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": "19d0436be488b2c04681a3df70e828f641be531fc0e4601e74671ee295a63fea",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "327cb8fff8f9cfd0e40105ba93b9c9cf5b... | by
rwa [H.ker_flip] | true | {
"chosen_tokens": 7,
"rejected_tokens": 3,
"token_jaccard": 0.111111,
"token_length_ratio": 0.428571
} | circular_dependency | circular_dependency | a6be3fd05195e6f878a64aeb8a696eeb1047161431e49a7709e89178422881e3 | 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.Basis.Basic
public import Mathlib.LinearAlgebra.BilinearMap
public import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
import Mathlib.Algebra.M... | by
exact ker_flip_eq_bot | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "61f421ed6f5d7dab3aafd8a69443d41e38c44d8201ae06d203c8fa6f3bd08f69",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/SesquilinearForm",
"family_id": "ker_flip_eq_bot",
"file_id": "mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Basic.lean",
"sample_id": "08fc9f4a5a976873fc6fe96edc2ccf033e6ac6f6554e9759e87483072bcf261d"
} | train |
{
"chosen_proof_sha256": "82023a5a0373446dea510bd79a74d7b917c1be62462ff0113fe4a4cf3e1520cd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by simp) :
∃ b : B, b ∉ s := by
simpa [-SetLike.coe_set_eq, SetLike.ext'_iff, h_top, Set.ne_univ_iff_exists_notMem] using hs | true | {
"chosen_tokens": 33,
"rejected_tokens": 8,
"token_jaccard": 0.107143,
"token_length_ratio": 0.242424
} | unsolved_goals | unsolved_goals | a6bf8381dc5784204fa7d80f0373bdd2e3b60ca28ad08e557734fd62d0f90608 | 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
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "a67f4427ae01713593cfaa5327c7aa720a8f6fce6519800c29c98aef13ec9b95",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/SetLike",
"family_id": "exists_not_mem_of_ne_top",
"file_id": "mathlib/Mathlib/Data/SetLike/Basic.lean",
"sample_id": "87399eed90a7a9d71500444beb6af36405b8a0740724037a49d189fcab7e9413"
} | train |
{
"chosen_proof_sha256": "5137effd66997cbfcff7aea301f3e52b8dc7efd3fad26dd29be5d337e1b39c36",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "71eeb2193c99aa08af51083f81cf199e0a... | by
rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp | false | {
"chosen_tokens": 16,
"rejected_tokens": 21,
"token_jaccard": 0.789474,
"token_length_ratio": 1.3125
} | markdown_preamble_trailing | trailing | a6c314f2114a80bc895a2931a261a91d262111c6a654964a00f8d52ff1dfb4a3 | 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.Basic
public import Mathlib.Algebra.Order.AbsoluteValue.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Anne Baanen. All rights r... | by
rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp
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/Int",
"family_id": "absolutevalue",
"file_id": "mathlib/Mathlib/Data/Int/AbsoluteValue.lean",
"sample_id": "0830875cbcc8e3b2fa3c485a8dfb76651d4e973c898e771f3c9e1024b82d5806"
} | train |
{
"chosen_proof_sha256": "ecfe891855dc120f1040ab8b4113f995f2ce6be82fd87d86c20a75c2f1e66b44",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ff05fe702b3e5167892ee225a104797e7e... | by
exact .tower_top (K := F⟮(X : F⟮X⟯)⟯) _
variable {y : K} | false | {
"chosen_tokens": 26,
"rejected_tokens": 31,
"token_jaccard": 0.818182,
"token_length_ratio": 1.192308
} | markdown_preamble_trailing | preamble | a6e8d9ac3f939c2590c037a083c71103f272572c5283744a978617e1480d2fb1 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.RatFunc.Degree
public import Mathlib.RingTheory.DedekindDomain.IntegralClosure
public import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
pu... | Here is the proof:
by
exact .tower_top (K := F⟮(X : F⟮X⟯)⟯) _
variable {y : K} | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory",
"family_id": "algebra",
"file_id": "mathlib/Mathlib/NumberTheory/FunctionField.lean",
"sample_id": "6381270961bbd859431dc91f4de11a424555613e987c739bd7c51cfea5347942"
} | train |
{
"chosen_proof_sha256": "9a1a0b80645445024d8bb550e3e8810e677e51b85aec8657ff2f9f66fe77107a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0e6aeac8925875b0126fdb46099ecbd051... | by
rw [isProperMap_iff_isClosedMap_and_compact_fibers]
refine ⟨f.continuous, f.isClosedMap, fun y ↦ ?_⟩
have := compactSpace_of_universallyClosed (pullback.snd f (Y.fromSpecResidueField y))
rw [← Scheme.range_fromSpecResidueField, ← Scheme.Pullback.range_fst]
exact isCompact_range (Scheme.Hom.continuous _) | false | {
"chosen_tokens": 60,
"rejected_tokens": 65,
"token_jaccard": 0.875,
"token_length_ratio": 1.083333
} | markdown_preamble_trailing | preamble | a6ea4037d7c0c5752a1d704af2ca4710d5af87718f428399d57368cfbd34695d | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
public import Mathlib.AlgebraicGeometry.PullbackCarrier
public import Mathlib.Topology.LocalAtTarget
Namespace:
Al... | Here is the proof:
by
rw [isProperMap_iff_isClosedMap_and_compact_fibers]
refine ⟨f.continuous, f.isClosedMap, fun y ↦ ?_⟩
have := compactSpace_of_universallyClosed (pullback.snd f (Y.fromSpecResidueField y))
rw [← Scheme.range_fromSpecResidueField, ← Scheme.Pullback.range_fst]
exact isCompact_range (Scheme.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": "AlgebraicGeometry/Morphisms",
"family_id": "scheme",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean",
"sample_id": "33d6750ddf63884a3db77c1ac0c0a14e4b5da0e25cd340d2ef99ffb5417edbe1"
} | train |
{
"chosen_proof_sha256": "a2aaf585a4a9909c497f3f34f264608d1efd5c7508490b5e6cf98acb7351c7d2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1bd1111a82684ac3770181e9ed442f50e6... | by
constructor
ext x'
simp_rw [LieIdeal.mem_killingCompl, LieModule.traceForm_comm]
refine ⟨fun hx' ↦ ?_, fun hx y _ ↦ hx ▸ LinearMap.map_zero₂ (killingForm R L') y⟩
suffices e.symm x' ∈ LinearMap.ker (killingForm R L) by
rw [IsKilling.ker_killingForm_eq_bot] at this
simpa [map_zero] using (e : L ≃ₗ[R... | true | {
"chosen_tokens": 130,
"rejected_tokens": 3,
"token_jaccard": 0.029851,
"token_length_ratio": 0.023077
} | circular_dependency | circular_dependency | a6ee258fa4a9dc8939b2ce74cef0692fffbfef8f09b471f5c628b481e84ea576 | 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.Lie.InvariantForm
public import Mathlib.Algebra.Lie.Semisimple.Basic
public import Mathlib.Algebra.Lie.TraceForm
Namespace:
LieAlgebra
Local context:... | by
exact isKilling_of_equiv | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "3d95404cc25321a319496f7f09e072a42f0a91641b11f41c4713bd1a11288f79",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Lie",
"family_id": "iskilling_of_equiv",
"file_id": "mathlib/Mathlib/Algebra/Lie/Killing.lean",
"sample_id": "e0c5330d2514224d882fef475861bb09dfae3c5924cf8a420d81d8d903a10d54"
} | train |
{
"chosen_proof_sha256": "68a591d0520d49abb38f3df228ef3dd797bb020e537249311751475dbc1d52d8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [max_def]; split_ifs with h
exacts [hb h, ha (le_of_not_ge h)] | true | {
"chosen_tokens": 20,
"rejected_tokens": 8,
"token_jaccard": 0.045455,
"token_length_ratio": 0.4
} | unsolved_goals | unsolved_goals | a7128f0600c5e39124039f1f51aebd89c9722f9e1da9460864b06b4d50f2fd90 | 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.Classes.Order
public import Batteries.Tactic.Trans
public import Mathlib.Data.Ordering.Basic
public import Mathlib.Tactic.ExtendDoc
public import Mathlib.Tac... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e23c888cba8807f65e06dfd4008129842890a6ecabebded29e3371b45091dd80",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Defs",
"family_id": "max_ind",
"file_id": "mathlib/Mathlib/Order/Defs/LinearOrder.lean",
"sample_id": "44ece7bf253c5f603c55e81415554957d2378036c79d5b76487dc473ce633518"
} | train |
{
"chosen_proof_sha256": "021873745f79b607afe2480d1a3ec9b92a92bc4d1a53d9616a37575a2e4fbf1f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by rw [← top_sdiff', ← top_sdiff', map_sdiff, map_top] | false | {
"chosen_tokens": 13,
"rejected_tokens": 3,
"token_jaccard": 0.090909,
"token_length_ratio": 0.230769
} | by_exact_placeholder | exact? | a74d81b107708f9ec4fc0d1b61568bdc29a4cc3adab3a13ee0d3175967f8f0c9 | 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.BoundedLattice
Namespace:
(root)
Local context:
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as descr... | 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": "Order/Heyting",
"family_id": "map_hnot",
"file_id": "mathlib/Mathlib/Order/Heyting/Hom.lean",
"sample_id": "88184057873a0a559cc6d5510520e38470290a3d57980287af7d4cf570e8c138"
} | train |
{
"chosen_proof_sha256": "e08724fba84126ab1ada747c46865f50f539e3028cb4c27da440863f309c14d1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "92784161800bc7703887ac5638e35236b0... | by
borelize G
rw [modularCharacterFun_eq_haarScalarFactor MeasureTheory.Measure.haar g]
exact haarScalarFactor_pos_of_isHaarMeasure _ _ | false | {
"chosen_tokens": 17,
"rejected_tokens": 22,
"token_jaccard": 0.789474,
"token_length_ratio": 1.294118
} | markdown_preamble_trailing | trailing | a76c53a866dfa50e9f8228aa2e2306f5da4746d6db35490b77167f10fc3c8a34 | 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.LocallyIntegrable
public import Mathlib.MeasureTheory.Group.Integral
public import Mathlib.MeasureTheory.Group.Measure
public import Mat... | by
borelize G
rw [modularCharacterFun_eq_haarScalarFactor MeasureTheory.Measure.haar g]
exact haarScalarFactor_pos_of_isHaarMeasure _ _
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": "MeasureTheory/Group",
"family_id": "modularcharacterfun_pos",
"file_id": "mathlib/Mathlib/MeasureTheory/Group/ModularCharacter.lean",
"sample_id": "5d6d44820ec47fa0236df113ccc6391b9ab3172c1aa71f5359c1564641b4264c"
} | train |
{
"chosen_proof_sha256": "22008102f7d34b93c9ef351ad767f156bb7824bb62e00233f134ea27d1c44cf6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [eq_geometrically, GeometricallyReduced.eq_geometrically,
GeometricallyIrreducible.eq_geometrically, ← geometrically_inf]
eta_expand
simp [isIntegral_iff_irreducibleSpace_and_isReduced, and_comm] | false | {
"chosen_tokens": 23,
"rejected_tokens": 2,
"token_jaccard": 0.0625,
"token_length_ratio": 0.086957
} | sorry_admit_sorryAx_proof_hole | admit | a78b35603ca4c7cb4ae39a5eb3c0e491d9d41b6027b8e2fd7ad441a719f86f11 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Geometrically.Reduced
public import Mathlib.AlgebraicGeometry.Geometrically.Irreducible
Namespace:
AlgebraicGeometry
Local context:
/-
Copy... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Geometrically",
"family_id": "geometricallyintegral",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Geometrically/Integral.lean",
"sample_id": "f1be1ab398e8e6f5f51fea76b3b1c755d097ca53fd2fb60f4b3eb6a00e32ab92"
} | train |
{
"chosen_proof_sha256": "f11a99acaff28e0866b0cf9264b8d87a92a312c57ace97d7c08c3310cdd9276a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2a683c5a45e5862414cab6197a75e17159... | by
rw [Finset.offDiag_filter_lt_eq_filter_le]
conv_rhs => rw [← Finset.sum_subtype_eq_sum_filter]
refine (Finset.sum_equiv Sym2.sortEquiv.symm ?_ ?_).symm
all_goals aesop | true | {
"chosen_tokens": 35,
"rejected_tokens": 3,
"token_jaccard": 0.041667,
"token_length_ratio": 0.085714
} | circular_dependency | circular_dependency | a7b86764b0bf626186de26d89f7c50b29f965d248e1f1954cae7c6e0376d56fb | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finset.Sym
public import Mathlib.Data.Sym.Sym2.Order
public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
Namespace:
Finset
Local context:
/-
C... | by
exact sum_sym2_filter_not_isDiag | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "4eedc8db06b12feaa92ce9e448b3ef1f7c512da87e83434cf9ad27506ca57378",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/BigOperators",
"family_id": "sum_sym2_filter_not_isdiag",
"file_id": "mathlib/Mathlib/Algebra/BigOperators/Sym.lean",
"sample_id": "5013414c846a303e2d7ae6cf16e87bd0d86e0af5364ad267930855e6631a1174"
} | train |
{
"chosen_proof_sha256": "0d95bb02e9e9a9b98bdd4fbdf7abc16ec6edaa269b3beda79e93adb84be4b936",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "930f23c5bc98778cfbbd2c8d89384e1922... | by
by_cases h : k = 0
· simp [h]
exact (isPrimitiveRoot_exp k h).card_primitiveRoots | true | {
"chosen_tokens": 20,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.1
} | invalid_tactic | invalid_tactic | a7b8db17588a9b6faeb1539523a8d6b5d3f308bb1f00c64ef82aa480a29c172c | 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.Complex.Log
public import Mathlib.RingTheory.Int.Basic
public import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
public import Mat... | by
__oprover_invalid_tactic_32b115c85276 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "3684b97611187606a3954cb932b3a6b2019b1c74af5e0005a6aad3cd2516644a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/RootsOfUnity",
"family_id": "card_primitiveroots",
"file_id": "mathlib/Mathlib/RingTheory/RootsOfUnity/Complex.lean",
"sample_id": "32b115c85276c93388c62a93ae6422efd911c7d0d06efd966d995ffe632b53e9"
} | train |
{
"chosen_proof_sha256": "f534ed3effb9755a614b851ef33384815c516806a3924c7986894d2e4e795cd3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3007d31a80186b26e9b8c1ae8d1dca5018... | by
rw [← isRegular_pairing_iff, Pairing.isRegular_iff_nonempty_weakRankFunction]
exact (P.weakRankFunctionEquiv ℕ).symm.nonempty_congr | true | {
"chosen_tokens": 21,
"rejected_tokens": 3,
"token_jaccard": 0.105263,
"token_length_ratio": 0.142857
} | unknown_identifier | unknown_identifier | a82f4f859a5e768f63cf9d1ab88ad7a78e3e6ee0939745cefd9466b50e2b8f83 | 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.SimplicialSet.AnodyneExtensions.Rank
public import Mathlib.Data.Finite.Sigma
Namespace:
SSet.Subcomplex.PairingCore
Local context:
/-
Copyr... | by
exact __oprover_missing_467d6115e45b | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d2b24cb184e9e1cf34f1d055c3cdaad9b7c56790093b3b8d88ff248695a993c5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/SimplicialSet",
"family_id": "isregular_iff_nonempty_weakrankfunction",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/RankNat.lean",
"sample_id": "467d6115e45bfeaa364bc7d6dc84e27f5374de312ed692a0238a4a35b99ebeeb"
} | train |
{
"chosen_proof_sha256": "caf7c1208514616ff661b06fa1617859cfde2ba543e0b78c240ce9e176c8198e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2afb39b3691bc4ab78a4e23759e1c825b6... | by
simpa using h [[]] [[], []] | false | {
"chosen_tokens": 15,
"rejected_tokens": 20,
"token_jaccard": 0.583333,
"token_length_ratio": 1.333333
} | markdown_preamble_trailing | preamble | a85abf74b32c8c10a31e79c7d5fcb0accd6bc00266cb63c14c57ae0d70d3edb1 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Set.Basic
Namespace:
InformationTheory
Local context:
/-
Copyright (c) 2026 Elazar Gershuni. All rights reserved.
