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": "7361ae6d5ff40c196b32b028010a5c56393ce19a055fbd12b76df3365e97ad61",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
refine ⟨fun h ↦ eq_univ_iff_forall.mpr h.mem_periodicPts, fun h ↦ ?_⟩
rw [Finite.injective_iff_surjective, ← range_eq_univ, ← univ_subset_iff, ← h]
apply periodicPts_subset_range | false | {
"chosen_tokens": 36,
"rejected_tokens": 3,
"token_jaccard": 0.038462,
"token_length_ratio": 0.083333
} | by_exact_placeholder | placeholder | 6cccae1b89db7521eed11bb5941dc56ba8caf5d9ee4a72d1085f431647e49944 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GCDMonoid.Finset
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.Data.PNat.Basic
public import Mathlib.Dynamics.PeriodicPts.Defs
publi... | by
exact 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": "Dynamics/PeriodicPts",
"family_id": "injective_iff_periodicpts_eq_univ",
"file_id": "mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean",
"sample_id": "108e499b9132e898676c8d66a09fbbcb3ff6a762c243e0f235bc6414a3079513"
} | train |
{
"chosen_proof_sha256": "bdd71d8bba5ab6bc0aadc26f2ebfd16f9c6d0a1ea732936b96c085c594ae8d5f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simp [pullSub, List.take_of_length_le hl, List.drop_eq_nil_iff.mpr hl] | false | {
"chosen_tokens": 17,
"rejected_tokens": 2,
"token_jaccard": 0.076923,
"token_length_ratio": 0.117647
} | by_exact_placeholder | by? | 6cdc61f01dcff11877381f602b1a234b9c8efc39079e4b9d8902c111598b6fa9 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.CompleteLattice.SetLike
Namespace:
Descriptive.Tree
Local context:
/-
Copyright (c) 2024 Sven Manthe. All rights reserved.
Released under Apache 2.0 li... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "SetTheory/Descriptive",
"family_id": "mem_pullsub_short",
"file_id": "mathlib/Mathlib/SetTheory/Descriptive/Tree.lean",
"sample_id": "4fba185756badbaa3d579461622f7431189517e5fb7071a627d47ad8323f43b0"
} | train |
{
"chosen_proof_sha256": "e1fc953f2e811d1c999db981871450abfdfcde88bd7349db65e45507d796d374",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
let ℬ := {s | s ∈ 𝒜} ∪ {s | sᶜ ∈ 𝒜}
have count_ℬ : ℬ.Countable := by
apply h.union
have : compl '' 𝒜 = {s | sᶜ ∈ 𝒜} := by
ext s
simpa using ⟨fun ⟨x, x_mem, hx⟩ ↦ by simp [← hx, x_mem], fun hs ↦ ⟨sᶜ, hs, by simp⟩⟩
exact this ▸ h.image compl
let f : Set (Set (Set α)) → Set α := fun A ... | false | {
"chosen_tokens": 246,
"rejected_tokens": 5,
"token_jaccard": 0.025974,
"token_length_ratio": 0.020325
} | sorry_admit_sorryAx_proof_hole | sorryAx | 6cf4f6b3bd66463364c12fdd1fd2931408a93654374ac8b5fad8735853c31a07 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finite.Prod
public import Mathlib.MeasureTheory.SetSemiring
Namespace:
MeasureTheory
Local context:
/-
Copyright (c) 2024 Etienne Marion. All rights res... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "MeasureTheory",
"family_id": "countable_generatesetalgebra",
"file_id": "mathlib/Mathlib/MeasureTheory/SetAlgebra.lean",
"sample_id": "c9b478f786784f9ac4583d69de5a7f68ad959e59303de1658982ea618e1c423e"
} | train |
{
"chosen_proof_sha256": "22533728b6f9b3f9588a530b9a97fe6bf76b4cb9f09b93a8510eba1da1bd326d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cc1a3c9a360e868e0958495a5e7eb164e9... | by
ext
apply Sieve.ext
simp [← NatTrans.naturality_apply] | false | {
"chosen_tokens": 13,
"rejected_tokens": 18,
"token_jaccard": 0.733333,
"token_length_ratio": 1.384615
} | markdown_preamble_trailing | trailing | 6cfda7002749ef81591ff3a1f47212d48e8594c71478dd59c276b52ab9c924ba | 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
ext
apply Sieve.ext
simp [← NatTrans.naturality_apply]
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Topos",
"family_id": "presheaf",
"file_id": "mathlib/Mathlib/CategoryTheory/Topos/Sheaf.lean",
"sample_id": "1709493a1e9d915fafb30165d87390b1f52391762f60fb9606bb3e33e9e6a8da"
} | train |
{
"chosen_proof_sha256": "82c27cb6e99a7d6e45601e4127587eb4a9e18558df87ff23a7cf9764fabc33f4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e35d2d0cef7ad182c12561cd06aa4dfc58... | by
simp [galRestrict', galRestrict', Subalgebra.algebraMap_eq] | true | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.090909,
"token_length_ratio": 0.272727
} | unknown_identifier | unknown_identifier | 6d1bd83e301130d85b93c07a713cf783b622b2e8156c7b756c1ee8e33ec8c0f7 | 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.IntegralClosure
public import Mathlib.RingTheory.RingHom.Finite
public import Mathlib.RingTheory.Localization.LocalizationLocalizatio... | by
exact __oprover_missing_12e4ed73b216 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "4b2c05a51566015007f9a673b4ce3036f25f1267fab174b95324e402b2aaeb03",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/IntegralClosure",
"family_id": "algebramap_galrestrict'_apply",
"file_id": "mathlib/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean",
"sample_id": "12e4ed73b2164c9034e09e737bb93b021152af848a112fdd9ed5bf5c79a07e22"
} | train |
{
"chosen_proof_sha256": "792cf305c49d49d53ed8828dada761a3db9769ab9dc9ac7edc92e61179edcded",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "dcee79aa8fce85c29907b9729bf9a202db... | by
constructor
· rw [Ne, eq_none_iff', not_not]
exact fun h => ⟨o.get h, eq_some_iff.2 (get_mem h)⟩
· rintro ⟨x, rfl⟩
apply some_ne_none | false | {
"chosen_tokens": 38,
"rejected_tokens": 43,
"token_jaccard": 0.852941,
"token_length_ratio": 1.131579
} | markdown_preamble_trailing | preamble | 6d243751e6f0f41d31118a5aac00c3dccfc5b19dde46d3cf95624ba14dfefdad | 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.Defs
public import Mathlib.Data.Set.Subsingleton
public import Mathlib.Logic.Equiv.Defs
Namespace:
Part
Local context:
/-
Copyright (c) 2017... | Here is the proof:
by
constructor
· rw [Ne, eq_none_iff', not_not]
exact fun h => ⟨o.get h, eq_some_iff.2 (get_mem h)⟩
· rintro ⟨x, rfl⟩
apply some_ne_none | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data",
"family_id": "ne_none_iff",
"file_id": "mathlib/Mathlib/Data/Part.lean",
"sample_id": "1d3d2dc551d90c30ba5f909bc69c9cc130fae372a274b0debe56a42386450ace"
} | train |
{
"chosen_proof_sha256": "242a43583e27fd27079936e1ce2e8cce7893c073e1ef4300e30f594b3e61646a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7af71cb8f41dbf3ff57b02e47f2e80d5b0... | by
rw [← Ideal.one_eq_top, ← Ideal.add_eq_sup, Ideal.add_eq_one_iff] at h
have ⟨y, hy, z, hz, eq⟩ := h
have ⟨p, hp⟩ := (mem_ideal_map_adjoin ..).mp hy
have ⟨w, hw⟩ := Ideal.mem_span_singleton.mp hz
have ⟨q, hq⟩ := adjoin_eq_exists_aeval R x w
use (1 - p - X * q).reverse
have : (1 - p - X * q).coeff 0 - 1 ... | true | {
"chosen_tokens": 179,
"rejected_tokens": 3,
"token_jaccard": 0.028571,
"token_length_ratio": 0.01676
} | circular_dependency | circular_dependency | 6d4a7398ba533895214e0e0d0c8ad8640a67dc0a9ce8c0c73669df7fa6879a03 | 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
exact exists_aeval_invOf_eq_zero_of_idealMap_adjoin_sup_span_eq_top | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "b5a3a42f11bd5da62fc5ded7564c9d18ded7fafced5bf19f6c3355b577846a38",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Polynomial",
"family_id": "exists_aeval_invof_eq_zero_of_idealmap_adjoin_sup_span_eq_top",
"file_id": "mathlib/Mathlib/RingTheory/Polynomial/Ideal.lean",
"sample_id": "91739cefd9b24c38b935a570789d9ce5a71a9f8d6f001da414f1877193c9705f"
} | train |
{
"chosen_proof_sha256": "e6c5467020ffc4f18638257b26b5e8e286b407b28f63974096a991066b7eb6a4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by rw [← div_eq_mul_inv, ← h3, mul_div_cancel_right₀ _ h2] | false | {
"chosen_tokens": 14,
"rejected_tokens": 2,
"token_jaccard": 0.076923,
"token_length_ratio": 0.142857
} | by_exact_placeholder | by? | 6d69c8225815643fa303e0d735cf6393b0272037e38458be8ac8c5582fe33312 | 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.Basic
public import Mathlib.Algebra.Order.Ring.Defs
public import Mathlib.Data.Tree.Basic
public import Mathlib.Tactic.NormNum.Core
public import... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Tactic/CancelDenoms",
"family_id": "inv_subst",
"file_id": "mathlib/Mathlib/Tactic/CancelDenoms/Core.lean",
"sample_id": "6be9545aedd7e6b94e2cba861b00ad8b06f63ac214c443b7f533604392c9fc8e"
} | train |
{
"chosen_proof_sha256": "de3f7865b87871f2b8374f7e15ee4575771052e51846b14aedf16b2bb0b0042a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "84c9e058563163df6246bc78ed6fd10279... | by
ext ⟨a, b⟩
simp only [mem_smulAntidiagonal, mem_singleton, Prod.ext_iff]
constructor
· rintro ⟨has, hat, hst⟩
obtain rfl :=
(hs.min_le hns has).eq_of_not_lt fun hlt =>
(SMul.smul_lt_smul_of_lt_of_le hlt <| ht.min_le hnt hat).ne' hst
exact ⟨rfl, IsCancelSMul.left_cancel _ _ _ hst⟩
· ri... | true | {
"chosen_tokens": 92,
"rejected_tokens": 3,
"token_jaccard": 0.042553,
"token_length_ratio": 0.032609
