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": "887cd45eb0ad0270c148f43d5c6fb5160031574c8c6c116689e7fb4bdff4ffd8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b26b39d9245fa55448b7c078ba618090e6... | by
rcases finite_or_infinite X with _ | _
· have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_right S X)
rwa [lift_umax.{v, u}, lift_id'] at this
have key (x : X) (s s' : S) (h : s • x = s' • x) (hc : Commute s s') : x /ₒ s = x /ₒ s' := by
rw [oreDiv_eq_iff]
refine ⟨s, s'.1, h,... | false | {
"chosen_tokens": 190,
"rejected_tokens": 195,
"token_jaccard": 0.948052,
"token_length_ratio": 1.026316
} | markdown_preamble_trailing | preamble | 3bd32c00b5c37bde1ffd4746746b5e3391cdbcf278af9ffbc3ede35edbc230f9 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Pigeonhole
public import Mathlib.GroupTheory.OreLocalization.Basic
public import Mathlib.SetTheory.Cardinal.Arithmetic
Namespace:
OreLocalization... | Here is the proof:
by
rcases finite_or_infinite X with _ | _
· have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_right S X)
rwa [lift_umax.{v, u}, lift_id'] at this
have key (x : X) (s s' : S) (h : s • x = s' • x) (hc : Commute s s') : x /ₒ s = x /ₒ s' := by
rw [oreDiv_eq_iff]
... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "GroupTheory/OreLocalization",
"family_id": "cardinalmk_le_lift_cardinalmk_of_commute",
"file_id": "mathlib/Mathlib/GroupTheory/OreLocalization/Cardinality.lean",
"sample_id": "00174d518458fef9a309864e49f69be9bd090110f25e5905b3ad9090d83909e4"
} | train |
{
"chosen_proof_sha256": "212d1f84cff2b596f7c1f93804fd9c282408c326066650fb7feba4fd785cf41f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c49a0b60eabd3506186f70e85a58714221... | by
simp_rw [max_def, val_le_val, ← apply_ite]
rfl | false | {
"chosen_tokens": 11,
"rejected_tokens": 16,
"token_jaccard": 0.666667,
"token_length_ratio": 1.454545
} | markdown_preamble_trailing | preamble | 3c047b19fcdba34445a6063b591186adf8ebf083cdb763379dca9c9e3743aadb | 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.Hom.Basic
public import Mathlib.Algebra.Group.Units.Defs
Namespace:
Units
Local context:
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Rele... | Here is the proof:
by
simp_rw [max_def, val_le_val, ← apply_ite]
rfl | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Order",
"family_id": "max_val",
"file_id": "mathlib/Mathlib/Algebra/Order/Monoid/Units.lean",
"sample_id": "a53b7c1c6dc7380167b7eef2678e6391bfc432a0b3abcb53f2f3e41f881be729"
} | train |
{
"chosen_proof_sha256": "afd4fe6929e62277a458a32fdf80154f965674b8199603407ac172cf56934b1a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "feb0ac643b6d6e33049a45eaff6295a840... | by
refine
le_trans ?_ <| ((Subobject.pullback (π(π₁, prod.map F.f (𝟙 _) ≫ G.F))).monotone F.le).trans <|
le_trans ?_ <| ((Subobject.pullback (prod.map F.f (𝟙 Z.tgt))).monotone G.le).trans ?_
<;> simp [← Subobject.pullback_comp] | true | {
"chosen_tokens": 90,
"rejected_tokens": 3,
"token_jaccard": 0.027778,
"token_length_ratio": 0.033333
} | unknown_identifier | unknown_identifier | 3c17ff7e5b9bfc7f99a0aef982097f9786d4b8bde18a9c5d705828b74dac9053 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Subobject.Basic
Namespace:
CategoryTheory.Dial
Local context:
/-
Copyright (c) 2024 Mario Carneiro. All rights reserved.
Released under Apache... | by
exact __oprover_missing_7b702b3f54a4 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "4c2b0791f2aa28619c5d5f056ac2aed2e8ced281ff1d586aab6d5bcad2bfbbba",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Dialectica",
"family_id": "comp_le_lemma",
"file_id": "mathlib/Mathlib/CategoryTheory/Dialectica/Basic.lean",
"sample_id": "7b702b3f54a49b24809e03e0a017a431793ec768bf4e679fe9f430ed6393e5c4"
} | train |
{
"chosen_proof_sha256": "3fa3b1641e3a223c1b1c2acb29844761f01c0429d5fdec1fe9c7e636fcc33dde",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e97cf30cee9ddef793712f354c924a5ad0... | by
suffices h₁ : n ≤ sInf {m | p (m - n)} by
convert! sInf_add h₁
simp_rw [Nat.add_sub_cancel_right]
obtain ⟨m, hm⟩ := nonempty_of_pos_sInf h
refine
le_csInf ⟨m + n, ?_⟩ fun b hb ↦
le_of_not_gt fun hbn ↦
ne_of_mem_of_not_mem ?_ (notMem_of_lt_sInf h) (Nat.sub_eq_zero_of_le hbn.le)
· dsi... | true | {
"chosen_tokens": 83,
"rejected_tokens": 3,
"token_jaccard": 0.038462,
"token_length_ratio": 0.036145
} | unknown_identifier | unknown_identifier | 3c9e8ad53dd79d8ba619d1bbe1715a818d4944e9d3302209978a9f4fed85562b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.ConditionallyCompleteLattice.Finset
public import Mathlib.Order.Interval.Finset.Nat
Namespace:
Nat
Local context:
/-
Copyright (c) 2018 Johannes Hölzl.... | by
exact __oprover_missing_717673c42ec9 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "65ae4901ffba7d5e92ccaaa270adc5cc0e665aa90f4c92ece65c19f6c6b6dbee",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Lattice",
"family_id": "sinf_add",
"file_id": "mathlib/Mathlib/Order/Lattice/Nat.lean",
"sample_id": "717673c42ec9cf536eab52497d443d7e188d9ccddc0b2dc34242dbb48af892e4"
} | train |
{
"chosen_proof_sha256": "801cfb0a412f7e9f521af786c135901e8a7681f3edb0eee3dd6e24550333a788",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "786cb7a4f6894264ed36b3120e49a20018... | by
constructor
· obtain a | a := a <;> obtain b | b := b
· rw [sumLexLift, mem_map]
rintro ⟨c, hc, rfl⟩
exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩
· refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_
· rintro ⟨c, hc, rfl⟩
exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)
· rintro ⟨c... | false | {
"chosen_tokens": 287,
"rejected_tokens": 291,
"token_jaccard": 0.93617,
"token_length_ratio": 1.013937
} | missing_wrong_import | import | 3cb5651472f70e5a2555b16bdf02f2a3ff2b28e2dd1e0d9c0e6422b5fd393164 | 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.Sum
public import Mathlib.Data.Sum.Order
public import Mathlib.Order.Interval.Finset.Defs
Namespace:
Finset
Local context:
/-
Copyright (c) 2022 ... | import Mathlib.WrongImport
by
constructor
· obtain a | a := a <;> obtain b | b := b
· rw [sumLexLift, mem_map]
rintro ⟨c, hc, rfl⟩
exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩
· refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_
· rintro ⟨c, hc, rfl⟩
exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, ... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Sum",
"family_id": "mem_sumlexlift",
"file_id": "mathlib/Mathlib/Data/Sum/Interval.lean",
"sample_id": "fcb3cb79819283ab318174f307cb21c4367decc586f83a6b3fb969ef48ad9494"
} | train |
{
"chosen_proof_sha256": "faf07d4b9f81d896a4dedf3d8bef12a774f393144e0c5f30d0f0b0458195b2b5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9aed89b367ed3619b282b85e2e0b0b3501... | by
introv W' hf hg hh hi w
simpa only [IsPushout.flip_iff, IsPullback.flip_iff, and_comm] using
H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip | true | {
"chosen_tokens": 42,
"rejected_tokens": 7,
"token_jaccard": 0.125,
"token_length_ratio": 0.166667
} | circular_dependency | circular_dependency | 3cc62d4238d5417ca23d555d365d225f5c526a10344d2df48da1c03bcc8e2fd1 | 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.Extensive
public import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
public import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
Names... | by
exact IsPushout.IsVanKampen.flip | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "b1d49eaa2ed29b14dcc02cf4b5722a5b80f1e2e4d2685c5083d7d8643a95c268",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Adhesive",
"family_id": "ispushout",
"file_id": "mathlib/Mathlib/CategoryTheory/Adhesive/Basic.lean",
"sample_id": "332d57c10825bff92a5f9c100471ccaa87a97a30c29a674c9e1339e0c80d29bd"
} | train |
{
"chosen_proof_sha256": "65460242af6486d44c1e55ce08a5b548dbca87ef42697a0dbcd7738054ac5b9c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0864def967194569451c916a4bd0e89264... | by
convert_to Tendsto (f ∘ (fun x ↦ c + x) ∘ Neg.neg ∘ Inv.inv) atTop l at h
· ext
simp [AddGroupWithOne.sub_eq_add_neg]
simpa [Tendsto, ← Filter.map_map] using h | true | {
"chosen_tokens": 46,
"rejected_tokens": 2,
"token_jaccard": 0.028571,
"token_length_ratio": 0.043478
} | invalid_tactic | invalid_tactic | 3ce4b3fbf4b32ed258d9807f73f0b5fae91570a0605e4acb716eddecab76db59 | 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.Algebra.Order.Field
public import Mathlib.Topology.Maps.Basic
public import Mathlib.Analysis.Asymptotics.Lemmas
Namespace:
Tactic.ComputeAsymptotics
... | by
__oprover_invalid_tactic_8415a8306797 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "60380dc2b99cd141789ee90f0acaca100e1fd4773b7102e1287d29e151a4a27a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/ComputeAsymptotics",
"family_id": "tendsto_nhdslt_of_tendsto_attop",
"file_id": "mathlib/Mathlib/Tactic/ComputeAsymptotics/Lemmas.lean",
"sample_id": "8415a8306797307a1399dfa46e7bc4bffb6c158dd15399817b536ab5989fba40"
} | train |
{
"chosen_proof_sha256": "d4e8af4e9759204cb08f477f781bb80c363e8cafacffe36b1eb6bdb2061a0215",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a5523b6e9518ffe55d198ff8a834dd6444... | by
refine ⟨?_, Submodule.projectionOnto_comp_surjective_of_exact hfg _ hmap⟩
rwa [LinearMap.coe_comp, Set.InjOn.injective_iff ↑(LinearMap.range f) _ subset_rfl]
simpa [← LinearMap.disjoint_ker_iff_injOn, ← hfg.linearMap_ker_eq] | true | {
"chosen_tokens": 46,
"rejected_tokens": 5,
"token_jaccard": 0.09375,
"token_length_ratio": 0.108696
} | circular_dependency | circular_dependency | 3cf9f2e4907291de3b141b90b5c812818f902ddc22515d404dbebeeee9470552 | 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.Exact.Basic
public import Mathlib.LinearAlgebra.Basis.Basic
public import Mathlib.LinearAlgebra.Projection
Namespace:
(root)
Local context:
/-
Copyri... | by
exact Submodule.projectionOnto_comp_bijective_of_exact | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "2ef5cf11c17f87bf0a7179215c448c7be9f9c4cf08ba0f4505033356cb16a365",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Basis",
"family_id": "submodule",
"file_id": "mathlib/Mathlib/LinearAlgebra/Basis/Exact.lean",
"sample_id": "4cb237f6549cf75bf0312f10927aa64c8d7dbadae8e86cca221ffb114034bffb"
} | train |
{
"chosen_proof_sha256": "9f76daa2d17794583c7f31dfaabd84b49dfce6e55e1f6c203ac803f9bd29220e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f212eda5800dde28c047e59ae432b8c938... | by rw [← subAt_pullSub S x]; gcongr
mpr _ := le_trans (by gcongr) (pullSub_subAt T x) | true | {
"chosen_tokens": 23,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.130435
} | unknown_identifier | unknown_identifier | 3cfc69799a4ef209d786f1ad1b775559aaae79ceced7a39346b2693969cd045f | 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
exact __oprover_missing_1e270d1bd1ee | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "bec847eafd109630d8de47c73aeee0c6d21cbb5a0f58c59988d31c6a33285397",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "dfefda437f41ddea0a1ee24ef23b40384ea8acbbe4173109f7ed5eb612dda632",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f660384a053241b3494fd1356d2eb560d0... | by
let f := dirichletSummandHom χ <| ne_zero_of_one_lt_re hs
have h n : term ↗χ s n = f n := by
rcases eq_or_ne n 0 with rfl | hn
· simp only [term_zero, map_zero]
· simp only [ne_eq, hn, not_false_eq_true, term_of_ne_zero, div_eq_mul_inv,
dirichletSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk,... | true | {
"chosen_tokens": 88,
"rejected_tokens": 5,
"token_jaccard": 0.038462,
"token_length_ratio": 0.056818
} | wrong_namespace_qualification | wrong_namespace_qualification | 3d16f71ff88635e8b8e9ec8ba3a19f836d157e2701d2d425ee7d80a8ab373bb8 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.NumberTheory.EulerProduct.ExpLog
public import Mathlib.NumberTheory.LSeries.Dirichlet
Namespace:
(root)
Local context:
/-
Copyright (c) 2023 Michael Stoll. A... | by
exact OProverMissing.missing_c48cf6260a27 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c5b82e864493151862c2c4b00482a603f251d2f2075317263a7e011eed2d2a6a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/EulerProduct",
"family_id": "dirichletcharacter",
"file_id": "mathlib/Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean",
"sample_id": "c48cf6260a277b10c2e23f28a0f369c22b3db624af2c1f400c3f2ea77b4dd0c6"
} | train |
{
"chosen_proof_sha256": "228e4c456772f8dbd4f2347154e877aedbab995a4c162aadcd1eff84790d42d5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8b02f39ef9581487d59d58b35216989c80... | by
rw [mul_app']
exact x.naturality (y.app X) | true | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.133333,
"token_length_ratio": 0.2
} | circular_dependency | circular_dependency | 3d343d1be63e7723c094f54ff312708399000705855cc60212de9fc8f2c83457 | 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.Endomorphism
Namespace:
CategoryTheory.CatCenter
Local context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.... | by
exact mul_app | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "8a39fdab4c6da6f2fc1a88b934c7e550826c810b163fcfd72eedd824d61d5425",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Center",
"family_id": "mul_app",
"file_id": "mathlib/Mathlib/CategoryTheory/Center/Basic.lean",
"sample_id": "d61440398863b603b3185bd9be1a7f02c573d18ffa317ffa26f3c3cf5dacdb05"
} | train |
{
"chosen_proof_sha256": "7222ee9e22761482f65e7ecb6d7dbb6c39c2c6b75d4bfee6e89b7661c7ff185a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d95a8f0403ed836e86d315e2fe09240f96... | by
refine le_antisymm ?_ subset_span
let s' : Submodule R M :=
{ carrier := s
add_mem' := add_mem
zero_mem' := zero_mem _
smul_mem' := SMulMemClass.smul_mem }
exact span_le (p := s') |>.mpr le_rfl | true | {
"chosen_tokens": 42,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.119048
} | wrong_namespace_qualification | wrong_namespace_qualification | 3d507e3b61dfeedd980b5997b6e6a6820851cb57bd1ff64a04d56180e74e0c43 | 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.Lattice
public import Mathlib.Algebra.Group.Pointwise.Set.Basic
Namespace:
Submodule
Local context:
/-
Copyright (c) 2017 Johannes H... | by
exact OProverMissing.missing_9a1583a6f100 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "29f15baa0da7df9b6f698b7e82b9606103e74eba5e8fb62b08622971e61073f8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Span",
"family_id": "coe_span_eq_self",
"file_id": "mathlib/Mathlib/LinearAlgebra/Span/Defs.lean",
"sample_id": "9a1583a6f100445b753a9ba00ceefda2c0d6b67968367089199c06ca3f1c14f0"
} | train |
{