Released under Apache 2.0 license as d... | Here is the proof:
by
simpa using 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": "InformationTheory/Coding",
"family_id": "uniquelydecodable",
"file_id": "mathlib/Mathlib/InformationTheory/Coding/UniquelyDecodable.lean",
"sample_id": "50e5e16cfcfd38d6568b5e49b24fb4dda95d8927c47ec6721aa44339e4393525"
} | train |
{
"chosen_proof_sha256": "5ca120de9531c196478bb85a2839de944f04348088dc99c356e217a11b64504f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "34db14d5b84ad45bdcf1fb3d5c33e842b3... | by
induction xs generalizing ys i with
| nil => cases h₂
| cons x' xs xs_ih =>
cases i
· simp only [length_cons, lt_add_iff_pos_left, add_pos_iff, Nat.lt_add_one, or_true,
getElem?_eq_getElem, getElem_cons_zero, Option.some.injEq] at h₂
subst h₂
cases ys
· cases h₁
· simp
... | false | {
"chosen_tokens": 129,
"rejected_tokens": 133,
"token_jaccard": 0.952381,
"token_length_ratio": 1.031008
} | missing_wrong_import | import | a861aaa4b13802099045e3372e2ac5f0118ab3cc7961bbe40c28b46cbc2929df | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Data.MLList.Basic
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.ZeroLEOne
public import Mathlib.Data.DFinsupp.Defs
public... | import Mathlib.WrongImport
by
induction xs generalizing ys i with
| nil => cases h₂
| cons x' xs xs_ih =>
cases i
· simp only [length_cons, lt_add_iff_pos_left, add_pos_iff, Nat.lt_add_one, or_true,
getElem?_eq_getElem, getElem_cons_zero, Option.some.injEq] at h₂
subst h₂
cases ys
... | {
"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": "Testing/Plausible",
"family_id": "list",
"file_id": "mathlib/Mathlib/Testing/Plausible/Functions.lean",
"sample_id": "e2cd3c1a8076b3af47b98c32f101f8a48aa6a72e4acab38020fc2356c4738a6b"
} | train |
{
"chosen_proof_sha256": "2139989fd770c5936c4b547ad3ae043d8606b002f3ccc3b366913c36130fefc8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1dd4201e2b46ac0426635902442743acfd... | by
filter_upwards [rnDeriv_compProd h_ac ν] with p hp
simp_rw [hp, ENNReal.toReal_mul]
by_cases h_zero1 : ((∂μ/∂ν) p.1).toReal = 0
· simp [h_zero1]
by_cases h_zero2 : ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal = 0
· simp [h_zero2]
simp [log_mul h_zero1 h_zero2] | true | {
"chosen_tokens": 76,
"rejected_tokens": 3,
"token_jaccard": 0.027027,
"token_length_ratio": 0.039474
} | circular_dependency | circular_dependency | a87df9a29cbf70ee58c5b300def1e0a51624bb89ab8e003b48da3dc1dc3397d6 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.InformationTheory.KullbackLeibler.Basic
public import Mathlib.Probability.Kernel.Composition.MeasureCompProd
public import Mathlib.Probability.Notation
import ... | by
exact rnDeriv_compProd_mul_log_eq_mul_add | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "da79b9c499ec0efbd5b73ce0827793aa31e143febc99a4f77847dd213dc393dd",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "InformationTheory/KullbackLeibler",
"family_id": "rnderiv_compprod_mul_log_eq_mul_add",
"file_id": "mathlib/Mathlib/InformationTheory/KullbackLeibler/ChainRule.lean",
"sample_id": "89245e196e8f1c3866de6d9410eff906de2a4f7ec9bd4db4b755d68b297fae0f"
} | train |
{
"chosen_proof_sha256": "c55c96a734207c6a61a412e0cab744c2c07b34209605ac6a4b05c1a6bd545ce9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a0803f253ced08eb971dedf8185abdf61b... | by
rw [tprod_apply, map_list_prod, List.map_ofFn]
simp_rw [Function.comp_def, toDirectSum_ι]
rw [DirectSum.list_prod_ofFn_of_eq_dProd]
apply DirectSum.of_eq_of_gradedMonoid_eq
rw [GradedMonoid.mk_list_dProd]
rw [TensorPower.list_prod_gradedMonoid_mk_single] | true | {
"chosen_tokens": 42,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.119048
} | wrong_namespace_qualification | wrong_namespace_qualification | a88ef5d0251a728382583723908a8f2e6d68c1c8c8fa8220647b931ca2b5057a | 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:
TensorAlgebra
Local context:
/-
Copyright (c) 2021 Eric Wi... | by
exact OProverMissing.missing_4d49f41524ca | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "2410e7311e2da84b2802f73a11dbd6ac88448e508bfc1ae122e9ae76303dd099",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/TensorAlgebra",
"family_id": "todirectsum_tensorpower_tprod",
"file_id": "mathlib/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean",
"sample_id": "4d49f41524ca78215a8298981fc8795e5f250086d26cc6a75dc11a9339f59a8c"
} | train |
{
"chosen_proof_sha256": "2d27cea8f3ca8caf10e9d8775b5b6885d300093e2302453f512d2957b166201a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
let : Field (A ⧸ p) := Ideal.Quotient.field p
rw [Ideal.isRadical_iff_quotient_reduced]
exact Algebra.FormallyUnramified.isReduced_of_field (A ⧸ p) (B ⧸ p.map (algebraMap A B)) | true | {
"chosen_tokens": 45,
"rejected_tokens": 8,
"token_jaccard": 0.103448,
"token_length_ratio": 0.177778
} | unsolved_goals | unsolved_goals | a8fe70e31fccd97cc78d9b9f8d9336c008c804bb4891fb281c5a333ef46c36cf | 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.Artinian.Ring
public import Mathlib.RingTheory.Unramified.Finite
public import Mathlib.Rin... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "bc74fb8be802f8eab74cd49d8f6d29d8f8313b18b5ea70ee4311c716d6e9129e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Unramified",
"family_id": "isradical_map_ismaximal",
"file_id": "mathlib/Mathlib/RingTheory/Unramified/Field.lean",
"sample_id": "4e9ba5114733e3b96be27c5b4d01de8367929d2bf280e4296d00c4e5e9fb55bb"
} | train |
{
"chosen_proof_sha256": "36e2743b3c488bb6e8cf2bf83c88a0f4aa1e0278b4cda351f399f8fa4b568278",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b387da58bcb61e7e635a1a81f3607072af... | by
refine OpenPartialHomeomorph.ext _ _ (fun x => rfl) (fun x => rfl) ?_
simp [e.open_source.interior_eq, ← inter_assoc] | true | {
"chosen_tokens": 32,
"rejected_tokens": 3,
"token_jaccard": 0.041667,
"token_length_ratio": 0.09375
} | unknown_identifier | unknown_identifier | a90eefd368bf7dde767f679839e2e224eb61ee544782833f3268eb64a10f33ee | 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.OpenPartialHomeomorph.Continuity
Namespace:
OpenPartialHomeomorph
Local context:
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Relea... | by
exact __oprover_missing_a3683d3fecf5 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "9c92b0b7255664f814ece45769a71af6937ed11bd12e371f8574781d60d32251",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/OpenPartialHomeomorph",
"family_id": "restr_source_inter",
"file_id": "mathlib/Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean",
"sample_id": "a3683d3fecf5a78a2e088915121e34b1d7028592a27520975260840e3084875a"
} | train |
{
"chosen_proof_sha256": "65d184d09b9f9abbbd2d10bfaea51c9a0bf698460f9d72b6fbec0c7ce1a8ad5d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
induction n with
| zero => rfl
| succ n ih =>
rw [replicate_succ]
conv => rhs; rw [replicate_succ]
rw [snoc_cons, ih] | false | {
"chosen_tokens": 31,
"rejected_tokens": 3,
"token_jaccard": 0.047619,
"token_length_ratio": 0.096774
} | by_exact_placeholder | placeholder | a95134a1fcf39e2d807349b5f66ca5156122aedc5ed47172b8840365e622ddab | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Vector.Basic
Namespace:
List.Vector
Local context:
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as describe... | 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/Vector",
"family_id": "replicate_succ_to_snoc",
"file_id": "mathlib/Mathlib/Data/Vector/Snoc.lean",
"sample_id": "2b26d3e25a280904d8577d75d632bf5a13763ff21493b59168968dcb1e0a5669"
} | train |
{
"chosen_proof_sha256": "88c077efecbeeb27504c4875f1630a5efd1eaa0f4e0d6e798f302270b541c3d4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "72da71e251950deea88fc047accda33cc2... | by rw [norm_apply, LinearMap.det]; split_ifs <;> trivial
variable {R} | false | {
"chosen_tokens": 19,
"rejected_tokens": 26,
"token_jaccard": 0.9,
"token_length_ratio": 1.368421
} | markdown_preamble_trailing | fence | a9a50f63d9d475fe67e33ac1dba2dbd60b957dc2eecbc0bcf26bd1f79574dfe9 | 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.Determinant
Namespace:
Algebra
Local context:
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as desc... | ```lean
by rw [norm_apply, LinearMap.det]; split_ifs <;> trivial
variable {R}
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Norm",
"family_id": "norm_eq_one_of_not_exists_basis",
"file_id": "mathlib/Mathlib/RingTheory/Norm/Defs.lean",
"sample_id": "5594b1e027625be77c21574d7888f7501782d701404f92b08cf130a4ddbabfb7"
} | train |
{
"chosen_proof_sha256": "6296f9ab27c166fefa518a334c1d06c276278c6742ae2b92e22ac06c24c52565",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c3d5b7f22e02615e5f628e9cf574732382... | by
conv_rhs => rw [← sum_single a, ← sum_single b]
simp_rw [mul_def, map_finsuppSum, liftNC_single, Finsupp.sum_mul, Finsupp.mul_sum]
refine Finset.sum_congr rfl fun y hy => Finset.sum_congr rfl fun x _hx => ?_
simp [mul_assoc, (h_comm hy).left_comm] | false | {
"chosen_tokens": 59,
"rejected_tokens": 64,
"token_jaccard": 0.9,
"token_length_ratio": 1.084746
} | markdown_preamble_trailing | trailing | a9bb9890fc5dab9ccfa25e726a57d03ab2a4a606082c1ffceff6378c0fffa565 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MonoidAlgebra.Defs
Namespace:
MonoidAlgebra
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 licen... | by
conv_rhs => rw [← sum_single a, ← sum_single b]
simp_rw [mul_def, map_finsuppSum, liftNC_single, Finsupp.sum_mul, Finsupp.mul_sum]
refine Finset.sum_congr rfl fun y hy => Finset.sum_congr rfl fun x _hx => ?_
simp [mul_assoc, (h_comm hy).left_comm]
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/MonoidAlgebra",
"family_id": "liftnc_mul",
"file_id": "mathlib/Mathlib/Algebra/MonoidAlgebra/Lift.lean",
"sample_id": "c88d0c7a08db8e0ee629366fac952f34c781e69b50a4902cc54b66d0c91187ae"
} | train |
{
"chosen_proof_sha256": "6f31eac0331d8119b977de849699fd5e0d1fb2f77898bc029e261ff94821e033",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "59450a4ffd61d7699b103e7b5e0a8e1339... | by simpa only [zero_mul, zero_add] using div_add_mod a 0 | true | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.25
} | unknown_identifier | unknown_identifier | a9d5f83fd593493db2947f918eb2ac5cb32a0c20121037786a30ed78ae1cde3b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.Order.RelClasses
Namespace:
EuclideanDomain
Local context:
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
R... | by
exact __oprover_missing_4793854c2e3a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "6193c093694f86b6eb2c5aa1536248f20666edd7902c1a654ace2dafabda826b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/EuclideanDomain",
"family_id": "mod_zero",
"file_id": "mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean",
"sample_id": "4793854c2e3af40b9d027f7b421519fea77582f2694be3b8462681fa81f0c2f6"
} | train |
{
"chosen_proof_sha256": "bdd86c88ef7416db2f30c34d0d939f5c26ce704f80d0007fe2f340c238437f0f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
refine IsPullback.of_forall_isPullback_app fun X => ?_
rw [Types.isPullback_iff]
refine ⟨congr(($(comp_χ_eq m)).app X), ?_, ?_⟩
· simpa using! (mono_iff_injective (m.app X)).mp (inferInstance)
· simp only [Functor.const_obj_obj, Functor.sieves_obj, χ_app, Opposite.op_unop,
TypeCat.hom_ofHom, TypeCat.... | false | {
"chosen_tokens": 123,
"rejected_tokens": 2,
"token_jaccard": 0.017241,
"token_length_ratio": 0.01626
} | sorry_admit_sorryAx_proof_hole | sorry | aa2527ee65976eebd73c63bb68dd0a3397e360ff95348059ffef0964d55e0f52 | 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... | 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/Topos",
"family_id": "presheaf",
"file_id": "mathlib/Mathlib/CategoryTheory/Topos/Sheaf.lean",
"sample_id": "e8c9dc5f379d2760fb24d04c1fd1b200b12c9f621f4607d9bb2924f7901f72b2"
} | train |
{
"chosen_proof_sha256": "8ea2eb314aa329962f79f410c370fc03bc8d9d4238b02a2baace82f999396bf2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f9ac29b8a38906cebe8f0ae27b17348c7d... | by simp [left_distrib, *] | true | {
"chosen_tokens": 7,
"rejected_tokens": 2,
"token_jaccard": 0.125,
"token_length_ratio": 0.285714
} | invalid_tactic | invalid_tactic | aa627b854ccae9aab3d94d37d054ab7ff50bde06c5a52a58fdb592a5a786876f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Field.Basic
public import Mathlib.Algebra.Order.Ring.Defs
public import Mathlib.Data.Tree.Basic
public import Mathlib.Tactic.NormNum.Core
public import... | by
__oprover_invalid_tactic_7b5b722d1bc1 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "68eabacc3e463870b47a1289e8507d28484f649756e74ba239131c2a3e597108",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/CancelDenoms",
"family_id": "add_subst",
"file_id": "mathlib/Mathlib/Tactic/CancelDenoms/Core.lean",
"sample_id": "7b5b722d1bc16ae3899d329eba58a7b2d1599243b0ea9acc8a97cda675c28cc8"
} | train |
{
"chosen_proof_sha256": "5fc5fcc6ed6270b255f9132aa985cf7070d173e497c862b80ffbe80e51562de8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5db3bcdbdea6a829d6f37e0e356b2ab39f... | by
simp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index]
exact forall_comm
-- `simp` normal form is `sized_iUnion`. | false | {
"chosen_tokens": 27,
"rejected_tokens": 34,
"token_jaccard": 0.95,
"token_length_ratio": 1.259259
} | markdown_preamble_trailing | fence | aa9ab3899a840f64b1d4246d99ddd70e0994c2514a99a39f70a20f39033fa0a2 | 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.Powerset
public import Mathlib.Order.Interval.Finset.Nat
public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
Namespace:
Set
Local cont... | ```lean
by
simp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index]
exact forall_comm
-- `simp` normal form is `sized_iUnion`.