} | unknown_identifier | unknown_identifier | 6d87707a649919115a07f8f5acedbd5228641db0b92c86df64af70fff9b032d2 | 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.Pointwise.Set.Scalar
public import Mathlib.Data.Set.SMulAntidiagonal
Namespace:
Finset
Local context:
/-
Copyright (c) 2024 Scott Carnahan. All... | by
exact __oprover_missing_986207edb34e | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "b8f654f6a70a7bd4a9b24154e4351207ca984629053ebbeed0411d9c84244e40",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Data/Finset",
"family_id": "smulantidiagonal_min_smul_min",
"file_id": "mathlib/Mathlib/Data/Finset/SMulAntidiagonal.lean",
"sample_id": "986207edb34e921fae14ac5de27f5721ed1936c5cf23f19974ccfb40895a7398"
} | train |
{
"chosen_proof_sha256": "b4cbd01d6c035dc37c5d6b8fd96656834089520d56a98eb2574c117f82f08ca7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simp [dens, card_biUnion h, sum_div] | false | {
"chosen_tokens": 10,
"rejected_tokens": 2,
"token_jaccard": 0.1,
"token_length_ratio": 0.2
} | by_exact_placeholder | by? | 6dcdcdafa5a2d032828cdc4f1d2da66942f09f7e14672a629716a2f9019355e8 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Ring.Finset
public import Mathlib.Algebra.Field.Defs
public import Mathlib.Data.Finset.Density
Namespace:
Finset
Local context:
/-
Copyr... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/BigOperators",
"family_id": "dens_biunion",
"file_id": "mathlib/Mathlib/Algebra/BigOperators/Field.lean",
"sample_id": "2e83f293bd34554caad077053dfa4b4856eabf4ae072c64cf35623b00558a124"
} | train |
{
"chosen_proof_sha256": "faf6bc027aba61a3300a23f2d313e28567ba4f3a90cdcf2560518c60032b7b01",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
classical
let ι := Module.Free.ChooseBasisIndex R S
let b : Module.Basis ι R S := Module.Free.chooseBasis R S
rw [trace_eq_matrix_trace b, trace_eq_matrix_trace (basisQuotient b), AddMonoidHom.map_trace]
congr 1
ext i j
simp only [leftMulMatrix_apply, coe_lmul_eq_mul, LinearMap.toMatrix_apply,
basi... | false | {
"chosen_tokens": 75,
"rejected_tokens": 3,
"token_jaccard": 0.022222,
"token_length_ratio": 0.04
} | by_exact_placeholder | placeholder | 6e32e5345b8e94445783f5568fecd3d7e871363a271f7662638cf3c0863140c1 | 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.Dvr
public import Mathlib.RingTheory.IntegralClosure.IntegralRestrict
public import Mathlib.RingTheory.LocalRing.Quotient
public impo... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Trace",
"family_id": "algebra",
"file_id": "mathlib/Mathlib/RingTheory/Trace/Quotient.lean",
"sample_id": "bb2be5794bc2bef47d57ee2509f8e94d0f35879b91907a617a92bba8e4b1af69"
} | train |
{
"chosen_proof_sha256": "03e21e37518786f0e8f480f42a07a2927cdb9237ca44f9d215146d88cd8ce9cd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
classical
obtain ⟨s, hs⟩ := hI
rw [← adjoin_monomial_eq_reesAlgebra, ← hs]
use s.image (monomial 1)
rw [Finset.coe_image]
change
_ =
Algebra.adjoin R
(Submodule.map (monomial 1 : R →ₗ[R] R[X]) (Submodule.span R ↑s) : Set R[X])
rw [Submodule.map_span, Algebra.adjoin_s... | true | {
"chosen_tokens": 83,
"rejected_tokens": 8,
"token_jaccard": 0.066667,
"token_length_ratio": 0.096386
} | unsolved_goals | unsolved_goals | 6e49e2cd2078731611b7de506027505ec3560f5e845d759c9814e918939b53bb | 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.Ideal.BigOperators
public import Mathlib.RingTheory.FiniteType
Namespace:
(root)
Local context:
/-
Copyright (c) 2022 Andrew Yang. All rights rese... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "c25a7977c628cdee59a2d045b1399722135e60f2926bfdebed852c938c348ed7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory",
"family_id": "reesalgebra",
"file_id": "mathlib/Mathlib/RingTheory/ReesAlgebra.lean",
"sample_id": "7f63bf8b080e808117284c406a2db8ef2f14a5c00152c31fb40b5d94668e2b76"
} | train |
{
"chosen_proof_sha256": "5a49187c184286dd02eeb6ca07b11443ba36cdee6d43f2c64edd8e661703bcce",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
simp only [mem_Icc, insertNth_le_iff, le_insertNth_iff, and_assoc, @and_left_comm (x ≤ q₂ i)] | false | {
"chosen_tokens": 22,
"rejected_tokens": 2,
"token_jaccard": 0.05,
"token_length_ratio": 0.090909
} | sorry_admit_sorryAx_proof_hole | admit | 6e66c6ac5795dde5aedc215ef6dfefc646ef4b569df750d7e7f42ed20af8e789 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Logic.Equiv.Fin.Basic
public import Mathlib.Order.Fin.Basic
public import Mathlib.Order.PiLex
public import Mathlib.... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/Fin",
"family_id": "insertnth_mem_icc",
"file_id": "mathlib/Mathlib/Order/Fin/Tuple.lean",
"sample_id": "394f27a561e1c227c4d6a3e91cad146754404323a259cba6efddc68715f8ebd8"
} | train |
{
"chosen_proof_sha256": "d99c571d1e3b54f4a7c947a3bff9621121704efdf01dd61bab5cf6b4a711ab1c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "116f8e1c9d0741f93e3292389186ed75e7... | by
simp [Functor.IsDense.leftKanExtensionIso] | true | {
"chosen_tokens": 9,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.333333
} | unknown_identifier | unknown_identifier | 6e786572ddae818e696bb13a8b65b1f22421c160ee57a1f6ef4b8466e3d2066a | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Functor.KanExtension.DenseAt
public import Mathlib.CategoryTheory.Limits.Presheaf
public import Mathlib.CategoryTheory.Generator.StrongGenerator... | by
exact __oprover_missing_052befed7742 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "5b40bdcd560b2e2de5bc2efee713385c2062a2b20c9d229094e132677744d401",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Functor",
"family_id": "isdense",
"file_id": "mathlib/Mathlib/CategoryTheory/Functor/KanExtension/Dense.lean",
"sample_id": "052befed77426c858368b90b76b9540e8aaef3d27651275eef8d687539481881"
} | train |
{
"chosen_proof_sha256": "3f320a505bb6b269ad39068f91895095fa32692c32719fd89a78bd89e6e3c623",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
classical
rw [LinearMap.trace_eq_matrix_trace 𝕜 b.toBasis T]
apply Fintype.sum_congr
intro i
rw [Matrix.diag_apply, T.toMatrix_apply, b.coe_toBasis, b.coe_toBasis_repr_apply,
b.repr_apply_apply]
variable [FiniteDimensional 𝕜 E]
variable {n : ℕ} (hn : Module.finrank 𝕜 E = n) | false | {
"chosen_tokens": 64,
"rejected_tokens": 5,
"token_jaccard": 0.02381,
"token_length_ratio": 0.078125
} | sorry_admit_sorryAx_proof_hole | sorryAx | 6eb085285944923e9453294de851a9b54c0f06217cb06f0d3d3c6d771c53e371 | 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.InnerProductSpace.PiL2
public import Mathlib.Analysis.InnerProductSpace.Spectrum
public import Mathlib.LinearAlgebra.Eigenspace.Charpoly
Namespace:
L... | 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": "Analysis/InnerProductSpace",
"family_id": "trace_eq_sum_inner",
"file_id": "mathlib/Mathlib/Analysis/InnerProductSpace/Trace.lean",
"sample_id": "308f6cf4c900ea2a1076236867cd7cbcb72edd4a9f392f88b684001d707ae5c6"
} | train |
{
"chosen_proof_sha256": "9346ea21709b5b4adcb76342b23a0fdfec46476e1c52d71821429c4b741b1882",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
obtain ⟨f, g, h⟩ := exists_pair_ne (α → β)
rw [ne_eq, funext_iff, Classical.not_forall] at h
obtain ⟨a, h⟩ := h
exact nontrivial_of_ne _ _ h | false | {
"chosen_tokens": 41,
"rejected_tokens": 3,
"token_jaccard": 0.068966,
"token_length_ratio": 0.073171
} | by_exact_placeholder | placeholder | 6f1a288c07c4d96a8b085094e2664682df61e706471a1c4c26fb17690de98816 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Tactic.Push.Attr
Namespace:
(root)
Local context:
/-
Copyright (c) 2020 Sébastien Gouëzel. 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": "Logic/Nontrivial",
"family_id": "function",
"file_id": "mathlib/Mathlib/Logic/Nontrivial/Defs.lean",
"sample_id": "3d58730a822f37dbe124ac3de8532bdf9e6b3a34db752189dda5130b2510e7eb"
} | train |
{
"chosen_proof_sha256": "df3d8d4ac17ee91cfc6c4a0a0a21cbff1216a809c22aaf032985f2bc0080f0a1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simpa using coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal' (I := ⊤) (f := f) (by simp) hg
variable [I.IsTwoSided] | false | {
"chosen_tokens": 25,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.08
} | by_exact_placeholder | by? | 6f35ce548dc8cebee7b14d3ad77827f59343ccdc4881d45ead3d8bd8a7afef75 | 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.Operations
public import Mathlib.RingTheory.Ideal.BigOperators
public import Mathlib.RingTheory.PowerSeries.Basic
Namespace:
PowerSeries
Loc... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/PowerSeries",
"family_id": "coeff_mul_mem_ideal_of_coeff_right_mem_ideal",
"file_id": "mathlib/Mathlib/RingTheory/PowerSeries/CoeffMulMem.lean",
"sample_id": "502c601dc704b016dbc827a9cd8efa7a22333a98a499271c49028df90994ce61"
} | train |
{
"chosen_proof_sha256": "7aefead92a5663164517806ecfcb8b153d186d51071288da4cb411af6d57db41",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
ext v