"chosen_proof_sha256": "b8f5229735b933ede0144ab9a48b8b3042b761eb7777cebb1d91a92fa7760495",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [Subgroup.rank_congr (commutator_eq_closure G)]
apply Subgroup.rank_closure_finite_le_nat_card
variable [Group.FG G] | false | {
"chosen_tokens": 22,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.136364
} | by_exact_placeholder | placeholder | 3d9ff7536d3ada5080bdbad6467ec1a18a13b5c923dd730cfffbec5b02c63761 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Subgroup.Finite
public import Mathlib.GroupTheory.Commutator.Basic
public import Mathlib.GroupTheory.Rank
public import Mathlib.GroupTheory.Index... | by
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": "GroupTheory/Commutator",
"family_id": "rank_commutator_le_card",
"file_id": "mathlib/Mathlib/GroupTheory/Commutator/Finite.lean",
"sample_id": "ad36c8eeea38b3617b2fa01b60e158adc224ea844614163c66de06f8b4d2fee4"
} | train |
{
"chosen_proof_sha256": "8e1e8082a0a84f2600d4003971ff82d46d02db0f4b2396697813b11973e2972c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f5f31c487c9867affa52b7440d78c3121f... | 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... | true | {
"chosen_tokens": 168,
"rejected_tokens": 3,
"token_jaccard": 0.028986,
"token_length_ratio": 0.017857
} | unknown_identifier | unknown_identifier | 3dbbe6c2a1df224f71f3ae2f345e3757b09334f0c34335e5cbaa5d541e4a89cb | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Action
public import Mathlib.Combinatorics.Quiver.Arborescence
public import Mathlib.Combinatorics.Quiver.ConnectedComponent
public import Mathl... | by
exact __oprover_missing_660e9e4b92da | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f863721a1e1ac4e6f631465e0938a20c3367c4d7bfa58dad02fa7c5902448e37",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "dca549e925c3a05e8b8bc1baae8e33cdf145c6f1d35ee1dc1c15d213d724ea07",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "97f596d6d90eb6f474110e8621a1788100... | by cat_disch) : f = g :=
Hom.ext w | false | {
"chosen_tokens": 12,
"rejected_tokens": 19,
"token_jaccard": 0.857143,
"token_length_ratio": 1.583333
} | markdown_preamble_trailing | fence | 3dc38da595ad5a94746cd27ae2d50fc5f4d4fda06f20773ebac8e9ef94fd9d48 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Shapes.IsTerminal
public import Mathlib.CategoryTheory.Functor.EpiMono
Namespace:
CategoryTheory.Endofunctor.Coalgebra
Local context:
/... | ```lean
by cat_disch) : f = g :=
Hom.ext w
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Endofunctor",
"family_id": "ext",
"file_id": "mathlib/Mathlib/CategoryTheory/Endofunctor/Algebra.lean",
"sample_id": "6aa245c658f8d36bf9da386e1af0245093910d695a91f4bad772966128f1d7d5"
} | train |
{
"chosen_proof_sha256": "dfefda437f41ddea0a1ee24ef23b40384ea8acbbe4173109f7ed5eb612dda632",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
let f := dirichletSummandHom χ <| ne_zero_of_one_lt_re hs
have h n : term ↗χ s n = f n := by
rcases eq_or_ne n 0 with rfl | hn
· simp only [term_zero, map_zero]
· simp only [ne_eq, hn, not_false_eq_true, term_of_ne_zero, div_eq_mul_inv,
dirichletSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk,... | false | {
"chosen_tokens": 88,
"rejected_tokens": 2,
"token_jaccard": 0.02,
"token_length_ratio": 0.022727
} | by_exact_placeholder | by? | 3dccb6049738626e34d7b81ec8264639896af9e5d100b048afa43790435319d3 | 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? | {
"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": "c48cf6260a277b10c2e23f28a0f369c22b3db624af2c1f400c3f2ea77b4dd0c6"
} | train |
{
"chosen_proof_sha256": "3939293b987d00efdc8989f961d8fe481cabdbec2fac5708f8344374642f9236",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "66509cfd3b611bcc3ba67f00ce88462a02... | by
rw [TopologicalSpace.nhds_generateFrom]
simp only [comap_iInf, comap_principal]
intro s hs
rw [← le_principal_iff]
refine iInf_le_of_le { u | s ∈ u } ?_
refine iInf_le_of_le ⟨hs, ⟨s, rfl⟩⟩ ?_
exact principal_mono.2 fun _ ↦ id | true | {
"chosen_tokens": 54,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.055556
} | unknown_identifier | unknown_identifier | 3df02329fafbe9629ebc4a4224fe722ab657f0b0963f6322be4c1cc6a5af1846 | 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.Bases
public import Mathlib.Topology.DenseEmbedding
public import Mathlib.Topology.Connected.TotallyDisconnected
Namespace:
(root)
Local context:
/-... | by
exact __oprover_missing_dd448d7bddd3 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "e147a58bd4b4460aefa4b06edb8fef3525c30b474f4922fe385d9e3a3cdfdcdd",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Compactification",
"family_id": "ultrafilter_comap_pure_nhds",
"file_id": "mathlib/Mathlib/Topology/Compactification/StoneCech.lean",
"sample_id": "dd448d7bddd39539db63b6005ceef026bb709e5272ecd6e5386e656fa08db32f"
} | train |
{
"chosen_proof_sha256": "4477bb12e5cc2a166f61746299c4ae29af03f6b7f73cd9a6d3ff697daf2ff447",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G
rw [← hS1, ← Fintype.card_coe, ← Nat.card_eq_fintype_card, ← Finset.coe_sort_coe, ← Nat.card_fun]
exact Finite.card_le_of_embedding (quotientCenterEmbedding hS2) | false | {
"chosen_tokens": 47,
"rejected_tokens": 5,
"token_jaccard": 0.060606,
"token_length_ratio": 0.106383
} | sorry_admit_sorryAx_proof_hole | sorryAx | 3e11b4af2d21ab27b87e43fe5d516e5e868ee72a8bb2899c772a35db1c4f772b | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Subgroup.Finite
public import Mathlib.GroupTheory.Commutator.Basic
public import Mathlib.GroupTheory.Rank
public import Mathlib.GroupTheory.Index... | by
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": "GroupTheory/Commutator",
"family_id": "index_center_le_pow",
"file_id": "mathlib/Mathlib/GroupTheory/Commutator/Finite.lean",
"sample_id": "865c2117838a5f9b6bdb0bfba027decbe6b709de52a194b8d1a351aa165f6f08"
} | train |
{
"chosen_proof_sha256": "a0b44731945f1ae77d84c174d4c4d596bfd46fdd862ded445d0f40690c121d90",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
have := h.hasNondegenerateEdges.two_le
by_contra! hlt
interval_cases n
exact (h 0).injective.ne (by decide : (0 : Fin 3) ≠ 2) (by simp) | false | {
"chosen_tokens": 39,
"rejected_tokens": 2,
"token_jaccard": 0.038462,
"token_length_ratio": 0.051282
} | sorry_admit_sorryAx_proof_hole | sorry | 3e1422ddee30d54e40877c67ae7a4a3d3b240bd8a0eddc3fd13783341c033ae8 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.Convex.Between
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.Tactic.Continuity
Namespace:
Polygon
Local context:
/-
Copyright (c) 20... | 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": "Geometry/Polygon",
"family_id": "hasnondegeneratevertices",
"file_id": "mathlib/Mathlib/Geometry/Polygon/Basic.lean",
"sample_id": "e5d640efe871327e6d7a85ba9e52dd3140e74fe8671f6a1d11428174bcd3a187"
} | train |
{
"chosen_proof_sha256": "fd4f7273c1abc3ee605f4507e24cb13680aff1584126fd18e76d89a937f7044b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
classical
exact Decidable.List.Lex.ne_iff H | true | {
"chosen_tokens": 11,
"rejected_tokens": 8,
"token_jaccard": 0.066667,
"token_length_ratio": 0.727273
} | unsolved_goals | unsolved_goals | 3e26ba83b0fa37c1a985506cdfa0bccd4e96811ef10705b18d2a058fceef37a9 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.List.Basic
public import Mathlib.Data.Nat.Basic
public import Mathlib.Order.RelClasses
Namespace:
List.Lex
Local context:
/-
Copyright (c) 2018 Mario Ca... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "f32c7f31abe05d24ffd62435c0aa53a68f663765217d99ecfe0ccf132431e40f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/List",
"family_id": "ne_iff",
"file_id": "mathlib/Mathlib/Data/List/Lex.lean",
"sample_id": "50332d0f41236ffc25367bcc287b5a5f97bce4b58e2df5be3cb1c6d4b44bee1a"
} | train |
{
"chosen_proof_sha256": "594293ce5d84984c59a5d88c876de5c8200038ce21c7d54ee21a2ba975b3d8b6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by rw [eq_bot_iff, sup_le_iff]; simp | false | {
"chosen_tokens": 9,
"rejected_tokens": 2,
"token_jaccard": 0.1,
"token_length_ratio": 0.222222
} | sorry_admit_sorryAx_proof_hole | admit | 3e4ac19298ec725a0f96e7f76e7052ba4a0ff43fa59cbbc95477d3c3d32ea310 | 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.BoundedOrder.Basic
public import Mathlib.Order.Lattice
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Rele... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/BoundedOrder",
"family_id": "sup_eq_bot_iff",
"file_id": "mathlib/Mathlib/Order/BoundedOrder/Lattice.lean",
"sample_id": "d7b302fa097a85ad40628a8309be13365dc90a026d8ed805296461111bd052b1"
} | train |
{
"chosen_proof_sha256": "faeb8949323faf61ca5ff806cf60d2c823a2170b0852dfc5d54df3a865ffc0fd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "64df50078f89549dc86aa8460f006bfc2d... | by
simp [zariskisMainProperty_iff, ← top_le_iff, SetLike.le_def,
Submonoid.mem_powers_iff, mem_integralClosure_iff] | true | {
"chosen_tokens": 18,
"rejected_tokens": 5,
"token_jaccard": 0.117647,
"token_length_ratio": 0.277778
} | wrong_namespace_qualification | wrong_namespace_qualification | 3e570820a241bc081149e3a3850d889496c72e288a64e272af8263655b2f25ba | 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.Algebraic.StronglyTranscendental
public import Mathlib.RingTheory.Conductor
public import Mathlib.RingTheory.Ideal.Quotient.Nilpotent
public import ... | by
exact OProverMissing.missing_988fd8cb4d8e | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "be396a1d6255c43c2383fef3588f29e2a2356a266cd92a0664681e1a6443fcb8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory",
"family_id": "zariskismainproperty_iff_exists_saturation_eq_top",
"file_id": "mathlib/Mathlib/RingTheory/ZariskisMainTheorem.lean",
"sample_id": "988fd8cb4d8e17e38f2a76981b98bfba51705b13e2c6dc4822f9d73a946a1e06"
} | train |
{
"chosen_proof_sha256": "61743a744bcd9aa71d7e4c51326303887dfd1d0a7a128d08960e283447fbb42c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d4232d0429bbf109d599498b2662b7bb33... | by
rw [numLeaves_eq_numNodes_succ]
exact x.numNodes.zero_lt_succ | true | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.181818,
"token_length_ratio": 0.272727
} | circular_dependency | circular_dependency | 3ecedd4aaaf8c951c48fc9fbab0d112c6a6c1f72cd3a5f8976febb9bdfe7fb7f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.Notation
public import Mathlib.Util.CompileInductive
import Batteries.Tactic.Alias
Namespace:
BinaryTree
Local context:
/-
Copyright (c) 2019 mathli... | by
exact numLeaves_pos | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "008a525c4b8018985d5916fe71a5543656f1d1500759a1c71f193d194a3c3883",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Tree",
"family_id": "numleaves_pos",
"file_id": "mathlib/Mathlib/Data/Tree/Basic.lean",
"sample_id": "06643b8b06bfa39bedc6e6457247fd95883e76318998fda10cf95f90e40967df"
} | train |
{
"chosen_proof_sha256": "b8dd807b486dcb43a0cfd46155c68422d627f5cb03d289ff5154b4c959da2515",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6150e96d7c55abb9f7848262359606a531... | by
simpa [← coe_subset_coe] using! Set.not_subset | true | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.181818
} | invalid_tactic | invalid_tactic | 3f072022e825bc41c97969f9ad9f2d7a4d4981c0e425b7518f21970bfce4a02a | 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.Monotonicity.Attr
public import Mathlib.Tactic.SetLike
public import Mathlib.Data.Set.Basic
Namespace:
SetLike
Local context:
/-
Copyright (c) 2021 Er... | by
__oprover_invalid_tactic_70091202abe0 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "1ca45044ccec6859f1ba9dd39aa479df5fc4b786c9682126f4cd39514593235c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/SetLike",
"family_id": "not_le_iff_exists",
"file_id": "mathlib/Mathlib/Data/SetLike/Basic.lean",
"sample_id": "70091202abe0a1d76505ded2eda1922a6e55f006910256559adb12050fa4d2a2"
} | train |
{
"chosen_proof_sha256": "53c62f34c0746917df7ca2e7b0151f3d1cc2f5a2e7dd8a4133dc4a6cf7d81680",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "42a4c36eb13ded6148878cebea2ece5533... | by
intro f g hfg
have : f.toAlgHom = g.toAlgHom := by
apply (hμ.powerBasis K).algHom_ext
rw [AlgEquiv.coe_algHom, AlgEquiv.coe_algHom, powerBasis_gen,
← autToPow_spec K hμ g, ← autToPow_spec K hμ f, hfg]
exact AlgEquiv.coe_algHom_injective this | true | {
"chosen_tokens": 59,
"rejected_tokens": 2,
"token_jaccard": 0.03125,
"token_length_ratio": 0.033898
} | invalid_tactic | invalid_tactic | 3f35029584b59c15e415f5d98bba7e4c2535a67565bed969284d86af78d34d23 | 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.Cyclotomic.PrimitiveRoots
public import Mathlib.FieldTheory.PolynomialGaloisGroup
Namespace:
IsPrimitiveRoot
Local context:
/-
Copyright (c) 202... | by
__oprover_invalid_tactic_21df6f68f868 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "94deba3e6bca2364d76a0900744d7e1cd329b86fc42df56259f2b88a68805cd8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Cyclotomic",
"family_id": "auttopow_injective",
"file_id": "mathlib/Mathlib/NumberTheory/Cyclotomic/Gal.lean",
"sample_id": "21df6f68f8686f6d7b4ab378fa3f7d2564d582425a9b504a6d726062d4362c74"
} | train |
{
"chosen_proof_sha256": "b7ecb19a982d835fa985827cc7886e0d4262e615155091b32c40d6c7877f5f38",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "42a9f1a907278d22812ff81dd94d8d168b... | by
simp only [minimalPeriod_piMap]
by_cases h : {n | 0 < n ∧ ∀ (i : ι), minimalPeriod (f i) (x i) ∣ n}.Nonempty
· exact (Nat.sInf_mem h).2 i
· simp [not_nonempty_iff_eq_empty.mp h] | false | {
"chosen_tokens": 56,
"rejected_tokens": 61,
"token_jaccard": 0.897436,
"token_length_ratio": 1.089286
} | markdown_preamble_trailing | preamble | 3f3af03442b8b49fc7366904d8c4e13b9cc84360b97f5c12bd2469121c57cc16 | 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... | Here is the proof:
by
simp only [minimalPeriod_piMap]
by_cases h : {n | 0 < n ∧ ∀ (i : ι), minimalPeriod (f i) (x i) ∣ n}.Nonempty
· exact (Nat.sInf_mem h).2 i
· simp [not_nonempty_iff_eq_empty.mp h] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Dynamics/PeriodicPts",
"family_id": "minimalperiod_single_dvd_minimalperiod_pimap",
"file_id": "mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean",
"sample_id": "5bfec5ae03bfd55ef1bbbff853e84f4a25242edd9b4ad0bd7f154c9852ef86aa"
} | train |
{
"chosen_proof_sha256": "a8bd42ecc90333f6177ba81b0d401c4a0d5b772d591ccefb5c358cae0906bf4e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "66a10d4bc2e644ca79ec68399eade5487c... | by
let e : (L ⊗[R] S) ≃ₐ[K] L ⊗[K] (K ⊗[R] S) :=
(Algebra.TensorProduct.cancelBaseChange R K K L S).symm
let e2 : L ⊗[K] (K ⊗[R] S) ≃ₐ[K] (K ⊗[R] S) ⊗[K] L := Algebra.TensorProduct.comm ..
