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": "08f7647cfd9ab05023d94bc2fcca4631870d5f71d7f24de348f6c58aa5102a1d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "49e29b7526fb817a02fc6d72904a3c4272... | by
rw [isPosSemidef_def, Matrix.posSemidef_iff_dotProduct_mulVec]
apply and_congr (B.isSymm_iff_isHermitian_toMatrix b)
rw [isNonneg_def]
refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
· rw [star_dotProduct_toMatrix₂_mulVec]
exact h _
· rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]
exact h _ | false | {
"chosen_tokens": 58,
"rejected_tokens": 63,
"token_jaccard": 0.864865,
"token_length_ratio": 1.086207
} | markdown_preamble_trailing | preamble | 006a588191b537987ce94f10d10cf124332130be53a18cec7aeaffc2d310f19f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Matrix.PosDef
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as ... | Here is the proof:
by
rw [isPosSemidef_def, Matrix.posSemidef_iff_dotProduct_mulVec]
apply and_congr (B.isSymm_iff_isHermitian_toMatrix b)
rw [isNonneg_def]
refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
· rw [star_dotProduct_toMatrix₂_mulVec]
exact h _
· rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]
exact ... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/SesquilinearForm",
"family_id": "linearmap",
"file_id": "mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Star.lean",
"sample_id": "9b67a823dcea17b7fc9288545c47193c34fe1b90d1bd9cf3ef9be1964ef22f5d"
} | train |
{
"chosen_proof_sha256": "343a166fea2e90fed4c3043781c3d8f443aa011a6cdb85e131c437406e24ed78",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1f5f28afd315b861373a4cc9f83e458915... | by
have : C a • D f = D (C a * f) := by simp
rw [this, C_mul', D.map_smul] | false | {
"chosen_tokens": 29,
"rejected_tokens": 36,
"token_jaccard": 0.916667,
"token_length_ratio": 1.241379
} | markdown_preamble_trailing | fence | 008161ec5aa1fb2766571828af325a0c8509ce89105a1ad79eadcdffbbbdc6f5 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Derivative
public import Mathlib.Algebra.Polynomial.Module.AEval
public import Mathlib.R... | ```lean
by
have : C a • D f = D (C a * f) := by simp
rw [this, C_mul', D.map_smul]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Polynomial",
"family_id": "c_smul_derivation_apply",
"file_id": "mathlib/Mathlib/Algebra/Polynomial/Derivation.lean",
"sample_id": "f98433b4daad566e797079ce130b71338bc97d14ae011a049aadb8ae222adbed"
} | train |
{
"chosen_proof_sha256": "667083b2ccb953eab7042f8125e1495f0c41208219db3ae4328356bea107af18",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e486eab1ccf683613dfe0b4c7f4076b42f... | by
rw [div_def, ← @Rat.round_cast ℝ _ _, ← @Rat.round_cast ℝ _ _]
simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, add_mul] | false | {
"chosen_tokens": 36,
"rejected_tokens": 40,
"token_jaccard": 0.857143,
"token_length_ratio": 1.111111
} | missing_wrong_import | import | 00b4d46a98db70243ec6d7266484f8f7eb851885d9aaa9d16dbcf198f8f67cd7 | 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.Archimedean.Real.Basic
public import Mathlib.Data.Complex.Basic
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.NumberTheory.Zsq... | import Mathlib.WrongImport
by
rw [div_def, ← @Rat.round_cast ℝ _ _, ← @Rat.round_cast ℝ _ _]
simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, add_mul] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/Zsqrtd",
"family_id": "tocomplex_im_div",
"file_id": "mathlib/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean",
"sample_id": "76ce3bf080d160d098641f9edded107effa5e468a4e154be6bb90faf62e605f0"
} | train |
{
"chosen_proof_sha256": "8f3a85b33714251a1a29f92568064949152f5e84c72c953dd09287cb0b6f1455",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2a102d88cbe43376559a5015393d625cba... | by
rw [NormNum.IsNat.to_eq h rfl]
exact Nat.cast_nonneg' n | false | {
"chosen_tokens": 16,
"rejected_tokens": 21,
"token_jaccard": 0.736842,
"token_length_ratio": 1.3125
} | markdown_preamble_trailing | preamble | 018ece668935a62be4abb3cf4d8d6f329c266da880680cdb88ac456c40f56ad6 | 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.Invertible
public import Mathlib.Algebra.Order.Ring.Cast
public import Mathlib.Tactic.HaveI
public import Mathlib.Tactic.NormNum.Core
Namespace:... | Here is the proof:
by
rw [NormNum.IsNat.to_eq h rfl]
exact Nat.cast_nonneg' n | {