simp [alternatizeUncurryFin_apply, ← map_insertNth]
variable {S : Type*} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] | false | {
"chosen_tokens": 32,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.0625
} | by_exact_placeholder | by? | 6f49b5efa959dbcff18e70db8cebfd582f9f967896300069799c9f6e115ef064 | 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.Alternating.Curry
public import Mathlib.GroupTheory.Perm.Fin
public import Mathlib.Data.Fin.Parity
Namespace:
AlternatingMap
Local context:
/-
... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Alternating",
"family_id": "alternatizeuncurryfin_curryleft",
"file_id": "mathlib/Mathlib/LinearAlgebra/Alternating/Uncurry/Fin.lean",
"sample_id": "c860402497f1b4dc3bdc93021a1e3e3c1ea1267241369ebbfdacfe5570fa2d4c"
} | train |
{
"chosen_proof_sha256": "74e6e717b4029a5bb695646d264a47c3c431da50efbba97f42340e634e0683b0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a15d943c4f0d40defc381407b0abb8cc65... | by
induction hs using Nonempty.cons_induction with
| singleton => simp [Hlt _ (mem_singleton_self _)]
| cons _ _ _ _ ih =>
simp only [sum_cons, forall_mem_cons] at Hlt ⊢
exact ENat.add_lt_add Hlt.1 (ih Hlt.2) | true | {
"chosen_tokens": 51,
"rejected_tokens": 2,
"token_jaccard": 0.030303,
"token_length_ratio": 0.039216
} | invalid_tactic | invalid_tactic | 6f6f8ec55d38042bdf3b4e1f651d62d1db40bd83d50a642fce317a22f72546ce | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Ring.Finset
public import Mathlib.Algebra.BigOperators.WithTop
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import... | by
__oprover_invalid_tactic_7736e8daaa8f | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "18bbe7497667610435b98cd62fe97151305f047eb40be51b061886d780052244",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/ENat",
"family_id": "sum_lt_sum_of_nonempty",
"file_id": "mathlib/Mathlib/Data/ENat/BigOperators.lean",
"sample_id": "7736e8daaa8f61cdcc7d061cb79e4a68846e752fbc7a81c150325816b7000c0c"
} | train |
{
"chosen_proof_sha256": "19d0436be488b2c04681a3df70e828f641be531fc0e4601e74671ee295a63fea",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "61f6ec391c59827903fc83f292d6ac8511... | by
rwa [H.ker_flip] | false | {
"chosen_tokens": 7,
"rejected_tokens": 11,
"token_jaccard": 0.7,
"token_length_ratio": 1.571429
} | missing_wrong_import | import | 6f972500216fbf0815cf77e393bf26fd1f42532696f74defa0acb35d57a24fb4 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Basis.Basic
public import Mathlib.LinearAlgebra.BilinearMap
public import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
import Mathlib.Algebra.M... | import Mathlib.WrongImport
by
rwa [H.ker_flip] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/SesquilinearForm",
"family_id": "ker_flip_eq_bot",
"file_id": "mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Basic.lean",
"sample_id": "08fc9f4a5a976873fc6fe96edc2ccf033e6ac6f6554e9759e87483072bcf261d"
} | train |
{
"chosen_proof_sha256": "fecffccc9af025141c64f561cc932cc9a7f6f3a1d76dd0660a012426d4c05839",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e4758bbc670928f41cb8dee16caeed3cac... | by
unfold xgcdAux
exact if_pos rfl | false | {
"chosen_tokens": 6,
"rejected_tokens": 11,
"token_jaccard": 0.545455,
"token_length_ratio": 1.833333
} | markdown_preamble_trailing | preamble | 6fa57c5f95bea5a1821df9c84c74977a6a7bf530eba351a423f25381c11f9773 | 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.Ring.Defs
public import Mathlib.Order.RelClasses
Namespace:
EuclideanDomain
Local context:
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
R... | Here is the proof:
by
unfold xgcdAux
exact if_pos rfl | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/EuclideanDomain",
"family_id": "xgcd_zero_left",
"file_id": "mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean",
"sample_id": "c3604586f2ac7368d223dc67d1199e013bd8135ef158fafca76c177f59a92d99"
} | train |
{
"chosen_proof_sha256": "a18d962dbf46cb1bc44d7afa74d89d68066f49e5784c2cb2792cd31f2ea27cd6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c4e84a42ec28d598074201ae4fc19260b8... | by
rw [H.apply_zero_path, H.apply_one_path, H.evalAt_eq]
erw [H.evalAt_eq]
dsimp only [prodToProdTopI]
constructor
· slice_lhs 2 4 => rw [eqToHom_trans, eqToHom_refl] -- Porting note: this ↓ `simp` didn't do this
slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]
rfl
· slice_lhs 2 4 => rw [eq... | false | {
"chosen_tokens": 103,
"rejected_tokens": 107,
"token_jaccard": 0.925,
"token_length_ratio": 1.038835
} | missing_wrong_import | import | 6faf74c3d717469da6af4b5b5a725c7cba7e6e3b58ed15f98e6e3b270dd9bb91 | 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.Homotopy.Equiv
public import Mathlib.CategoryTheory.Equivalence
public import Mathlib.AlgebraicTopology.FundamentalGroupoid.Product
Namespace:
Contin... | import Mathlib.WrongImport
by
rw [H.apply_zero_path, H.apply_one_path, H.evalAt_eq]
erw [H.evalAt_eq]
dsimp only [prodToProdTopI]
constructor
· slice_lhs 2 4 => rw [eqToHom_trans, eqToHom_refl] -- Porting note: this ↓ `simp` didn't do this
slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]
rfl
... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/FundamentalGroupoid",
"family_id": "eq_diag_path",
"file_id": "mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean",
"sample_id": "5b6616e952970d213dc2af95acd62437ab9f4ee924f0692e7518320229a74eaa"
} | train |
{
"chosen_proof_sha256": "ebb83eb68b98b09837a3cd4cf60703741f66ac05dee2a408489caf38cd22110c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "dc649d8b167a8fedb6377057872d96f163... | by
unfold rieszContentAux
gcongr
apply rieszContentAux_image_nonempty | false | {
"chosen_tokens": 6,
"rejected_tokens": 11,
"token_jaccard": 0.545455,
"token_length_ratio": 1.833333
} | markdown_preamble_trailing | preamble | 6fccc1678e04898a08c4493ae77f0a3f76d5c5a2fc5225d76d1f7b3a4516a559 | 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.Measure.Content
public import Mathlib.Topology.ContinuousMap.CompactlySupported
public import Mathlib.Topology.PartitionOfUnity
Namespace:
(root... | Here is the proof:
by
unfold rieszContentAux
gcongr
apply rieszContentAux_image_nonempty | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Integral",
"family_id": "rieszcontentaux_mono",
"file_id": "mathlib/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Basic.lean",
"sample_id": "9fd02cd1275b92cba5b435d3686fa58fec1fc7b2cad79d852c9e074db6daabbc"
} | train |
{
"chosen_proof_sha256": "eaff7377a9ad125a2190653d93c4626decbfd214d6bde989ce3ba7f1259b5118",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3c420338d39a0c37f216d1672eac3275cd... | by
cases nonempty_fintype α
simpa using exists_max_image univ f univ_nonempty | false | {
"chosen_tokens": 10,
"rejected_tokens": 14,
"token_jaccard": 0.714286,
"token_length_ratio": 1.4
} | missing_wrong_import | import | 6ff9f9b5eb5c04889ef76297a1c45df506b92a7c44d0594d5ccc0146939ab879 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finset.Max
public import Mathlib.Data.Fintype.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released... | import Mathlib.WrongImport
by
cases nonempty_fintype α
simpa using exists_max_image univ f univ_nonempty | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Fintype",
"family_id": "finite",
"file_id": "mathlib/Mathlib/Data/Fintype/Lattice.lean",
"sample_id": "b748ffc79ac3338e3482458996caeaad3f245a010466e177b0f35271384d2cf9"
} | train |
{
"chosen_proof_sha256": "08218aabda46401fe50c2dc9de3ff5011d981b5e7f148f3d58bd8c99abd66739",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0848e9cfd63256eb77b8f8cfb09985072e... | 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 _ _⟩ | true | {
"chosen_tokens": 40,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.075
} | unknown_identifier | unknown_identifier | 705afe824e58477343667ebd317fe2449f001c9c760fd1a8bc5c8e69a42ac565 | 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.FundamentalGroupoid.FundamentalGroup
public import Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps
public import Mathlib.Topology.H... | by
exact __oprover_missing_631dd733f339 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "60fd3419fb59c04f92b71f03cdaee0d354b07e39135ac3737b9444ffa36e397a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "04ff734dba3e12dd755f41758fae7a61d3a513148d7d550f593f25ac71b54468",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
subst coeff_eq deg_eq_deg coeff_eq_deg
exact natDegree_eq_of_le_of_coeff_ne_zero ‹_› ‹_› | true | {
"chosen_tokens": 13,
"rejected_tokens": 8,
"token_jaccard": 0.0625,
"token_length_ratio": 0.615385
} | unsolved_goals | unsolved_goals | 709ab7efafab3645b7ee4c0887f3218b55c26e2bdcfad4201e1b9808c2c18b1f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.Degree.Lemmas
Namespace:
Mathlib.Tactic.ComputeDegree
Local context:
/-
Copyright (c) 2023 Damiano Testa. All rights reserved.