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Finset",
"family_id": "sized_iunion",
"file_id": "mathlib/Mathlib/Data/Finset/Slice.lean",
"sample_id": "0f82b3064f5d92a16f748b7b934596ecec0fdc7677d78faf22eb29829f183996"
} | train |
{
"chosen_proof_sha256": "8cf4a63a26dc0e5b281fd2da3d3e4050e7d2b711edcf9188ce1a11ef8a2b0753",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "07ea5168f4a5f0cbe84ebdfb15ab18e849... | by
unfold mkFinCons
exact coe_mk (v := Fin.cons y (N.subtype ∘ b)) _ _
/-- 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.cons y b`. -/
noncomputable def mkFinCon... | true | {
"chosen_tokens": 262,
"rejected_tokens": 3,
"token_jaccard": 0.022472,
"token_length_ratio": 0.01145
} | circular_dependency | circular_dependency | ab59a70b514264ba51cc2251127873feceec15be7021f74105cd74dc15d24c4f | 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.Basis.Basic
public import Mathlib.LinearAlgebra.Pi
Namespace:
Module.Basis
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights rese... | by
exact coe_mkFinCons | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "4a7764f3ab31d965bf33dd6357092b657777d0754e02c92947f328cd62b225c8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Basis",
"family_id": "coe_mkfincons",
"file_id": "mathlib/Mathlib/LinearAlgebra/Basis/Fin.lean",
"sample_id": "4814cbf7f3508d036f5e5a6a7155650097bbec6785ba9dace923119e89bbbe59"
} | train |
{
"chosen_proof_sha256": "3e9d48f580176762fa7a5e0ebfd29db0eb21f956c84ea97eca970101095c5878",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
letI : MeasurableSpace (ℕ × Ω) := 𝓕.predictable
letI : MeasurableSpace Ω := 𝓕 n
let X m := (Function.curry (h𝓕.approx m) (n + 1))
refine ⟨(fun m ↦ SimpleFunc.mk (X m) ?_ ?_), (fun ω ↦ h𝓕.tendsto_approx ⟨(n + 1), ω⟩)⟩
· intro s
rw [(by aesop : X m ⁻¹' {s} = {ω | (n + 1, ω) ∈ h𝓕.approx m ⁻¹' {s}})]
... | false | {
"chosen_tokens": 154,
"rejected_tokens": 2,
"token_jaccard": 0.035088,
"token_length_ratio": 0.012987
} | by_exact_placeholder | by? | ab7fc8aeab3490a6ee50dd3a6194778463d48d84a5275842966070cc00bf2ff4 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Probability.Process.Adapted
Namespace:
MeasureTheory.IsStronglyPredictable
Local context:
/-
Copyright (c) 2025 Kexing Ying. All rights reserved.
Released un... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Probability/Process",
"family_id": "measurable_add_one",
"file_id": "mathlib/Mathlib/Probability/Process/Predictable.lean",
"sample_id": "57ea28a7b08d8f885b958267bc30048df940f453f6ddd0117c4636da80751a9c"
} | train |
{
"chosen_proof_sha256": "e68fac77a9fc516d7c10bdaed9dcefeaf4db7121c9ed0829fb3a283ec7ee3c03",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [range, List.range_succ, ← coe_add, Multiset.add_comm, range, coe_singleton, singleton_add] | false | {
"chosen_tokens": 22,
"rejected_tokens": 3,
"token_jaccard": 0.058824,
"token_length_ratio": 0.136364
} | by_exact_placeholder | placeholder | ac198000550733419f064fb751817a48cb38b4c8103814152f676aca57562dca | 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.Multiset.UnionInter
Namespace:
Multiset
Local context:
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 licen... | 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/Multiset",
"family_id": "range_succ",
"file_id": "mathlib/Mathlib/Data/Multiset/Range.lean",
"sample_id": "735bf5dce5b321e535055fa4c86fa49603a8028a22626fad8900aee15b3514d6"
} | train |
{
"chosen_proof_sha256": "42ee086aa0660c3672751c6ea4362a4dc2000a25c82c17c805dafacc7e49fe69",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cd37bf158342299f6da986fa2bd70e6d1e... | by
have eq₁ : _ = (π₁ i, j, k).base := PreservesPullback.iso_hom_fst (forget C) _ _
have eq₂ : _ = (π₂ i, j, k).base := PreservesPullback.iso_hom_snd (forget C) _ _
rw [← eq₁, ← eq₂, TopCat.coe_comp, Set.image_comp, TopCat.coe_comp, Set.preimage_comp,
Set.image_preimage_eq]
· simp only [forget_obj, forget_m... | true | {
"chosen_tokens": 105,
"rejected_tokens": 5,
"token_jaccard": 0.045455,
"token_length_ratio": 0.047619
} | wrong_namespace_qualification | wrong_namespace_qualification | ac479104f0653b9f118303bb920662f773988851ef38eb8260cedea96f63a646 | 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.Gluing
public import Mathlib.Geometry.RingedSpace.OpenImmersion
public import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits
Namespace:
... | by
exact OProverMissing.missing_141687aaa229 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "562437279b8862124dae3df3a30096b846c56d73ea89694590bac7368768b816",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Geometry/RingedSpace",
"family_id": "pullback_base",
"file_id": "mathlib/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean",
"sample_id": "141687aaa229bc38944346a4f56eb62c28baec7cf1b9f600772bfcea49a063a4"
} | train |
{
"chosen_proof_sha256": "95de6bb494dcefbe243626eb340736b29bf3b291e155dac01267e899f1bb69c2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [of_eq, Prefunctor.comp_assoc, Prefunctor.comp_assoc, Functor.toPrefunctor_comp]
dsimp [lift]
rw [Quotient.lift_spec, Paths.lift_spec, Quiver.Symmetrify.lift_spec] | false | {
"chosen_tokens": 37,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.081081
} | by_exact_placeholder | placeholder | ac55e5156f89a8dc337214916a9aaa262e95b5a44d813f906847a2139202dc29 | 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.Groupoid
public import Mathlib.CategoryTheory.PathCategory.Basic
Namespace:
Quiver.FreeGroupoid
Local context:
/-
Copyright (c) 2022 Rémi Bott... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Groupoid",
"family_id": "lift_spec",
"file_id": "mathlib/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean",
"sample_id": "c031c9432732d05cd237126cbcac6a473645115eab86e0eb5195fb77c4512d50"
} | train |
{
"chosen_proof_sha256": "2f0e35fce05624abb1b88e07b05ba59ec3da85bd198bf90cf2c93b8503cfca8a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
let g : X ⟶ (⨆ i, U i : Y.Opens) := IsOpenImmersion.lift (Scheme.Opens.ι _) f (by simpa using H)
rw [← IsOpenImmersion.lift_fac (⨆ i, U i).ι f (by simpa using H)]
apply MorphismProperty.RespectsRight.postcomp (Q := @IsOpenImmersion) _ inferInstance
rw [iff_of_iSup_eq_top (P := P) (U := fun i : ι ↦ (⨆ i, U i)... | false | {
"chosen_tokens": 294,
"rejected_tokens": 2,
"token_jaccard": 0.013158,
"token_length_ratio": 0.006803
} | sorry_admit_sorryAx_proof_hole | sorry | ac6f33e286ce6690c295765e6cf8d3c77022dd6a5df88c1ea7b3dea3524a33e7 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Limits
public import Mathlib.CategoryTheory.MorphismProperty.Local
public import Mathlib.Data.List.TFAE
Namespace:
AlgebraicGeometry.IsZaris... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Morphisms",
"family_id": "of_range_subset_isup",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Morphisms/Basic.lean",
"sample_id": "93a0ad9cdbe52633a2a6476e78bca93ac766df3668d5e36441bd4a7ce8340870"
} | train |
{
"chosen_proof_sha256": "af8967d58c125507dfdd8a870d8e2d61acb151c1a146b922c5673f800fa162b8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "89069437bed98817c329262f0d45f97272... | by
rw [mem_disjSum, or_iff_right]
· simp only [inr.injEq, exists_eq_right]
rintro ⟨a, _, ha⟩
exact inl_ne_inr ha | false | {
"chosen_tokens": 28,
"rejected_tokens": 35,
"token_jaccard": 0.916667,
"token_length_ratio": 1.25
} | markdown_preamble_trailing | fence | ac79195f2a25646b1e73b23d0f80d4270cb41a4cbf19d36e68f55e31d17751a8 | 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.Multiset
Namespace:
Multiset
Local context:
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as... | ```lean
by
rw [mem_disjSum, or_iff_right]
· simp only [inr.injEq, exists_eq_right]
rintro ⟨a, _, ha⟩
exact inl_ne_inr ha
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Multiset",
"family_id": "inr_mem_disjsum",
"file_id": "mathlib/Mathlib/Data/Multiset/Sum.lean",
"sample_id": "2f8dba92a8be5eab58edb05807e69424a9fcd6a33e1126c48358e2bed2791279"
} | train |
{
"chosen_proof_sha256": "55840963317e3a0d749bfa4ab4253cf6959feaf799aad1f633d62acc28d49b26",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by valid) : F.map' i i = 𝟙 _ := F.map_id _ | false | {
"chosen_tokens": 17,
"rejected_tokens": 2,
"token_jaccard": 0.071429,
"token_length_ratio": 0.117647
} | sorry_admit_sorryAx_proof_hole | admit | ac84000199581a56459e1f2d1dfd752af7c78c3d9cd0dafb411bf97dae304fd3 | 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.Nat.Defs
public import Mathlib.CategoryTheory.Category.Preorder
public import Mathlib.CategoryTheory.Comma.Arrow
public import Mathlib.Data.Finty... | 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/ComposableArrows",
"family_id": "map'_self",
"file_id": "mathlib/Mathlib/CategoryTheory/ComposableArrows/Basic.lean",
"sample_id": "a1e462f08764c6d42623fb769d4457124c492b34cbacfef092d7509dfdd31de7"
} | train |
{
"chosen_proof_sha256": "02be6bd02ac451c737251c72bcfefcd2bda23393af373ca35cd3c4b0bb93ffd2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a09da6d7ec6d1cfa34656994fd49c4f73c... | by
rw [coe_inv, inv_def, normSq_coe, inv_one, ofReal_one, mul_one] | true | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.2
} | unknown_identifier | unknown_identifier | aca33954fc274c6e79fc7d53eabeccec17ec0aca0f15cd5938a87616decde659 | 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.Trigonometric.Basic
public import Mathlib.Analysis.Normed.Field.UnitBall
Namespace:
Circle
Local context:
/-
Copyright (c) 2021 Hea... | by
exact __oprover_missing_4dda6d8eed8a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "209e5553c0e8f45f8f144b7c10b0380a9f59fb90fca9d1700761562c9f3a5e68",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Complex",
"family_id": "coe_inv_eq_conj",
"file_id": "mathlib/Mathlib/Analysis/Complex/Circle.lean",
"sample_id": "4dda6d8eed8aa9990b9ce99d74443691206268bc0aae2f322f2853e45e3b954a"
} | train |
{
"chosen_proof_sha256": "1fd411efd0adcfd9a0a62aeb48dc422a81d7841cde6fbd3cbad674180785c839",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "324ea35ce6179c7986048fb9f8e478d89d... | by
simp +instances only [r_iff_exists, Subtype.exists, exists_prop, OreLocalization.oreEqv,
smul_eq_mul, Submonoid.mk_smul]
constructor
· rintro ⟨u, hu, e⟩
exact ⟨_, mul_mem hu x.2.2, u * y.2, by rw [mul_assoc, mul_assoc, ← e], mul_right_comm _ _ _⟩
· rintro ⟨u, hu, v, e₁, e₂⟩
exact ⟨u, hu, by rw [←... | false | {
"chosen_tokens": 112,
"rejected_tokens": 117,
"token_jaccard": 0.913043,
"token_length_ratio": 1.044643
} | markdown_preamble_trailing | trailing | aca4e195617fff08a2cc2c2d5da5f511add2ecf2ca787d6964f105706d4ee584 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
public import Mathlib.Algebra.Group.Submonoid.Operations
public import Mathlib.Algebra.Regular.Basic
public import Math... | by
simp +instances only [r_iff_exists, Subtype.exists, exists_prop, OreLocalization.oreEqv,
smul_eq_mul, Submonoid.mk_smul]
constructor
· rintro ⟨u, hu, e⟩
exact ⟨_, mul_mem hu x.2.2, u * y.2, by rw [mul_assoc, mul_assoc, ← e], mul_right_comm _ _ _⟩
· rintro ⟨u, hu, v, e₁, e₂⟩
exact ⟨u, hu, by rw [←... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "GroupTheory/MonoidLocalization",
"family_id": "r_iff_oreeqv_r",
"file_id": "mathlib/Mathlib/GroupTheory/MonoidLocalization/Basic.lean",
"sample_id": "bdf9d42bef05adc3f26afabef77136e641ea973a1c33f1f631c7b584f6c81787"
} | train |
{
"chosen_proof_sha256": "22008102f7d34b93c9ef351ad767f156bb7824bb62e00233f134ea27d1c44cf6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2aad5e99e9d08105118c7fe3a6e9502304... | by
rw [eq_geometrically, GeometricallyReduced.eq_geometrically,
GeometricallyIrreducible.eq_geometrically, ← geometrically_inf]
eta_expand
simp [isIntegral_iff_irreducibleSpace_and_isReduced, and_comm] | true | {
"chosen_tokens": 23,
"rejected_tokens": 2,
"token_jaccard": 0.0625,
"token_length_ratio": 0.086957
} | invalid_tactic | invalid_tactic | aca994ad1a8871b19352d41d1bb2ae6b4010df66d27f04fed4ce0fb24d39136b | 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.Geometrically.Reduced
public import Mathlib.AlgebraicGeometry.Geometrically.Irreducible
Namespace:
AlgebraicGeometry
Local context:
/-
Copy... | by
__oprover_invalid_tactic_f1be1ab398e8 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "511f381c93065221d7065f1c7b1a341ca5020a824aff117afff08123d34556f4",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Geometrically",
"family_id": "geometricallyintegral",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Geometrically/Integral.lean",
"sample_id": "f1be1ab398e8e6f5f51fea76b3b1c755d097ca53fd2fb60f4b3eb6a00e32ab92"
} | train |
{
"chosen_proof_sha256": "69cb17d2f43d6924d494a8be87b9a52df302f4a4d1c3a416993a289830e67b47",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
by_cases hR : R = 0
· simp_all
have H : ContinuousOn (circleAverage f c) (Set.Ioc 0 |R|) := by