"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/Positivity",
"family_id": "nonneg_of_isnat",
"file_id": "mathlib/Mathlib/Tactic/Positivity/Core.lean",
"sample_id": "309747378fad33151e6310726ff0174aa0d05877efaa42bd524d8d5cf49ca2f9"
} | train |
{
"chosen_proof_sha256": "3d1a7e2cbfbf665394764afad32089ad753a926b64cde1fb10b4f271be3fc87d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ac00d8bac021edc80086045826c5231fb9... | by
obtain (hs | hs) := hs
· exact hs.2
· exact hs.1.isCompact | false | {
"chosen_tokens": 21,
"rejected_tokens": 25,
"token_jaccard": 0.8125,
"token_length_ratio": 1.190476
} | missing_wrong_import | import | 01e20b3186c068c1f21ac0d461eb91dad6eacf8557baadfe969cd4871e4b9f7b | 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.Spectral.Basic
public import Mathlib.Topology.WithTopology
public import Mathlib.Topology.JacobsonSpace
public import Mathlib.Data.Set.Card
Namespace... | import Mathlib.WrongImport
by
obtain (hs | hs) := hs
· exact hs.2
· exact hs.1.isCompact | {
"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/Spectral",
"family_id": "iscompact_of_mem_constructibletopologysubbasis",
"file_id": "mathlib/Mathlib/Topology/Spectral/ConstructibleTopology.lean",
"sample_id": "0f5b96fc655dafea499598a2e613044abf5991cec77163c042656873851c8051"
} | train |
{
"chosen_proof_sha256": "255334aceee88256ccde5808751019138f069b260876b485bc11ea482a1aed8d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e0497951fa64a3a8bf8993417b48930a17... | by
intro b hb
have hg' := hg b hb
obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hg'⟩ := hg'
refine ⟨c₁, hc₁_mem, c₂, hc₂_mem, ?_⟩
filter_upwards [hg', (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg, hfg]
with x hx₁ hx₂ hx₃
intro u hu
rw [hx₂ u hu.1, hx₃]
exact hx₁ u hu | true | {
"chosen_tokens": 95,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.031579
} | circular_dependency | circular_dependency | 01f45cf449314a1f396201cad53d979f4eb7eb8729d86df2e208459d7c12a39e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecialFunctions.Pow.Real
public import Mathlib.Algebra.Order.ToIntervalMod
public import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Al... | by
exact congr_of_eventuallyEq | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "5e5c1b87bdcc6b043dfed46c9a7c541bf3d9a6766666dc2a8e16b19fdb4ccfca",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Computability/AkraBazzi",
"family_id": "congr_of_eventuallyeq",
"file_id": "mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean",
"sample_id": "04f65d5b50a300cc6227733efdcd3f39e044ddd8767d10f8047fdbe408fc2df7"
} | train |
{
"chosen_proof_sha256": "61dd4a4b69745b7e0fa68c4103cf026764280161c8ce4459f05b5e5caa88e47f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ff92774dd6570cff0844a95accc32cba20... | by
simp_rw [univLE_iff_cardinal_le]; apply le_total | false | {
"chosen_tokens": 8,
"rejected_tokens": 13,
"token_jaccard": 0.615385,
"token_length_ratio": 1.625
} | markdown_preamble_trailing | trailing | 0208076cd7af211c6929eeda1e2a5d13d9881f1f09cbf03935e09d4a89f55e1b | 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.UnivLE
public import Mathlib.SetTheory.Ordinal.Univ
Namespace:
(root)
Local context:
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released und... | by
simp_rw [univLE_iff_cardinal_le]; apply le_total
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": "SetTheory/Cardinal",
"family_id": "univle_total",
"file_id": "mathlib/Mathlib/SetTheory/Cardinal/UnivLE.lean",
"sample_id": "7de96c1ef7802d716a78d8f34f82e51e820ae7e56b3d2af666ad2f6428d02fa0"
} | train |
{
"chosen_proof_sha256": "1c6409609dd76697d622025294c8bc5145bcfe414ffbcf9cd053bd8d9c3b4785",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
let w'' : V₂.op ⋙ eB.op.functor ≅ eT.op.functor ⋙ V₃.op := NatIso.op w'
have : (w ≫ₕ w'.hom).op = (w.op ≫ᵥ w''.hom) := by ext; simp [w'']
rw [← guitartExact_op_iff, ← guitartExact_op_iff w,
← vComp_iff_of_equivalences _ _ _ w'', this]
rfl | false | {
"chosen_tokens": 80,
"rejected_tokens": 2,
"token_jaccard": 0.025,
"token_length_ratio": 0.025
} | sorry_admit_sorryAx_proof_hole | sorry | 0258a2a508961b1612849636bcad3885eb05b24688e91001df56fe74d3e18880 | 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.GuitartExact.Opposite
Namespace:
CategoryTheory.TwoSquare.GuitartExact
Local context:
/-
Copyright (c) 2026 Joël Riou. All rights reserved.