Released un... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "81ae73f5c87bdbe878e9591419658790651faf62d37a64e02b720b209863c8b7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic",
"family_id": "natdegree_eq_of_le_of_coeff_ne_zero",
"file_id": "mathlib/Mathlib/Tactic/ComputeDegree.lean",
"sample_id": "9e767d0b834339320d21f2fe6faa7f7606cdac3e6fc203de41750386839ebf3b"
} | train |
{
"chosen_proof_sha256": "bb1aa1df4a3618112e80043cd006efbcdf4906086196e285cd31ba1676129c32",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "46f6f8415ade511d6c5719e771f8a7d1ca... | by
rw [← closure_openCell_eq_closedCell, E.closed.closure_subset_iff, ← E.union]
apply subset_union_of_subset_right
exact subset_iUnion_of_subset n
(subset_iUnion (fun (j : ↑(E.I n)) ↦ openCell (C := C) n j) ⟨i, hi⟩) | true | {
"chosen_tokens": 53,
"rejected_tokens": 2,
"token_jaccard": 0.029412,
"token_length_ratio": 0.037736
} | invalid_tactic | invalid_tactic | 70d262f3eb7cb27a461d247c90f663d95aa2f2b97c4a989d8ee137bbf304e2cb | 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
__oprover_invalid_tactic_26a28f745aa7 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "baf62482116c32afa88114c6c653d725b96cd4611e7a0263d244159888447578",
"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": "26a28f745aa7034105aa52a93b588fb63e8e92ad6d8d99ffd3be3558367e64ea"
} | train |
{
"chosen_proof_sha256": "4fff200b3969b2099ac39b5df9f12ed101d4778e55341e30c66e404ab2007cba",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8c9c68d8ea4801f822185f3282b0bfffb0... | by
obtain ⟨k, rfl⟩ := hn
rw [pow_add, pow_one, pow_mul, sq]
cases s <;> simp | true | {
"chosen_tokens": 25,
"rejected_tokens": 5,
"token_jaccard": 0.038462,
"token_length_ratio": 0.2
} | wrong_namespace_qualification | wrong_namespace_qualification | 7108b244e01f4fb241f31eea24ddfb5904edd1be2ffc2742e3ece4ab09afa418 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GroupWithZero.Units.Lemmas
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import Mathlib.Algebra.Order.Ring.Cast
public import Ma... | by
exact OProverMissing.missing_68d51948bcb2 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "68f76d11d3c9ee5a1dd860c2a62b6cc335392300d9d976b7d8b17ab7ba24d5bc",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Sign",
"family_id": "pow_odd",
"file_id": "mathlib/Mathlib/Data/Sign/Basic.lean",
"sample_id": "68d51948bcb20f9102720ae7d20bb860c7d2a1e39fd0c3d403bf97713844dd99"
} | train |
{
"chosen_proof_sha256": "2fa6d6384b3fc37bc32726c7d0dfd505d0e32dc595d3c34173c19db9a5d115ee",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
have := IsClosedUnderBinaryCoproducts.closedUnderIsomorphisms P
have := hasFiniteCoproducts_of_has_binary_and_initial (C := P.FullSubcategory)
have := PreservesFiniteCoproducts.of_preserves_binary_and_initial P.ι
exact ⟨fun J _ ↦ P.isClosedUnderColimitsOfShape_of_preservesColimitsOfShape_ι _⟩ | false | {
"chosen_tokens": 37,
"rejected_tokens": 5,
"token_jaccard": 0.12,
"token_length_ratio": 0.135135
} | sorry_admit_sorryAx_proof_hole | sorryAx | 710c3ab28a5b75a6174a98a9fd32910d4685cf896c0fb0b79049f549c7563792 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
public import Mathlib.CategoryTheory.Limits.FullSubcategory
public import Mathlib.CategoryTh... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/ObjectProperty",
"family_id": "isclosedunderfinitecoproducts",
"file_id": "mathlib/Mathlib/CategoryTheory/ObjectProperty/FiniteProducts.lean",
"sample_id": "d26cc5be978dead2b623c3a4790cc21af0d92aeaa8715f5d06e969086e47e5d8"
} | train |
{
"chosen_proof_sha256": "8beab1ef316edfb8523c7f6d8a094e22bcc81e68fa42d1a455d344f9254b12c0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "51c303a48a29878142e6b9a5fc566752c5... | by
simp only [MonomialOrder.lex_lt_iff, Finsupp.Lex.lt_iff_of_unique, ofLex_toLex] | true | {
"chosen_tokens": 16,
"rejected_tokens": 5,
"token_jaccard": 0.125,
"token_length_ratio": 0.3125
} | wrong_namespace_qualification | wrong_namespace_qualification | 7111c363f29e854ab2c68ba9341c8701557619ead5eeb6de8c8487bf22dfc307 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finsupp.Lex
public import Mathlib.Data.Finsupp.WellFounded
public import Mathlib.Data.List.TFAE
Namespace:
(root)
Local context:
/-
Copyright (c) 2024 A... | by
exact OProverMissing.missing_be98270d4e64 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a102d739eac1f3dd7ed21bb61800a3923f35ee6d358e86e1001f89e0dc92a11a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Finsupp",
"family_id": "monomialorder",
"file_id": "mathlib/Mathlib/Data/Finsupp/MonomialOrder.lean",
"sample_id": "be98270d4e641660001524cb4a6a1e9cfae5fb603cff7ae0acf7c6d8fed02aa2"
} | train |
{
"chosen_proof_sha256": "9a5efd93ec86ca2b78ac7e681d48ea70e75ceb2212f655294f2322b5b525cad8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
apply isCyclic_of_card_dvd_prime (p := 2)
simpa [card_perm] using factorial_dvd_factorial hα | false | {
"chosen_tokens": 16,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.1875
} | by_exact_placeholder | exact? | 711b93fe2d8be1452273f6c56fd69626c7ce8427b002529cf474d9ffa2cbae6d | 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.Perm
public import Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic
public import Mathlib.SetTheory.Cardinal.Finite
Namespace:
Equiv.Perm
Local c... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Finite",
"family_id": "iscyclic_of_card_le_two",
"file_id": "mathlib/Mathlib/Data/Finite/Perm.lean",
"sample_id": "aad3a3baeb6dacb034a541f0080f4a6bbc685cc200a408a1414f15d0a2cae0b7"
} | train |
{
"chosen_proof_sha256": "ebb5a2a7a52df61da4119d7165c562d64bfda64efb43081a7067ed8d5938555e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [toComplex_def₂, toComplex_def₂]
exact congr_arg₂ _ rfl (Int.cast_neg _) | false | {
"chosen_tokens": 20,
"rejected_tokens": 2,
"token_jaccard": 0.058824,
"token_length_ratio": 0.1
} | sorry_admit_sorryAx_proof_hole | admit | 714c12810245961ba2256119eadc3a82adcdab15bb591b97a452458e355e4255 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Archimedean.Real.Basic
public import Mathlib.Data.Complex.Basic
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.NumberTheory.Zsq... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/Zsqrtd",
"family_id": "tocomplex_star",
"file_id": "mathlib/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean",
"sample_id": "40b2cc76633014d1e5dccc9d2cd9aa43dc95f8c9291dfcfb08d96f928e0a655e"
} | train |
{
"chosen_proof_sha256": "a6a0730acc445e18a71efd0b82b6e6d57aa672e42e9dfc4ec8e49f142e23c559",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree, ← not_exists]
simp only [hr, not_true, false_or, and_false] | true | {
"chosen_tokens": 19,
"rejected_tokens": 8,
"token_jaccard": 0.05,
"token_length_ratio": 0.421053
} | unsolved_goals | unsolved_goals | 723b7a339aad6bb628855fecf2ce6e5f1cac375832de65f03ff840d6326a56cc | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Coprime.Lemmas
public import Mathlib.RingTheory.Nilpotent.Basic
public import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid
public import M... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "3eeb917ca4946ca631feaf1139eeb44cb63e6768c58d836ebc2806e74847831d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Squarefree",
"family_id": "squarefree_iff_irreducible_sq_not_dvd_of_exists_irreducible",
"file_id": "mathlib/Mathlib/Algebra/Squarefree/Basic.lean",
"sample_id": "510546c198df0e51ff7856781031b1d5536b836338687232036e8eccc1052100"
} | train |
{
"chosen_proof_sha256": "5e5876ae05a155ef3db2c120345927956dab48f54ef044b4e48039170ed59c7d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
cases f; cases g
dsimp at hfg₁ hfg₂
cat_disch | false | {
"chosen_tokens": 13,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.153846
} | sorry_admit_sorryAx_proof_hole | admit | 724348b5bba5c16b637bfc2a008231021f39e73a6adab3a9af6059f65f56fdab | 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.Bicategory.LocallyDiscrete
public import Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo
Namespace:
CategoryTheory.Pseudofunctor... | 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/Bicategory",
"family_id": "hom",
"file_id": "mathlib/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean",
"sample_id": "b8a6fc429e8fab84bafaec67f5918625193c8c32c8b4d06caaef778ad29f2b44"
} | train |
{
"chosen_proof_sha256": "61c7560cab56bf0e5d3f545a1a32c8f50733fcb31780bdea8dc6b7263c7d7328",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9d2bf5ac2ac3150f033b3aa1e7ec2bd054... | by
haveI : NumberField K := @NumberField.mk _ _ _ (IsCyclotomicExtension.finiteDimensional {n} ℚ K)
have H₁ : (aeval (hζ.powerBasis ℚ).gen) (X - 1 : ℤ[X]) = (hζ.subOnePowerBasis ℚ).gen := by simp
have H₂ : (aeval (hζ.subOnePowerBasis ℚ).gen) (X + 1 : ℤ[X]) = (hζ.powerBasis ℚ).gen := by simp
refine discr_eq_disc... | true | {
"chosen_tokens": 211,
"rejected_tokens": 3,
"token_jaccard": 0.036364,
"token_length_ratio": 0.014218
} | unknown_identifier | unknown_identifier | 729481b6fc06b267d2b7f4a5aa1137432021d685ce9385836a75b793169fa0f3 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
public import Mathlib.RingTheory.DedekindDomain.Dvr
public import Mathlib.NumberTheory.NumberField.Discriminant.Defs
Na... | by
exact __oprover_missing_35877c6cf79f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "3634919b7b582a179393e98c6e2abeeeaffdb548451d95d5ed3d333ebbffe312",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Cyclotomic",
"family_id": "discr_zeta_eq_discr_zeta_sub_one",
"file_id": "mathlib/Mathlib/NumberTheory/Cyclotomic/Discriminant.lean",
"sample_id": "35877c6cf79fa66db90e5991ad29f2b3f093444f285a8e92079a5bb760e5281a"
} | train |
{
"chosen_proof_sha256": "097944a1a41dd54c4d08892020794d36ebd2632227e02e3aff759ac09d3dd6c6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f7b0ddcb6f37e2b9d3ae884fa2def989d9... | by
haveI : Nonempty ι := Fintype.card_pos_iff.mp (hn.symm ▸ Nat.succ_pos _)
haveI := Classical.decEq ι
-- Choose an arbitrary element `x : ι` and `y : t x`.