refine (h₁f.2.mono ?_).circleAverage (fun z hz ↦ hz.1.le)
intro x hx
rw [closure_ball _ (by aesop), mem_closedBall_iff_norm]
exact hx.2
rw [← circleAverage_abs_radius]
apply H.eq_of_eqOn_Ioo (by a... | false | {
"chosen_tokens": 114,
"rejected_tokens": 2,
"token_jaccard": 0.035714,
"token_length_ratio": 0.017544
} | by_exact_placeholder | by? | acc4f3b35322e7c127477ee14383aac88fba3c495c8c1fccde10ba670ca99791 | 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.Complex.Harmonic.Analytic
public import Mathlib.Analysis.Complex.MeanValue
public import Mathlib.Analysis.InnerProductSpace.Harmonic.HarmonicContOnCl
... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/Complex",
"family_id": "harmoniccontoncl",
"file_id": "mathlib/Mathlib/Analysis/Complex/Harmonic/MeanValue.lean",
"sample_id": "2c9653db60a3ddae63cedc1cc9e2ba6e8d6ce8800e7284483f64df67132c5de6"
} | train |
{
"chosen_proof_sha256": "73ba32450f7eaa32ac31b9569cb16af401e9b0fc7b4daddcdd761838e6b70efb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2299db0e612a8842de7ca07c9ac6d10446... | by
letI : TopologicalSpace Y := ⊥; haveI := discreteTopology_bot Y
exact (isCompact_range hf.continuous).finite_of_discrete | true | {
"chosen_tokens": 21,
"rejected_tokens": 5,
"token_jaccard": 0.15,
"token_length_ratio": 0.238095
} | wrong_namespace_qualification | wrong_namespace_qualification | ace213d4ac65d212b0bda6500924bda111f27ebb2415ba025192577707662db2 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Notation.Indicator
public import Mathlib.Topology.Connected.LocallyConnected
public import Mathlib.Topology.Sets.Closeds
Namespace:
IsLocallyConstant
... | by
exact OProverMissing.missing_8138a062ed7a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "9513c60c0e0eec4da6f7664da9fb896178298d8a8b4a18c57fa777604db848cd",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/LocallyConstant",
"family_id": "range_finite",
"file_id": "mathlib/Mathlib/Topology/LocallyConstant/Basic.lean",
"sample_id": "8138a062ed7aa5e0b5f610f4aeff3a70100ca75f6fdd0cf8c052c2cf58487e47"
} | train |
{
"chosen_proof_sha256": "d409d608f76c2016efc5fdd42930cab75c91e811c1cdb05b5a33c8b551b1384f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
intro i
have h_eq_iUnion : {ω | τ ω ≤ i} = ⋃ k ≤ i, {ω | τ ω = k} := by
ext ω
simp only [Set.mem_setOf_eq, Set.mem_iUnion, exists_prop]
cases τ ω with
| top => simp
| coe a => norm_cast; simp
rw [h_eq_iUnion]
refine MeasurableSet.biUnion (Set.to_countable _) fun k hk => ?_
exact f.mono ... | false | {
"chosen_tokens": 91,
"rejected_tokens": 3,
"token_jaccard": 0.06,
"token_length_ratio": 0.032967
} | by_exact_placeholder | exact? | ad1c17c702b77db041118e3cfca2b85bcf6d16e78c9915a76ea8c91bee70941c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Probability.Process.Adapted
public import Mathlib.MeasureTheory.Constructions.BorelSpace.WithTop
public import Mathlib.Data.ENat.Lattice
Namespace:
MeasureThe... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Probability/Process",
"family_id": "isstoppingtime_of_measurableset_eq",
"file_id": "mathlib/Mathlib/Probability/Process/Stopping.lean",
"sample_id": "63d4c6f7bdfaad78a31356f5390bcb2b6c5c4366f0f8ad12785fcd4685b1c7e7"
} | train |
{
"chosen_proof_sha256": "5cfc8f87ea330fcf416830673402289862153b1294b56a5e4d961dfa30463860",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ae5b4930d39b46c6599bc819ee08d84d74... | by
refine ⟨fun h => .of_comp (f := ({·} : β → Compacts β)) ?_, map_injective hf⟩
simp_rw [Function.comp_def, ← map_singleton hf]
exact h.comp singleton_injective | false | {
"chosen_tokens": 43,
"rejected_tokens": 48,
"token_jaccard": 0.897436,
"token_length_ratio": 1.116279
} | markdown_preamble_trailing | preamble | adb86cc875c6377ada0cf61e174f3ca4c0b8f1e8c91bdac97aab8eb6b33796e2 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.Sets.Closeds
public import Mathlib.Topology.QuasiSeparated
Namespace:
TopologicalSpace.Compacts
Local context:
/-
Copyright (c) 2020 Floris van Door... | Here is the proof:
by
refine ⟨fun h => .of_comp (f := ({·} : β → Compacts β)) ?_, map_injective hf⟩
simp_rw [Function.comp_def, ← map_singleton hf]
exact h.comp singleton_injective | {
"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/Sets",
"family_id": "map_injective_iff",
"file_id": "mathlib/Mathlib/Topology/Sets/Compacts.lean",
"sample_id": "a931f0b3f033fc32c017444ea0eb1a5592318ecdbcdc093591ad2758f58c4528"
} | train |
{
"chosen_proof_sha256": "d110cfbf729fa1ead7d2f77345ddd3124e402db97adfe146f40e24d032256864",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ebbba5759c869c22b0a2b53d7b28f55271... | by
have := RingHom.toMorphismProperty_respectsIso_iff.mp RingHom.isIntegral_respectsIso
rw [HasAffineProperty.iff_of_isAffine (P := @IsIntegralHom), and_iff_right]
exacts [MorphismProperty.arrow_mk_iso_iff (RingHom.toMorphismProperty RingHom.IsIntegral)
(arrowIsoΓSpecOfIsAffine φ).symm, inferInstance] | true | {
"chosen_tokens": 49,
"rejected_tokens": 3,
"token_jaccard": 0.030303,
"token_length_ratio": 0.061224
} | unknown_identifier | unknown_identifier | ae43b96cf08f0c8939f7150d82c133ffbe1d64219c741cad04c9ffe1f8f2fd23 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.Separated
public import Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed
public import Mathlib.RingTheory.RingHom.Integral
Na... | by
exact __oprover_missing_099a68aecd4c | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f85e18ecc80b27190dbeafa6808df25a09153b6e2171842ef242f03abc507a40",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Morphisms",
"family_id": "specmap_iff",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Morphisms/Integral.lean",
"sample_id": "099a68aecd4c932bd014347b7efcc8259e886e53c407723e803e298e292ed9b6"
} | train |
{
"chosen_proof_sha256": "8b4eba4a7d76ab6115b8794619b18b2fd26643e7d931d0fcde4744d29b34df3e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3eb24946d7afb6ed4e9945af45b2ccb512... | 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 | true | {
"chosen_tokens": 32,
"rejected_tokens": 3,
"token_jaccard": 0.034483,
"token_length_ratio": 0.09375
} | circular_dependency | circular_dependency | aed6d954183d7f2b7be63285f408ee1fb6d9c57f1aab635927b150d35c5b39f6 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Basic
public import Mathlib.Data.Fintype.Powerset
public import Mathlib.Data.Set.Finite.Basic
Namespace:
HallMarriageTheorem
Local context:
/-
C... | by
exact hall_cond_of_restrict | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "282836e62cc7826b4d35289f8c2a38c8213b53d0f6dd968e4198fe188b33193e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "b54a12dda5879cff4ee4a1d4e164d0430717d0b1ed9dc7072b52b58f4f7965cc",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
refine eq_bot_iff.2 (le_trans jacobson_bot_polynomial_le_sInf_map_maximal ?_)
refine fun f hf => (Submodule.mem_bot R[X]).2 <| Polynomial.ext fun n =>
Trans.trans (?_ : coeff f n = 0) (coeff_zero n).symm
suffices f.coeff n ∈ Ideal.jacobson ⊥ by rwa [h, Submodule.mem_bot] at this
exact mem_sInf.2 fun j hj... | false | {
"chosen_tokens": 108,
"rejected_tokens": 2,
"token_jaccard": 0.037736,
"token_length_ratio": 0.018519
} | by_exact_placeholder | by? | aef03cd48d0139ad7f0a3eeaddbeb26c9213a004649c509b56a1681e4541ad96 | 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.Jacobson.Ideal
public import Mathlib.RingTheory.Polynomial.Quotient
Namespace:
Ideal
Local context:
/-
Copyright (c) 2020 Devon Tuma. All rights r... | 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/Jacobson",
"family_id": "jacobson_bot_polynomial_of_jacobson_bot",
"file_id": "mathlib/Mathlib/RingTheory/Jacobson/Polynomial.lean",
"sample_id": "cc46ae0cd6fbf8b5c482692a262adbee52fa600011e9053140abc53478aa5ddd"
} | train |
{
"chosen_proof_sha256": "9a1a0b80645445024d8bb550e3e8810e677e51b85aec8657ff2f9f66fe77107a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [isProperMap_iff_isClosedMap_and_compact_fibers]
refine ⟨f.continuous, f.isClosedMap, fun y ↦ ?_⟩
have := compactSpace_of_universallyClosed (pullback.snd f (Y.fromSpecResidueField y))
rw [← Scheme.range_fromSpecResidueField, ← Scheme.Pullback.range_fst]
exact isCompact_range (Scheme.Hom.continuous _) | true | {
"chosen_tokens": 60,
"rejected_tokens": 8,
"token_jaccard": 0.076923,
"token_length_ratio": 0.133333
} | unsolved_goals | unsolved_goals | af03e155d216c8fb668e10c22cdfe0eaf90435cb353ddbfbc5398e15ecf92e42 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
public import Mathlib.AlgebraicGeometry.PullbackCarrier
public import Mathlib.Topology.LocalAtTarget
Namespace:
Al... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "ec2f02bf97c18a5bded120090c52d3efb2036960ae95fe7f5b72ffed08280a37",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Morphisms",
"family_id": "scheme",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean",
"sample_id": "33d6750ddf63884a3db77c1ac0c0a14e4b5da0e25cd340d2ef99ffb5417edbe1"
} | train |
{
"chosen_proof_sha256": "2139989fd770c5936c4b547ad3ae043d8606b002f3ccc3b366913c36130fefc8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
filter_upwards [rnDeriv_compProd h_ac ν] with p hp
simp_rw [hp, ENNReal.toReal_mul]
by_cases h_zero1 : ((∂μ/∂ν) p.1).toReal = 0
· simp [h_zero1]
by_cases h_zero2 : ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal = 0
· simp [h_zero2]
simp [log_mul h_zero1 h_zero2] | false | {
"chosen_tokens": 76,
"rejected_tokens": 3,
"token_jaccard": 0.027027,
"token_length_ratio": 0.039474
} | by_exact_placeholder | exact? | af113fa099ea8205eb6ac77f9dd39c64b916806def9ab73576cc9cd0439e095b | 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
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": "InformationTheory/KullbackLeibler",
"family_id": "rnderiv_compprod_mul_log_eq_mul_add",
"file_id": "mathlib/Mathlib/InformationTheory/KullbackLeibler/ChainRule.lean",
"sample_id": "89245e196e8f1c3866de6d9410eff906de2a4f7ec9bd4db4b755d68b297fae0f"
} | train |
{
"chosen_proof_sha256": "a3707e1c83ef260ac94852111d4f8128b1e6f031babe4d0a83c426fc4ebac9e6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
dsimp [Part.fix]
rw [assert_neg h'] | false | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.25
} | by_exact_placeholder | exact? | af2390a997ed2e52c9f5ae2d97cad750fc86e8c5d2fa402d8e0d9a483df496a3 | 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.Part
public import Mathlib.Data.Nat.Find
public import Mathlib.Data.Nat.Upto
public import Mathlib.Data.Stream.Defs
Namespace:
Part
Local context:
/-
Co... | 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": "Control",
"family_id": "fix_def",
"file_id": "mathlib/Mathlib/Control/Fix.lean",
"sample_id": "9944f4492e2edc232793a5f7e57320d19cd2217d990686c0e9b0e255481105bf"
} | train |
{
"chosen_proof_sha256": "760a46c45735c2286ac991019ea42b32b81c6e8ac4991d80d094da0aa71c32a8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
dsimp [destruct]
cases f0 : s.1 0 <;> intro h
· contradiction
· apply Subtype.ext
funext n
induction n with
| zero => injection h with h'; rwa [h'] at f0
| succ n IH => exact s.2 IH | false | {
"chosen_tokens": 53,
"rejected_tokens": 2,
"token_jaccard": 0.026316,
"token_length_ratio": 0.037736
} | sorry_admit_sorryAx_proof_hole | admit | af3d53f9f2332874af61370edf7d012bb72e84e32ebed15fd47719bcfc5b1d67 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.Find
public import Mathlib.Data.Stream.Init
public import Mathlib.Logic.Relator
public import Mathlib.Tactic.Common
public import Batteries.Tactic.Lin... | by
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/Seq",
"family_id": "destruct_eq_pure",
"file_id": "mathlib/Mathlib/Data/Seq/Computation.lean",
"sample_id": "5ec0b0e55183ff152ce79ab5d2c02cf74a8634036c497bc83a0db9d263f074b7"
} | train |
{
"chosen_proof_sha256": "1a29d536dd839cb2ac804ef83ab5abb79ba3d4ec297a0fa4ca91c3477c2aaaa0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "45ea66a4976169e006a11c1610afe29ec4... | 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... | false | {
"chosen_tokens": 165,
"rejected_tokens": 170,
"token_jaccard": 0.938462,
"token_length_ratio": 1.030303
} | markdown_preamble_trailing | trailing | af4a092b8d38ca7b20fe1444e35c85799e62230915f1f587f15eba9863de4975 | 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... | 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... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | 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": "f0fe75d1087a41898b65fcdcdb5116eac360500295f3e4e2a2c616668dd8129d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7fe45320e9d9e2a8740d56b6ea8a40e7b3... | by
nth_rw 2 [← Quiver.Path.reverse_reverse p]
apply congr_comp_reverse | false | {
"chosen_tokens": 14,
"rejected_tokens": 19,
"token_jaccard": 0.722222,
"token_length_ratio": 1.357143
} | markdown_preamble_trailing | preamble | af82112c37087d1a8fb10a343478e315ae9f1146a8ffe6aa3dc998f132c4d15b | 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.Groupoid