have heq : Algebra.TensorProduct.map (Algebra.ofId K L) (AlgHom.id R S) =
(e.symm.toAlgHom.comp e2.symm.toAlgHom).c... | true | {
"chosen_tokens": 226,
"rejected_tokens": 3,
"token_jaccard": 0.038462,
"token_length_ratio": 0.013274
} | unknown_identifier | unknown_identifier | 3f62b7423568eb97eade30c7f53ed16ddc5871078a750a11cf768939c6793698 | 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.PurelyInseparable.Basic
public import Mathlib.RingTheory.Flat.Basic
public import Mathlib.RingTheory.Spectrum.Prime.Topology
Namespace:
(root)
Lo... | by
exact __oprover_missing_389d6b6e4066 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "008529422e97546d74652288c6977e48f8e59d3a573d905e1e78b431e5bcd8ab",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Spectrum",
"family_id": "primespectrum",
"file_id": "mathlib/Mathlib/RingTheory/Spectrum/Prime/Homeomorph.lean",
"sample_id": "389d6b6e4066fc2cabec7aefc51cac6e21581a5a3e730cc7243010a6f145a966"
} | train |
{
"chosen_proof_sha256": "bcf496a337f71ad6a5e3548ca780931dad5f52b6127e92076ca6f453d0ce3736",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
refine le_trans (measure_iUnion_le _) (le_trans
(ENNReal.tsum_le_tsum <| notConvergentSeqLTIndex_spec (half_pos hε) hf hsm hs hfg) ?_)
simp_rw [ENNReal.ofReal_mul (half_pos hε).le]
rw [ENNReal.tsum_mul_left, ← ENNReal.ofReal_tsum_of_nonneg, inv_eq_one_div, tsum_geometric_two,
← ENNReal.ofReal_mul (half... | false | {
"chosen_tokens": 86,
"rejected_tokens": 3,
"token_jaccard": 0.045455,
"token_length_ratio": 0.034884
} | by_exact_placeholder | placeholder | 3f9a3dab90c7c3bc1749dd82fc7754b2199ebea3f2baa34867ed9fd7884fce8f | 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.Function.StronglyMeasurable.Basic
Namespace:
MeasureTheory.Egorov
Local context:
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Releas... | 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": "MeasureTheory/Function",
"family_id": "measure_iunionnotconvergentseq",
"file_id": "mathlib/Mathlib/MeasureTheory/Function/Egorov.lean",
"sample_id": "444126766a6d3867417667c40dd37f00d68870fb70c1031516cd452dd27d2227"
} | train |
{
"chosen_proof_sha256": "3b000d981c87ecc371d9ed638c299bf8ec24374d90b8958428a050b3089d276d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6b8295f80bbfd1f35f38e61aa6f8084c8... | by
rcases ofSeq_surjective x with ⟨f, rfl⟩
rw [tendsto_ofSeq, (nhds_basis_Ioo _).tendsto_right_iff]
simp_rw [Set.mem_Ioo, eventually_and, ← ofSeq_lt_ofSeq]
refine ⟨fun H ↦ ⟨fun s hs ↦ ?_, fun s hs ↦ ?_⟩, fun H ⟨s, t⟩ ⟨hs, ht⟩ ↦ ⟨?_, ?_⟩⟩
· obtain ⟨t, ht⟩ := exists_gt r
exact (H ⟨s, t⟩ ⟨hs, ht⟩).1.le
· o... | true | {
"chosen_tokens": 186,
"rejected_tokens": 2,
"token_jaccard": 0.018868,
"token_length_ratio": 0.010753
} | invalid_tactic | invalid_tactic | 3faaf69e6802aa84354387cba4bbfb3bfcfdf27ee27ea284b5b5030113b56803 | 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.Ring.StandardPart
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Order.Filter.FilterProduct
Namespace:
Hyperreal
Loc... | by
__oprover_invalid_tactic_c8ff660e3cb0 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "21fdcf88254979f047f517e27c5dbbe277b2e92ef5713174fdfe6e774a5d6e04",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Real",
"family_id": "tendsto_iff_forall",
"file_id": "mathlib/Mathlib/Analysis/Real/Hyperreal.lean",
"sample_id": "c8ff660e3cb075cd21a289a8fe0e908d06e3a660d3865e6edea15898913062b8"
} | train |
{
"chosen_proof_sha256": "a0b44731945f1ae77d84c174d4c4d596bfd46fdd862ded445d0f40690c121d90",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d9c98e23776b09be391189ac538d2208b8... | by
have := h.hasNondegenerateEdges.two_le
by_contra! hlt
interval_cases n
exact (h 0).injective.ne (by decide : (0 : Fin 3) ≠ 2) (by simp) | true | {
"chosen_tokens": 39,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.076923
} | unknown_identifier | unknown_identifier | 40850515c35b73afd9c6fddb4d13e762283825c2543e457b6a967d6d61cebde3 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.Convex.Between
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.Tactic.Continuity
Namespace:
Polygon
Local context:
/-
Copyright (c) 20... | by
exact __oprover_missing_e5d640efe871 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "028bed2aa68d24b3604bb8238e46d52e315ff4a220e82e0ed0e8b682a757ce7e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Geometry/Polygon",
"family_id": "hasnondegeneratevertices",
"file_id": "mathlib/Mathlib/Geometry/Polygon/Basic.lean",
"sample_id": "e5d640efe871327e6d7a85ba9e52dd3140e74fe8671f6a1d11428174bcd3a187"
} | train |
{
"chosen_proof_sha256": "d0424b22b522a25f022cebb1346824da37ff580ade98ae91173e442bd2f6ba06",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f78ddd49796f36d4b728f9170045f34f37... | by
apply IsPullback.of_right (t := (inf_isPullback ((«exists» f).obj A') B').flip)
(p := by simp [frobeniusMorphism])
simpa [frobeniusMorphism, IsPullback.lift_fst, ← imageFactorisation_F_m,
(isPullback f B').paste_horiz_iff] using
(inf_isPullback A' ((Subobject.pullback f).obj B')).flip | true | {
"chosen_tokens": 72,
"rejected_tokens": 3,
"token_jaccard": 0.028571,
"token_length_ratio": 0.041667
} | circular_dependency | circular_dependency | 409e23a7e9e1f849d416f1317cd9c65b096e53f63d93d4dc0c771f033fa9c906 | 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.ExtremalEpi
public import Mathlib.CategoryTheory.MorphismProperty.Limits
public import Mathlib.CategoryTheory.Sites.Coherent.Basic
Namespace:
C... | by
exact frobeniusMorphism_isPullback | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "95c11f5057ea83aae1bae672ce95f1b298fbcc3ceb1de9912c81f822227d2730",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/RegularCategory",
"family_id": "frobeniusmorphism_ispullback",
"file_id": "mathlib/Mathlib/CategoryTheory/RegularCategory/Basic.lean",
"sample_id": "093333b00fcbe2feec5384aaaba84c83717794caa26f2c6955f67a6005b04fa4"
} | train |
{
"chosen_proof_sha256": "81a9d05bf0ed54d4fa50cc7311fe12bd2332adf9f758867f731c0d0d23db9024",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cb0401825ad6d1e1b558ca9c25c51b4a8e... | by
ext i
by_cases h : i = i₀
· subst h
simp
· rw [subtypeNeLift_of_neq _ _ _ _ h] | false | {
"chosen_tokens": 24,
"rejected_tokens": 28,
"token_jaccard": 0.8,
"token_length_ratio": 1.166667
} | missing_wrong_import | import | 40b71e0f45fedf8cbe1826c33ae76fdc78b2339d3989c264820ccc6f3613b8f6 | 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.Option
Namespace:
Function
Local context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in... | import Mathlib.WrongImport
by
ext i
by_cases h : i = i₀
· subst h
simp
· rw [subtypeNeLift_of_neq _ _ _ _ h] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data",
"family_id": "subtypenelift_restriction",
"file_id": "mathlib/Mathlib/Data/SubtypeNeLift.lean",
"sample_id": "15f253266f8e1f550df58301525f21097e11cc98ed777b7a946120509154235e"
} | train |
{
"chosen_proof_sha256": "2cd47198bc2f9b64bb28556c6ab7308c7324a109d2a3891b6dff94f0ce2c3121",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9ba3ca3c063a13ab1bd3991a202b055b4... | by
simp_rw [Nat.lt_add_one_iff, biSup_le_eq_sup] | true | {
"chosen_tokens": 9,
"rejected_tokens": 3,
"token_jaccard": 0.090909,
"token_length_ratio": 0.333333
} | circular_dependency | circular_dependency | 412ec17cbcd87d1977e8216dc98818c7e4904c9cb4c879617067b58f2a88b3af | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.ConditionallyCompleteLattice.Finset
public import Mathlib.Order.Interval.Finset.Nat
Namespace:
Nat
Local context:
/-
Copyright (c) 2018 Johannes Hölzl.... | by
exact iSup_lt_succ | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "b910b8c480ed9e9b033051ea554b60079013651fd3fe80596df977ae866fce57",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Lattice",
"family_id": "isup_lt_succ",
"file_id": "mathlib/Mathlib/Order/Lattice/Nat.lean",
"sample_id": "2b6b462ef4cebc5cf1be9ae0f138ed0a50ba4a689e61f2c8626b10e8cc24c3b7"
} | train |
{
"chosen_proof_sha256": "7749888414d06cd3f5ca86c4419d34b5759e4af27af9cf5a4232c6e45cfd9c6a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "36b8d32675fdb938bd83c305167a8b369e... | by
simpa only [add_comm _ (n : ℝ)] using f.commute_add_int n | true | {
"chosen_tokens": 17,
"rejected_tokens": 2,
"token_jaccard": 0.058824,
"token_length_ratio": 0.117647
} | invalid_tactic | invalid_tactic | 413625687de6e6eba503b6192443285c3af8347f5e29f50a245268adc30911ce | 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
__oprover_invalid_tactic_099bd67010bd | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "7233c5be510663c049a4b3b52e65f41e1840c00dfd8ba69e41c5a9871c5438aa",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "62c33a874b3140b3e0d71815632c5a3a67b39c9eeeb9eb02b51c7aa26e877fd4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "bf9223aaabfc90647618733bf83627a635... | by
constructor <;> grind [IsZero] | false | {
"chosen_tokens": 9,
"rejected_tokens": 14,
"token_jaccard": 0.642857,
"token_length_ratio": 1.555556
} | markdown_preamble_trailing | trailing | 4151f0bfbe5d215e252eaaadc98de727d1b233371bf45ae4e22f1c36f6dab589 | 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.ComputeAsymptotics.Multiseries.Defs
Namespace:
Tactic.ComputeAsymptotics.MultiseriesExpansion.IsZero
Local context:
/-
Copyright (c) 2026 Vasilii Nest... | by
constructor <;> grind [IsZero]
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Tactic/ComputeAsymptotics",
"family_id": "const_iff",
"file_id": "mathlib/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Trimming.lean",
"sample_id": "b7d8bbdbc675ed1b504fd154e9b2c440cdd8d44f35c4f66773b08147245061fc"
} | train |
{
"chosen_proof_sha256": "badd85084558f6da2abee0803d9cfe41d37ed7236c401bd880c419ea46047156",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7b470604d9e322074df2adc2ca1effcd92... | by
rw [lt_univ']
constructor
· rintro ⟨β, e⟩
exact ⟨#β, lift_mk_eq.{u, _, v + 1}.2 e⟩
· rintro ⟨c, hc⟩
exact ⟨⟨c.out, lift_mk_eq.{u, _, v + 1}.1 (hc.trans (congr rfl c.mk_out.symm))⟩⟩ | true | {
"chosen_tokens": 76,
"rejected_tokens": 5,
"token_jaccard": 0.081081,
"token_length_ratio": 0.065789
} | wrong_namespace_qualification | wrong_namespace_qualification | 418d287f6bb67962af655be24a04191f0a4c55c72dc5859f9559bb4f93ed64ba | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.SetTheory.Ordinal.Basic
Namespace:
Cardinal
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as de... | by
exact OProverMissing.missing_73b36450d7be | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "313bb6754713b0d9c9e719b9d2f04612099b39d13035f3fb89659e2872359669",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "SetTheory/Ordinal",
"family_id": "small_iff_lift_mk_lt_univ",
"file_id": "mathlib/Mathlib/SetTheory/Ordinal/Univ.lean",
"sample_id": "73b36450d7beabc8b542d5bd6e54c7d1928d98c7ccc3d417ffbb153021170e98"
} | train |
{
"chosen_proof_sha256": "363f3ee37f17de1a17c56efabc2dfd6a0b7c412da4a7733db969356b0041dbda",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "55e5c34623a2e778c9d9153c0ad047a585... | by
apply SetLike.coe_injective
simp only [coe_centralizer, Set.centralizer_centralizer_centralizer]
variable {M} in | true | {
"chosen_tokens": 19,
"rejected_tokens": 5,
"token_jaccard": 0.095238,
"token_length_ratio": 0.263158
} | wrong_namespace_qualification | wrong_namespace_qualification | 41905867a23c2ab967075a2663c218509f7d9bd078a0eb73b697e53b9a45cbff | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.GroupTheory.Subsemigroup.Centralizer
public import Mathlib.GroupTheory.Submonoid.Center
Namespace:
Submonoid
Local context:
/-
Copyright (c) 2021 Thomas Brow... | by
exact OProverMissing.missing_a921660930bc | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "2ba0c7b76fc548bb992dcea9353fc73ddd4024956c677acbcd5ef72a0eb96706",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "GroupTheory/Submonoid",
"family_id": "centralizer_centralizer_centralizer",
"file_id": "mathlib/Mathlib/GroupTheory/Submonoid/Centralizer.lean",
"sample_id": "a921660930bcd0f4c9bd0e614c1768bb72ffc642b6565ee11d368d38df7c680d"
} | train |
{
"chosen_proof_sha256": "b77e7e870da0ed2ed6f8b7ac1969e927be1488555d4e8afb15051655e1aaced2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
simp [← Function.comp_def, Tendsto, ← map_map, map_inv_atTop] | false | {
"chosen_tokens": 15,
"rejected_tokens": 2,
"token_jaccard": 0.076923,
"token_length_ratio": 0.133333
} | sorry_admit_sorryAx_proof_hole | sorry | 41aaa5590fb5c2f7a5763e39b7c2d516b3ff11baab056f0de0eff84cf2faec91 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Group.MinMax
public import Mathlib.Order.Filter.AtTopBot.Basic
public import Mathlib.Order.Filter.AtTopBot.Map
public import Mathlib.Order.Filter... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/Filter",
"family_id": "tendsto_comp_inv_attop_iff",
"file_id": "mathlib/Mathlib/Order/Filter/AtTopBot/Group.lean",
"sample_id": "105c96e643efa1f749e66cefffdea4f338def0d05aec0167af11a6685aed3ac3"
} | train |
{
"chosen_proof_sha256": "d3371ad1ae3b9f3bec6391019a5fb1c62af2682915c2a167e06751160d75c560",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
have hb' : (0 : ℝ) < ↑(abs b) := Int.cast_pos.mpr (abs_pos.mpr hb)
have hbε : 0 < abs b • ε := by
rw [Algebra.smul_def]
exact mul_pos hb' hε
have hfloor : ∀ i, 0 ≤ floor ((A i % b : ℤ) / abs b • ε : ℝ) :=
fun _ ↦ floor_nonneg.mpr (div_nonneg (cast_nonneg (emod_nonneg _ hb)) hbε.le)
refine ⟨fun i ... | true | {
"chosen_tokens": 376,
"rejected_tokens": 8,
"token_jaccard": 0.033333,
"token_length_ratio": 0.021277
} | unsolved_goals | unsolved_goals | 41bf18869ead390331de6d70aa9fe0df126ffb58db84de53e086454f37eda47b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Basic
public import Mathlib.Algebra.Order.Archimedean.Real.Basic
public import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
Namesp... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "40599cf467faf304b330c65bb7be1aeda3b8acded303696c63cc6650ecda71a7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/ClassNumber",
"family_id": "exists_partition_int",
"file_id": "mathlib/Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean",