Re... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/GuitartExact",
"family_id": "hcomp_iff_of_equivalences",
"file_id": "mathlib/Mathlib/CategoryTheory/GuitartExact/HorizontalComposition.lean",
"sample_id": "de181494cf6df2f46264b7108f8b5fe8ade01b1f3c5c162ddf30528ffc055308"
} | train |
{
"chosen_proof_sha256": "2933327708396b2202fe5681ecb1091bcb221fad36d86c22e4ed4f768e81a87c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
ext <;> simp | true | {
"chosen_tokens": 6,
"rejected_tokens": 8,
"token_jaccard": 0.083333,
"token_length_ratio": 1.333333
} | unsolved_goals | unsolved_goals | 02667132161f29e3e1dba5dd662419fad71580543d17984d1f9c0b446c0abaec | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Bicategory.LocallyDiscrete
public import Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo
Namespace:
CategoryTheory.Pseudofunctor... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "3a85eef59907097f2969ba69d7196a93b4304624baf94c3312cb153590fd8dc3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Bicategory",
"family_id": "map_id_map",
"file_id": "mathlib/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean",
"sample_id": "77f925e6d7fb9f855b12d8cba8f5cb36cbebbc68e7684132b1a6af65dd6fe312"
} | train |
{
"chosen_proof_sha256": "306e987910ff023d43690d85687bf5ae9a7b00372cbbe2300a8443541f4fbbe4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3b8de9776eeb7f86a13700a0710fcf6249... | by
intro Y f hf
simp only [Presheaf.χ_app, Opposite.op_unop] at hf ⊢
choose a ha using fun Z (g : Z ⟶ Y) (hg : (Sieve.pullback f ((χ m).app X x)).arrows g) => hg
refine ⟨(hF _ hf).amalgamate a ?_, ?_⟩
· introv Y₁ h
apply (mono_iff_injective (m.app (.op Z))).mp inferInstance
simp_rw [NatTrans.naturalit... | false | {
"chosen_tokens": 165,
"rejected_tokens": 169,
"token_jaccard": 0.958333,
"token_length_ratio": 1.024242
} | missing_wrong_import | import | 02b4094cf6a650f96ac5a6e3e983121316efa3c3c4fd9d7c36f69d7a3987b7e3 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Sites.Closed
public import Mathlib.CategoryTheory.Sites.Equivalence
public import Mathlib.CategoryTheory.Subobject.Classifier.Defs
public import... | import Mathlib.WrongImport
by
intro Y f hf
simp only [Presheaf.χ_app, Opposite.op_unop] at hf ⊢
choose a ha using fun Z (g : Z ⟶ Y) (hg : (Sieve.pullback f ((χ m).app X x)).arrows g) => hg
refine ⟨(hF _ hf).amalgamate a ?_, ?_⟩
· introv Y₁ h
apply (mono_iff_injective (m.app (.op Z))).mp inferInstance
... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Topos",
"family_id": "grothendiecktopology",
"file_id": "mathlib/Mathlib/CategoryTheory/Topos/Sheaf.lean",
"sample_id": "20b48f91a7f4ad693944c040d924608c454aa3ef8bafb869719f264b357fd194"
} | train |
{
"chosen_proof_sha256": "665f9b24ed35095c9258c237fa3353b33a4af46379e1ca75f0d890346defc187",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f402ded8ec1b3fd49c322fa61f9e9685ec... | by
apply EqualizerCondition.mk
intro Z B π _ _
refine ⟨fun a b h ↦ ?_, fun ⟨a, ha⟩ ↦ ?_⟩
· simp only [yonedaPresheaf, comp, Quiver.Hom.unop_op, TypeCat.Fun.coe_mk,
Set.coe_setOf, mapToEqualizer, Set.mem_setOf_eq, ConcreteCategory.hom_ofHom, Subtype.mk.injEq,
mk.injEq] at h
simp only [yonedaPresh... | true | {
"chosen_tokens": 325,
"rejected_tokens": 2,
"token_jaccard": 0.009615,
"token_length_ratio": 0.006154
} | invalid_tactic | invalid_tactic | 031569d80dcce1eeb2a0ab012dc89afd319979620e1bf1a8d9b4bcce58098c02 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Preserves.Opposites
public import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
public import Mathlib.Condensed.Basic
public impo... | by
__oprover_invalid_tactic_df00ef8b543a | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "7e4c508849f8e6007a94799fa3f1a0eb791e8921b5c410b8fd282a7b2b82ab3e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Condensed",
"family_id": "equalizercondition_yonedapresheaf",
"file_id": "mathlib/Mathlib/Condensed/TopComparison.lean",
"sample_id": "df00ef8b543a3838a8dcfd15337dcf5374c33e276a9693b751546ead28ce4bfb"
} | train |
{
"chosen_proof_sha256": "a7995287f4a56e7a4f65ad1427f3f2d4f2887e143a6beb10ba5ae796bad00236",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [← subsingleton_iff_zero_eq_one, OreLocalization.one_def,