public import Mathlib.CategoryTheory.PathCategory.Basic
Namespace:
Quiver.FreeGroupoid
Local context:
/-
Copyright (c) 2022 Rémi Bott... | Here is the proof:
by
nth_rw 2 [← Quiver.Path.reverse_reverse p]
apply congr_comp_reverse | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Groupoid",
"family_id": "congr_reverse_comp",
"file_id": "mathlib/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean",
"sample_id": "5ae786a0f2c5c57019f4c1e4edc7524aa65574003fae2570dc839740f449cb98"
} | train |
{
"chosen_proof_sha256": "5c5e46221d8b229ee79bbd41b4c1bdba3112695c6156a1e23176053e1c54427e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "23be217614af4f9bf2fdcf78fa95677947... | by subst h; rfl | true | {
"chosen_tokens": 5,
"rejected_tokens": 3,
"token_jaccard": 0.142857,
"token_length_ratio": 0.6
} | circular_dependency | circular_dependency | af915a31132bd8857f6299a643f4c99837855fb061e101d6ba87c1815a1a9a94 | 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.EqToHom
public import Mathlib.CategoryTheory.Functor.Const
public import Mathlib.CategoryTheory.Opposites
public import Mathlib.Data.Prod.Basic
... | by
exact eqToHom_snd | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "7d357bb85d9cbc2292a0c338bcaeb556cef462b3a65e91a472fb92f95d8d62df",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Products",
"family_id": "eqtohom_snd",
"file_id": "mathlib/Mathlib/CategoryTheory/Products/Basic.lean",
"sample_id": "cb00d28e03305d658f5fd091ef6af272ded07e63d8901ab1f776e0190fa772a7"
} | train |
{
"chosen_proof_sha256": "1656a7f84780a26168e97da85997b830e60c4a9c21bd50667f35238ce2172d26",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
let U := f ⁻¹' {f y}
suffices x ∉ Uᶜ from Classical.not_not.1 this
intro hxV
specialize hs U Uᶜ (hf {f y}) (hf {f y}ᶜ) _ ⟨y, ⟨hy, rfl⟩⟩ ⟨x, ⟨hx, hxV⟩⟩
· simp only [union_compl_self, subset_univ]
· simp only [inter_empty, Set.not_nonempty_empty, inter_compl_self] at hs | false | {
"chosen_tokens": 87,
"rejected_tokens": 2,
"token_jaccard": 0.020833,
"token_length_ratio": 0.022989
} | sorry_admit_sorryAx_proof_hole | admit | afbc40eeb3212726ba09fd48d70f1a1557e8a415303ff43d2a4348eed39482b4 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Notation.Indicator
public import Mathlib.Topology.Connected.LocallyConnected
public import Mathlib.Topology.Sets.Closeds
Namespace:
IsLocallyConstant
... | by
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": "Topology/LocallyConstant",
"family_id": "apply_eq_of_ispreconnected",
"file_id": "mathlib/Mathlib/Topology/LocallyConstant/Basic.lean",
"sample_id": "0e4b1666a2df18b8f82460264078354bcb621eb24caecc149f02733eb9e19be1"
} | train |
{
"chosen_proof_sha256": "5af7d1d4d55c3a73fbff3849f778662494382e96b2f978d7cb9fc32bd567bdc1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
intro f
obtain ⟨⟨f⟩, rfl⟩ := (bijective_leftHomotopyClassToHom L X Y).2 f
exact ⟨f, rfl⟩ | false | {
"chosen_tokens": 27,
"rejected_tokens": 2,
"token_jaccard": 0.052632,
"token_length_ratio": 0.074074
} | sorry_admit_sorryAx_proof_hole | admit | afcb29c71a84c8b9c87ac9be774228de3f63418389ead84e0787370e04cf4f29 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy
Namespace:
HomotopicalAlgebra
Local context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Re... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/ModelCategory",
"family_id": "map_surjective_of_islocalization",
"file_id": "mathlib/Mathlib/AlgebraicTopology/ModelCategory/FundamentalLemma.lean",
"sample_id": "17c8f688ffbfe3f08997e2e441397e109ef966a6e5472eeef0ec671a2f11c5b3"
} | train |
{
"chosen_proof_sha256": "5137effd66997cbfcff7aea301f3e52b8dc7efd3fad26dd29be5d337e1b39c36",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "70992bd7daf4c7526afc1418802c19b19b... | by
rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp | true | {
"chosen_tokens": 16,
"rejected_tokens": 5,
"token_jaccard": 0.111111,
"token_length_ratio": 0.3125
} | wrong_namespace_qualification | wrong_namespace_qualification | b000f4d05fa2d070326b4a65a06a6bf5289161500c96fcbda8ca83540b2a9324 | 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.Basic
public import Mathlib.Algebra.Order.AbsoluteValue.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Anne Baanen. All rights r... | by
exact OProverMissing.missing_0830875cbcc8 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "0e0464254a41f8d70a8a2c2073ed364198414fe763c551396cd8b4f9523dc7d3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Int",
"family_id": "absolutevalue",
"file_id": "mathlib/Mathlib/Data/Int/AbsoluteValue.lean",
"sample_id": "0830875cbcc8e3b2fa3c485a8dfb76651d4e973c898e771f3c9e1024b82d5806"
} | train |
{
"chosen_proof_sha256": "f11fe3945a9851f302dfa7c95843f91ea71198e392581ee0ee3ba699adf230d7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
simp only [toPrincipalIdeal]; exact Units.ext_iff | false | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.166667,
"token_length_ratio": 0.272727
} | by_exact_placeholder | exact? | b00d899fb071b143c8beba00a5f12b533ece721404e20704e853074e6f9eb2e2 | 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.Ideal.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 lice... | 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/ClassGroup",
"family_id": "toprincipalideal_eq_iff",
"file_id": "mathlib/Mathlib/RingTheory/ClassGroup/Basic.lean",
"sample_id": "9757f7a78bda71d8e7517609a4d26326ae1122e116d5aaf449c4d98b5651cdcf"
} | train |
{
"chosen_proof_sha256": "36b9bfe8ecf433634e45203e8b895f570c57f0adf9b70b7ca8ace15a79e21d3e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "be90198cd5d1c08c4c9d1cf652ee121e57... | by
ext
simp [single, ← Finsupp.smul_single] | false | {
"chosen_tokens": 11,
"rejected_tokens": 15,
"token_jaccard": 0.785714,
"token_length_ratio": 1.363636
} | missing_wrong_import | import | b00ef8a1028ff3b4d037b60a1ad4f00701c8039c95a90f0fa6fe4e5573615f11 | 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.GroupWithZero.Action.TransferInstance
public import Mathlib.Algebra.Module.Defs
public import Mathlib.Data.Finsupp.Basic
public import Mathlib.Data.Fin... | import Mathlib.WrongImport
by
ext
simp [single, ← Finsupp.smul_single] | {
"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/MonoidAlgebra",
"family_id": "smul_single",
"file_id": "mathlib/Mathlib/Algebra/MonoidAlgebra/Defs.lean",
"sample_id": "bcfecfd0c0cb2071cab40553cc6bdd48d75c67302392e5f99abe6ae29aa62d18"
} | train |
{
"chosen_proof_sha256": "4a140838d46443a223247967ef703140b4be6ea2808c8d4f93f75968349bd809",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
have := h.out
simp [this]
/- evalCast -/ | false | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.2
} | by_exact_placeholder | exact? | b0108481ac663956f3a661c3f8f9327eac7375b084f845f516b5b57355b39b63 | 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? | {
"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": "isnat_zero_eq",
"file_id": "mathlib/Mathlib/Tactic/Algebra/Lemmas.lean",
"sample_id": "b579425a25a44dbf03d7aec5fcfc9f633b6db243ba8c3f42fbd8c2796e7b54f2"
} | train |
{
"chosen_proof_sha256": "73aa3a7c73d07e3e5e0d2c9a0dc0fab61b3371c9b2e729b99cd088422d654f85",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rcases Finite.exists_equiv_fin β with ⟨n, ⟨e⟩⟩
classical exact .of_equiv (Set.range (e ∘ f)) (Equiv.ofInjective _ (e.injective.comp H)).symm
-- see Note [lower instance priority] | false | {
"chosen_tokens": 53,
"rejected_tokens": 2,
"token_jaccard": 0.027027,
"token_length_ratio": 0.037736
} | by_exact_placeholder | by? | b021b760ce8bf23cb553f9595796c2bc8b7145bc493171c3dcb1f2fb40b785c4 | 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.Card
public import Mathlib.Data.List.NodupEquivFin
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
R... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Fintype",
"family_id": "finite",
"file_id": "mathlib/Mathlib/Data/Fintype/EquivFin.lean",
"sample_id": "ba99dd8f6392afa104bfc6ac9136b281cb75f223a9c829f9749723664ecb6a1c"
} | train |
{
"chosen_proof_sha256": "cd750d60c8401fd18ff81d5b091050a985f85f02e4e58110a29f7aecd6f9f0ac",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp [card_eq_card_quotient_mul_card_subgroup s, @dvd_mul_right ℕ]
variable {H : Type*} [Group H] | true | {
"chosen_tokens": 21,
"rejected_tokens": 8,
"token_jaccard": 0.086957,
"token_length_ratio": 0.380952
} | unsolved_goals | unsolved_goals | b061d0f7784ceb23a4dd3a7f223126f171f34c40ace73f6afdd43410473d0006 | 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.Coset.Basic
public import Mathlib.SetTheory.Cardinal.Finite
Namespace:
Subgroup
Local context:
/-
Copyright (c) 2018 Mitchell Rowett. All rights ... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "507fabdaaf3c0886cf347528bd5cb83a491b6979b19e352c734731a872a2f8f9",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "GroupTheory/Coset",
"family_id": "card_quotient_dvd_card",
"file_id": "mathlib/Mathlib/GroupTheory/Coset/Card.lean",
"sample_id": "f168373cfd1f31b36e20d360cfbb18b1a6084a7395ef31c4c3eb93947132babd"
} | train |
{
"chosen_proof_sha256": "c1ea19ea49c645e697b6d5466464113860db944a733b14a16a41636e7325b4ec",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
obtain ⟨x, rfl⟩ := x.mk_surjective
obtain ⟨y, rfl⟩ := y.mk_surjective
obtain rfl : x = y := h
rfl | true | {
"chosen_tokens": 30,
"rejected_tokens": 8,
"token_jaccard": 0.166667,
"token_length_ratio": 0.266667
} | unsolved_goals | unsolved_goals | b071e4f739699b10dc7463c73093a908e8f23fb1beb679191c8ba6b563cff1b7 | 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.SimplicialObject.Basic
public import Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal
public import Mathlib.AlgebraicTopology.SimplicialSet... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "23efcc095954488a8537405ce81963fcfbb3152d5a7ed3e257f2eff7539a2f84",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/SimplicialSet",
"family_id": "ext",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplicialSet/HomotopyCat.lean",
"sample_id": "3065ddea640d150a719f7ca546348a960da065f54e8ab9a9d7476b502ae8f9af"
} | train |
{
"chosen_proof_sha256": "d110cfbf729fa1ead7d2f77345ddd3124e402db97adfe146f40e24d032256864",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
have := RingHom.toMorphismProperty_respectsIso_iff.mp RingHom.isIntegral_respectsIso
rw [HasAffineProperty.iff_of_isAffine (P := @IsIntegralHom), and_iff_right]
exacts [MorphismProperty.arrow_mk_iso_iff (RingHom.toMorphismProperty RingHom.IsIntegral)
(arrowIsoΓSpecOfIsAffine φ).symm, inferInstance] | false | {
"chosen_tokens": 49,
"rejected_tokens": 2,
"token_jaccard": 0.03125,
"token_length_ratio": 0.040816
} | sorry_admit_sorryAx_proof_hole | sorry | b0ffaeadda617610b297d58de5e49a2ea9cee929a1d0e410bf34dfbe8c370857 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.Separated
public import Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed
public import Mathlib.RingTheory.RingHom.Integral
Na... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Morphisms",
"family_id": "specmap_iff",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Morphisms/Integral.lean",
"sample_id": "099a68aecd4c932bd014347b7efcc8259e886e53c407723e803e298e292ed9b6"
} | train |
{
"chosen_proof_sha256": "0f3c5c923c48eece1af3a7c60f1fa94f43ad2e6d58c10a4a8ef6046b414d8933",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sSup_empty, Real.sInf_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csSup_singleton 0
by_cases h : BddBelow s
· exact ((OrderIso.smulRightDual ℝ ha').map_csInf' hs h).symm
· rw [Real.sS... | false | {
"chosen_tokens": 89,
"rejected_tokens": 3,
"token_jaccard": 0.045455,
"token_length_ratio": 0.033708
} | by_exact_placeholder | exact? | b12757db08e9c7fb991df5ce809ac076468d3b8e98f206d0acbb21bf3fa3a3a9 | 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.GroupWithZero.Action.Pointwise.Set
public import Mathlib.Algebra.Order.Archimedean.Real.Basic
public import Mathlib.Algebra.Order.Module.Pointwise
publ... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Real",
"family_id": "real",
"file_id": "mathlib/Mathlib/Data/Real/Pointwise.lean",
"sample_id": "d0df3140da7fa364792a7ce4f3fa0fb068f28c3068cdfb76bb78b78e0d7a5c60"
} | train |
{
"chosen_proof_sha256": "9c89a7a96cac539d3af0940d447c59a7152af477d925ba54bdc0d7f1377d3647",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7bc5d50d79474e2eb00b8d1589e6b3098d... | by
ext; simp [mem_coe, mem_extended_iff] | true | {
"chosen_tokens": 9,
"rejected_tokens": 2,
"token_jaccard": 0.1,
"token_length_ratio": 0.222222
} | invalid_tactic | invalid_tactic | b1c31560e411c4c18d0b7e0ad40f4b33df7fdb657a43b98ab2be4d26ea30ba9f | 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.Lemmas
Namespace:
FractionalIdeal
Local context:
/-
Copyright (c) 2024 James Sundstrom. All rights reserved.