"sample_id": "f5c3570decc7db710ef7c794c68d9c0b62e8e6967dc4c3bfe5c338cfbcf456b8"
} | train |
{
"chosen_proof_sha256": "0cde5a74deeec0335214ee6e0227e2c1e15f0001cffe80b1335cabf0968874c8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
have : (p.map C).LiesOver p := ⟨IsMaximal.eq_of_le inferInstance IsPrime.ne_top' le_comap_map⟩
simp [height_eq_height_add_of_liesOver_of_hasGoingDown p] | false | {
"chosen_tokens": 28,
"rejected_tokens": 2,
"token_jaccard": 0.041667,
"token_length_ratio": 0.071429
} | sorry_admit_sorryAx_proof_hole | sorry | 421fee0bd506a99ef01333645913b8967970e234d8674a8be7167254f046f498 | 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.FieldDivision
public import Mathlib.RingTheory.KrullDimension.PID
public import Mathlib.RingTheory.LocalRing.ResidueField.Fiber
public impor... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/KrullDimension",
"family_id": "height_map_c",
"file_id": "mathlib/Mathlib/RingTheory/KrullDimension/Polynomial.lean",
"sample_id": "80af364b54e4f9d5bc6e01d40bc9d559f947616a68e18a5600b526c2e866a061"
} | train |
{
"chosen_proof_sha256": "e1906f3243bcdc4427e534120589c5393cc550f48188ef4c67cb950df5b01321",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
have h : whiskerLeft F (add F s i a b).hom = _ :=
((whiskeringLeft C D D).obj F).map_preimage _
exact (NatTrans.congr_app h X).trans (by simp) | false | {
"chosen_tokens": 47,
"rejected_tokens": 2,
"token_jaccard": 0.033333,
"token_length_ratio": 0.042553
} | sorry_admit_sorryAx_proof_hole | admit | 42244ad7dbff3768e8368a8e7395ddc7343cb9206d4f9c76434b5c059e1f2c87 | 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.Shift.CommShift
Namespace:
CategoryTheory.HasShift.Induced
Local context:
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Shift",
"family_id": "add_hom_app_obj",
"file_id": "mathlib/Mathlib/CategoryTheory/Shift/Induced.lean",
"sample_id": "2640774717b5cc5cf0106fb4ef5d3f69a7c6539d5084db036f9ef862ac01288e"
} | train |
{
"chosen_proof_sha256": "7cabac5c0510d840ec215a55b2c97b8bd4ff4bb3ba7ca644c381e4c621c7c4ab",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
induction rs generalizing M N with
| nil => simp
| cons x _ ih =>
simp only [List.map_cons, isWeaklyRegular_cons_iff] at reg ⊢
have e := (QuotSMulTop.algebraMapTensorEquivTensorQuotSMulTop x M S).symm ≪≫ₗ
QuotSMulTop.congr ((algebraMap R S) x) hf.equiv
have hg : IsBaseChange S <|
e.to... | false | {
"chosen_tokens": 112,
"rejected_tokens": 2,
"token_jaccard": 0.016129,
"token_length_ratio": 0.017857
} | by_exact_placeholder | by? | 4237a088b715916ec909bae34c81cafba67a8d66f249d8e1c436aa703a6fb56c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
public import Mathlib.RingTheory.Flat.Localization
public import Mathlib.RingTheory.Regular.RegularSequence
Namespace:
Ri... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Regular",
"family_id": "isweaklyregular",
"file_id": "mathlib/Mathlib/RingTheory/Regular/Flat.lean",
"sample_id": "b76866134268ce8c1e930e844ff2aaa363a473acc169ff6b4db1a719899e0bd6"
} | train |
{
"chosen_proof_sha256": "9c3ded6353915dd072b1066cf2ce119a7132edcb22b959bc51e04f3927cf0477",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "def6a45549df833e35870bc3cedcd463aa... | by
have h1 (y : ℝ) : 𝐞 y = fourier 1 (y : UnitAddCircle) := by
rw [fourierChar_apply, fourier_coe_apply]
push_cast
ring_nf
simpa only [h1, Int.cast_one, ofReal_one, div_one, mul_one] using hasDerivAt_fourier 1 1 x | true | {
"chosen_tokens": 49,
"rejected_tokens": 2,
"token_jaccard": 0.029412,
"token_length_ratio": 0.040816
} | invalid_tactic | invalid_tactic | 427b404de34fe85f67a0576d38af56f0cab238b2cb99d4bcb670fd4d4899ba40 | 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.Calculus.FDeriv.Analytic
public import Mathlib.Analysis.Fourier.AddCircle
public import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analy... | by
__oprover_invalid_tactic_5adb15e5706a | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "f62c287cad842a967ad265077e93d0382dde31fad821d5f07974688d59ec3726",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Fourier",
"family_id": "hasderivat_fourierchar",
"file_id": "mathlib/Mathlib/Analysis/Fourier/FourierTransformDeriv.lean",
"sample_id": "5adb15e5706aefda52cadd04c16407f0f1f64e18c9972fb156247c44ba666c48"
} | train |
{
"chosen_proof_sha256": "443eea2a8757775ee8a442c763e06d326c2fd7f65ba73927f6537e2b3fa75567",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "63406a1df7aba3e4d515c54c2b6910fb07... | by
rw [finite_iff_subgroup_quotient]; constructor <;> assumption | true | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.181818
} | invalid_tactic | invalid_tactic | 42afa11a4574b933e082d57a47661c8b6e6329d27a320999fa48a1a5dfa34f9f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Subgroup.Finite
public import Mathlib.Data.Finite.Prod
public import Mathlib.GroupTheory.QuotientGroup.Basic
Namespace:
(root)
Local context:
/... | by
__oprover_invalid_tactic_36013d3b030a | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "8d8d2b17aa2978443abf69aead49998c8c2a9378fc7c577f892442c1a20f3703",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "GroupTheory/QuotientGroup",
"family_id": "finite",
"file_id": "mathlib/Mathlib/GroupTheory/QuotientGroup/Finite.lean",
"sample_id": "36013d3b030a4beac323ba03c7b15bdae8a250c801f46d872bcc359cdf27248b"
} | train |
{
"chosen_proof_sha256": "b4230b54afe23a3c3ef25af1bbb191756411be86b06eb1ac6af58052e937afcd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5be88ee7294e42cf554af7ccb31b4028b4... | by
simp [PresheafedSpace.congr_app (F.map_id j)] | true | {
"chosen_tokens": 13,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.230769
} | circular_dependency | circular_dependency | 42bc257cccf504a46028be6d493122c7bfe94ede9ad83935e960f2f8ff13591c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Geometry.RingedSpace.PresheafedSpace
public import Mathlib.Topology.Category.TopCat.Limits.Basic
public import Mathlib.Topology.Sheaves.Limits
public import Ma... | by
exact map_id_c_app | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "d6d3a55f8bf79e2314a6e5a78536172922b0e24dff67a01c324663d7634c0bfe",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Geometry/RingedSpace",
"family_id": "map_id_c_app",
"file_id": "mathlib/Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean",
"sample_id": "d0d0056d6763aa2dc8a9996175ddeb4b3a077afcfff88c7bba8165c2216d8b23"
} | train |
{
"chosen_proof_sha256": "c311ad4b581caa0a3b895b5e4dcbcd3cc30063d829252ea41d678e32db406435",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f3389f96c5e2aa369eaf83ff655a44ee24... | by
tfae_have 1 ↔ 2 := Sheaf.isConstant_iff_isIso_counit_app _ _ _
tfae_have 1 ↔ 3 := ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩
tfae_have 1 ↔ 4 := Sheaf.isConstant_iff_mem_essImage _ LightProfinite.isTerminalPUnit adjunction X
tfae_have 1 ↔ 5 :=
have : functor.Faithful := inferInstance
have : functor.Full := inferInsta... | true | {
"chosen_tokens": 149,
"rejected_tokens": 3,
"token_jaccard": 0.016667,
"token_length_ratio": 0.020134
} | circular_dependency | circular_dependency | 42c981131762e6f0836ac9a3cd49e07a810490e54d3fb1bfdaf235a3228600c4 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Condensed.Discrete.Colimit
public import Mathlib.Condensed.Discrete.Module
Namespace:
LightCondSet
Local context:
/-
Copyright (c) 2024 Dagur Asgeirsson. All... | by
exact isDiscrete_tfae | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "f0ce0420ae2db90c54c78b6cceacc46310171dc4aa362a5844cc160ba76fcf78",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Condensed/Discrete",
"family_id": "isdiscrete_tfae",
"file_id": "mathlib/Mathlib/Condensed/Discrete/Characterization.lean",
"sample_id": "4edfa6f1dfc9284bddb186d5cd60c15b919148603ac348edfd11fc15fed97087"
} | train |
{
"chosen_proof_sha256": "6217ca4bcfee77224af82f13005a67bfc164dca4064e34175a675cb4920a7928",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5c22024a34a06e016e56b56461bf861c46... | by
simp only [List.getD_eq_getElem?_getD]
exact option_getD.comp₂ list_getElem? (const _) | true | {
"chosen_tokens": 21,
"rejected_tokens": 5,
"token_jaccard": 0.142857,
"token_length_ratio": 0.238095
} | wrong_namespace_qualification | wrong_namespace_qualification | 42ebd72c7dbadd77d1b508eac24ad71e68f693ee3708ac09b30b3525f6a6dec4 | 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.Computability.Primrec.Basic
public import Mathlib.Logic.Encodable.Pi
Namespace:
Primrec
Local context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reser... | by
exact OProverMissing.missing_c422540f094e | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "6a1bb39b68ad5369fe7c1e60cc8c2b1eee6398a743be347d1335c1c38c8d2b48",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Computability/Primrec",
"family_id": "list_getd",
"file_id": "mathlib/Mathlib/Computability/Primrec/List.lean",
"sample_id": "c422540f094e4df2b71b39ea0c0e9f6334865968fac379ba2de62eb381a0a862"
} | train |
{
"chosen_proof_sha256": "6124388c54e3601f5b392168c48f6a150dee4519aaf6dbd646732ffc8061300b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "366f97549100ba64f200cb187b78f8d0bb... | by
ext x <;> simp | false | {
"chosen_tokens": 7,
"rejected_tokens": 11,
"token_jaccard": 0.636364,
"token_length_ratio": 1.571429
} | missing_wrong_import | import | 42f73a2bc8d62a71682deae578c650538a5d9258fc7bd84cbbc9cf55b6d32972 | 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.RootSystem.Basic
public import Mathlib.LinearAlgebra.RootSystem.Defs
Namespace:
RootPairing.Hom
Local context:
/-
Copyright (c) 2024 Scott Carn... | import Mathlib.WrongImport
by
ext x <;> simp | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/RootSystem",
"family_id": "comp_id",
"file_id": "mathlib/Mathlib/LinearAlgebra/RootSystem/Hom.lean",
"sample_id": "84f48056d33f497341acedeaa75d75fead8d9c6cb4f1d4df07a45458a9009e00"
} | train |
{
"chosen_proof_sha256": "2bb1a47c41a1654399bf9970e1dd3f153a3e08560ed360131ac9536f9ce93d74",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
induction s using Finset.cons_induction with
| empty => simp
| cons a s ha ihs =>
simp_rw [Finset.sum_cons, ihs]
refine iSup_add_iSup fun i j ↦ (hf i j).imp fun k hk ↦ ?_
gcongr
exacts [(hk a).1, (hk _).2] | true | {
"chosen_tokens": 63,
"rejected_tokens": 8,
"token_jaccard": 0.022222,
"token_length_ratio": 0.126984
} | unsolved_goals | unsolved_goals | 4314373bc9e76418578dbe986a91ec7ffc724e28a63a92118a4c7f20de99e3d5 | 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.Ring.Finset
public import Mathlib.Algebra.BigOperators.WithTop
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "67d4a1fc4b3abafd7606f3d1ac1adeb2bc3c73b6f5c30d42313a15c003af2cd8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/ENat",
"family_id": "sum_isup",
"file_id": "mathlib/Mathlib/Data/ENat/BigOperators.lean",
"sample_id": "d14998e2d236d5892c9ed492b6063a072a0ec193e356245fa60104974e67b50c"
} | train |
{
"chosen_proof_sha256": "d0709903e2751d33af1193290b412e76b1fe0c171376fb60a72f4abe9461ccc4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [sub_eq_add_neg, ofCauchy_add, ofCauchy_neg]
rfl | false | {
"chosen_tokens": 10,
"rejected_tokens": 3,
"token_jaccard": 0.090909,
"token_length_ratio": 0.3
} | by_exact_placeholder | placeholder | 43270389f6dff85cce58dcaae21c59c2e5ac18066570f242419a03d53633d632 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.CauSeq.Completion
public import Mathlib.Algebra.Order.Ring.Rat
public import Mathlib.Data.Rat.Cast.Defs
Namespace:
Real
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/Real",
"family_id": "ofcauchy_sub",
"file_id": "mathlib/Mathlib/Data/Real/Basic.lean",
"sample_id": "ea366b04c54fa2d802b07a4ed0e9ad19edd8f2e0e916dddae3f37ee74b249b0d"
} | train |
{
"chosen_proof_sha256": "79e94f24c2f6775d53dff748682f833bc2540c933d9b7a7fd1509ddae3499adc",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [← ofSupportFinite_coe (f := a) (hf := Set.toFinite _),
prod_congr (fun r _ ↦ rfl), ← coeff_linearCombination_X_pow]
simp [linearCombination_apply, sum_of_support_subset (s := Finset.univ)] | false | {
"chosen_tokens": 44,
"rejected_tokens": 2,
"token_jaccard": 0.033333,
"token_length_ratio": 0.045455
} | by_exact_placeholder | by? | 43399dbcc53e05d762408b22db24f31f2fea58c81f98d6f6dd8244b5b1ca6f9a | 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.Basic
public import Mathlib.Data.Nat.Choose.Multinomial
Namespace:
MvPolynomial
Local context:
/-
Copyright (c) 2026 Antoine Chambert-Lo... | 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/MvPolynomial",
"family_id": "coeff_linearcombination_x_pow_of_fintype",
"file_id": "mathlib/Mathlib/Algebra/MvPolynomial/Coeff.lean",
"sample_id": "0fc65d8000935f5344eab88fab4e2c3713d4a97efe4538dc3faf285b863f943a"
} | train |
{
"chosen_proof_sha256": "d9b98cd109749b4d841c4e5ec47310732f89bc8df99f9f6e87e3aa11beb61e37",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | 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 (... | false | {
"chosen_tokens": 56,
"rejected_tokens": 2,
"token_jaccard": 0.029412,
"token_length_ratio": 0.035714
} | by_exact_placeholder | by? | 43db951c40561c3a9b04b48cd002bec5689d669066453a8455597ff2c678829e | 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.InformationTheory.KullbackLeibler.Basic
public import Mathlib.Probability.Kernel.Composition.MeasureCompProd
public import Mathlib.Probability.Notation
import ... | 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": "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": "9c6b8baf38f61be33d84aa02acad26a64e1f3aa8d253f05349f099a61ea5244e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "370d662bc0a91476deac5c89915d2f05cb... | 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) | false | {
"chosen_tokens": 79,
"rejected_tokens": 84,
"token_jaccard": 0.904762,
"token_length_ratio": 1.063291
} | markdown_preamble_trailing | trailing | 43dee5eb4f9b1b70ac165d43236017772b80a86bf36a420bee2da4e81e6741f3 | 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... | 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)
This comple... | {