OreLocalization.zero_def, oreDiv_eq_iff]
simp | false | {
"chosen_tokens": 17,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.176471
} | by_exact_placeholder | placeholder | 0339380785d74e10a0a891ffeb27765d4a0522dac924f45152effc53d4080fc6 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Submonoid.DistribMulAction
public import Mathlib.GroupTheory.OreLocalization.Basic
public import Mathlib.Algebra.GroupWithZero.Defs
Namespace:
O... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/OreLocalization",
"family_id": "subsingleton_iff",
"file_id": "mathlib/Mathlib/RingTheory/OreLocalization/Basic.lean",
"sample_id": "cc3728dc6e72e50fd255371dff3e5f651d3ce2d9c638da326516235746fae11d"
} | train |
{
"chosen_proof_sha256": "4525ed279745c1a825a2d966ade6667dde0e9859d5d810c453aa56b8f6012282",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4ae32efd86a43fc03720d3484c6c7ad3df... | by
simpa only using! congr_fun (f.isCompat' φ) x | true | {
"chosen_tokens": 13,
"rejected_tokens": 5,
"token_jaccard": 0.125,
"token_length_ratio": 0.384615
} | circular_dependency | circular_dependency | 03461c44c88cb9a1380cb0dfcdd1fdd3b1fa120bf5139ca5cd6765379579bf3e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorProduct.RightExactness
public import Mathlib.RingTheory.Congruence.Hom
public import Mathlib.RingTheory.FiniteType
public import Mathlib.Ri... | by
exact PolynomialLaw.isCompat_apply | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "09b14f7b49ae29a5a60e049189e78be13343a428b834c70791d0b5537b9de052",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/PolynomialLaw",
"family_id": "polynomiallaw",
"file_id": "mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean",
"sample_id": "06f153f627e2ad13220f5061c7800c30d1bdd6c1da9c7ac7f9963af2c8d4a748"
} | train |
{
"chosen_proof_sha256": "faf07d4b9f81d896a4dedf3d8bef12a774f393144e0c5f30d0f0b0458195b2b5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "79e65930fe73a626b39d9d7e987131fdfa... | 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 | false | {
"chosen_tokens": 42,
"rejected_tokens": 46,
"token_jaccard": 0.909091,
"token_length_ratio": 1.095238
} | missing_wrong_import | import | 03669718c3d3faaf071366f6cd0ec55acbcead621fe0c78cd84310a1c4f84c3c | 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.Extensive
public import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
public import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
Names... | import Mathlib.WrongImport
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 | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Adhesive",
"family_id": "ispushout",
"file_id": "mathlib/Mathlib/CategoryTheory/Adhesive/Basic.lean",
"sample_id": "332d57c10825bff92a5f9c100471ccaa87a97a30c29a674c9e1339e0c80d29bd"
} | train |
{
"chosen_proof_sha256": "8f9fde22f4f5e3c6a342c774b51f2e9a3888b3631f985c3a4bfd65b729742429",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9f5d681bdf15caa79e692f23201793dd5e... | by
ext : 1
simp [derivative'] | true | {
"chosen_tokens": 8,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.375
} | unknown_identifier | unknown_identifier | 03988e09076bf188ebdad8f32cae975a6b519a883bea2ca78ba354b3ef45f06b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Derivative
public import Mathlib.Algebra.Polynomial.Module.AEval
public import Mathlib.R... | by
exact __oprover_missing_bbc6b49baeed | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1bd067824a37314782da7518e377e1fd33ad11304939633f5b47140ea3e90fd6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Polynomial",
"family_id": "mkderivation_one_eq_derivative",
"file_id": "mathlib/Mathlib/Algebra/Polynomial/Derivation.lean",
"sample_id": "bbc6b49baeedf3e55b05cb3f8d67c144875e1981fa10f0ed9325586b2eeede01"
} | train |
{
"chosen_proof_sha256": "5416e2b8428ca9ff8cc9626edea3896864ca77c250a86dc31c56a1ba22b8daf5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a2c1aaa8d0f8e5dae27717b6de1757cb5f... | by
unfold Measure.mconv
rw [integral_map (by fun_prop) hf.1, integral_prod]
exact (integrable_map_measure hf.1 (by fun_prop)).mp hf | true | {