Released under A... | by
__oprover_invalid_tactic_70071016d6f9 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "94a470b8317c21461ef0556c5f357f4049d9a43f6115a46158f4519a68161ff3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/FractionalIdeal",
"family_id": "coe_extended_eq_span",
"file_id": "mathlib/Mathlib/RingTheory/FractionalIdeal/Extended.lean",
"sample_id": "70071016d6f956ee807fdebd39b7e38611eccc815e374ddf4ad147924380a172"
} | train |
{
"chosen_proof_sha256": "2e876053dd524313a985f03953e4b880c7de87fe925bf70214a79832334fbbdd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2478a8e6254a32b82717f4f3721b328383... | by
obtain ⟨F, hF⟩ := hc.exists_eq_range (hn.image _)
simp only [Set.ext_iff, Quotient.forall, mem_image, mem_range] at hF
simp_rw [Quotient.eq_mk_iff_out] at hF
have hF' : ∀ n : ℕ, (F n).out ∈ K := by
intro n
obtain ⟨P, hP1, hP2⟩ := (hF (F n).out).2 ⟨n, Setoid.refl _⟩
-- Porting note: fix hP2 becaus... | false | {
"chosen_tokens": 480,
"rejected_tokens": 487,
"token_jaccard": 0.99187,
"token_length_ratio": 1.014583
} | markdown_preamble_trailing | fence | b2167ae7201e8d27f946fc899bcbe77c5ad5fe1f5fb62477c3012366789c7791 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.ModelTheory.FinitelyGenerated
public import Mathlib.ModelTheory.PartialEquiv
public import Mathlib.ModelTheory.Bundled
public import Mathlib.Algebra.Order.Arch... | ```lean
by
obtain ⟨F, hF⟩ := hc.exists_eq_range (hn.image _)
simp only [Set.ext_iff, Quotient.forall, mem_image, mem_range] at hF
simp_rw [Quotient.eq_mk_iff_out] at hF
have hF' : ∀ n : ℕ, (F n).out ∈ K := by
intro n
obtain ⟨P, hP1, hP2⟩ := (hF (F n).out).2 ⟨n, Setoid.refl _⟩
-- Porting note: fix hP... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "ModelTheory",
"family_id": "exists_cg_is_age_of",
"file_id": "mathlib/Mathlib/ModelTheory/Fraisse.lean",
"sample_id": "3b9738526fb66d231f0474bd9878a45063a3e22be465efe04b4fbbe9d62f823a"
} | train |
{
"chosen_proof_sha256": "b1c7cebb6805403539cf95bc62a5c22e029b7da315a1e90239408f66439637c6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d7f033ea05a77516a16a1eb1a4d807d052... | by
rw [superpolynomialDecay_iff_abs_tendsto_zero] at hf ⊢
refine fun z =>
tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds (hf z)
(Eventually.of_forall fun x => abs_nonneg _) (hfg.mono fun x hx => ?_)
calc
|k x ^ z * g x| = |k x ^ z| * |g x| := abs_mul (k x ^ z) (g x)
_ ≤ |k x ^ z| ... | true | {
"chosen_tokens": 123,
"rejected_tokens": 5,
"token_jaccard": 0.069767,
"token_length_ratio": 0.04065
} | wrong_namespace_qualification | wrong_namespace_qualification | b23e569f379f81a586c80741403aca03ad3607a6b3ce607fa27fabc277a3aed4 | 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.Eval.Defs
public import Mathlib.Analysis.Asymptotics.Lemmas
Namespace:
Asymptotics
Local context:
/-
Copyright (c) 2021 Devon Tuma. All ri... | by
exact OProverMissing.missing_f97d07cceec9 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "8be3c6327935bda09fd2da3c8a57759bf4c7a29302587a1221e704619ddeee44",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Asymptotics",
"family_id": "superpolynomialdecay",
"file_id": "mathlib/Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean",
"sample_id": "f97d07cceec981c168c5f2eca445cd4e95e1981b7df2ab5f563b675d89d0e28f"
} | train |
{
"chosen_proof_sha256": "f8689006584708e8f5f2ea5a3447138e87630e033d57129a1fcb9fca25aae75e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
let h := MorphismProperty.factorizationData (trivialCofibrations C) (fibrations C)
(terminal.from X.obj)
have := isCofibrant_of_cofibration h.i
have : IsFibrant h.Z := by
rw [isFibrant_iff_of_isTerminal h.p terminalIsTerminal]
infer_instance
exact ⟨BifibrantObject.mk h.Z, homMk h.i, inferInstan... | false | {
"chosen_tokens": 180,
"rejected_tokens": 5,
"token_jaccard": 0.030303,
"token_length_ratio": 0.027778
} | sorry_admit_sorryAx_proof_hole | sorryAx | b25216169f7611bacc187d47990993c081d5decbab23d3dd2b33e2d3648cabe3 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy
public import Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy
public import Mathlib.Categ... | 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": "AlgebraicTopology/ModelCategory",
"family_id": "exists_bifibrant",
"file_id": "mathlib/Mathlib/AlgebraicTopology/ModelCategory/BifibrantObjectHomotopy.lean",
"sample_id": "1b257d253a1d99e1ebaccfa098b639ce8bc9949faf2e55ad6ca74bb6e26b343e"
} | train |
{
"chosen_proof_sha256": "9d077355c8aafccdf2f765766657b2aef7013299b24845a61f82fd2eda23f5d5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "eeee9288bac943d40417c3f56fb23df234... | by
zify
convert! Rat.num_mul_num_eq_num_mul_gcd q₁ q₂ <;> norm_cast | false | {
"chosen_tokens": 15,
"rejected_tokens": 22,
"token_jaccard": 0.875,
"token_length_ratio": 1.466667
} | markdown_preamble_trailing | fence | b2a6a27982231f34c0259c3bc5ec4ed4b798124e625a892f70896f8acbf51030 | 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.Field.Rat
public import Mathlib.Algebra.Group.Indicator
public import Mathlib.Algebra.Order.Field.Rat
public import Mathlib.Data.Rat.Lemmas
public impo... | ```lean
by
zify
convert! Rat.num_mul_num_eq_num_mul_gcd q₁ q₂ <;> norm_cast
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/NNRat",
"family_id": "num_mul_num_eq_num_mul_gcd",
"file_id": "mathlib/Mathlib/Data/NNRat/Lemmas.lean",
"sample_id": "471ab45e244c5046c4d6db5db33762cae30b42d81f33c3c388fbdb7ee1366b37"
} | train |
{
"chosen_proof_sha256": "6932312c5f36cec3a14fb1d6fd3d6a1837663ad65e9f1ee82ae3d44d58f2d03c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
let _ : Nontrivial N := by simp [← Module.supportDim_ne_bot_iff_nontrivial R, dim]
rw [PrimeSpectrum.zeroLocus_eq_singleton]
apply le_antisymm
· intro p hp
by_contra nmem
push _ ∈ _ at nmem
have : p < ⟨maximalIdeal R, IsMaximal.isPrime' (maximalIdeal R)⟩ :=
lt_of_le_of_ne (IsLocalRing.le_ma... | false | {
"chosen_tokens": 135,
"rejected_tokens": 2,
"token_jaccard": 0.015152,
"token_length_ratio": 0.014815
} | sorry_admit_sorryAx_proof_hole | admit | b2aee270782a042c5aede4eaf0e2a53d9ce091da448f0e09807b4935dfef650c | 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.KrullDimension.NonZeroDivisors
public import Mathlib.RingTheory.Spectrum.Prime.Module
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Naili... | 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/KrullDimension",
"family_id": "support_of_supportdim_eq_zero",
"file_id": "mathlib/Mathlib/RingTheory/KrullDimension/Module.lean",
"sample_id": "062acd11f87739fb15d069f3d86b913dd7cb13fa854b155bc79d97daea43f761"
} | train |
{
"chosen_proof_sha256": "e13f4e0fbff74fe329154bb27fe6656b950cc8d302959791efca88cd4237a5ab",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "17969d4630b562cbde413c26c21cf5e308... | by
refine ⟨fun h => ?_, fun ⟨h₁, h₂⟩ => Equiv.antisymm h₁ h₂⟩
obtain - | ⟨h₁, h₂⟩ := h
· simp [Lists'.Subset.refl]
· exact ⟨h₁, h₂⟩ | true | {
"chosen_tokens": 56,
"rejected_tokens": 5,
"token_jaccard": 0.103448,
"token_length_ratio": 0.089286
} | wrong_namespace_qualification | wrong_namespace_qualification | b2c0f7cd9dbe977cf68e6e075ae305d92f51eeb43fd63c04bcecacea4c5ac5ba | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.Monoid.NatCast
public import Mathlib.Algebra.Ring.Nat
public import Mathlib.Data.Sigma.Basic
public... | by
exact OProverMissing.missing_727321c81fb4 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "493cd1762115e34cafdf59884185112811c261b16b38124fc5fc44f684c749e1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "SetTheory",
"family_id": "islist_of_mem",
"file_id": "mathlib/Mathlib/SetTheory/Lists.lean",
"sample_id": "727321c81fb49fb44e787715ddf3c0aafbedfa69ef549affbe7e565a44f0be51"
} | train |
{
"chosen_proof_sha256": "882f9cd65c324f45860677174f52bfb0dee89c309337831b30132904e328a1fb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
lift s to Finset α using hs
classical
rw [stabilizer_coe_finset, mem_stabilizer_finset_iff_smul_finset_subset, ← Finset.coe_smul_finset,
Finset.coe_subset] | false | {
"chosen_tokens": 24,
"rejected_tokens": 3,
"token_jaccard": 0.047619,
"token_length_ratio": 0.125
} | by_exact_placeholder | exact? | b2dbb29721b7dbdee80ab92d474c45c927c68299a45f9c7cd9dabdeeb7e58c36 | 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.Action.Pointwise.Finset
public import Mathlib.GroupTheory.QuotientGroup.Defs
public import Mathlib.Order.ConditionallyCompleteLattice.Basic
Name... | 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": "Algebra/Pointwise",
"family_id": "mem_stabilizer_set_iff_smul_set_subset",
"file_id": "mathlib/Mathlib/Algebra/Pointwise/Stabilizer.lean",
"sample_id": "5ad3f4c1074336dbfd17b6d6041c87d9308d36a6123c6ad702c726184bd0c5fe"
} | train |
{
"chosen_proof_sha256": "5a49187c184286dd02eeb6ca07b11443ba36cdee6d43f2c64edd8e661703bcce",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "262a69776ad7e9ea0f6337da04d3acff30... | by
simp only [mem_Icc, insertNth_le_iff, le_insertNth_iff, and_assoc, @and_left_comm (x ≤ q₂ i)] | true | {
"chosen_tokens": 22,
"rejected_tokens": 2,
"token_jaccard": 0.05,
"token_length_ratio": 0.090909
} | invalid_tactic | invalid_tactic | b3018b782d5d1976cf0542376bd5284262da5c3e89f6fbd44e33f097a49f9964 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Logic.Equiv.Fin.Basic
public import Mathlib.Order.Fin.Basic
public import Mathlib.Order.PiLex
public import Mathlib.... | by
__oprover_invalid_tactic_394f27a561e1 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "15aa71346e634dec7a8f214ce48684ce23f853180708f069c88dda272cc8ec98",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Fin",
"family_id": "insertnth_mem_icc",
"file_id": "mathlib/Mathlib/Order/Fin/Tuple.lean",
"sample_id": "394f27a561e1c227c4d6a3e91cad146754404323a259cba6efddc68715f8ebd8"
} | train |
{
"chosen_proof_sha256": "d4714964e668b31f0718a64704bfc3c462347f6aac0cd63c4a504fd6654a6944",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4f54315a32700ee5c1139f169b34823e5e... | by
subst s_card
simp [basis_apply]
variable (s : powersetCard I m) (t : powersetCard I n) | true | {
"chosen_tokens": 22,
"rejected_tokens": 3,
"token_jaccard": 0.052632,
"token_length_ratio": 0.136364
} | unknown_identifier | unknown_identifier | b33fced4ee87b0cce33d82f8999a74500691a74fd6ef581ea5a39be643d94dd0 | 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.ExteriorAlgebra.Grading
public import Mathlib.LinearAlgebra.ExteriorPower.Basis
Namespace:
ExteriorAlgebra
Local context:
/-
Copyright (c) 2026... | by
exact __oprover_missing_bcfe71f4669f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "b918206dd12ac3d3424feb43ed65c3eb6b3495c71471b5a1957a553db184b17f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/ExteriorAlgebra",
"family_id": "basis_apply_ofcard",
"file_id": "mathlib/Mathlib/LinearAlgebra/ExteriorAlgebra/Basis.lean",
"sample_id": "bcfe71f4669f39454a991e49045a95b381eb505965239b851eaa93006f10ef03"
} | train |
{
"chosen_proof_sha256": "bc0a4d32839ad3481e8623af1efc75f8290857aa2e5f1191a4c4a6c87919cec5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b84e567048f9867d2afcbc469aa5345705... | by
apply fun s ↦ (residue_ne_zero_iff_isUnit s).mp
rw [map_aeval_eq_aeval_map (ψ := residue S) (φ := residue R) rfl,
← derivative_map, minpoly_map_residue hadj]
exact (Algebra.IsSeparable.isSeparable _ _).aeval_derivative_ne_zero (minpoly.aeval _ _) | false | {
"chosen_tokens": 53,
"rejected_tokens": 60,
"token_jaccard": 0.944444,
"token_length_ratio": 1.132075
} | markdown_preamble_trailing | fence | b40d10cdca82b64d5ef018944718daf379b4b52cb46125b34be96a46b28244da | 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.IsAdjoinRoot
public import Mathlib.RingTheory.LocalRing.Quotient
public import Mathlib.RingTheory.Smooth.Flat
public import Mathlib.RingTheory.Unram... | ```lean
by
apply fun s ↦ (residue_ne_zero_iff_isUnit s).mp
rw [map_aeval_eq_aeval_map (ψ := residue S) (φ := residue R) rfl,
← derivative_map, minpoly_map_residue hadj]
exact (Algebra.IsSeparable.isSeparable _ _).aeval_derivative_ne_zero (minpoly.aeval _ _)
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/LocalRing",
"family_id": "isunit_aeval_derivative_minpoly_of_adjoin_eq_top",
"file_id": "mathlib/Mathlib/RingTheory/LocalRing/Etale.lean",
"sample_id": "0e6ca74f069156e403ca37b6a55058b4c1f3d343ff3c45b25117782dc63111e3"
} | train |
{
"chosen_proof_sha256": "afd4fe6929e62277a458a32fdf80154f965674b8199603407ac172cf56934b1a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | 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] | false | {
"chosen_tokens": 90,
"rejected_tokens": 3,
"token_jaccard": 0.027778,
"token_length_ratio": 0.033333
} | by_exact_placeholder | placeholder | b4385880b93569abe7193d668dcaab35826fcd95d3f5e94b230c4f558fc8ab8c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Subobject.Basic
Namespace:
CategoryTheory.Dial
Local context:
/-
Copyright (c) 2024 Mario Carneiro. All rights reserved.