"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/Squarefree",
"family_id": "squarefree_iff_no_irreducibles",
"file_id": "mathlib/Mathlib/Algebra/Squarefree/Basic.lean",
"sample_id": "ff5f243c31598fd2aadb8dc36b39e21f486eb66be282ba3b720cbff07478671d"
} | train |
{
"chosen_proof_sha256": "da195f4abae7ab0a2c1c536bf1f2974db5c532e362792146fd16a70b703b9aa9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf
obtain ⟨cg, rfl⟩ := Code.exists_code.1 hg
have : Nat.Partrec fun n => Nat.rfindOpt fun k => cf.evaln k n <|> cg.evaln k n :=
Partrec.nat_iff.1
(Partrec.rfindOpt <|
Primrec.option_orElse.to_comp.comp
(Code.primrec_evaln.to_comp.comp <| (snd.p... | true | {
"chosen_tokens": 350,
"rejected_tokens": 8,
"token_jaccard": 0.05,
"token_length_ratio": 0.022857
} | unsolved_goals | unsolved_goals | 43fbb68d8cf74e083d6d85d1fec91975f1e3922051bad8e402c9c94ada4f4e48 | 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.Computability.PartrecCode
Namespace:
Nat.Partrec
Local context:
/-
Copyright (c) 2018 Mario Carneiro. 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": "644436e402e5479e5e92ff08568e4adad83df762908082eda5f53e5632900b4e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Computability",
"family_id": "merge",
"file_id": "mathlib/Mathlib/Computability/RE.lean",
"sample_id": "f4c589a7f4818f438b23af657c2db224655a94001f25229b505dee753deb8fe1"
} | train |
{
"chosen_proof_sha256": "99a954ce5f0b66b4d53e9b41552229a51be60d1da33548a2621f7dcf8fed671d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8908242e67265bda7ad427e8c08cbaa7bc... | by
use 1, S.one_mem
intro b hb
rw [one_smul]
obtain ⟨b', b'_mem, rfl⟩ := mem_one.mp (h hb)
exact Set.mem_range_self b' | true | {
"chosen_tokens": 35,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.085714
} | circular_dependency | circular_dependency | 443b2062bc4ef7eebc3867958ff1b04b4b850bc2b88b5440b840137cb1448001 | 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.Localization.Integer
public import Mathlib.RingTheory.Localization.Submodule
Namespace:
FractionalIdeal
Local context:
/-
Copyright (c) 2020 Anne ... | by
exact isFractional_of_le_one | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "f23acf797a473e5c80bcb0cc47f71b4940f2238c84932eadb1e018f45336dfc5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/FractionalIdeal",
"family_id": "isfractional_of_le_one",
"file_id": "mathlib/Mathlib/RingTheory/FractionalIdeal/Basic.lean",
"sample_id": "4fa9f7ffaee479f67dcc3f60683ec20977b8a4ed853209875ba4ba0d88e8525f"
} | train |
{
"chosen_proof_sha256": "8f3b01149f94ae9d113cd2afe1cda1c376344890d5f097114694fdbd6fd42eb6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "624824fed1f8ed60e18c6dc25bea0252c9... | by
rcases s.eq_empty_or_nonempty with (rfl | hs); · simp
rcases t.eq_empty_or_nonempty with (rfl | ht); · simp
simp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or] | false | {
"chosen_tokens": 48,
"rejected_tokens": 52,
"token_jaccard": 0.888889,
"token_length_ratio": 1.083333
} | missing_wrong_import | import | 4443a4d5840f418065b914a8e00090a38533f3915f9789fc4c54d9a17c70d9bb | 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.TypeTags.Basic
public import Mathlib.Topology.Bornology.Basic
Namespace:
Bornology
Local context:
/-
Copyright (c) 2022 Yury Kudryashov. All ri... | import Mathlib.WrongImport
by
rcases s.eq_empty_or_nonempty with (rfl | hs); · simp
rcases t.eq_empty_or_nonempty with (rfl | ht); · simp
simp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/Bornology",
"family_id": "isbounded_prod",
"file_id": "mathlib/Mathlib/Topology/Bornology/Constructions.lean",
"sample_id": "c7abf66d2d061cad451c9a404bc80bcd09d739b8aa88cc477da70de900f24a00"
} | train |
{
"chosen_proof_sha256": "b232bd6f68b5950f3c5e2ed9ba68ca285d64e0f598a20d32ae0f8c799f5bfc41",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
let M' := algebraMapSubmonoid S M
have hM' : algebraMapSubmonoid S M ≤ S⁰ := algebraMapSubmonoid_le_nonZeroDivisors_of_faithfulSMul
_ hM
let f₁ : Rₘ →+* K := map _ (T := R⁰) (RingHom.id R) hM
let f₂ : Sₘ →+* L := map _ (T := S⁰) (RingHom.id S) hM'
algebraize [f₁, f₂]
have := localization_isScalarTowe... | false | {
"chosen_tokens": 238,
"rejected_tokens": 3,
"token_jaccard": 0.015625,
"token_length_ratio": 0.012605
} | by_exact_placeholder | exact? | 44522ed7d3eecdd9540e4b48e25a0457b28b4dbf3d6a4a377bdb7e059762f911 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.DedekindDomain.PID
public import Mathlib.FieldTheory.Separable
public import Mathlib.RingTheory.RingHom.Finite
Namespace:
(root)
Local context:
/-... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/DedekindDomain",
"family_id": "fractionring",
"file_id": "mathlib/Mathlib/RingTheory/DedekindDomain/Instances.lean",
"sample_id": "b14089476197b86942d51dd07351b5cf69d062ba1bdb2cb6a8c8c4b42ae306f4"
} | train |
{
"chosen_proof_sha256": "3d1a7e2cbfbf665394764afad32089ad753a926b64cde1fb10b4f271be3fc87d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "29350a05453bd19d7ef84c08e0437d9c4f... | by
obtain (hs | hs) := hs
· exact hs.2
· exact hs.1.isCompact | true | {
"chosen_tokens": 21,
"rejected_tokens": 3,
"token_jaccard": 0.142857,
"token_length_ratio": 0.142857
} | circular_dependency | circular_dependency | 445cb03a8b8a3c4caca14fcdce043fb8b437bafdf262c1e0a51c7cd96ae6c7aa | 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.Spectral.Basic
public import Mathlib.Topology.WithTopology
public import Mathlib.Topology.JacobsonSpace
public import Mathlib.Data.Set.Card
Namespace... | by
exact isCompact_of_mem_constructibleTopologySubbasis | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "cee7b61a688398a79f854a07bfc4284d79892216bef2800461145b9ca25b869a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Spectral",
"family_id": "iscompact_of_mem_constructibletopologysubbasis",
"file_id": "mathlib/Mathlib/Topology/Spectral/ConstructibleTopology.lean",
"sample_id": "0f5b96fc655dafea499598a2e613044abf5991cec77163c042656873851c8051"
} | train |
{
"chosen_proof_sha256": "04ff734dba3e12dd755f41758fae7a61d3a513148d7d550f593f25ac71b54468",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "57c3ec1326de5e5cf945d22e65a425ed48... | by
subst coeff_eq deg_eq_deg coeff_eq_deg
exact natDegree_eq_of_le_of_coeff_ne_zero ‹_› ‹_› | false | {
"chosen_tokens": 13,
"rejected_tokens": 18,
"token_jaccard": 0.666667,
"token_length_ratio": 1.384615
} | markdown_preamble_trailing | preamble | 44970deeb22ba5843fc7e05ffa095e2ca77a6dc89aff4296a4575f8daee5110b | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.Degree.Lemmas
Namespace:
Mathlib.Tactic.ComputeDegree
Local context:
/-
Copyright (c) 2023 Damiano Testa. All rights reserved.
Released un... | Here is the proof:
by
subst coeff_eq deg_eq_deg coeff_eq_deg
exact natDegree_eq_of_le_of_coeff_ne_zero ‹_› ‹_› | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Tactic",
"family_id": "natdegree_eq_of_le_of_coeff_ne_zero",
"file_id": "mathlib/Mathlib/Tactic/ComputeDegree.lean",
"sample_id": "9e767d0b834339320d21f2fe6faa7f7606cdac3e6fc203de41750386839ebf3b"
} | train |
{
"chosen_proof_sha256": "3f00e9105f1d5704619d01594b14b1a2c0b0c68fe25f9bbf22b6c6f972b81c9f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [moduleCat_exact_iff_function_exact] at hC ⊢
exact rTensor_exact M hC | false | {
"chosen_tokens": 12,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.166667
} | sorry_admit_sorryAx_proof_hole | admit | 449a1c68066140eb5949a1a19a0cf946b71ddb684ad734d9df343ea208d3a031 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Flat.Basic
public import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
public import Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed
Namespace... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Flat",
"family_id": "rtensor_shortcomplex_exact",
"file_id": "mathlib/Mathlib/RingTheory/Flat/CategoryTheory.lean",
"sample_id": "a26f956f637c764b2f59cef3d4d51fd447c50773705cdd84c865b39ccbebb619"
} | train |
{
"chosen_proof_sha256": "332fb89312684e7786baadfdf197587e5cdba892591555a524aedd43ed4113c6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6f14ad8f52f1b968fcf983bc79e14954c8... | by
induction n with
| zero =>
simp
| succ n IH =>
simp_rw [iteratedLineDerivOp_succ_left, IH, lineDerivOp_add] | false | {
"chosen_tokens": 21,
"rejected_tokens": 25,
"token_jaccard": 0.8,
"token_length_ratio": 1.190476
} | missing_wrong_import | import | 44c57f9801266bf37e860450a49958c1a5a11070241ef83db3a933cf2b98c716 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Module.Equiv.Defs
public import Mathlib.Data.Fin.Tuple.Basic
public import Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
public import Math... | import Mathlib.WrongImport
by
induction n with
| zero =>
simp
| succ n IH =>
simp_rw [iteratedLineDerivOp_succ_left, IH, lineDerivOp_add] | {
"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": "Analysis/Distribution",
"family_id": "iteratedlinederivop_add",
"file_id": "mathlib/Mathlib/Analysis/Distribution/DerivNotation.lean",
"sample_id": "e31fb78612321e73c41c554f6e35840ea8a109f4f7b6f9fddcff3cf0f701ebac"
} | train |
{
"chosen_proof_sha256": "98bbccba21050a97745f14c454bcd7300ae6caac0c9cdef2d7b7857a2d6f0324",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [← totallyDisconnectedSpace_subtype_iff]
have : Countable s := hs
infer_instance | false | {
"chosen_tokens": 13,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.230769
} | by_exact_placeholder | exact? | 44d7ca051765c1b692857da001cd0bab371164b96f3e1dd1b046fd0f9a3cc100 | 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.GDelta.MetrizableSpace
public import Mathlib.Topology.Separation.CompletelyRegular
public import Mathlib.Topology.Separation.Profinite
Namespace:
(ro... | 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": "Topology/Separation",
"family_id": "set",
"file_id": "mathlib/Mathlib/Topology/Separation/Lemmas.lean",
"sample_id": "28dab043652a1525adfd5ddc52e6243324bae2d9c4d94e5b8f18a41fa2a7d11f"
} | train |
{
"chosen_proof_sha256": "1a29d536dd839cb2ac804ef83ab5abb79ba3d4ec297a0fa4ca91c3477c2aaaa0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "539a7c9270ba302e32332d578382773109... | by
refine OreLocalization.ind fun x s ↦ ?_
obtain ⟨i, j, hne, heq⟩ := Finite.exists_ne_map_eq_of_infinite (α := ℕ) (s ^ · • x)
wlog! hlt : j < i generalizing i j
· exact this j i hne.symm heq.symm (hne.lt_of_le hlt)
use s ^ (i - (j + 1)) • x
rw [oreDiv_eq_iff]
refine ⟨s ^ j, (s ^ (j + 1)).1, ?_, ?_⟩
· c... | true | {
"chosen_tokens": 165,
"rejected_tokens": 5,
"token_jaccard": 0.047619,
"token_length_ratio": 0.030303
} | wrong_namespace_qualification | wrong_namespace_qualification | 4518950c919b80b9300c2dcb446a82150b72c276ff66f4ff2f0d3c618718526b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Pigeonhole
public import Mathlib.GroupTheory.OreLocalization.Basic
public import Mathlib.SetTheory.Cardinal.Arithmetic
Namespace:
OreLocalization... | by
exact OProverMissing.missing_a793e6cc882c | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c91fee88ad1ed12eba6e009039b69dd7aee2e448ff9d91e392161ba9b48d9406",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "GroupTheory/OreLocalization",
"family_id": "orediv_one_surjective_of_finite_right",
"file_id": "mathlib/Mathlib/GroupTheory/OreLocalization/Cardinality.lean",
"sample_id": "a793e6cc882c9887527edba47d8bc663c406cd6c0a53d249021c37e71fed657c"
} | train |
{
"chosen_proof_sha256": "b7ecb19a982d835fa985827cc7886e0d4262e615155091b32c40d6c7877f5f38",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp only [minimalPeriod_piMap]
by_cases h : {n | 0 < n ∧ ∀ (i : ι), minimalPeriod (f i) (x i) ∣ n}.Nonempty
· exact (Nat.sInf_mem h).2 i
· simp [not_nonempty_iff_eq_empty.mp h] | true | {
"chosen_tokens": 56,
"rejected_tokens": 8,
"token_jaccard": 0.05,
"token_length_ratio": 0.142857
} | unsolved_goals | unsolved_goals | 4549190c469bc55b3512f4988ce585891baf93f6ba6eee53863200e8a1009e27 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GCDMonoid.Finset
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.Data.PNat.Basic
public import Mathlib.Dynamics.PeriodicPts.Defs
publi... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "59d633f642b6cb2164b6d1c43a7c438f0a2966b7435cb98112c3bf8684251005",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Dynamics/PeriodicPts",
"family_id": "minimalperiod_single_dvd_minimalperiod_pimap",
"file_id": "mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean",
"sample_id": "5bfec5ae03bfd55ef1bbbff853e84f4a25242edd9b4ad0bd7f154c9852ef86aa"
} | train |
{
"chosen_proof_sha256": "3a259627f5a5b32fa98a969454c4436a807645c0fdfa0bcf717382246cd616fd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d2824e2c18d86e102371bf1923e5cd4376... | by
apply Stream'.ext; intro n
cases n <;> rfl | false | {
"chosen_tokens": 14,
"rejected_tokens": 19,
"token_jaccard": 0.705882,
"token_length_ratio": 1.357143
} | markdown_preamble_trailing | preamble | 4608cd25c866a01d302c793bf0d5d65101cf3eb929904f973da70b8245e93bdd | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Stream.Defs
public import Mathlib.Logic.Function.Basic
public import Mathlib.Data.Nat.Basic
public import Mathlib.Tactic.Common
Namespace:
Stream'
Local... | Here is the proof:
by
apply Stream'.ext; intro n
cases n <;> rfl | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Stream",
"family_id": "const_eq",
"file_id": "mathlib/Mathlib/Data/Stream/Init.lean",
"sample_id": "89ecc5dec26210b39010cde2e335b6d4cf0a4c1bf7eb4d3bd4ba691ff88b6aba"
} | train |
{
"chosen_proof_sha256": "1f1ea0e13f9e5b99d77e0d90716326ac0d10ebd31e81aeaa4cd98f72f2f5c63c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5a66cefb33286413a5fb6e69c074a044e6... | 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... | false | {
"chosen_tokens": 286,
"rejected_tokens": 291,
"token_jaccard": 0.946667,
"token_length_ratio": 1.017483
} | markdown_preamble_trailing | preamble | 462c157226fc5a39159b42cf3af0c16a56927ceac1aea7407a13612ed30bf77c | 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.ExpDeriv
Namespace:
(root)
Local context:
/-
Copyright (c) 2026 Dennj Osele. All rights reserved.