let x := Classical.arbitrary ι
have tx_ne : (t x).Nonempty := by
rw [← Finset.card_pos]
calc
0 < 1 := Nat.one_pos
_ ≤ #(.biUnion {x} ... | true | {
"chosen_tokens": 367,
"rejected_tokens": 2,
"token_jaccard": 0.008197,
"token_length_ratio": 0.00545
} | invalid_tactic | invalid_tactic | 72b3eb706509af3ab55c19e72abe4d777c44a86a4af0ab6e578833c623d9f6e8 | 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
__oprover_invalid_tactic_b15d779c1537 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "79b6cb0d082e5f0f9bb4a732a80eef3002e5fe15b6e3012c3b4a72cb5a112dcf",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/Hall",
"family_id": "hall_hard_inductive_step_a",
"file_id": "mathlib/Mathlib/Combinatorics/Hall/Finite.lean",
"sample_id": "b15d779c153730a548af1bcb1242b64da3cdb5f65c04ae8beeb21d386329dcf8"
} | train |
{
"chosen_proof_sha256": "d69154e0d9492b7c81193d7944884dcb10496ed41d6438469a8fe12e15c51a76",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "214a60175ac5e85f8cd7bac259c887da20... | by
refine ⟨fun h => ?_, fun ⟨⟨a', m, e⟩, s⟩ => Subset.cons e m s⟩
generalize h' : Lists'.cons a l₁ = l₁' at h
obtain - | @⟨a', _, _, _, e, m, s⟩ := h
· cases a
cases h'
cases a; cases a'; cases h'; exact ⟨⟨_, m, e⟩, s⟩ | true | {
"chosen_tokens": 91,
"rejected_tokens": 3,
"token_jaccard": 0.054054,
"token_length_ratio": 0.032967
} | circular_dependency | circular_dependency | 72e9a62dc1a3921b53bf4b1f349c404504d34a92aad7fdb44286311051c8eb06 | 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 cons_subset | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "fbe7bf6cc9d05a9574443f0e80e225b84d5f7aa72975cbdc09b46367c805e5f4",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "SetTheory",
"family_id": "cons_subset",
"file_id": "mathlib/Mathlib/SetTheory/Lists.lean",
"sample_id": "6a8521660d4ef846f6eee10f1b6a7111248906a80446120366f799acf8507e31"
} | train |
{
"chosen_proof_sha256": "bc0a4d32839ad3481e8623af1efc75f8290857aa2e5f1191a4c4a6c87919cec5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7c154fd643bb633d6de220155a96922617... | 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 _ _) | true | {
"chosen_tokens": 53,
"rejected_tokens": 2,
"token_jaccard": 0.028571,
"token_length_ratio": 0.037736
} | invalid_tactic | invalid_tactic | 73038170996a2c96cef1dd4a9f719857b0936c52e8eeec00a9b44b293f174714 | 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.IsAdjoinRoot
public import Mathlib.RingTheory.LocalRing.Quotient
public import Mathlib.RingTheory.Smooth.Flat
public import Mathlib.RingTheory.Unram... | by
__oprover_invalid_tactic_0e6ca74f0691 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "32b73c6080d4ab0a707397b21cf0a8042dda92368c6bc7f07d843ae3b5656ff0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "d7e6aa18e95f9b73927d19d13b21d8b699befd36d25410b98b9a340426e1233b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b4e99ad4afa8ca8b8a63a7f500ca31ddf4... | by
obtain rfl | hn := eq_or_ne n 0
· simpa [ssubset_iff_subset_not_subset, hX₁, -Finset.subset_singleton_iff]
using! hX.not_subset_singleton
refine (pow_subset_pow_right hX₁ <| n.le_add_right _).ssubset_of_ne ?_
contrapose hXclosure with hXn
rw [← closure_pow (mod_cast hX₁) hn]
wlog hn₁ : n = 1
· si... | true | {
"chosen_tokens": 307,
"rejected_tokens": 3,
"token_jaccard": 0.018018,
"token_length_ratio": 0.009772
} | circular_dependency | circular_dependency | 732b975916060b779192d7889a54911aecf001716a59de5ad38fef92ada24267 | 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.Action.Pointwise.Finset
public import Mathlib.Algebra.Group.Subgroup.Pointwise
public import Mathlib.Data.Nat.SuccPred
Namespace:
Finset
Local ... | by
exact pow_ssubset_pow_succ_of_pow_ne_closure | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "06c8cac30534d0abea709c353e66c0f59f4a94fce155bf36b91bdfc8351f0811",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Geometry/Group",
"family_id": "pow_ssubset_pow_succ_of_pow_ne_closure",
"file_id": "mathlib/Mathlib/Geometry/Group/Growth/LinearLowerBound.lean",
"sample_id": "2f91e46e3433b86a1e3699a32e11802a7183cc285fe3179f472be8008081dcf8"
} | train |
{
"chosen_proof_sha256": "d4714964e668b31f0718a64704bfc3c462347f6aac0cd63c4a504fd6654a6944",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
subst s_card
simp [basis_apply]
variable (s : powersetCard I m) (t : powersetCard I n) | false | {
"chosen_tokens": 22,
"rejected_tokens": 3,
"token_jaccard": 0.052632,
"token_length_ratio": 0.136364
} | by_exact_placeholder | placeholder | 733441983324e5f25e03f63cd4ee9898fe2f501939be6abd620a64de9c0903e0 | 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.ExteriorAlgebra.Grading
public import Mathlib.LinearAlgebra.ExteriorPower.Basis
Namespace:
ExteriorAlgebra
Local context:
/-
Copyright (c) 2026... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/ExteriorAlgebra",
"family_id": "basis_apply_ofcard",
"file_id": "mathlib/Mathlib/LinearAlgebra/ExteriorAlgebra/Basis.lean",
"sample_id": "bcfe71f4669f39454a991e49045a95b381eb505965239b851eaa93006f10ef03"
} | train |
{
"chosen_proof_sha256": "a3b30f98b08bb268a88816f205f3d07c5c34faaa5cf59c628e0cdbe5b5a6a78c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "bf06058cb3cfa5ca349bc783b48b8f8528... | by
rw [mul_comm]
exact mul_left_not_lt b h | false | {
"chosen_tokens": 9,
"rejected_tokens": 16,
"token_jaccard": 0.818182,
"token_length_ratio": 1.777778
} | markdown_preamble_trailing | fence | 7355e5cb055628c8164914fff035164c22b437301bda2f2a7c3d8ff0b78a2e5f | 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.Ring.Defs
public import Mathlib.Order.RelClasses
Namespace:
EuclideanDomain
Local context:
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
R... | ```lean
by
rw [mul_comm]
exact mul_left_not_lt b h
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/EuclideanDomain",
"family_id": "mul_right_not_lt",
"file_id": "mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean",
"sample_id": "e87fe6db087248fc861a35d6fd880f370abae53489bf00891ff2b411146d68fa"
} | train |
{
"chosen_proof_sha256": "055208fdff31df9d8fb76bf05bb6a68b4de72a41035c9aa9cc8d72c2b3b25d67",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simp [Algebra.smul_def, ← nat_rawCast_2] | false | {
"chosen_tokens": 10,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.2
} | by_exact_placeholder | by? | 73abfdbe560cdbb26633ee5b11db55425d30641d69065182ca351928bf2bbeca | 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? | {
"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": "rat_ofnat_smul",
"file_id": "mathlib/Mathlib/Tactic/Algebra/Lemmas.lean",
"sample_id": "029e00c3f6a53264f7ab33ca9285b495e3986d3f7052bd853ac23e6d759b17de"
} | train |
{
"chosen_proof_sha256": "3d02c82bbfbe829ee560bc14711178f8e05f9f6e82051c0e8f4a63b14e8d38c0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8a646e481e3d4448b1b3d15a701d2ad658... | by
let φ (i : ι) := separatingFamily c (fun (_ : Unit) ↦ X) ⟨⟨⟩, i⟩
refine isSeparator_of_isColimit_cofan
(isSeparating_separatingFamily c (X := fun (_ : Unit) ↦ X) (by simpa using! hX))
(c := Cofan.mk (∐ φ) (fun ⟨_, i⟩ ↦ Sigma.ι φ i)) ?_
exact IsColimit.ofWhiskerEquivalence
(Discrete.equivalence (E... | true | {
"chosen_tokens": 104,
"rejected_tokens": 2,
"token_jaccard": 0.022222,
"token_length_ratio": 0.019231
} | invalid_tactic | invalid_tactic | 7440a937d4b1cfcf174438ecb381329039be73cb489f0b83c3846c61a1322b0f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Homology.Double
public import Mathlib.Algebra.Homology.HomologicalComplexLimits
public import Mathlib.CategoryTheory.Generator.Basic
Namespace:
Homolo... | by
__oprover_invalid_tactic_595e91c0da49 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e4be2211094dc4b10904b5e40f9d20199f28992057e828a5a9f8fa9f92cc629f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Generator",
"family_id": "isseparator_coproduct_separatingfamily",
"file_id": "mathlib/Mathlib/CategoryTheory/Generator/HomologicalComplex.lean",
"sample_id": "595e91c0da49b46a91c3741f0089b51f2875fc10f72a63c9a56e6dda6a7ca6c2"
} | train |
{
"chosen_proof_sha256": "9347b24919ebf6db54344437810a6552ffb014d358370aa936d28051c4570548",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "81c21537640f160fa2f5e82543efd90215... | by
rw [eq_comm, ← Ultrafilter.coe_le_coe]
change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α | s ∈ v } ∈ u
simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff,
mem_setOf_eq]
constructor
· intro h a ha
exact h _ ⟨ha, a, rfl⟩
· rintro h a ⟨xi, a, rfl⟩
exact ... | true | {
"chosen_tokens": 80,
"rejected_tokens": 3,
"token_jaccard": 0.041667,
"token_length_ratio": 0.0375
} | circular_dependency | circular_dependency | 74a9a8897f4eb65ee88d4ea73e28c5385dad876258bfe3af20d90f707977f21c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.Bases
public import Mathlib.Topology.DenseEmbedding
public import Mathlib.Topology.Connected.TotallyDisconnected
Namespace:
(root)
Local context:
/-... | by
exact ultrafilter_converges_iff | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "230a632b0943838c0947592f77cdba6e07961bd01d5f6c829d750204550d4ef3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Compactification",
"family_id": "ultrafilter_converges_iff",
"file_id": "mathlib/Mathlib/Topology/Compactification/StoneCech.lean",
"sample_id": "02e40ac0605619b9d35a8cf4395e31d8750e7cb631f2db38ba43f21adcd702b7"
} | train |
{
"chosen_proof_sha256": "f6f5bbf3230230f3a0428b8083f535c36ff227e3eca17b3d7dbf4a73cb1bc39d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [weaklyEtale_iff, weaklyEtale_iff, Module.Flat.ulift_left_iff, Module.Flat.ulift_right_iff]
congr!