"chosen_tokens": 32,
"rejected_tokens": 2,
"token_jaccard": 0.05,
"token_length_ratio": 0.0625
} | invalid_tactic | invalid_tactic | 03a2139af4db8c546781b6d035205f2757bd776fd1019d5707999d001d400ac5 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Group.Convolution
public import Mathlib.MeasureTheory.Integral.Prod
Namespace:
MeasureTheory
Local context:
/-
Copyright (c) 2025 Rémy Degenne.... | by
__oprover_invalid_tactic_91fa9e01efb7 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "a7719d524d4f92f6ff0164abd392f193e5fa01840c627a209ed9ff60798ea72a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Group",
"family_id": "integral_mconv",
"file_id": "mathlib/Mathlib/MeasureTheory/Group/IntegralConvolution.lean",
"sample_id": "91fa9e01efb72d4bb46c66a3c3321fad75748aa4aba9170a103de81feb5b067d"
} | train |
{
"chosen_proof_sha256": "91dab1cd2e2e53fc5f8d17a5c93b18cccb419f8155621f1a414c77fee50249a1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
have : x < 0 := lt_of_tendsto_atBot 0 hx
intro n
simpa [abs_of_neg this, lt_neg] using! lt_of_tendsto_atBot (-n) hx | true | {
"chosen_tokens": 27,
"rejected_tokens": 8,
"token_jaccard": 0.153846,
"token_length_ratio": 0.296296
} | unsolved_goals | unsolved_goals | 03b4b63db8e58f06409315b64b5d42c967e16e1402ac2260952d2a9b06f48669 | 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
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "39839f0bf7bb15e2c15464e4b0aebb811f80512edef7ba8a27da7a2779825d38",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Real",
"family_id": "archimedeanclassmk_neg_of_tendsto_atbot",
"file_id": "mathlib/Mathlib/Analysis/Real/Hyperreal.lean",
"sample_id": "9f0aba5e8c958a1721ba7a16d006099909f296bcaa86d09dd7b80102ebf2e9d4"
} | train |
{
"chosen_proof_sha256": "2f6287791c3459636988da88fa63c514ca770a07a4d1537e256b0a039a8246d2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rcases l.exists_antitone_basis with ⟨u, hu⟩
rcases (Filter.hasBasis_self.mpr hl'.exists_measurable_subset).exists_antitone_subbasis with
⟨v, v_meas, hv⟩
simp only [hu.tendsto_iff hv.toHasBasis, true_imp_iff, true_and, setOf_forall, setOf_exists]
exact .iInter fun n ↦ .iUnion fun _ ↦ .biInter (to_countabl... | false | {
"chosen_tokens": 83,
"rejected_tokens": 2,
"token_jaccard": 0.021739,
"token_length_ratio": 0.024096
} | by_exact_placeholder | by? | 03c4c28dae193b43ef87179c737549426d04fa775fcbfd7ace25e33dfdabf23e | 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.MeasurableSpace.Constructions
public import Mathlib.Order.Filter.AtTopBot.CompleteLattice
public import Mathlib.Order.Filter.AtTopBot.CountablyGe... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "MeasureTheory/MeasurableSpace",
"family_id": "measurableset_tendsto",
"file_id": "mathlib/Mathlib/MeasureTheory/MeasurableSpace/MeasurablyGenerated.lean",
"sample_id": "f6e07b9557819a58269647c6b5189639d71a69d808a07410892e1fb416291719"
} | train |
{
"chosen_proof_sha256": "318c2cb9db795a2dd46d3f5a27d1f76a3d76756692909ad6bddc78fa03561671",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7ad57aa84d793cc3401f5f4a8dc65aac7c... | by
constructor
rw [← trans_dualMap_symm_flip e]
exact LinearEquiv.bijective _ | true | {
"chosen_tokens": 13,
"rejected_tokens": 3,
"token_jaccard": 0.142857,
"token_length_ratio": 0.230769
} | circular_dependency | circular_dependency | 03e83ffab3b24511547a5c1a3e2a00b485a60092b86eeaa712e710a6b54d8fe7 | 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:
LinearEquiv
Local context:
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as ... | by
exact isReflexive_of_equiv_dual_of_isReflexive | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "cdb778f32decca19a855e9a5065b72a08874ee34b39c0ee27b4c189e9421d036",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/PerfectPairing",
"family_id": "isreflexive_of_equiv_dual_of_isreflexive",
"file_id": "mathlib/Mathlib/LinearAlgebra/PerfectPairing/Basic.lean",
"sample_id": "7da1d9db371103784bbc6daf0c80df8acd3ad5e9a823b6012d8ef842a94b24f4"
} | train |
{
"chosen_proof_sha256": "7361ae6d5ff40c196b32b028010a5c56393ce19a055fbd12b76df3365e97ad61",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6729f70f1e32fd9d52ce0ff9275d6acf57... | by