Released under Apache 2.0 license... | Here is the proof:
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, m... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/ODE",
"family_id": "discrete_gronwall_prod_general",
"file_id": "mathlib/Mathlib/Analysis/ODE/DiscreteGronwall.lean",
"sample_id": "19249b12421b93ef9aca4c5c6690238801a0a01bd844a6075bfc5c80d4237b2d"
} | train |
{
"chosen_proof_sha256": "ddb596461bd0c4bd75d4fb0cc2c7a82f17820be10b8f87f0c46c723e2b555f0f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
have hg : DifferentiableAt ℂ (fun z ↦ log (1 - z)) 0 := by
have : 1 - 0 ∈ slitPlane := (sub_zero (1 : ℂ)).symm ▸ one_mem_slitPlane
fun_prop
have : (fun z ↦ log (1 - z)) =O[𝓝 0] id := by
simpa only [sub_zero, log_one] using! hg.isBigO_sub
exact this.comp_summable hsum | false | {
"chosen_tokens": 80,
"rejected_tokens": 5,
"token_jaccard": 0.045455,
"token_length_ratio": 0.0625
} | sorry_admit_sorryAx_proof_hole | sorryAx | 463c792303982d2516b86bdebd3033e45a5fb0ca38bc68da9b73b19dad89415a | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.Normed.Module.FiniteDimension
public import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
public import Mathlib.LinearAlgebra.Complex.FiniteDimen... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/EulerProduct",
"family_id": "summable",
"file_id": "mathlib/Mathlib/NumberTheory/EulerProduct/ExpLog.lean",
"sample_id": "fe21963d2aeeba6cb61599867ca841b158d2155ae815d66b4331eaa8e5ce4e3b"
} | train |
{
"chosen_proof_sha256": "0d4d6dccf2c84c2f8c5355bc4fd7dec43e2d86ae85215b172de94088a2f48bcb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f874782215c2982072ce26a83604cb1ed6... | by
ext x
simp only [spectrum.mem_iff, isUnit_iff_forall_ne_zero, not_forall, sub_apply,
Classical.not_not, Set.mem_range,
sub_eq_zero, @eq_comm _ (x • 1 : R) _, Set.mem_preimage, Algebra.algebraMap_eq_smul_one,
smul_apply, one_apply] | false | {
"chosen_tokens": 50,
"rejected_tokens": 55,
"token_jaccard": 0.891892,
"token_length_ratio": 1.1
} | markdown_preamble_trailing | trailing | 466b04cf1d13f2e2ae9b5dd42fe730941d7c173793514d7917691485e8fa6fbf | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.Normed.Ring.Units
public import Mathlib.Algebra.Algebra.Spectrum.Basic
public import Mathlib.Topology.ContinuousMap.Algebra
Namespace:
ContinuousMap
... | by
ext x
simp only [spectrum.mem_iff, isUnit_iff_forall_ne_zero, not_forall, sub_apply,
Classical.not_not, Set.mem_range,
sub_eq_zero, @eq_comm _ (x • 1 : R) _, Set.mem_preimage, Algebra.algebraMap_eq_smul_one,
smul_apply, one_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": "Topology/ContinuousMap",
"family_id": "spectrum_eq_preimage_range",
"file_id": "mathlib/Mathlib/Topology/ContinuousMap/Units.lean",
"sample_id": "dd8a4792fd8238e946d75aea51e934ebc1fab1139603667bd836f2fb5b0acf87"
} | train |
{
"chosen_proof_sha256": "fd3b7a2ce076a9542b33ce25f06638c50f65a2500253da07e8afbc3c349b995c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "39df060b78e1e695bb13dca8ed0e62daa8... | by
induction m with
| zero => simpa
| succ m ih =>
simp_rw [pow_succ _ m, pow_mul]
exact ih.pow_add_one hpI m.succ_ne_zero | true | {
"chosen_tokens": 29,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.068966
} | invalid_tactic | invalid_tactic | 468c36fbb6b109c4d1e4335a3a0a063208dbd6dfd028ef8a60550f5a0ceeb992 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.CharP.Lemmas
public import Mathlib.LinearAlgebra.SModEq.Basic
public import Mathlib.RingTheory.Ideal.Operations
Namespace:
SModEq
Local context:
/-
C... | by
__oprover_invalid_tactic_f2813c1994d0 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "22f31c6c6cb622fcceff3b4a2289445422846e31fdcc6eca71ab63a9d7712d14",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/SModEq",
"family_id": "pow_pow_add_one",
"file_id": "mathlib/Mathlib/LinearAlgebra/SModEq/Pow.lean",
"sample_id": "f2813c1994d0014dcc0cbbe6ef4d234abd05fcb0aaba7d50c3b4450586130664"
} | train |
{
"chosen_proof_sha256": "4967cf04180cb0748d25ecdf64a662a7f6a1d6bdd998e61ce6f3d7d2c6f082a5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [← map_add]
exact m.toSyn_monotone le_self_add | false | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.181818
} | sorry_admit_sorryAx_proof_hole | admit | 4699d4773f8622959c8e22375a66dcf62b79560afe2577aab0a229e17b6e9796 | 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.Lex
public import Mathlib.Data.Finsupp.WellFounded
public import Mathlib.Data.List.TFAE
Namespace:
MonomialOrder
Local context:
/-
Copyright (c)... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Finsupp",
"family_id": "le_add_right",
"file_id": "mathlib/Mathlib/Data/Finsupp/MonomialOrder.lean",
"sample_id": "756685a57c1c40f2d410134b5fc2580aacd3ec04b425aa2b56ef0fc77bf58bf8"
} | train |
{
"chosen_proof_sha256": "f11fe3945a9851f302dfa7c95843f91ea71198e392581ee0ee3ba699adf230d7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6f236f3198461aabd72d37b97a5e2b5e82... | by
simp only [toPrincipalIdeal]; exact Units.ext_iff | true | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.166667,
"token_length_ratio": 0.272727
} | circular_dependency | circular_dependency | 47916be85ef61c5281f8a87f23e798534748b12a7ec195b21ae3ebb2aa6a9930 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.DedekindDomain.Ideal.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 lice... | by
exact toPrincipalIdeal_eq_iff | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "61c619d88b56c6afe0decbceed7ed363ee114dda3ea91469b8f148ba3d4099fd",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/ClassGroup",
"family_id": "toprincipalideal_eq_iff",
"file_id": "mathlib/Mathlib/RingTheory/ClassGroup/Basic.lean",
"sample_id": "9757f7a78bda71d8e7517609a4d26326ae1122e116d5aaf449c4d98b5651cdcf"
} | train |
{
"chosen_proof_sha256": "7605cac82936a34e51b347b493387fd67f6c60e4f590ba9df4da446fc48c38a4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "059486db01226e7fb2c9a1e273235fb925... | by
let l := hi.lift ((pullback.fst αX f) ≫ h') ((pullback.snd αX f) ≫ g)
(by simp only [Category.assoc, hh.toCommSq.w, pullback.condition_assoc, ← H.w])
use (pullback αX f), (pullback.fst αX f), l, (pullback.snd αX f)
refine ⟨IsPullback.of_hasPullback αX f, ?_, ?_⟩
· refine IsPullback.of_right' ?_ hi
rw... | true | {
"chosen_tokens": 219,
"rejected_tokens": 3,
"token_jaccard": 0.039216,
"token_length_ratio": 0.013699
} | unknown_identifier | unknown_identifier | 47c8bd73ad0c3e992b5d632b5a1a0ca97d22bfec50099a1c647c645010c1f6b1 | 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.Extensive
public import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
public import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
Names... | by
exact __oprover_missing_1e9f94a5050c | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "0b627c7602bf1ad0488ae8c863103641b650ddaf0c810931559c4a71ff85535d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Adhesive",
"family_id": "ispushout",
"file_id": "mathlib/Mathlib/CategoryTheory/Adhesive/Basic.lean",
"sample_id": "1e9f94a5050c3f631d6ef4550fc4148994113fabe92f719d1a84b2cac8998b0b"
} | train |
{
"chosen_proof_sha256": "f4514d3004c77d49e14e5cb3065d22f4527708fcd18c9c1d909c571ef396f6f1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
obtain ⟨n, hn⟩ := hnp
obtain ⟨k, hk⟩ := IsNilpotent.nilpotent R I M
have hk' : I.lcs M k = ⊥ := by
simp only [← toSubmodule_inj, I.coe_lcs_eq, hk, bot_toSubmodule]
suffices ∀ l, lowerCentralSeries R L M (l * n) ≤ I.lcs M l by
rw [isNilpotent_iff R]
use k * n
simpa [hk'] using this k
intro l... | false | {
"chosen_tokens": 112,
"rejected_tokens": 3,
"token_jaccard": 0.032787,
"token_length_ratio": 0.026786
} | by_exact_placeholder | placeholder | 47fc992b83f050545243413d13ccb7226de7fc1e73009880c9e8461a7c0e2255 | 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.Lie.AdjointAction.Basic
public import Mathlib.Algebra.Lie.Nilpotent
public import Mathlib.Algebra.Lie.Normalizer
Namespace:
LieSubmodule
Local contex... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Lie",
"family_id": "isnilpotentofisnilpotentspansupeqtop",
"file_id": "mathlib/Mathlib/Algebra/Lie/Engel.lean",
"sample_id": "e96a9f2e6f2cd142ce46e8c5c7977a2fb22f0ea1c0daabf15e8cbf9d7c647459"
} | train |
{
"chosen_proof_sha256": "31a79f6998a821c6e1751ba23882d5c421febc08424475fbe1cf68e517fa33e6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "75ec7c738aaa4368c89d8dfb1f412cc867... | by
cases L <;> grind | true | {
"chosen_tokens": 7,
"rejected_tokens": 3,
"token_jaccard": 0.111111,
"token_length_ratio": 0.428571
} | circular_dependency | circular_dependency | 48082a975f8d67251c063815152b2f46689e0316ebcb5ec10b20066e5292be94 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono
Namespace:
SimplexCategoryGenRel
Local context:
/-
Copyright (c) 2025 Robin Carlier. All rights... | by
exact isAdmissible_cons_iff | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "69789abbe8338ba4f64b45438a4926152ece4153683f4c16b89a33a6da6cec97",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/SimplexCategory",
"family_id": "isadmissible_cons_iff",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/NormalForms.lean",
"sample_id": "6c68bdabc3643e303bfd886e5b6ece2de0d41819d4c4fb2cea020fa9d7f7d7c8"
} | train |
{
"chosen_proof_sha256": "9ed2efcc5bc8272d2a70d89cfb0ba3dc939ae6d8c6a39b0504302e5c2afc9b8d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
obtain ⟨x, v₁, v₂⟩ := x
obtain ⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩ := h
simp [Prod.toFun', Prod.invFun', h₁, h₂] | false | {
"chosen_tokens": 53,
"rejected_tokens": 5,
"token_jaccard": 0.038462,
"token_length_ratio": 0.09434
} | sorry_admit_sorryAx_proof_hole | sorryAx | 481296d7ceb149b5ed19cb7a106ebed9c4f91f66c76a67967e99b87abc75347a | 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.FiberBundle.Basic
Namespace:
Bundle.Trivialization
Local context:
/-
Copyright (c) 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache... | 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/FiberBundle",
"family_id": "prod",
"file_id": "mathlib/Mathlib/Topology/FiberBundle/Constructions.lean",
"sample_id": "b765d1892775d5e6e6a62ecd85051d3ac26c69fedc8cb19fe0955539acfca9f3"
} | train |
{
"chosen_proof_sha256": "0ec8ef0b04cc9e360fe54d83ec9b445dee17ea05e7abf3a9b4757fa21a5fb78e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5cf02018572ea2a48c79b73191cb5822bd... | by
simp [Irrational, Rat.forall, eq_comm] | true | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.272727
} | unknown_identifier | unknown_identifier | 482c56b5cd119129b12043dad116decd800531b148fd7edef8496e9739657eb1 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Rat
public import Mathlib.Data.Nat.Prime.Int
public import Mathlib.Data.Rat.Sqrt
public import Mathlib.Analysis.Real.Sqrt
public import Mathlib... | by
exact __oprover_missing_71f1b32ec46d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "74c9ec3867b26ad3dc254c1ba114e63a10ea00360df83b58e3eb2cbb3a745aad",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Real",
"family_id": "irrational_iff_ne_rational",
"file_id": "mathlib/Mathlib/NumberTheory/Real/Irrational.lean",
"sample_id": "71f1b32ec46df06988407bec9fec4c171851cda268d1134ad7fa0191f2142682"
} | train |
{
"chosen_proof_sha256": "6986b7de3b811139639af9103eba6dc70b8f13e73607506a5de302cee24db96a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [BilinForm.toMatrixAux, Matrix.toBilin'Aux, toLinearMap₂'Aux_toMatrix₂Aux] | false | {
"chosen_tokens": 18,
"rejected_tokens": 2,
"token_jaccard": 0.0625,
"token_length_ratio": 0.111111
} | sorry_admit_sorryAx_proof_hole | sorry | 482f8820fe9468f5ac36722732b4434fb6efe4e8e2b98e55873e1b71de28547d | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.BilinearForm.Properties
public import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
Namespace:
(root)
Local context:
/-
Copyright (c) 2020 Anne... | 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": "LinearAlgebra/Matrix",
"family_id": "linearmap",
"file_id": "mathlib/Mathlib/LinearAlgebra/Matrix/BilinearForm.lean",
"sample_id": "1e74a857d727dcb19bacc65e1dd39ea3f06dc3533cc251ea326cf338c51e0a18"
} | train |
{
"chosen_proof_sha256": "0520ccfd6af7e55e37364ebbda9a263098edd7a0ec6ad9af960d91cce2f0e2d9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "541da4b081e2e34b78b074868e478c10ad... | by
rw [← ClassGroup.mk_mk0 (FractionRing A), extendedHom_mk]
variable [IsDedekindDomain B] | true | {