conv_rhs => rw [← RingHom.Flat.ulift_iff.{u₁, u₂}]
rw [TensorProduct.lmul'_ulift, AlgHom.toRingHom_eq_coe, AlgHom.comp_toRingHom]
exact RingHom.Flat.comp_iff_of_bijective_right (Equiv.bijective _) | false | {
"chosen_tokens": 66,
"rejected_tokens": 5,
"token_jaccard": 0.081081,
"token_length_ratio": 0.075758
} | sorry_admit_sorryAx_proof_hole | sorryAx | 74d7bd6d85f89af9b663fd617b42617c14ad918b8743d30c81f1566277f2a0f5 | 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.RingHom.Flat
public import Mathlib.RingTheory.Etale.Basic
public import Mathlib.RingTheory.Smooth.Flat
Namespace:
Algebra.WeaklyEtale
Local contex... | 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": "RingTheory/Etale",
"family_id": "ulift_iff",
"file_id": "mathlib/Mathlib/RingTheory/Etale/Weakly.lean",
"sample_id": "ea51963a627d887e53897ff9660cd0137a6ca3fccf9a879fe29c6140902dfe65"
} | train |
{
"chosen_proof_sha256": "8c5d22170da6aa84840f7344a2c31c2097b39c782de19947a86a24798df1b587",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rintro ⟨f, a, hf⟩ ⟨g, b, hg⟩ rfl
congr
refine extensional_of_trichotomous_of_irrefl s fun x ↦ ?_
rw [← hf, hg] | false | {
"chosen_tokens": 33,
"rejected_tokens": 3,
"token_jaccard": 0.037037,
"token_length_ratio": 0.090909
} | by_exact_placeholder | placeholder | 760516a94ff607ee7433aa3e4fcfded531922f7507862fb87dfbc70c7ce7e6d9 | 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.Sum.Order
public import Mathlib.Order.Hom.Lex
public import Mathlib.Order.RelIso.Set
public import Mathlib.Order.UpperLower.Basic
public import Mathlib.Or... | 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": "Order",
"family_id": "torelembedding_injective",
"file_id": "mathlib/Mathlib/Order/InitialSeg.lean",
"sample_id": "c360ffd931955f77da36029b2c2c6057afeb7e861c5fc96268f3b26c2470d808"
} | train |
{
"chosen_proof_sha256": "d4e8af4e9759204cb08f477f781bb80c363e8cafacffe36b1eb6bdb2061a0215",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "19e049bd84946cd04ffe0d191841761afe... | by
refine ⟨?_, Submodule.projectionOnto_comp_surjective_of_exact hfg _ hmap⟩
rwa [LinearMap.coe_comp, Set.InjOn.injective_iff ↑(LinearMap.range f) _ subset_rfl]
simpa [← LinearMap.disjoint_ker_iff_injOn, ← hfg.linearMap_ker_eq] | false | {
"chosen_tokens": 46,
"rejected_tokens": 50,
"token_jaccard": 0.909091,
"token_length_ratio": 1.086957
} | missing_wrong_import | import | 764b930a80816f3f1fd9ac6872042589b47d9dafced7e4c496b48f970a1a9ce5 | 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.Exact.Basic
public import Mathlib.LinearAlgebra.Basis.Basic
public import Mathlib.LinearAlgebra.Projection
Namespace:
(root)
Local context:
/-
Copyri... | import Mathlib.WrongImport
by
refine ⟨?_, Submodule.projectionOnto_comp_surjective_of_exact hfg _ hmap⟩
rwa [LinearMap.coe_comp, Set.InjOn.injective_iff ↑(LinearMap.range f) _ subset_rfl]
simpa [← LinearMap.disjoint_ker_iff_injOn, ← hfg.linearMap_ker_eq] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Basis",
"family_id": "submodule",
"file_id": "mathlib/Mathlib/LinearAlgebra/Basis/Exact.lean",
"sample_id": "4cb237f6549cf75bf0312f10927aa64c8d7dbadae8e86cca221ffb114034bffb"
} | train |
{
"chosen_proof_sha256": "0e10a34c52b1aa6caddc8087552eba2fdbebe27ef42d67264d88c7c0b6ced53f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
induction H with
| basic h =>
cases h
· exact Or.inr ⟨Set.univ, cond.univ_mem, by simpa⟩
· exact Or.inl (by assumption)
| univ => exact Or.inl cond.univ_mem
| @inter T1 T2 _ _ h1 h2 =>
rcases h1 with (h | ⟨Q, hQ, rfl⟩) <;> rcases h2 with (i | ⟨R, hR, rfl⟩)
· exact Or.inl (cond.inter_mem h... | true | {
"chosen_tokens": 183,
"rejected_tokens": 8,
"token_jaccard": 0.015625,
"token_length_ratio": 0.043716
} | unsolved_goals | unsolved_goals | 766cf4c6cf29027f1ffcf23a849ba93aa65998d9f624cd26aee3b2f64cd5c511 | 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.Insert
public import Mathlib.Data.Set.Lattice
Namespace:
FiniteInter
Local context:
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Releas... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "dc1b773b5abda086c7650ef3df3463d8c51c37214f3423c2eaba91e3ec85e619",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Set",
"family_id": "finiteinterclosure_insert",
"file_id": "mathlib/Mathlib/Data/Set/Constructions.lean",
"sample_id": "b69b8cf11d81b81760b7e2d01790b8cb08e792859b8026925d68c9b27e00c6b1"
} | train |
{
"chosen_proof_sha256": "186eee1641eb21a1d1c5d6e7242727e956490464bf744e56de75eb2d692d16ff",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2fa13ea3ad40a789f283ed35615353cfe1... | by
rw [← singleton_vadd]; exact (h₁ x).add h₂ | true | {
"chosen_tokens": 17,
"rejected_tokens": 5,
"token_jaccard": 0.166667,
"token_length_ratio": 0.294118
} | wrong_namespace_qualification | wrong_namespace_qualification | 767026fd7fc5c319c882fca799ab7680c88ed05aaf0d594ac5bd4d9da2e47dd4 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
public import Mathlib.Algebra.Ring.Action.Pointwise.Set
public import Mathlib.Topology.Bornology.Basic
Namespace:
A... | by
exact OProverMissing.missing_88a199b64f05 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "60e06e1d1ea35c6f69ca0321d47428be6844b68ddbf92c23fef098d8d6c86f9f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Bornology",
"family_id": "vadd_absorbs",
"file_id": "mathlib/Mathlib/Topology/Bornology/Absorbs.lean",
"sample_id": "88a199b64f05b0ccb332981cf02b5c961ab80a822fd1dae1175ba359f20a3e82"
} | train |
{
"chosen_proof_sha256": "dbcf29961f4c4a120ac8530a8dcc28a30bfa6b6db329e1836f9e00211fe2e46d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f8430c85d561b56ecd3591aebb22bff125... | by
simpa [CochainComplex.of.d] using (inhomogeneousCochains A).d_comp_d n (n + 1) (n + 2) | false | {
"chosen_tokens": 27,
"rejected_tokens": 34,
"token_jaccard": 0.9,
"token_length_ratio": 1.259259
} | markdown_preamble_trailing | fence | 769949689ce1580f593341d4bd24d7c7e0a3fae5951eaa317579c0b5034ae0e7 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Homology.Opposite
public import Mathlib.Algebra.Homology.ConcreteCategory
public import Mathlib.RepresentationTheory.Homological.Resolution
public impo... | ```lean
by
simpa [CochainComplex.of.d] using (inhomogeneousCochains A).d_comp_d n (n + 1) (n + 2)
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RepresentationTheory/Homological",
"family_id": "inhomogeneouscochains",
"file_id": "mathlib/Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean",
"sample_id": "405cc27ec849e33c860448f43291f011dd4708a3dd857ace59279e3e185b0743"
} | train |
{
"chosen_proof_sha256": "a577b07c0d33a825fd2224d959c4010ad6df89972c66448b93e986876b718d2f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
calc
Var[X; μ | m]
_ = μ[X ^ 2 - 2 * X * μ[X | m] + μ[X | m] ^ 2 | m] := by rw [condVar, sub_sq]
_ =ᵐ[μ] μ[X ^ 2 | m] - 2 * μ[X | m] ^ 2 + μ[X | m] ^ 2 := by
have aux₀ : Integrable (X ^ 2) μ := hX.integrable_sq
have aux₁ : Integrable (2 * X * μ[X | m]) μ := by
rw [mul_assoc]
... | false | {
"chosen_tokens": 288,
"rejected_tokens": 3,
"token_jaccard": 0.031746,
"token_length_ratio": 0.010417
} | by_exact_placeholder | placeholder | 769a9f9a7e0c717f9e434be18c7e99bbc8dd0a838bc01ca595bb87d04e868ce8 | 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.ConditionalExpectation.PullOut
public import Mathlib.MeasureTheory.Integral.Average
public import Mathlib.Probability.Moments.Variance
... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Probability",
"family_id": "condvar_ae_eq_condexp_sq_sub_sq_condexp",
"file_id": "mathlib/Mathlib/Probability/CondVar.lean",
"sample_id": "56bb01dbdf1213a5571dfdae532b08e414ed78cc621e447a3cc9e4531d26ca79"
} | train |
{
"chosen_proof_sha256": "8c34499c0ebf931c33761750b78ccc8342159b860f73a3673d688956dbdeb67f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "29c7c255fa95ebdb4ea48bea160ffb43d6... | by
constructor <;> rintro ⟨u, h₂⟩
· refine ⟨f <$$> u, ?_⟩
apply (h₁ u).mp h₂
· refine ⟨g <$$> u, ?_⟩
rw [h₁]
simp only [MvFunctor.map_map, h₀, LawfulMvFunctor.id_map, h₂] | false | {
"chosen_tokens": 70,
"rejected_tokens": 74,
"token_jaccard": 0.921053,
"token_length_ratio": 1.057143
} | missing_wrong_import | import | 769b6febc48a54ca2353f7d71fe0e4110ea25bb5a11850a295bb93d61b3e7250 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fin.Fin2
public import Mathlib.Data.TypeVec
public import Mathlib.Logic.Equiv.Defs