refine ⟨fun h ↦ eq_univ_iff_forall.mpr h.mem_periodicPts, fun h ↦ ?_⟩
rw [Finite.injective_iff_surjective, ← range_eq_univ, ← univ_subset_iff, ← h]
apply periodicPts_subset_range | true | {
"chosen_tokens": 36,
"rejected_tokens": 3,
"token_jaccard": 0.038462,
"token_length_ratio": 0.083333
} | unknown_identifier | unknown_identifier | 049191310a0dddf7afcd055460d88623cc382f2ac12ebb46054d2e2d91633a24 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GCDMonoid.Finset
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.Data.PNat.Basic
public import Mathlib.Dynamics.PeriodicPts.Defs
publi... | by
exact __oprover_missing_108e499b9132 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "e9369201a8ae13c9e08dbb556769b0fb5ac5f4aa659c3b39e30c09ae5f57502a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Dynamics/PeriodicPts",
"family_id": "injective_iff_periodicpts_eq_univ",
"file_id": "mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean",
"sample_id": "108e499b9132e898676c8d66a09fbbcb3ff6a762c243e0f235bc6414a3079513"
} | train |
{
"chosen_proof_sha256": "c66eb3df5899011ef7909b5b28a8dbc7ead7f1cd7984dec7b597a92d174e6203",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e4be22c9b9981e432ef9024a9996296ed8... | by
rw [SModEq.sub_mem] at hxy ⊢
rw [← smul_sub]
exact smul_mem_smul hc hxy | true | {
"chosen_tokens": 19,
"rejected_tokens": 3,
"token_jaccard": 0.125,
"token_length_ratio": 0.157895
} | unknown_identifier | unknown_identifier | 0497dfba73125893b1e63c80e2a829a70b784bd03d43948f3bedeea907ff7248 | 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.Operations
public import Mathlib.LinearAlgebra.SModEq.Basic
Namespace:
SModEq
Local context:
/-
Copyright (c) 2025 Jiedong Jiang. All rights ... | by
exact __oprover_missing_bf0246d237d6 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "998eeff469aca5126fc003e01a64a753056863204f1986fc278ccae4f1bdc251",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/SModEq",
"family_id": "smul",
"file_id": "mathlib/Mathlib/LinearAlgebra/SModEq/Pointwise.lean",
"sample_id": "bf0246d237d6d9933a85d317c9082d92e62a7779e9ca287e29ec9e8f9ef938b0"
} | train |
{
"chosen_proof_sha256": "a0b20474523f493304d2e5c3c4796a60a88d5f5165ad276749d342179c8ed04e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0f8900d3e899ee934779d7ba7219c40f26... | by
classical
rw [separatingLeft_iff_ker_eq_bot, ker_eq_bot]
let e := LinearEquiv.prodComm R _ _ ≪≫ₗ Module.dualProdDualEquivDual R (Module.Dual R M) M
let h_d := e.symm.toLinearMap.comp (dualProd R M)
refine (Function.Injective.of_comp_iff e.symm.injective
(dualProd R M)).symm.trans ?_
rw [← LinearEquiv... | true | {
"chosen_tokens": 174,
"rejected_tokens": 5,
"token_jaccard": 0.046875,
"token_length_ratio": 0.028736
} | wrong_namespace_qualification | wrong_namespace_qualification | 04a1f03b51301ea5b5c4fc1791ce61e223b3c20ca1450cf564e127aa61be91fa | 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
public import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
public import Mathlib.LinearAlgebra.QuadraticForm.Prod
Namespace:
Li... | by
exact OProverMissing.missing_6f6ae3b60a22 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a1cd792e41f8264e0873890de7e6e0c0c697cec886773af634497dc1fb85445c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/QuadraticForm",
"family_id": "separatingleft_dualprod",
"file_id": "mathlib/Mathlib/LinearAlgebra/QuadraticForm/Dual.lean",
"sample_id": "6f6ae3b60a22d3ee8598639d07edd15dc214dc93b32db436d37ba0364ed7ec2a"
} | train |
{
"chosen_proof_sha256": "5fc5fcc6ed6270b255f9132aa985cf7070d173e497c862b80ffbe80e51562de8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "36e6a6a064db8ebbe5935e08c6a7f9aae4... | by
simp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index]
exact forall_comm
-- `simp` normal form is `sized_iUnion`. | true | {
"chosen_tokens": 27,
"rejected_tokens": 2,
"token_jaccard": 0.05,
"token_length_ratio": 0.074074
} | invalid_tactic | invalid_tactic | 04a5fd50bb702262dca90065fecf02aad0d4b646d83b5ae4421cb8cb17361903 | 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.Powerset
public import Mathlib.Order.Interval.Finset.Nat