Released under Apache... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Dialectica",
"family_id": "comp_le_lemma",
"file_id": "mathlib/Mathlib/CategoryTheory/Dialectica/Basic.lean",
"sample_id": "7b702b3f54a49b24809e03e0a017a431793ec768bf4e679fe9f430ed6393e5c4"
} | train |
{
"chosen_proof_sha256": "a76192e154cb0c91a316e618a2d1566a7760c095bf189498c16a175bdc2813bb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "84988f78a63dd6457f39f9a79617a3487d... | by
generalize h' : [] = L'
intro h
rcases h with - | ⟨L₁, L₂⟩
simp at h' | false | {
"chosen_tokens": 25,
"rejected_tokens": 29,
"token_jaccard": 0.846154,
"token_length_ratio": 1.16
} | missing_wrong_import | import | b4b40aff18267f3f1d44d3d36c4be6a1f0bb62d7ecd52295691185c91c44ff46 | 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.Pi.Basic
public import Mathlib.Algebra.Group.Subgroup.Ker
public import Mathlib.Data.List.Chain
public import Mathlib.Algebra.Group.Int.Defs
publ... | import Mathlib.WrongImport
by
generalize h' : [] = L'
intro h
rcases h with - | ⟨L₁, L₂⟩
simp at 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": "GroupTheory/FreeGroup",
"family_id": "step",
"file_id": "mathlib/Mathlib/GroupTheory/FreeGroup/Basic.lean",
"sample_id": "e4f8810a3c97776fcd6674079eb5c88b07ed5c16b257876471689d8a0da4bfcc"
} | train |
{
"chosen_proof_sha256": "0378e91a096e555db6c2dcff0f9d43073fa97735128ddc7daf75d45b02b151ac",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
apply le_antisymm
· refine iSup₂_le fun i hi y hy => ⟨DFinsupp.single i ⟨y, hy⟩, ?_⟩
rw [AddMonoidHom.comp_apply, filterAddMonoidHom_apply, filter_single_pos _ _ hi]
exact sumAddHom_single _ _ _
· rintro x ⟨v, rfl⟩
refine dfinsuppSumAddHom_mem _ _ _ fun i _ => ?_
refine AddSubmonoid.mem_iSup_of... | false | {
"chosen_tokens": 87,
"rejected_tokens": 2,
"token_jaccard": 0.022727,
"token_length_ratio": 0.022989
} | sorry_admit_sorryAx_proof_hole | sorry | b4f42aa2a088b0bf5f539e1b4a4e51526b5286faff376ae83408575ada8a9552 | 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.BigOperators
public import Mathlib.Algebra.Group.Submonoid.Membership
public import Mathlib.Data.DFinsupp.BigOperators
public import Ma... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/DFinsupp",
"family_id": "addsubmonoid",
"file_id": "mathlib/Mathlib/Data/DFinsupp/Submonoid.lean",
"sample_id": "f6367fab7503d969f0bae44dba3d63ca6fddd75a99ad5b0e2f65909eedb272b7"
} | train |
{
"chosen_proof_sha256": "7d2f77b8c1bb155392db6b9e50f050fefd652d8cabc59188758f6cc33ac80619",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "682d48ad8b9e90576e74171f482cee42e8... | by convert! aeval_algebraMap_apply A x p | true | {
"chosen_tokens": 7,
"rejected_tokens": 5,
"token_jaccard": 0.090909,
"token_length_ratio": 0.714286
} | wrong_namespace_qualification | wrong_namespace_qualification | b4f5286e1bbd5caa5a11829f2cd53e1f0836e087eb59ae8a07171a56c061f3dd | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Subalgebra.Tower
public import Mathlib.Algebra.MvPolynomial.Eval
Namespace:
Subalgebra
Local context:
/-
Copyright (c) 2022 Yuyang Zhao. All ... | by
exact OProverMissing.missing_bb469c3613f9 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c873ee2aca5ae636fdca5c53d085bd704c25b4cd75dc9004e3f1047a9a94e084",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/MvPolynomial",
"family_id": "mvpolynomial_aeval_coe",
"file_id": "mathlib/Mathlib/RingTheory/MvPolynomial/Tower.lean",
"sample_id": "bb469c3613f92d8cabf0255e6c012999e3bdcfb6d61f1c8b6b5d3a6dc8c383d5"
} | train |
{
"chosen_proof_sha256": "dca549e925c3a05e8b8bc1baae8e33cdf145c6f1d35ee1dc1c15d213d724ea07",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e912fa5d231de1681e276afa664abc4b41... | by cat_disch) : f = g :=
Hom.ext w | true | {
"chosen_tokens": 12,
"rejected_tokens": 2,
"token_jaccard": 0.076923,
"token_length_ratio": 0.166667
} | invalid_tactic | invalid_tactic | b50f3b67cdc1d8179ac9a024c59b10168a6dfd0f45e9933589d3ce251054c185 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Shapes.IsTerminal
public import Mathlib.CategoryTheory.Functor.EpiMono
Namespace:
CategoryTheory.Endofunctor.Coalgebra
Local context:
/... | by
__oprover_invalid_tactic_6aa245c658f8 | {
"accepted": false,
"failure_type": "unknown_tactic",
"output_sha256": "b2f6e75651ce1c07631343cb324f01a6566eee8a4fd1e94b8a4566aa4689da09",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "6db23e1003596f7373b150a65cdc1843f9cceb615009a5e13fae5265b46dd136",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f362ce03318e89ad5084cd4764e967b35a... | by
intro L₁ L₂ hflat
apply Subtype.ext
exact h L₁.val L₂.val L₁.prop L₂.prop hflat | false | {
"chosen_tokens": 30,
"rejected_tokens": 35,
"token_jaccard": 0.777778,
"token_length_ratio": 1.166667
} | markdown_preamble_trailing | trailing | b51623415e585f54efe78cebe5e355db0aff9ecb355dd22a469048cf37136b98 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Set.Basic
Namespace:
InformationTheory
Local context:
/-
Copyright (c) 2026 Elazar Gershuni. All rights reserved.
Released under Apache 2.0 license as d... | by
intro L₁ L₂ hflat
apply Subtype.ext
exact h L₁.val L₂.val L₁.prop L₂.prop hflat
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": "InformationTheory/Coding",
"family_id": "uniquelydecodable",
"file_id": "mathlib/Mathlib/InformationTheory/Coding/UniquelyDecodable.lean",
"sample_id": "70ee751da2d1168fe9d7a11059342b3a3f15c210604e29d540ce2dd98fb9afcf"
} | train |
{
"chosen_proof_sha256": "5668706c687646eb1a543fe99da9ca45de8dce6aa203fed4f6e41809be74067a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8ec2135d00a06c0bdde5ffa67ec3d4e80d... | by
obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos
have hk₀' : (0 : ℝ) < k := Int.cast_pos.mpr hk₀
have hden : ((j / k : ℚ).den : ℤ) ≤ k := by
convert! le_of_dvd hk₀ (Rat.den_dvd j k)
exact Rat.intCast_div_eq_divInt _ _
refine ⟨j / k, ?_, Nat.cast_le.mp (hden.trans hk₁)⟩
rw [← div_... | true | {
"chosen_tokens": 192,
"rejected_tokens": 3,
"token_jaccard": 0.029851,
"token_length_ratio": 0.015625
} | circular_dependency | circular_dependency | b51625a5a6878b6ddc5559abc8a5e2e2f40bb6fb285adc67f84356ef2fb1fcf0 | 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.Real.Irrational
public import Mathlib.RingTheory.Coprime.Lemmas
public import Mathlib.RingTheory.Int.Basic
public import Mathlib.Tactic.Basic
Nam... | by
exact exists_rat_abs_sub_le_and_den_le | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "d52c7cdd2bd404deb9a1bc208ae6e61ce682e00c89dc18d04785079eb47b1656",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/DiophantineApproximation",
"family_id": "exists_rat_abs_sub_le_and_den_le",
"file_id": "mathlib/Mathlib/NumberTheory/DiophantineApproximation/Basic.lean",
"sample_id": "f7c063a4e5099ef9d392d9d1440be2d9799586ee41a1ffac00a85cda728bea8f"
} | train |
{
"chosen_proof_sha256": "31b27ce5f99a3bf78bf18a3e524b1460389b6e7ac4458283ab4e0eee4a7b4805",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1c5375cf4fad0c1095d3773e5d3eececd8... | by
unfold unfolds; rw [corec_eq] | true | {
"chosen_tokens": 8,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.625
} | wrong_namespace_qualification | wrong_namespace_qualification | b531b8837ebdb8b4f5a462c898794a4490f023239ae9bfe90637257df017c8a6 | 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.Stream.Defs
public import Mathlib.Logic.Function.Basic
public import Mathlib.Data.Nat.Basic
public import Mathlib.Tactic.Common
Namespace:
Stream'
Local... | by
exact OProverMissing.missing_767863c4f273 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "88df87fef37fa62737ab5369eec4a3819ceb6e8eea2deb54603aa2b7cb0ddb22",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Stream",
"family_id": "unfolds_eq",
"file_id": "mathlib/Mathlib/Data/Stream/Init.lean",
"sample_id": "767863c4f273ffd20ab08a8048315c1504cb395ceb5b970fd18d5025077b58cc"
} | train |
{
"chosen_proof_sha256": "909daa16cd181b78a203d073bc4331c1f30425ac8b8431620e4cd7f332d4a635",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "21cb92a17308c1ecba7cf2fdd675eece83... | by
constructor
· intro H
induction H using Submodule.span_induction with
| mem a ha =>
obtain ⟨a, ha, rfl⟩ := ha
exact ⟨C a, fun i ↦ by rw [coeff_C]; aesop, aeval_C _ _⟩
| zero => exact ⟨0, by simp, aeval_zero _⟩
| add a b ha hb ha' hb' =>
obtain ⟨a, ha, ha'⟩ := ha'
obtain ⟨b... | true | {
"chosen_tokens": 312,
"rejected_tokens": 2,
"token_jaccard": 0.010638,
"token_length_ratio": 0.00641
} | invalid_tactic | invalid_tactic | b556bf1c369b40dfd91e7a88f69331bc08aa6133313376a903aca4569f09a89b | 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.RingDivision
public import Mathlib.RingTheory.Adjoin.Polynomial.Basic
public import Mathlib.RingTheory.Ideal.Maps
Namespace:
Algebra
Local... | by
__oprover_invalid_tactic_a13d8df10b31 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "50dd4c9cd7569858f6c7ce052cff48d4a7509c4568bf47a765575af3b4e78faa",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Polynomial",
"family_id": "mem_ideal_map_adjoin",
"file_id": "mathlib/Mathlib/RingTheory/Polynomial/Ideal.lean",
"sample_id": "a13d8df10b3157064561e6f5629998f83e9b759c9ef6010480db3810fec50f71"
} | train |
{
"chosen_proof_sha256": "6ed251e98bcc09fc840097a1704c71dc97a2385a1bbef8d0c694a48f689a68b3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b482550037179e53cdaa84a61ee7777641... | by
rw [OneSkeletonGraph_isLink, closedCell_zero_eq_singleton, closedCell_zero_eq_singleton,
singleton_union] | false | {
"chosen_tokens": 11,
"rejected_tokens": 15,
"token_jaccard": 0.666667,
"token_length_ratio": 1.363636
} | missing_wrong_import | import | b55f2f4072dda5af039bd4a80814b1c4213415d26f25ff5dbd5ae22610658c45 | 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.CWComplex.Classical.Finite
public import Mathlib.Combinatorics.Graph.Basic
Namespace:
Topology.CWComplex.OneSkeletonGraph
Local context:
/-
Copyrigh... | import Mathlib.WrongImport
by
rw [OneSkeletonGraph_isLink, closedCell_zero_eq_singleton, closedCell_zero_eq_singleton,
singleton_union] | {
"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/CWComplex",
"family_id": "islink_iff_pair",
"file_id": "mathlib/Mathlib/Topology/CWComplex/Classical/Graph.lean",
"sample_id": "0d7956b6ebdcf8c47b68d95a35dc56b9f93a664feaf120a85ef061f559b47042"
} | train |
{
"chosen_proof_sha256": "75765532e9f719b9017c00ea0609cd6aeeb9a4a2bf7c2d14d50fca0b0687a396",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "19c41640b1c800a77df051f9d2a4e84180... | by
obtain ⟨x, hx⟩ := h.mul_inv_cancel ha
simpa [← mul_assoc, h.mul_comm, hx] using congr_arg (x * ·) hb
mul_right_cancel_of_ne_zero ha _ _ hb := by
obtain ⟨x, hx⟩ := h.mul_inv_cancel ha
simpa [mul_assoc, hx] using congr_arg (· * x) hb
exists_pair_ne := h.exists_pair_ne | true | {
"chosen_tokens": 68,
"rejected_tokens": 2,
"token_jaccard": 0.034483,
"token_length_ratio": 0.029412
} | invalid_tactic | invalid_tactic | b5625bde5ab6dcbbae7aa99f0a65ed3c6ff80518560623dfbd6b94761f593242 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Field.Defs
public import Mathlib.Tactic.Common
Namespace:
(root)
Local context:
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released ... | by
__oprover_invalid_tactic_e5781a280c95 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "5d2938203b0b8c1eb2df6767b3693e8328df755d04270f35f82858cd253cbade",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Field",
"family_id": "isfield",
"file_id": "mathlib/Mathlib/Algebra/Field/IsField.lean",
"sample_id": "e5781a280c95429bef6967a38411ca2235131ecb05d9745326c5155130e9a4ab"
} | train |
{
"chosen_proof_sha256": "8a67a77843a75ba583cb43ac0e535963aaf96ee59db334d6516f871f6e4da8bd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [← @isBounded_univ, isBounded_induced, image_univ] | false | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.181818
} | sorry_admit_sorryAx_proof_hole | sorry | b5666fe90493a29321e4e941036eb0c188e8ec20bb1f4a9028fed1533765d6f7 | 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:
(root)
Local context:
/-
Copyright (c) 2022 Yury Kudryashov. All right... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/Bornology",
"family_id": "boundedspace_induced_iff",
"file_id": "mathlib/Mathlib/Topology/Bornology/Constructions.lean",
"sample_id": "9f9a70e78a68c5afe844cbabb734e8a42046f9373491df56a598c7ff77b730eb"
} | train |
{
"chosen_proof_sha256": "2fe6742b90ce6b6b787aeae843d8b7476e8ec34c6e8c54b97436e93c09e1ac78",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
have : IsNoetherianRing A := Algebra.FiniteType.isNoetherianRing R A
rw [finite_iff_isArtinianRing, isArtinianRing_iff_isNoetherianRing_krullDimLE_zero,
and_iff_right this] | false | {
"chosen_tokens": 22,
"rejected_tokens": 3,
"token_jaccard": 0.047619,
"token_length_ratio": 0.136364
} | by_exact_placeholder | exact? | b56b454402599c5959b0ec7f09ea09ade70068ddd6f350240aca62598ff80d5c | 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.HopkinsLevitzki
public import Mathlib.RingTheory.Jacobson.Ring
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Andrew Yang. All rights rese... | 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/Jacobson",
"family_id": "module",
"file_id": "mathlib/Mathlib/RingTheory/Jacobson/Artinian.lean",
"sample_id": "9edf83ad0b7479676e136f7279c23e835e46a349748bddabb519ed9ff699acc7"
} | train |
{
"chosen_proof_sha256": "ca01e87830c1462b512c1bad10dd02834fa50907bb2a15cdffba13757dee4a07",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
refine mt ((injective_iff_map_eq_zero _).mp bS.algebraMap_injective _) ?_
simp only [Finset.prod_eq_zero_iff, not_exists]
rintro x ⟨hx, rfl⟩
exact finsetApprox.zero_notMem bS adm hx | false | {
"chosen_tokens": 40,
"rejected_tokens": 2,
"token_jaccard": 0.066667,
"token_length_ratio": 0.05
} | by_exact_placeholder | by? | b5789e37ebbdbe9370f3f392555d742b65c21135ed23aedf208525c81079b22c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecialFunctions.Pow.Real
public import Mathlib.LinearAlgebra.FreeModule.PID
public import Mathlib.LinearAlgebra.Matrix.AbsoluteValue
public import Ma... | 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/ClassNumber",
"family_id": "prod_finsetapprox_ne_zero",
"file_id": "mathlib/Mathlib/NumberTheory/ClassNumber/Finite.lean",
"sample_id": "32a4aa28ca53ec5f375880fb329a6c92a05a17c1c9d5e8c664a77a4bde982211"
} | train |
{