Namespace:
MvFunctor
Local context:
/-
Copyright (c) 2018 Jeremy Avig... | import Mathlib.WrongImport
by
constructor <;> rintro ⟨u, h₂⟩
· refine ⟨f <$$> u, ?_⟩
apply (h₁ u).mp h₂
· refine ⟨g <$$> u, ?_⟩
rw [h₁]
simp only [MvFunctor.map_map, h₀, LawfulMvFunctor.id_map, h₂] | {
"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": "Control/Functor",
"family_id": "exists_iff_exists_of_mono",
"file_id": "mathlib/Mathlib/Control/Functor/Multivariate.lean",
"sample_id": "267c884cb305e890be2d32bd1dd81644507811de08af6ce289cc15754404e6fb"
} | train |
{
"chosen_proof_sha256": "6c44e0bc1aac850175730ade0c94468670de6fadad8303d7beb2df51cfcf9e9e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ca377602111c20dcbd45558310dc275698... | by
cases a <;> cases b
exacts [map_subset_map.2 (hf₁ _ _), disjSum_mono (hg₁ _ _) (hg₂ _ _), Subset.rfl,
map_subset_map.2 (hf₂ _ _)] | false | {
"chosen_tokens": 48,
"rejected_tokens": 55,
"token_jaccard": 0.923077,
"token_length_ratio": 1.145833
} | markdown_preamble_trailing | fence | 76a74d1f75f257a951cce24505fc91343e077154afc9a9b4a226eb34b01c9d27 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finset.Sum
public import Mathlib.Data.Sum.Order
public import Mathlib.Order.Interval.Finset.Defs
Namespace:
Finset
Local context:
/-
Copyright (c) 2022 ... | ```lean
by
cases a <;> cases b
exacts [map_subset_map.2 (hf₁ _ _), disjSum_mono (hg₁ _ _) (hg₂ _ _), Subset.rfl,
map_subset_map.2 (hf₂ _ _)]
``` | {
"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/Sum",
"family_id": "sumlexlift_mono",
"file_id": "mathlib/Mathlib/Data/Sum/Interval.lean",
"sample_id": "94c18dd81e6ce7afdea3016639b32636d0e629ff55fd5405a516ea90027133c6"
} | train |
{
"chosen_proof_sha256": "97f9e88375230bfbd4354802c179b8f0d71d6327ad14ee730bd5022e5fe5beee",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f38bfad9e7315903887b1c13bd5bbedf01... | by
rw [center_eq_bot, Algebra.mem_bot]
simp [eq_comm] | true | {
"chosen_tokens": 13,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.153846
} | invalid_tactic | invalid_tactic | 76d7669704ea192a8a46c7f5e863d61e4ccd2d3505bb25290ffae7cfada4a26f | 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.Central.Defs
import Mathlib.Algebra.Module.Torsion.Field
Namespace:
Algebra.IsCentral
Local context:
/-
Copyright (c) 2024 Jujian Zhang. All rights r... | by
__oprover_invalid_tactic_2a418ca60834 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "ab9a2f582d2df4e1b0208d4ea3460428eef7a79ba049950e32dbe97d8598c9d3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Central",
"family_id": "mem_center_iff",
"file_id": "mathlib/Mathlib/Algebra/Central/Basic.lean",
"sample_id": "2a418ca608341e871713a18e1c34bfa17c3846a9b9d6a3f1d347ce4be3076bae"
} | train |
{
"chosen_proof_sha256": "edfba660bad11fda81e3675a3dfc917ea4d8a1ad555ebc02b9f4efe9064c3712",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "93f185ad5f3a4564fc85cbc6bdef840bfa... | by
intro y hy
obtain ⟨x, (rfl : f x = y), h'⟩ := exists_ne_zero_of_finsum_mem_ne_zero hy
grind [mem_support] | true | {
"chosen_tokens": 26,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.076923
} | invalid_tactic | invalid_tactic | 772ed1c46d917f0135afb77534fd8f73c673bbc6aa8d4ed53580b9b18e8a0d33 | 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.LocallyFinsupp
public import Mathlib.Topology.Spectral.Basic
Namespace:
Function.locallyFinsupp
Local context:
/-
Copyright (c) 2026 Raphael Douglas... | by
__oprover_invalid_tactic_3b27666f8db3 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "d30545b10931b8493c0d75457a70cbd499d3def0dc543a6654369515ba9f266e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/LocallyFinsupp",
"family_id": "support_map_subset_of_forall_mem",
"file_id": "mathlib/Mathlib/Topology/LocallyFinsupp/Pushforward.lean",
"sample_id": "3b27666f8db369265222b785fd8e88c116b6148890f43b6b80939bde09ac3038"
} | train |
{
"chosen_proof_sha256": "89a379c0af7b46c3f249bf357aea100510d92246e204cc5b623f3d9530166237",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6808201512d88e8601fe27d1642ad64d66... | by
rw [← LieSubalgebra.toSubmodule_inj, range_toSubmodule, LieIdeal.toLieSubalgebra_toSubmodule,
ker_toSubmodule]
exact LinearMap.range_inl R L₁ L₂ | false | {
"chosen_tokens": 25,
"rejected_tokens": 32,
"token_jaccard": 0.909091,
"token_length_ratio": 1.28
} | markdown_preamble_trailing | fence | 773329383b851aa154549cf3cdd74922b642b81a49a3edd53a2a6ab68e89dea0 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Lie.Ideal
Namespace:
LieHom
Local context:
/-
Copyright (c) 2026 Leonid Ryvkin. All rights reserved.
Released under Apache 2.0 license as described i... | ```lean
by
rw [← LieSubalgebra.toSubmodule_inj, range_toSubmodule, LieIdeal.toLieSubalgebra_toSubmodule,
ker_toSubmodule]
exact LinearMap.range_inl R L₁ L₂
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Lie",
"family_id": "range_inl",
"file_id": "mathlib/Mathlib/Algebra/Lie/Prod.lean",
"sample_id": "58255fb9a08e3ea4853832b4b2e6754cd8d2484fe722ffff05ba10373b5631a3"
} | train |
{
"chosen_proof_sha256": "36011e8dfa494a26101903f706aa37335eb673289a71da56c05014c7180fb544",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f1fe02c934b4c86a0442cf4e7b420b4af8... | by
induction n generalizing g with
| zero =>
funext (x : Fin 1)
simp [diagonalSuccIsoTensorTrivial, diagonalOneIsoLeftRegular, Subsingleton.elim x 0,
ofMulAction_V, types_tensorObj_def, types_tensorUnit_def]
| succ n hn =>
funext x
induction x using Fin.cases with
| zero => simp; rfl
... | true | {
"chosen_tokens": 91,
"rejected_tokens": 5,
"token_jaccard": 0.042553,
"token_length_ratio": 0.054945
} | wrong_namespace_qualification | wrong_namespace_qualification | 77c6df0f5fa30300c1aee2e5a0fdb7092e617bfc89072e63027081531c520a97 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Fin
public import Mathlib.CategoryTheory.Monoidal.Linear
public import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory
public import... | by
exact OProverMissing.missing_e1e83a15a254 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ae5030265a4ed6b8ab61fa2a066d996ed34bb244c22ca6e54f855f2c85999331",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Action",
"family_id": "diagonalsuccisotensortrivial_inv_hom_apply",
"file_id": "mathlib/Mathlib/CategoryTheory/Action/Monoidal.lean",
"sample_id": "e1e83a15a2543ab2c7734d45b0801536ee9f8edb36b69899defb9d3f1b0cfbea"
} | train |
{
"chosen_proof_sha256": "dfb0cfa54bf916f968be75edf3a87f502ebb8bc73b72d17a24c14c29c6f3e807",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
convert! sup_mem_subalgebra_closure A f g
apply SetLike.ext'
simp | true | {
"chosen_tokens": 12,
"rejected_tokens": 8,
"token_jaccard": 0.055556,
"token_length_ratio": 0.666667
} | unsolved_goals | unsolved_goals | 77cebed937e7b6a7708952fa1b2e136c827021a41a80593c9f8486ba9984d1c6 | 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.Analysis.RCLike.Basic
public import Mathlib.Topology.Algebra.Star.Real
public import Mathlib.Topology.Al... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "2a8f205f52a4e03d4c0662e35d4724ca07c42a56f9d74550af5099c739d1d40f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/ContinuousMap",
"family_id": "sup_mem_closed_subalgebra",
"file_id": "mathlib/Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean",
"sample_id": "5718bd96cf9f59ca4c83944f21497324dfdb689e13608aa103fc1d5ac63a1635"
} | train |
{
"chosen_proof_sha256": "839d48065eb916980f8e3d750584330734f1d7501b43b9c8e7cc3615d69e6548",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
classical
exact if a0 : a = 0 then
a0.symm ▸ H0 b
else
have _ := mod_lt b a0
H1 _ _ a0 (GCD.induction (b % a) a H0 H1)
termination_by a | false | {
"chosen_tokens": 42,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.071429
} | by_exact_placeholder | exact? | 77f93ac84efdb30ee989fa4d74a070befb44c5dbbd62c2303eaf2f8b1ae6ca89 | 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.Ring.Defs
public import Mathlib.Order.RelClasses
Namespace:
EuclideanDomain
Local context:
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
R... | 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/EuclideanDomain",
"family_id": "gcd",
"file_id": "mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean",
"sample_id": "04cd638bd1240f2fe7fe9709e6cdc0eae054c396fecbe87a5f13fcb8dac94627"
} | train |
{