"chosen_tokens": 19,
"rejected_tokens": 3,
"token_jaccard": 0.052632,
"token_length_ratio": 0.157895
} | unknown_identifier | unknown_identifier | 48517f2220c82cb31768b141753f55ef0959b070279bb311b8f0e291c1a23938 | 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.FractionalIdeal.Extended
public import Mathlib.RingTheory.ClassGroup.Basic
Namespace:
ClassGroup
Local context:
/-
Copyright (c) 2026 Riccardo Bra... | by
exact __oprover_missing_b16015c2c8cb | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "89a6928a37b9590ad665943f05e8e25c828348e19982a1bbefaa574957ecb9d5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/ClassGroup",
"family_id": "extendedhom_mk",
"file_id": "mathlib/Mathlib/RingTheory/ClassGroup/ExtendedHom.lean",
"sample_id": "b16015c2c8cb752b26e08d47468536e58a1c1a2c7cc95ef8b2185f3dd54359d8"
} | train |
{
"chosen_proof_sha256": "bd9e89042700066e34db2eae2a5bdca2cc788a55ff318687cbca751f9272e192",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]
split_ifs <;> rfl | true | {
"chosen_tokens": 27,
"rejected_tokens": 8,
"token_jaccard": 0.04,
"token_length_ratio": 0.296296
} | unsolved_goals | unsolved_goals | 48880fabbd2f921943ec9b8d75691d0088f5fc23d4308e5f8cf1f1e61e20ee3c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Data.MLList.Basic
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.ZeroLEOne
public import Mathlib.Data.DFinsupp.Defs
public... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "072f58d4d9323e32ce9b0bd3261f93220e3af16282fde86ec58c222727cf35bc",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Testing/Plausible",
"family_id": "list",
"file_id": "mathlib/Mathlib/Testing/Plausible/Functions.lean",
"sample_id": "25e33e2ad73b99a239389fcab2e1bb17f4373e01742868a9ff04d6e9bc1274a4"
} | train |
{
"chosen_proof_sha256": "8b4eba4a7d76ab6115b8794619b18b2fd26643e7d931d0fcde4744d29b34df3e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c5d49b24160a98ef5dde849af319186f4d... | by
classical
rw [← card_image_of_injective s' Subtype.coe_injective]
convert! ht (s'.image fun z => z.1) using 1
apply congr_arg
ext y
simp | false | {
"chosen_tokens": 32,
"rejected_tokens": 36,
"token_jaccard": 0.9,
"token_length_ratio": 1.125
} | missing_wrong_import | import | 488f6387366706ce14512350ac0b3218a31bed8ca127ad6eb0e177a9ca559bb7 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Basic
public import Mathlib.Data.Fintype.Powerset
public import Mathlib.Data.Set.Finite.Basic
Namespace:
HallMarriageTheorem
Local context:
/-
C... | import Mathlib.WrongImport
by
classical
rw [← card_image_of_injective s' Subtype.coe_injective]
convert! ht (s'.image fun z => z.1) using 1
apply congr_arg
ext y
simp | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/Hall",
"family_id": "hall_cond_of_restrict",
"file_id": "mathlib/Mathlib/Combinatorics/Hall/Finite.lean",
"sample_id": "40d4ba4e788dd7fbc85ecb01598bd57d742b372ed1f0e1d36e21ca910229b975"
} | train |
{
"chosen_proof_sha256": "3c1f07b625c847d49cc43a89c05d3139e535c28f43f07276611fd8f7f1db78de",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0c4813e84ac03a598437ff70e6508e2117... | by trunc) (hb : b.len ≤ n := by trunc)
(hc : c.len ≤ n := by trunc) :
tr (f ≫ g) = tr f ≫ tr g :=
rfl | false | {
"chosen_tokens": 42,
"rejected_tokens": 47,
"token_jaccard": 0.833333,
"token_length_ratio": 1.119048
} | markdown_preamble_trailing | trailing | 48ec5534fbd07f864eeeeb6740f31ec73673670c82f80e0e2b824f5f84d0f00e | 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.Category.Preorder
public import Mathlib.CategoryTheory.Opposites
public import Mathlib.Order.Fin.Basic
public import Mathlib.Util.Superscript
N... | by trunc) (hb : b.len ≤ n := by trunc)
(hc : c.len ≤ n := by trunc) :
tr (f ≫ g) = tr f ≫ tr g :=
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": "AlgebraicTopology/SimplexCategory",
"family_id": "hom",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplexCategory/Defs.lean",
"sample_id": "cb9967aa5d54ae74d6b7c334cf672946e1ac267085a38bfbf8b9d07e6fea6e09"
} | train |
{
"chosen_proof_sha256": "6dfab6bad543a2d9c97fdd7bad870518b9b6feb74bbecb3fcb1c2ddbf6b41bd4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "99db565f5bf6a82c918f0e5129424afa9b... | by
have : IsReflexive R N := e.isReflexive_of_equiv_dual_of_isReflexive
suffices
(p.map (e.symm : Dual R M →ₗ[R] N)).dualAnnihilator.map (e.flip.symm : Dual R N →ₗ[R] M) =
(p.dualCoannihilator.map (e.flip : M →ₗ[R] Dual R N)).map (e.flip.symm : Dual R N →ₗ[R] M)
from (Submodule.map_injective_of_in... | true | {
"chosen_tokens": 135,
"rejected_tokens": 5,
"token_jaccard": 0.045455,
"token_length_ratio": 0.037037
} | wrong_namespace_qualification | wrong_namespace_qualification | 48f1969b216dbd95031c3b48ec4e85623403157065635f327862e043e91270eb | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Dual.Lemmas
Namespace:
Submodule
Local context:
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as de... | by
exact OProverMissing.missing_44eed7b98300 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "9d4013c29b476e968b384a48f875a1490053a0e3881ddb4af45ad8c5170c048a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/PerfectPairing",
"family_id": "map_dualcoannihilator_linearequiv_flip",
"file_id": "mathlib/Mathlib/LinearAlgebra/PerfectPairing/Basic.lean",
"sample_id": "44eed7b983007fa4c1d9c73cb7fb1872722dad2bf7f08f9f9ca305f717904a0a"
} | train |
{
"chosen_proof_sha256": "b925b94bb1cfb3ff34ceacbe250abd23143fbe7540078964bfa526bb642e3c9f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
classical
simpa [dotProduct_comm] using! (isSymmetric_toEuclideanLin_iff.mpr hA).im_inner_self_apply _ | true | {
"chosen_tokens": 17,
"rejected_tokens": 8,
"token_jaccard": 0.045455,
"token_length_ratio": 0.470588
} | unsolved_goals | unsolved_goals | 490ea10d3bc19ff6e1430cf39619d1a15a51e8866b65121442418c21bc5f4646 | 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.InnerProductSpace.PiL2
public import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.Analysis.InnerProductSpace.Adjoint
Namespace:
Matrix
Loca... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "85ae0ff883f1bceaaf5429811a4454c8f50827d09e85c1e537a7723d25e5300e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Matrix",
"family_id": "ishermitian",
"file_id": "mathlib/Mathlib/Analysis/Matrix/Hermitian.lean",
"sample_id": "f7bbb727f1ea351360c7052b952d4fc3113423ef399aac4de4fe2b1d680daf36"
} | train |
{
"chosen_proof_sha256": "b33ee369bdfae6b8f7816eb5661ea638ec7dc5ec9125117e507e6aaa3923f7b0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1da3d2e9dd76dc2f56adcba04528e39b3d... | by
intro a b h
refine le_of_lt_or_eq (or_iff_not_imp_right.2 fun hne ↦ ?_)
classical
exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),
fun j hj ↦ notMem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_gt hj,
(h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩ | false | {
"chosen_tokens": 72,
"rejected_tokens": 77,
"token_jaccard": 0.9,
"token_length_ratio": 1.069444
} | markdown_preamble_trailing | trailing | 4928049a9146bf2342c8c2325ad661352892774ef229ee483ac53b80a3b1a328 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Group.PiLex
public import Mathlib.Data.DFinsupp.Order
public import Mathlib.Data.DFinsupp.NeLocus
public import Mathlib.Order.WellFoundedSet
Nam... | by
intro a b h
refine le_of_lt_or_eq (or_iff_not_imp_right.2 fun hne ↦ ?_)
classical
exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),
fun j hj ↦ notMem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_gt hj,
(h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/DFinsupp",
"family_id": "tolex_monotone",
"file_id": "mathlib/Mathlib/Data/DFinsupp/Lex.lean",
"sample_id": "cfce958e665e03e21c1db8b838eb32bd3a0604379292b7cebef2b66be097f2db"
} | train |
{
"chosen_proof_sha256": "e79e7e0dc18cc849a7cc458c4f413b9fd00b3758b5b3041d108cd5fc7c5dd925",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "43a95204387fcb917327a44f5490f1b702... | by
refine ⟨fun _ ↦ ?_, fun _ ↦ inferInstance⟩
let e : G.inv ⋙ G ⋙ F ≅ F := (associator _ _ _).symm ≪≫
isoWhiskerRight (G.asEquivalence.counitIso) _ ≪≫ F.leftUnitor
exact of_iso e | true | {
"chosen_tokens": 54,
"rejected_tokens": 5,
"token_jaccard": 0.088235,
"token_length_ratio": 0.092593
} | circular_dependency | circular_dependency | 493c3d055e550b3d98ba02927645c7b31c70375b6a0e3bf8514c1d058f370249 | 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.Functor.KanExtension.DenseAt
public import Mathlib.CategoryTheory.Limits.Presheaf
public import Mathlib.CategoryTheory.Generator.StrongGenerator... | by
exact IsDense.comp_left_iff_of_isEquivalence | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "01f19691fd40e213a02c079d707e728b6e329f4561624e61745877ccf57be5af",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Functor",
"family_id": "isdense",
"file_id": "mathlib/Mathlib/CategoryTheory/Functor/KanExtension/Dense.lean",
"sample_id": "05a81ca678f684a0e635006515f905d4791271bb1386b472a3d4d7bf0d040e70"
} | train |
{
"chosen_proof_sha256": "6dbb4280aff812e88179a089ca7e30391af8146bdc02e8489a28db8f88a2918d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
ext a
apply Subtype.ext
ext ⟨i, hi⟩
obtain ⟨⟨n, a, ⟨b, hb'⟩, i, rfl : _ = b⟩, rfl⟩ := mk_surjective a
simp only [homOfLE_leOfHom, CommRingCat.hom_comp, RingHom.coe_comp, Function.comp_apply]
erw [ProjectiveSpectrum.Proj.awayToSection_apply]
rw [CommRingCat.hom_ofHom, val_awayMap_mk, Localization.mk_eq_... | true | {
"chosen_tokens": 107,
"rejected_tokens": 8,
"token_jaccard": 0.052632,
"token_length_ratio": 0.074766
} | unsolved_goals | unsolved_goals | 497637912598f337141e1b86c2a8b5aa8303640a4d6ced13db40675c1e2642d6 | 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.ProjectiveSpectrum.Scheme
public import Mathlib.AlgebraicGeometry.AffineScheme
public import Mathlib.AlgebraicGeometry.Gluing
Namespace:
Alg... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "c06c447ac10e48d8e2833722fad093807ac47177780c23119db9fe5c4f22ddc5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/ProjectiveSpectrum",
"family_id": "awaymap_awaytosection",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean",
"sample_id": "f56b4dc966c61de3f4b4118e398dfcc1e44b0ddfc9b9e3ab8b9799975c8612d7"
} | train |
{
"chosen_proof_sha256": "f144c9feaed254a6570ff3fe42b4f3dd16c597e6f392545a7ebcf73899e7bd04",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a7cf948f3b6fdc8d09936e8976199a5252... | by
classical exact computable_iff_re_compl_re | true | {
"chosen_tokens": 4,
"rejected_tokens": 2,
"token_jaccard": 0.2,
"token_length_ratio": 0.5
} | invalid_tactic | invalid_tactic | 49a3383574538cf7706fa8db894494a2c48dcd177c2ddc8b5313f4b99fadcc2c | 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.Computability.PartrecCode
Namespace:
ComputablePred
Local context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 licen... | by
__oprover_invalid_tactic_c5c582457298 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "1afb3554effcd8d9a859a71ae2f4ec43ef495b60eb5b31f20811b66773e11dd3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Computability",
"family_id": "computable_iff_re_compl_re",
"file_id": "mathlib/Mathlib/Computability/RE.lean",
"sample_id": "c5c582457298af65c0d36d7c0a352351e64ed55551e90cd77c5b767876e6c965"
} | train |
{
"chosen_proof_sha256": "5bc8e347ea35f5e54b983043b84f723d4bb439695fcffd6e054bbe0bc3c13d4f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
classical
constructor
· simp only [basis, TwoSidedIdeal.le_iff, TwoSidedIdeal.coe_mk', setOf_subset_setOf]
intro h
constructor
· intro x hx
have (d' : _) : coeff d' (C (σ := σ) x) ∈ J := by
rw [coeff_C]; split_ifs <;> [exact hx; exact J.zero_mem]
simpa using h (C x) (fun _ _ ↦ t... | true | {
"chosen_tokens": 185,
"rejected_tokens": 8,
"token_jaccard": 0.05,
"token_length_ratio": 0.043243
} | unsolved_goals | unsolved_goals | 49f9dd29a62867eb5d79cefb4522701eb2f74d8f0400dcdff911ad022772b489 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finsupp.Interval
public import Mathlib.RingTheory.Ideal.Quotient.Defs
public import Mathlib.RingTheory.MvPowerSeries.PiTopology
public import Mathlib.Topo... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "fb4dc1caa077054f40a0a4acf45c5cefbc8d0baf1ffff33e96cb1ce32b9d275e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/MvPowerSeries",
"family_id": "basis_le_iff",
"file_id": "mathlib/Mathlib/RingTheory/MvPowerSeries/LinearTopology.lean",
"sample_id": "0ccc7aefbf796c9bc413bd34f90036d969463ddb4ebea0b7afe1f034f8ff048d"