"chosen_proof_sha256": "a3b30f98b08bb268a88816f205f3d07c5c34faaa5cf59c628e0cdbe5b5a6a78c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "10a45c3e9911e5b7b3fc6816e3e6db463f... | by
rw [mul_comm]
exact mul_left_not_lt b h | true | {
"chosen_tokens": 9,
"rejected_tokens": 2,
"token_jaccard": 0.1,
"token_length_ratio": 0.222222
} | invalid_tactic | invalid_tactic | b58a35579fec5583888a353a943f4675d0443910cd36bb092fb95a28a678d0bf | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.Order.RelClasses
Namespace:
EuclideanDomain
Local context:
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
R... | by
__oprover_invalid_tactic_e87fe6db0872 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "5b45c37cfb1c4d5e9d2bf2812af809dbb59514f5517dd2d6f4d802304d1c3b6f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/EuclideanDomain",
"family_id": "mul_right_not_lt",
"file_id": "mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean",
"sample_id": "e87fe6db087248fc861a35d6fd880f370abae53489bf00891ff2b411146d68fa"
} | train |
{
"chosen_proof_sha256": "a6d320117b4f0031ed5a5af663cf8c7397e183bff5fdfb69cb2aa2ba76b6c4c2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4966b3585ecb4af08a122a7051ce365cbf... | by
rw [← mul_right_inj (x : M), mul_inv_cancel, mul_leftInvEquiv_symm] | false | {
"chosen_tokens": 15,
"rejected_tokens": 22,
"token_jaccard": 0.875,
"token_length_ratio": 1.466667
} | markdown_preamble_trailing | fence | b5ae2932e37eb0ec40caff371064a0bd2054e1255c2405d7d5eb4e81b0d4f709 | 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.Pointwise
Namespace:
Submonoid
Local context:
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 licen... | ```lean
by
rw [← mul_right_inj (x : M), mul_inv_cancel, mul_leftInvEquiv_symm]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "GroupTheory/Submonoid",
"family_id": "leftinvequiv_symm_eq_inv",
"file_id": "mathlib/Mathlib/GroupTheory/Submonoid/Inverses.lean",
"sample_id": "a81f25208ce01c9c1e94e1a7e16ac4440ac59daf394e99a5ec3de20f35dc63a8"
} | train |
{
"chosen_proof_sha256": "ef657d6d5fc4b7d8887ac9017ba9ce06e5bcee28dc51338cd8d6ad2502a8eccb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "88998f6d1d9f42ffd0de1fc17033025060... | by
simp_rw [← union, disjoint_union_right, disjoint_iUnion_right]
exact ⟨disjointBase n i , fun _ _ ↦ disjoint_openCell_of_ne (by lia)⟩ | true | {
"chosen_tokens": 26,
"rejected_tokens": 3,
"token_jaccard": 0.086957,
"token_length_ratio": 0.115385
} | unknown_identifier | unknown_identifier | b5b644305a4bc30f47ab5d2ab05b81300e09cb52741c70993b7b243b6e0fc12d | 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.CWComplex.Classical.Finite
public import Mathlib.Analysis.Normed.Module.RCLike.Real
Namespace:
Topology
Local context:
/-
Copyright (c) 2025 Floris ... | by
exact __oprover_missing_4b6a657545ab | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "308561b0193d05667e50dbbdaeaf05078510337353c6133486b5e57abe419859",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/CWComplex",
"family_id": "relcwcomplex",
"file_id": "mathlib/Mathlib/Topology/CWComplex/Classical/Subcomplex.lean",
"sample_id": "4b6a657545abaaf7eebda573d07dd91e0d6097e9312e9cce6c12048585c3e4e7"
} | train |
{
"chosen_proof_sha256": "ff1ae62e27377baaefea54f50e782d5e8647a507a59b2f78b75ee566cf2b6bd5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [iff_comp_injective]
intro B _ _ I hI f₁ f₂ e
ext x
have : f₁ x - f₂ x ∈ I := by
simpa [Ideal.Quotient.mk_eq_mk_iff_sub_mem] using AlgHom.congr_fun e x
have := Polynomial.eval_add_of_sq_eq_zero ((minpoly K x).map (algebraMap K B)) (f₂ x)
(f₁ x - f₂ x) (show (f₁ x - f₂ x) ^ 2 ∈ ⊥ from hI ▸ Ideal.... | false | {
"chosen_tokens": 187,
"rejected_tokens": 2,
"token_jaccard": 0.013889,
"token_length_ratio": 0.010695
} | by_exact_placeholder | by? | b60c025fc5f730573cd95f50e625985832e27f5c94a9dcaac7a7cc889fcd27b3 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.PurelyInseparable.Basic
public import Mathlib.RingTheory.Artinian.Ring
public import Mathlib.RingTheory.Unramified.Finite
public import Mathlib.Rin... | 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/Unramified",
"family_id": "of_isseparable",
"file_id": "mathlib/Mathlib/RingTheory/Unramified/Field.lean",
"sample_id": "5c3559b3cf2ee9a241bb81b7e23213b68dd9f8307cc2252bb99a753e0c04c076"
} | train |
{
"chosen_proof_sha256": "faeb8949323faf61ca5ff806cf60d2c823a2170b0852dfc5d54df3a865ffc0fd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8befb9059d33074287276d870632867722... | by
simp [zariskisMainProperty_iff, ← top_le_iff, SetLike.le_def,
Submonoid.mem_powers_iff, mem_integralClosure_iff] | false | {
"chosen_tokens": 18,
"rejected_tokens": 23,
"token_jaccard": 0.777778,
"token_length_ratio": 1.277778
} | markdown_preamble_trailing | trailing | b62d138266ed1e5f9934e927550d5f8bdfc5b66548d6ed304421b760723ef6d6 | 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.Algebraic.StronglyTranscendental
public import Mathlib.RingTheory.Conductor
public import Mathlib.RingTheory.Ideal.Quotient.Nilpotent
public import ... | by
simp [zariskisMainProperty_iff, ← top_le_iff, SetLike.le_def,
Submonoid.mem_powers_iff, mem_integralClosure_iff]
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory",
"family_id": "zariskismainproperty_iff_exists_saturation_eq_top",
"file_id": "mathlib/Mathlib/RingTheory/ZariskisMainTheorem.lean",
"sample_id": "988fd8cb4d8e17e38f2a76981b98bfba51705b13e2c6dc4822f9d73a946a1e06"
} | train |
{
"chosen_proof_sha256": "0378e91a096e555db6c2dcff0f9d43073fa97735128ddc7daf75d45b02b151ac",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "90b3c016e6178b281419875d8a8be9e9c3... | by
apply le_antisymm
· refine iSup₂_le fun i hi y hy => ⟨DFinsupp.single i ⟨y, hy⟩, ?_⟩
rw [AddMonoidHom.comp_apply, filterAddMonoidHom_apply, filter_single_pos _ _ hi]
exact sumAddHom_single _ _ _
· rintro x ⟨v, rfl⟩
refine dfinsuppSumAddHom_mem _ _ _ fun i _ => ?_
refine AddSubmonoid.mem_iSup_of... | true | {
"chosen_tokens": 87,
"rejected_tokens": 3,
"token_jaccard": 0.045455,
"token_length_ratio": 0.034483
} | unknown_identifier | unknown_identifier | b64e1f3a32fcaeabda1692f23578b2639372c35373a6c15da73564591d014eab | 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.Submonoid.BigOperators
public import Mathlib.Algebra.Group.Submonoid.Membership
public import Mathlib.Data.DFinsupp.BigOperators
public import Ma... | by
exact __oprover_missing_f6367fab7503 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a2726fc91447bd8d1d4fb8a3ff11f09e2f1bc98dbbe62658502108d99dd3b80a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/DFinsupp",
"family_id": "addsubmonoid",
"file_id": "mathlib/Mathlib/Data/DFinsupp/Submonoid.lean",
"sample_id": "f6367fab7503d969f0bae44dba3d63ca6fddd75a99ad5b0e2f65909eedb272b7"
} | train |
{
"chosen_proof_sha256": "ace5857b04711e3f6e5ec35f7db5255eacbf0110e349e471daaf3e18ca693ed4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "81a2ff22c892d0579ef9c1611ce99af9ab... | by
by_cases! hn : n = 0
· rw [pow_eq_top_iff.mpr (Or.inr hn)]
exact isUnit_of_subsingleton _
exact (isUnit_mk_pow_iff_notMem I hn).mpr hx | false | {
"chosen_tokens": 33,
"rejected_tokens": 37,
"token_jaccard": 0.892857,
"token_length_ratio": 1.121212
} | missing_wrong_import | import | b6563503c39397490182642400e447a3ec41931e5c453d3b1ccf73ea252b0b59 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Ideal.Quotient.Operations
public import Mathlib.RingTheory.Nilpotent.Lemmas
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Oliver Nash. Al... | import Mathlib.WrongImport
by
by_cases! hn : n = 0
· rw [pow_eq_top_iff.mpr (Or.inr hn)]
exact isUnit_of_subsingleton _
exact (isUnit_mk_pow_iff_notMem I hn).mpr hx | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Ideal",
"family_id": "ideal",
"file_id": "mathlib/Mathlib/RingTheory/Ideal/Quotient/Nilpotent.lean",
"sample_id": "22290c022264f0f12d52570adb279eb6ca130b8e7ce84838db022df1840fc31a"
} | train |
{
"chosen_proof_sha256": "ea63d2911a78e27e3d0efd6769a583eaec219edad30fe20857b3d050d16b7158",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d05263c7bf55a58e8c8de9c28c2d1ec245... | by
letI := (IntermediateField.inclusion H.le).toAlgebra
have : IsScalarTower F E L := .of_algebraMap_eq' rfl
refine lt_of_le_of_ne ?_ ?_
· exact Module.finrank_top_le_finrank_of_isScalarTower _ _ _
· exact .symm (mt (eq_of_le_of_finrank_eq' H.le) H.ne) | true | {
"chosen_tokens": 53,
"rejected_tokens": 2,
"token_jaccard": 0.03125,
"token_length_ratio": 0.037736
} | invalid_tactic | invalid_tactic | b66988e7eb714cfcdb8c43abf10b1551f3f24909939445c93c9743c83053eccb | 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.IntermediateField.Basic
public import Mathlib.FieldTheory.Minpoly.Basic
public import Mathlib.FieldTheory.Tower
public import Mathlib.LinearAlgebra... | by
__oprover_invalid_tactic_f611f13c4698 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "aacfb5302f375a74a50816b0dbc1a8cef7fd24c09b08f05ba6286e386e1b7630",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/IntermediateField",
"family_id": "finrank_lt_of_gt",
"file_id": "mathlib/Mathlib/FieldTheory/IntermediateField/Algebraic.lean",
"sample_id": "f611f13c4698d3e99009d189350bca949401d584539fd12a5756b6e5b77a146b"
} | train |
{
"chosen_proof_sha256": "08218aabda46401fe50c2dc9de3ff5011d981b5e7f148f3d58bd8c99abd66739",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
simp only [simplyConnectedSpace_iff, equiv_punit_iff_unique,
FundamentalGroupoid.nonempty_iff X, and_congr_right_iff, Nonempty.forall]
intros
exact ⟨fun h _ _ => h _ _, fun h _ _ => h _ _⟩ | false | {
"chosen_tokens": 40,
"rejected_tokens": 2,
"token_jaccard": 0.041667,
"token_length_ratio": 0.05
} | sorry_admit_sorryAx_proof_hole | sorry | b66c6d427bca768a114f217572f7bbc1097ca2977d6b33a362cfaa9b12d8c09c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
public import Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps
public import Mathlib.Topology.H... | 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": "AlgebraicTopology/FundamentalGroupoid",
"family_id": "simply_connected_iff_unique_homotopic",
"file_id": "mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean",
"sample_id": "631dd733f33921d58948a0df45ceabb189d48a1055bf4f8fc988734c4839b6cb"
} | train |
{
"chosen_proof_sha256": "31a2975efba554cce04bfa852066668eb27b98da82a9d33652d228e97ecec653",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [removeFactor, natDegree_divByMonic _ (monic_X_sub_C _), natDegree_map, natDegree_X_sub_C] | false | {
"chosen_tokens": 16,
"rejected_tokens": 2,
"token_jaccard": 0.071429,
"token_length_ratio": 0.125
} | by_exact_placeholder | by? | b66f439d309434f1998e9d144d7db1dc2ab1be527ace70537efb2deb489d8777 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.CharP.Algebra
public import Mathlib.FieldTheory.SplittingField.IsSplittingField
public import Mathlib.RingTheory.Algebraic.Basic
Namespace:
Polynomial... | 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": "FieldTheory/SplittingField",
"family_id": "natdegree_removefactor",
"file_id": "mathlib/Mathlib/FieldTheory/SplittingField/Construction.lean",
"sample_id": "7e8c17610a3baa53fbdd954115a7d7f9ea7c655e0a0e548683cb4072c1194356"
} | train |
{
"chosen_proof_sha256": "768478b669f1b0c48360ff5cb94ef6db06e9b1b6432c6287e4928bb1cb2132ad",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [eq_comm, ← ltb_cons_addChar' c]
simp | false | {
"chosen_tokens": 10,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.2
} | by_exact_placeholder | by? | b689f922ce9b0cddf688b98239699931002cdf4999808b36a301f5f63d48158c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Data.String.Lemmas
public import Mathlib.Data.List.Lex
public import Mathlib.Data.Char
public import Mathlib.Algebra.Order.Group.Nat
import all Init.Data.Str... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/String",
"family_id": "ltb_cons_addchar",
"file_id": "mathlib/Mathlib/Data/String/Basic.lean",
"sample_id": "2ad48a41750f9ee7416f3ed47f9d4bdc606c25a12a288f7161359dab4138e50a"
} | train |
{
"chosen_proof_sha256": "bbb068c1b254f9a62b1fe49e6c1cd52f49bd7cbde7d3a811059b2617b8439517",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7887373d14e0ce8b2a10aa48b0884c760c... | by
have h0 : v πᵥ ≠ 0 := by simpa using uniformizingPolynomial_ne_zero hle
induction f using RatFunc.induction_on with
| f p q hq =>
use (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {p})).factors -
(Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {q})).factors
simp only... | true | {
"chosen_tokens": 116,
"rejected_tokens": 3,
"token_jaccard": 0.018868,
"token_length_ratio": 0.025862
} | unknown_identifier | unknown_identifier | b699dca0b8715f1070646516ab65da3e81f9db24bec979dd721ddf6ae5b705dc | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.Finite.Valuation
public import Mathlib.NumberTheory.FunctionField
public import Mathlib.RingTheory.Valuation.Discrete.Basic
Namespace:
RatFunc
Lo... | by
exact __oprover_missing_c87ad3ae2402 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c3c4f4aadecd04077ed58cc7d8706d04245d9a690e48822b76699b5cb154be57",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/RatFunc",
"family_id": "exists_zpow_uniformizingpolynomial",
"file_id": "mathlib/Mathlib/NumberTheory/RatFunc/Ostrowski.lean",
"sample_id": "c87ad3ae24029059963e03348c139139f0d135dcb45808f3d2427d4e578980f8"
} | train |
{
"chosen_proof_sha256": "2b60f1251ce380c3c88892d3140d40e19399072a863e960ad4ea10cd3795106f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rcases exists_pair_ne α with ⟨x, y, hxy⟩
cases lt_or_gt_of_ne hxy <;> exact ⟨_, _, ‹_›⟩ | true | {
"chosen_tokens": 28,
"rejected_tokens": 8,
"token_jaccard": 0.038462,
"token_length_ratio": 0.285714
} | unsolved_goals | unsolved_goals | b6aa1915879309ab5bbf519805ba77ed2bd9bd2cbbd01fe32ad90a30fa7a0ece | 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.Prod.Basic
public import Mathlib.Logic.Function.Basic
public import Mathlib.Logic.Nontrivial.Defs
public import Mathlib.Logic.Unique
public import Mathlib... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "d19e3b2ffed3bc088683f6ee7555b22a069dfda563d38856803dcb2388a43300",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Logic/Nontrivial",
"family_id": "exists_pair_lt",
"file_id": "mathlib/Mathlib/Logic/Nontrivial/Basic.lean",
"sample_id": "9a258eabbca29a5c3771ee588ea1b0dec5082cf5ef32ba82797313921b1c69f8"
} | train |
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