public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
Namespace:
Set
Local cont... | by
__oprover_invalid_tactic_0f82b3064f5d | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "ccbccaa3b15516808c6e7c4993eb990ce90eaaad1e5262b3004b3cb74c374b7e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Finset",
"family_id": "sized_iunion",
"file_id": "mathlib/Mathlib/Data/Finset/Slice.lean",
"sample_id": "0f82b3064f5d92a16f748b7b934596ecec0fdc7677d78faf22eb29829f183996"
} | train |
{
"chosen_proof_sha256": "e79d2c183534cfffc99b615577f1db04cdc59929d66d45cf7905520be00385d4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2a949889ffca8542db6fbc433458938646... | by
classical
obtain ⟨F₁, hF₁, hAF₁⟩ := hA.sq_covBySMul
obtain ⟨F₂, hF₂, hBF₂⟩ := hB.sq_covBySMul
have := hA.one_le
choose f hf using exists_smul_inter_smul_subset_smul_inv_mul_inter_inv_mul A B
refine ⟨.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1)), ?_, ?_⟩
· calc
(#(.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1))... | false | {
"chosen_tokens": 429,
"rejected_tokens": 436,
"token_jaccard": 0.977011,
"token_length_ratio": 1.016317
} | markdown_preamble_trailing | fence | 04eaba62f4e2be0fad56c2419db1bd1553812e384969072ab892f566a8dcf757 | 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.Pointwise
public import Mathlib.Combinatorics.Additive.CovBySMul
public import Mathlib.Combinatorics.Additive.RuzsaCovering
public impor... | ```lean
by
classical
obtain ⟨F₁, hF₁, hAF₁⟩ := hA.sq_covBySMul
obtain ⟨F₂, hF₂, hBF₂⟩ := hB.sq_covBySMul
have := hA.one_le
choose f hf using exists_smul_inter_smul_subset_smul_inv_mul_inter_inv_mul A B
refine ⟨.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1)), ?_, ?_⟩
· calc
(#(.image₂ f (F₁ ^ (m - 1)) (F₂ ^ ... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/Additive",
"family_id": "pow_inter_pow_covbysmul_sq_inter_sq",
"file_id": "mathlib/Mathlib/Combinatorics/Additive/ApproximateSubgroup.lean",
"sample_id": "5627650f4b1175b07bdab889615a32ef5fc6133ef66c020ba65b75f9c09ed833"
} | train |
{
"chosen_proof_sha256": "9ea2d48ebe00bb73f1eea97bd737a2f6b8f66a4067141833800162c91765c532",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp [GradedMonoid.GOne.one, TensorPower.toTensorAlgebra_tprod] | true | {
"chosen_tokens": 13,
"rejected_tokens": 8,
"token_jaccard": 0.058824,
"token_length_ratio": 0.615385
} | unsolved_goals | unsolved_goals | 05239f75c87635d84490583f3f9a8c44b6f1a90bcc8585dd96bf1b926ab86034 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorAlgebra.Basic
public import Mathlib.LinearAlgebra.TensorPower.Basic
Namespace:
TensorPower
Local context:
/-
Copyright (c) 2021 Eric Wies... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "98aefb3cadd5f1aade6e34aae778184241b0dff9f78ebac25e4dfc51ce971434",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/TensorAlgebra",
"family_id": "totensoralgebra_gone",
"file_id": "mathlib/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean",
"sample_id": "dc8404380687b2e37f6ac350d18b83560147a6545b4e3c6396ca1a3bc04d0633"
} | train |
{
"chosen_proof_sha256": "389a8937a3f2af4740351819b93e71736e399c56154cd81ab2e2e646a0bfda28",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [eq_comm, prod_eq_prod_of_nonempty h, eq_comm, @eq_comm _ eY'] | false | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.2
} | by_exact_placeholder | exact? | 052eb700b25f8527bd38890cecc9319b535d5dc2c6e56d3ff48a7b865dd7a646 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.OpenPartialHomeomorph.Composition
Namespace:
OpenPartialHomeomorph
Local context:
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Rele... | 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/OpenPartialHomeomorph",
"family_id": "prod_eq_prod_of_nonempty",
"file_id": "mathlib/Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean",
"sample_id": "b0b58afdb4cef1d362a08061526d43713fac33d2fd2c151db38d446b2e4f58cd"
} | train |
{
"chosen_proof_sha256": "0e11e10bbaaaed2dc873b84e165eba0bc5466c4e31acb825aeb49b92a5fe94d8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
-- Let `Z` be a minimal set of prime ideals such that their product is contained in `J`.