} | train |
{
"chosen_proof_sha256": "b232bd6f68b5950f3c5e2ed9ba68ca285d64e0f598a20d32ae0f8c799f5bfc41",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "285ac08c42ed5e3ea6a0bc84afe24ee439... | by
let M' := algebraMapSubmonoid S M
have hM' : algebraMapSubmonoid S M ≤ S⁰ := algebraMapSubmonoid_le_nonZeroDivisors_of_faithfulSMul
_ hM
let f₁ : Rₘ →+* K := map _ (T := R⁰) (RingHom.id R) hM
let f₂ : Sₘ →+* L := map _ (T := S⁰) (RingHom.id S) hM'
algebraize [f₁, f₂]
have := localization_isScalarTowe... | true | {
"chosen_tokens": 238,
"rejected_tokens": 5,
"token_jaccard": 0.046875,
"token_length_ratio": 0.021008
} | circular_dependency | circular_dependency | 4a147518f54534069e68b3beddcdf163a8d59f0beb48758b885e2d4cccabdac4 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.DedekindDomain.PID
public import Mathlib.FieldTheory.Separable
public import Mathlib.RingTheory.RingHom.Finite
Namespace:
(root)
Local context:
/-... | by
exact FractionRing.isSeparable_of_isLocalization | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "de136c05feed13899ff87bc574cd2851d498acde159eba003f0885cdc7826e17",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/DedekindDomain",
"family_id": "fractionring",
"file_id": "mathlib/Mathlib/RingTheory/DedekindDomain/Instances.lean",
"sample_id": "b14089476197b86942d51dd07351b5cf69d062ba1bdb2cb6a8c8c4b42ae306f4"
} | train |
{
"chosen_proof_sha256": "7f711a73112f71461efde233fe8c7cca840501a653c0391902a7be231d8cd075",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
induction l with
| nil => simp
| cons head tail ih => simp [← ih, smul_mul_smul_comm, pow_succ'] | false | {
"chosen_tokens": 23,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.130435
} | by_exact_placeholder | placeholder | 4a169661d08d0e1795e69ce39176cb889a7821c78ef58b4724918e9ea69351ec | 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.Finprod
public import Mathlib.Algebra.GroupWithZero.Action.Defs
public import Mathlib.Algebra.Order.Group.Multiset
public import Mathlib.D... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/BigOperators",
"family_id": "smul_prod",
"file_id": "mathlib/Mathlib/Algebra/BigOperators/GroupWithZero/Action.lean",
"sample_id": "6a93f461afc68c4564b026d64e62d3228a07b26e1dd6066974bf51a8514702ac"
} | train |
{
"chosen_proof_sha256": "893f9954189be14b6650b8e11183442e164c9e72d10c7e09ce63fc59ebc05f3e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d56eafe3c4252de8949772acbb02a5dc70... | by
ext; simp [equiv] | false | {
"chosen_tokens": 7,
"rejected_tokens": 12,
"token_jaccard": 0.583333,
"token_length_ratio": 1.714286
} | markdown_preamble_trailing | trailing | 4a2a14743e7cbac8ba974fd6f72cd9ae030cfd3cc412237d98d7ae97a4cc753a | 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.Colimit.Module
public import Mathlib.RingTheory.Finiteness.Basic
Namespace:
Module.fgSystem
Local context:
/-
Copyright (c) 2024 Junyan Xu. All right... | by
ext; simp [equiv]
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/Colimit",
"family_id": "equiv_comp_of",
"file_id": "mathlib/Mathlib/Algebra/Colimit/Finiteness.lean",
"sample_id": "e5021bc99d39f6a7ba2ff9754e2b8736041ba7f4cf828fd50b3785d42ee868a1"
} | train |
{
"chosen_proof_sha256": "7f54226990013dd5c5400459b377a357ecf590ee9b93a656a04179c0ea205d15",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "fe8a92a902f5ce0e14ed5c2fddfd650067... | by
rw [isUnit_iff_coprime]
exact (Prime.dvd_iff_not_coprime <| prime_of_mem_primeFactors h).mp <| dvd_of_mem_primeFactors h | false | {
"chosen_tokens": 21,
"rejected_tokens": 26,
"token_jaccard": 0.772727,
"token_length_ratio": 1.238095
} | markdown_preamble_trailing | preamble | 4a2e65127e4c3fb0d1d24d1f5ea4ac7eb5df9f7b0ebce25e57878b5fc082c2d8 | 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... | Here is the proof:
by
rw [isUnit_iff_coprime]
exact (Prime.dvd_iff_not_coprime <| prime_of_mem_primeFactors h).mp <| dvd_of_mem_primeFactors h | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/ZMod",
"family_id": "not_isunit_of_mem_primefactors",
"file_id": "mathlib/Mathlib/Data/ZMod/Units.lean",
"sample_id": "4e7c68a0ae1706e7fbfa84a0b8e103da95b624a1edb0fcbc393efbb9763f79d8"
} | train |
{
"chosen_proof_sha256": "3fa4e6a02899a81962f514978e80ca0d79529b0b1a773bcbf2dd1c22e7f2b19e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [Order.covBy_iff_add_one_eq, Order.covBy_iff_add_one_eq]
exact Int.natCast_inj | true | {
"chosen_tokens": 15,
"rejected_tokens": 8,
"token_jaccard": 0.058824,
"token_length_ratio": 0.533333
} | unsolved_goals | unsolved_goals | 4a7c478931aad4647762863e031a70f36b2c32df5133c08acf3b1024fd23df0e | 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.Ring.Int
public import Mathlib.Data.Nat.SuccPred
Namespace:
Int
Local context:
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Release... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "c6c8a02e22119901ef4648f31170b8bcb81e8923b31c22280cd140bf1bd2b748",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Int",
"family_id": "natcast_covby",
"file_id": "mathlib/Mathlib/Data/Int/SuccPred.lean",
"sample_id": "6402417a187a534989eab3a2c52dac736091b114295a76d321088d940f0ef62a"
} | train |
{
"chosen_proof_sha256": "a76aedca09d06687f23aabb71a976948a3f5261d4be2092b4354b20a250ef4d4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
simp [h]
/- ExProd.equateZero -/ | false | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.25
} | by_exact_placeholder | placeholder | 4aa4753f2d907c1750f160a2a0f3efc6fb9e73bfed28c5c83018eb27201bd56e | 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.Algebra.Basic
public import Mathlib.Tactic.Ring.RingNF
Namespace:
Mathlib.Tactic.Algebra
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": "Tactic/Algebra",
"family_id": "smul_one_eq_zero",
"file_id": "mathlib/Mathlib/Tactic/Algebra/Lemmas.lean",
"sample_id": "c124e4499748820a17df65d5a170a2889e6ded7f6f676ef140d015398e461d38"
} | train |
{
"chosen_proof_sha256": "4de448a7d71906907793d1dc7af096bf74aa03a276d6cf774b8ce324536dc08e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "03c570aaf8f111c107d758da0bd5e98a1f... | by
refine tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => ?_
rw [Set.indicator_of_notMem]
simp only [not_le, Set.mem_setOf_eq]
refine lt_of_le_of_lt (Nat.le_ceil _) ?_
refine lt_of_lt_of_le (lt_add_one _) ?_
norm_cast
variable {p : ℝ≥0∞} | false | {
"chosen_tokens": 71,
"rejected_tokens": 78,
"token_jaccard": 0.959184,
"token_length_ratio": 1.098592
} | markdown_preamble_trailing | fence | 4abffdce2654b5dd3e238172fe02bc0a2bdbd1dff1ff272140cb26170e39fceb | 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.Function.ConvergenceInMeasure
public import Mathlib.MeasureTheory.Function.L1Space.Integrable
Namespace:
MeasureTheory
Local context:
/-
Copyri... | ```lean
by
refine tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => ?_
rw [Set.indicator_of_notMem]
simp only [not_le, Set.mem_setOf_eq]
refine lt_of_le_of_lt (Nat.le_ceil _) ?_
refine lt_of_lt_of_le (lt_add_one _) ?_
norm_cast
variable {p : ℝ≥0∞}
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Function",
"family_id": "tendsto_indicator_ge",
"file_id": "mathlib/Mathlib/MeasureTheory/Function/UniformIntegrable.lean",
"sample_id": "e762d0b76da2ca7a3ba42e4fe5df7c426f0bb2153a670147e34342a186a3e955"
} | train |
{
"chosen_proof_sha256": "2c1ae7a6d731fe18987c9a16e015e76772289e20b04a8b5d9fc3cf3d2c6b5c61",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
unfold mkFinSnoc
exact coe_mk (v := Fin.snoc (N.subtype ∘ b) y) _ _
/-- Let `b` be a basis for a submodule `N ≤ O`. If `y ∈ O` is linear independent of `N`
and `y` and `N` together span the whole of `O`, then there is a basis for `O`
whose basis vectors are given by `Fin.snoc b y`. -/
noncomputable def mkFinSno... | false | {
"chosen_tokens": 262,
"rejected_tokens": 2,
"token_jaccard": 0.011236,
"token_length_ratio": 0.007634
} | by_exact_placeholder | by? | 4ae27756f716074d2045181f1258cbabe74c4011df5883ed418ce765c93ff95a | 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.Basic
public import Mathlib.LinearAlgebra.Pi
Namespace:
Module.Basis
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights rese... | 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/Basis",
"family_id": "coe_mkfinsnoc",
"file_id": "mathlib/Mathlib/LinearAlgebra/Basis/Fin.lean",
"sample_id": "6eee5d2ff91023d609bd493224edd9b87b2fa60a603a305b0c65f8936a71c40f"
} | train |
{
"chosen_proof_sha256": "002c7d89e4368c0249a9cb4cfc7e30d967572004732ef0400a272686c72ce645",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
simpa [basis_repr_apply] using ιMultiDual_apply_nondiag R n b t s hst.symm | false | {
"chosen_tokens": 16,
"rejected_tokens": 2,
"token_jaccard": 0.058824,
"token_length_ratio": 0.125
} | sorry_admit_sorryAx_proof_hole | sorry | 4b08ea0b081d1f430bdce7a8c28f1ac7ade56b4d2514b0e01e9f0534c864c834 | 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.ExteriorPower.Basic
public import Mathlib.LinearAlgebra.ExteriorPower.Pairing
public import Mathlib.RingTheory.Finiteness.Subalgebra
public impor... | 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": "LinearAlgebra/ExteriorPower",
"family_id": "basis_repr_ne",
"file_id": "mathlib/Mathlib/LinearAlgebra/ExteriorPower/Basis.lean",
"sample_id": "8c1abff1d102ab8663f817566b5f1e0679aded535e3b61cc03bd6dcfd4532f76"
} | train |
{
"chosen_proof_sha256": "abdd5d1c7dac8f0ca69859030dcc532b57d2b6a28ac7caf0a9109fccea1058bd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]
apply sdiff_unique
· calc
x ⊓ y \ z ⊔ (z ⊓ x ⊔ x \ y) = (x ⊔ (z ⊓ x ⊔ x \ y)) ⊓ (y \ z ⊔ (z ⊓ x ⊔ x \ y)) := by
rw [sup_inf_right]
_ = (x ⊔ x ⊓ z ⊔ x \ y) ⊓ (y \ z ⊔ (x ⊓ z ⊔ x \ y)) := by ac_rfl
_ = x ⊓ (y \ z ⊔ (x ⊓ z ⊔ x ⊓ y... | false | {
"chosen_tokens": 334,
"rejected_tokens": 2,
"token_jaccard": 0.021739,
"token_length_ratio": 0.005988
} | sorry_admit_sorryAx_proof_hole | admit | 4b31703d672509e84c008e56dee9469a7908f79dedcc86e97867842f79f70d8d | 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.BooleanAlgebra.Defs
public import Mathlib.Tactic.GRewrite
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
R... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/BooleanAlgebra",
"family_id": "sdiff_sdiff_right",
"file_id": "mathlib/Mathlib/Order/BooleanAlgebra/Basic.lean",
"sample_id": "5b70195082dfe681d1970751f7a4885c8fdbaa9cac549078cb4a1812112d4d46"
} | train |
{
"chosen_proof_sha256": "82e688aad755eac6f657be89199e8e6fa7e63dacb10bb688605e04b2c9309872",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
suffices IsSquare (-x) → x = 0 by aesop
exact fun hc ↦ le_antisymm (nonpos_of_neg_nonneg (IsSquare.nonneg hc)) h
mpr := IsSquare.nonneg
alias ⟨_root_.IsSquare.of_nonneg, _⟩ := nonneg_iff_isSquare | false | {
"chosen_tokens": 45,
"rejected_tokens": 2,
"token_jaccard": 0.032258,
"token_length_ratio": 0.044444
} | sorry_admit_sorryAx_proof_hole | admit | 4b5862fa895c466abf9019621fad19d0a32eb81493407242a73c5b5cb0f06e63 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.Degree.Domain
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.Algebra.Ring.Semireal.Defs
public import Mathlib.Tact... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "FieldTheory/IsRealClosed",
"family_id": "nonneg_iff_issquare",
"file_id": "mathlib/Mathlib/FieldTheory/IsRealClosed/Basic.lean",
"sample_id": "d3b554a7490cfc938221d5d7fff9423a8ad834bad53b65c4db2638f83535b57e"
} | train |
{
"chosen_proof_sha256": "3973b528cf86a6a3105014eb45539f7af155ea83fbda961dfd591ee49e04f82a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
obtain ⟨p₁, p₂, n, e⟩ := P.cond
have : aeval x P.f.derivative ∣ aeval x P.g ^ n :=
⟨_, by simpa [h.1] using congr(aeval x $e.symm)⟩
exact isUnit_of_dvd_unit this (.pow _ h.2) | false | {
"chosen_tokens": 67,
"rejected_tokens": 3,
"token_jaccard": 0.04878,
"token_length_ratio": 0.044776
} | by_exact_placeholder | exact? | 4b839d175445379ea5fe88ec52fd89cc193b18e25470f022d7879888e9c297b8 | 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.Bivariate
public import Mathlib.Algebra.Polynomial.Taylor
public import Mathlib.RingTheory.Etale.Basic
public import Mathlib.RingTheory.Exte... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Etale",
"family_id": "hasmap",
"file_id": "mathlib/Mathlib/RingTheory/Etale/StandardEtale.lean",
"sample_id": "a1d25de80740477526c09d1872d0447c8000fb20b6d0ca785a16aeb63ebdfb95"
} | train |
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