1.078125
} | markdown_preamble_trailing | trailing | 6c9003c36ce82bbfe994aa2099238fb105a507356857eef5de44825a2266f4c8 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
public import Mathlib.LinearAlgebra.Matrix.BilinearForm
public import Mathlib.LinearAlgebra.Trace
Namespace:
Algebra
L... | by
let f := (lmul R S).toLinearMap.prodMap (lmul R T).toLinearMap
have : (lmul R (S × T)).toLinearMap = (prodMapLinear R S T S T R).comp f :=
LinearMap.ext₂ Prod.mul_def
simp_rw [trace, this]
exact trace_prodMap' _ _
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Trace",
"family_id": "trace_prod_apply",
"file_id": "mathlib/Mathlib/RingTheory/Trace/Defs.lean",
"sample_id": "5cac71067938ac4bef77108bfac10db01fab6c38979ad732ebaa5982c97b9231"
} | train |
{
"chosen_proof_sha256": "667083b2ccb953eab7042f8125e1495f0c41208219db3ae4328356bea107af18",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b1b23128020afe0708f69c540c35c829ec... | by
rw [div_def, ← @Rat.round_cast ℝ _ _, ← @Rat.round_cast ℝ _ _]
simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, add_mul] | true | {
"chosen_tokens": 36,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.083333
} | circular_dependency | circular_dependency | 6c9fabc5a718462041620288093484af2a1dae83875e6f5bdce5d0e2dd5f8783 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Archimedean.Real.Basic
public import Mathlib.Data.Complex.Basic
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.NumberTheory.Zsq... | by
exact toComplex_im_div | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "f245cf6f88286ec979e42f060a828c3dbf251248eda49d47534f93a072f59898",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Zsqrtd",
"family_id": "tocomplex_im_div",
"file_id": "mathlib/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean",
"sample_id": "76ce3bf080d160d098641f9edded107effa5e468a4e154be6bb90faf62e605f0"
} | train |
{
"chosen_proof_sha256": "6bc6459c3657e743f6b65ab2f495aaf259c49629a6dfb3dcfa555bfa29fc68bb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
have hf : Monotone fun n => ⨅ i ≥ n, f i := fun n m h => biInf_mono fun i => h.trans
rw [← Monotone.iSup_nat_add hf k]
· simp_rw [iInf_ge_eq_iInf_nat_add, ← Nat.add_assoc]
-- Not `@[simp]` since the subterm `?f (i + ?k)` produces an ugly higher-order unification problem.
-- (Although the `simpNF` linter does ... | false | {
"chosen_tokens": 143,
"rejected_tokens": 2,
"token_jaccard": 0.025316,
"token_length_ratio": 0.013986
} | by_exact_placeholder | by? | 6caa21cea7054c22cab76a355d75bd6a77c89a6b3d004604d78ba2cb400e424e | 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.Bool.Set
public import Mathlib.Data.Nat.Set
public import Mathlib.Order.CompleteLattice.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Joh... | 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/CompleteLattice",
"family_id": "isup_iinf_ge_nat_add",
"file_id": "mathlib/Mathlib/Order/CompleteLattice/Lemmas.lean",
"sample_id": "56582689395b3fd4d53bdd9aa4d7c0b70282e024f7b7737ed578d8946fb979bf"
} | train |
{
"chosen_proof_sha256": "5e0e8b084461b68f53dd226528eec6d4daa60789f672fc8f9edb2d39518f89fb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "73b61e1aea22a359d649ed44c1a7748117... | by
apply c.isPushout.hom_ext h₀
apply Cofan.IsColimit.hom_ext c.isColimit₂
simpa [cell_def] using h | true | {
"chosen_tokens": 25,
"rejected_tokens": 5,
"token_jaccard": 0.1,
"token_length_ratio": 0.2
} | wrong_namespace_qualification | wrong_namespace_qualification | 6cb1943243d158949aa2734503e5b5f5bb33c0df588558990fe08884a6ac429d | 1 | 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
exact OProverMissing.missing_f5cf3adf5b2f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "be2ea9014a88548e9b8f4a61cb0933db339bd78a9aad1e2f24f318c4bdc82922",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/RelativeCellComplex",
"family_id": "hom_ext",
"file_id": "mathlib/Mathlib/AlgebraicTopology/RelativeCellComplex/AttachCells.lean",
"sample_id": "f5cf3adf5b2f007282e6afc982afeb8adfc7d1ca08c6a7becb4328354365d479"
} | train |
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