obtain ⟨Z₀, hZ₀⟩ := exists_primeSpectrum_prod_le_and_ne_bot_of_domain hNF hI0
obtain ⟨Z, ⟨hZI, hprodZ⟩, h_eraseZ⟩ :=
wellFounded_lt.has_min
{Z | (Z.map asIdeal).prod ≤ I ∧ (Z.map asIdeal).prod ≠ ⊥}
⟨Z₀, hZ₀... | false | {
"chosen_tokens": 346,
"rejected_tokens": 2,
"token_jaccard": 0.008197,
"token_length_ratio": 0.00578
} | sorry_admit_sorryAx_proof_hole | admit | 05511de79a89488a1357faf2a592ef25671df129fd76d0f508cfe5bc6eeb30fa | 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.Subalgebra.Pointwise
public import Mathlib.RingTheory.DedekindDomain.Basic
public import Mathlib.RingTheory.FractionalIdeal.Inverse
public impo... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/DedekindDomain",
"family_id": "primespectrum",
"file_id": "mathlib/Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean",
"sample_id": "0996acbca200c0dccec01a46d4e81205ed46c98fd950cb9f91af86182dcfbe7d"
} | train |
{
"chosen_proof_sha256": "843286c15dd0e5c3687da59e28ddbf32542077397522d384d533e4f852e034a9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "96285e98a90c2e00636f856577b756ddde... | by
rw [map_comp]
exact hg.trans <| map_mono _ hf | true | {
"chosen_tokens": 14,
"rejected_tokens": 3,
"token_jaccard": 0.133333,
"token_length_ratio": 0.214286
} | circular_dependency | circular_dependency | 05578208c6008563b61054e814ca53c19b5bccf2e619f3de237f20f838328eea | 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.GradedAlgebra.Homogeneous.Ideal
public import Mathlib.RingTheory.GradedAlgebra.RingHom
Namespace:
HomogeneousIdeal
Local context:
/-
Copyright (c)... | by
exact irrelevant_le_map_comp | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "fa9e40f9ac772bc5aa5a2780dca6e94d3723a408946d5514dcb8cd1bae663a85",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/GradedAlgebra",
"family_id": "irrelevant_le_map_comp",
"file_id": "mathlib/Mathlib/RingTheory/GradedAlgebra/Homogeneous/Maps.lean",
"sample_id": "2c90f2e32be3da6fb260df3bb94bf0c15d7de35d2f1513e19789e1136caf26c6"
} | train |
{
"chosen_proof_sha256": "de67f9b6451b5db3ba017889f07c8f6879cb98f3fc2736a86445d62c14d1c664",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3d470e7ad773073099cd7eb08ccd0ead6e... | by
rw [Squarefree, not_forall]
exact ⟨0, by simp⟩ | false | {
"chosen_tokens": 14,
"rejected_tokens": 18,
"token_jaccard": 0.75,
"token_length_ratio": 1.285714
} | missing_wrong_import | import | 05ff9d4452c635482cdef1b657e4ec716ce9b8154a9b10402ed87294aeedbfe7 | 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... | import Mathlib.WrongImport
by
rw [Squarefree, not_forall]
exact ⟨0, by 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": "Algebra/Squarefree",
"family_id": "not_squarefree_zero",
"file_id": "mathlib/Mathlib/Algebra/Squarefree/Basic.lean",
"sample_id": "ba8a1ee7158309c944c89fb3fff36b668ca7628b981cba7184536316677b8d1f"
} | train |
{
"chosen_proof_sha256": "dfb0cfa54bf916f968be75edf3a87f502ebb8bc73b72d17a24c14c29c6f3e807",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "dfa2375d52b2512c773d6c8fddc24e78da... | by
convert! sup_mem_subalgebra_closure A f g
apply SetLike.ext'
simp | false | {
"chosen_tokens": 12,
"rejected_tokens": 17,
"token_jaccard": 0.705882,
"token_length_ratio": 1.416667
} | markdown_preamble_trailing | preamble | 060d74d0d090822121affa5c6d6fbf46b7f7ce76bc162e83c6a261b674b0c44d | 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.Subalgebra.Tower
public import Mathlib.Analysis.RCLike.Basic
public import Mathlib.Topology.Algebra.Star.Real
public import Mathlib.Topology.Al... | Here is the proof:
by
convert! sup_mem_subalgebra_closure A f g
apply SetLike.ext'
simp | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/ContinuousMap",
"family_id": "sup_mem_closed_subalgebra",
"file_id": "mathlib/Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean",
"sample_id": "5718bd96cf9f59ca4c83944f21497324dfdb689e13608aa103fc1d5ac63a1635"
} | train |
End of preview. Expand in Data Studio
Kakeya OProver Training
Apache-2.0 training and evaluation assets for the four-stage Kakeya OProver LoRA journey.
Configs and split counts
sft-90: 63 train / 9 validation / 18 holdoutsft-1000: 700 train / 100 validation / 200 holdoutdpo-2000: 1,400 train / 200 validation / 400 holdout preference pairsrepair-2800: 2,800 train / 400 validation (train: 2,100 repair + 700 rehearsal; validation: 300 + 100)strict-evaluation-22: 22 public-safe task identifiers and split metadata only (18 holdout + 4 canary); aggregate outcomes are instrict-evaluation-22/results.json
Schema
SFT records include the exact public Lean context, target, and verified proof body. DPO records include prompt, chosen, rejected, rejection metadata, and source bindings. Repair records include the public context, failed proof, structured compiler diagnostic, and corrected proof. Strict evaluation records intentionally omit prompts and candidate proof bodies.
Provenance and compile validation
All proof-bearing samples are pure Mathlib material or derived negatives/repairs bound to Mathlib revision 360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56 and Lean v4.32.0-rc1. Source manifests report compile validation and zero cross-split family overlap where applicable. Provenance-incomplete OProofs mixed samples were excluded.
Leakage and release policy
Holdout splits were excluded from training. The strict 22-task release contains metadata and aggregates only. No production Jensen prompt, raw candidate proof body, private chain-of-thought, credential, or local path is included.
License and attribution
Apache-2.0. Mathlib copyright and author notices remain embedded in source contexts. See LICENSE and NOTICE.
Release Manifest
The content-addressed public/private boundary and immutable asset references are in release-manifest.json.
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