"chosen_proof_sha256": "f4514d3004c77d49e14e5cb3065d22f4527708fcd18c9c1d909c571ef396f6f1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "56d1d90b79ded4c653dd4d1569f516fe3d... | by
obtain ⟨n, hn⟩ := hnp
obtain ⟨k, hk⟩ := IsNilpotent.nilpotent R I M
have hk' : I.lcs M k = ⊥ := by
simp only [← toSubmodule_inj, I.coe_lcs_eq, hk, bot_toSubmodule]
suffices ∀ l, lowerCentralSeries R L M (l * n) ≤ I.lcs M l by
rw [isNilpotent_iff R]
use k * n
simpa [hk'] using this k
intro l... | true | {
"chosen_tokens": 112,
"rejected_tokens": 3,
"token_jaccard": 0.032787,
"token_length_ratio": 0.026786
} | unknown_identifier | unknown_identifier | 7809726635ea05f7f73b6575daef09d00dfe5f7012f95baecf033134b35e4c57 | 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.AdjointAction.Basic
public import Mathlib.Algebra.Lie.Nilpotent
public import Mathlib.Algebra.Lie.Normalizer
Namespace:
LieSubmodule
Local contex... | by
exact __oprover_missing_e96a9f2e6f2c | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "72262c7d8e2f20094c49fdd0b0ad2dbb2f532a8cf3441b40d13a57ddc9990029",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Lie",
"family_id": "isnilpotentofisnilpotentspansupeqtop",
"file_id": "mathlib/Mathlib/Algebra/Lie/Engel.lean",
"sample_id": "e96a9f2e6f2cd142ce46e8c5c7977a2fb22f0ea1c0daabf15e8cbf9d7c647459"
} | train |
{
"chosen_proof_sha256": "e9b659a1781e41131bb39de2a0211b972065e3468a0c84eab21865825ba5a5a7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7f3aefe15b0c458cb353fa8b5e262f1eaf... | by
rw [← Rat.isSquare_intCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg (mod_cast hz),
Rat.cast_intCast] | false | {
"chosen_tokens": 19,
"rejected_tokens": 26,
"token_jaccard": 0.882353,
"token_length_ratio": 1.368421
} | markdown_preamble_trailing | fence | 784a6cd8456315bf9fbd106fe97999d6027dff661b78faceb2cabce960374c0d | 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.Rat
public import Mathlib.Data.Nat.Prime.Int
public import Mathlib.Data.Rat.Sqrt
public import Mathlib.Analysis.Real.Sqrt
public import Mathlib... | ```lean
by
rw [← Rat.isSquare_intCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg (mod_cast hz),
Rat.cast_intCast]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/Real",
"family_id": "irrational_sqrt_intcast_iff_of_nonneg",
"file_id": "mathlib/Mathlib/NumberTheory/Real/Irrational.lean",
"sample_id": "9c259b24270f51c0a7679a42e8272d4047f65364bf40859a29277a457689ac84"
} | train |
{
"chosen_proof_sha256": "93aa954ac038910f3cc0d692da80205ae5a7230c0f479de23b1a6ec2e526a44a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a1c310e6cbed0abac9a29428b8024284b... | by
rw [FractionalOperation.tt, Multiset.mem_map] at rin
rw [show r = ![r 0, r 1] by simp [← List.ofFn_inj]]
apply lt_of_le_of_ne (mcf.right (r 0) (r 1)).left
intro equ
have asymm : r 0 ≠ r 1 := by
rcases (mcf.right (r 0) (r 1)).right equ with ⟨ha0, hb1⟩ | ⟨ha1, hb0⟩
· rw [ha0, hb1] at hab
exact ... | true | {
"chosen_tokens": 237,
"rejected_tokens": 7,
"token_jaccard": 0.036585,
"token_length_ratio": 0.029536
} | circular_dependency | circular_dependency | 785bf7be35b27ca08a8cb20294d2641d1097b4a33657c0096757c9e17417f85a | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Fin
public import Mathlib.Algebra.Order.BigOperators.Group.Multiset
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Linea... | by
exact Function.HasMaxCutPropertyAt.rows_lt_aux | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "13bb5694a04c3658ab08a9768a64349de82812aed3f04b5206d6c377a31c8316",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/Optimization",
"family_id": "function",
"file_id": "mathlib/Mathlib/Combinatorics/Optimization/ValuedCSP.lean",
"sample_id": "4158bc4533e6e85b1b86e67b31a3d6eca02e3c8f701cb75f73a81c4775a53885"
} | train |
{
"chosen_proof_sha256": "974123677968aef5cec49d697aa3b1eb6faa233c3680735631e7ed936621e266",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [integral_fintype .of_finite]
congr with x
rw [measureReal_def]
congr 2
exact PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _) | false | {
"chosen_tokens": 28,
"rejected_tokens": 2,
"token_jaccard": 0.045455,
"token_length_ratio": 0.071429
} | sorry_admit_sorryAx_proof_hole | sorry | 7867391291474dece28c8cf0604786c759ff179160f6900a956f6e010f2fce7f | 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.ProbabilityMassFunction.Basic
public import Mathlib.Probability.ProbabilityMassFunction.Constructions
public import Mathlib.MeasureTheory.Integral.... | 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": "Probability/ProbabilityMassFunction",
"family_id": "integral_eq_sum",
"file_id": "mathlib/Mathlib/Probability/ProbabilityMassFunction/Integrals.lean",
"sample_id": "aab605044ac1a0cc179875c4807508cccde8f4829eb26975d9537a751ad57521"
} | train |
{
"chosen_proof_sha256": "5e5876ae05a155ef3db2c120345927956dab48f54ef044b4e48039170ed59c7d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0f51432df11e05e50b8d812ab933e7980d... | by
cases f; cases g
dsimp at hfg₁ hfg₂
cat_disch | true | {
"chosen_tokens": 13,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.153846
} | invalid_tactic | invalid_tactic | 78a2b846914e32b87245c710debec5081424b1241e9ff94111f603e25b322da7 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Bicategory.LocallyDiscrete
public import Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo
Namespace:
CategoryTheory.Pseudofunctor... | by
__oprover_invalid_tactic_b8a6fc429e8f | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "f11705708e0fe26b1af2fca31fde499b7af56a2b6718f99ad116c1a8060ec8d0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Bicategory",
"family_id": "hom",
"file_id": "mathlib/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean",
"sample_id": "b8a6fc429e8fab84bafaec67f5918625193c8c32c8b4d06caaef778ad29f2b44"
} | train |
{
"chosen_proof_sha256": "7727f8d1a6315f3170b243f902f0e2ff87bac72c93a1e273349cdbafdd693264",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
dsimp [homEquiv]
ext j
simp [← uncurry_natural_left] | false | {
"chosen_tokens": 12,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.166667
} | sorry_admit_sorryAx_proof_hole | sorry | 78a95fb6532aa1d5cefea55265257fbb8fcc883354e5fa588ce4d5bc05683d4e | 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.Monoidal.Closed.Enrichment
public import Mathlib.CategoryTheory.Enriched.FunctorCategory
Namespace:
CategoryTheory.MonoidalClosed.FunctorCatego... | 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/Monoidal",
"family_id": "homequiv_naturality_two_symm",
"file_id": "mathlib/Mathlib/CategoryTheory/Monoidal/Closed/FunctorCategory/Basic.lean",
"sample_id": "1205eb9e1bd541c4f97a6b6d682d3d9f05b7c621a6ffc31f4b5fc185468fec86"
} | train |
{
"chosen_proof_sha256": "d74b30d157e4d96b787ba5699f29a97d3d77e2405ded9e8ea9925e08645a5660",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [coeff_eq_ncoeff, ncoeff_eq_zero_of_lt_order A (-n - 1) x]
lia
/-- Given an endomorphism-valued function on integers satisfying a pointwise bounded-pole condition,
we produce a vertex operator. -/
noncomputable def of_coeff (f : ℤ → Module.End R V)
(hf : ∀ x, BddBelow (Function.support fun y ↦ f y x)) : ... | false | {
"chosen_tokens": 99,
"rejected_tokens": 3,
"token_jaccard": 0.016667,
"token_length_ratio": 0.030303
} | by_exact_placeholder | exact? | 78d5208459f8be00b5184cd7b88a491fa71f5430b613a3de97bcd69ad5d3158c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Vertex.HVertexOperator
public import Mathlib.Data.Int.Interval
Namespace:
VertexOperator
Local context:
/-
Copyright (c) 2024 Scott Carnahan. All rig... | by
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/Vertex",
"family_id": "coeff_eq_zero_of_lt_order",
"file_id": "mathlib/Mathlib/Algebra/Vertex/VertexOperator.lean",
"sample_id": "aad755a0feeacee18eda311e51c7bdd7c4de3325eadaa0977eddb936e76e9ea5"
} | train |
{
"chosen_proof_sha256": "2fa6d6384b3fc37bc32726c7d0dfd505d0e32dc595d3c34173c19db9a5d115ee",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
have := IsClosedUnderBinaryCoproducts.closedUnderIsomorphisms P
have := hasFiniteCoproducts_of_has_binary_and_initial (C := P.FullSubcategory)
have := PreservesFiniteCoproducts.of_preserves_binary_and_initial P.ι
exact ⟨fun J _ ↦ P.isClosedUnderColimitsOfShape_of_preservesColimitsOfShape_ι _⟩ | true | {
"chosen_tokens": 37,
"rejected_tokens": 8,
"token_jaccard": 0.111111,
"token_length_ratio": 0.216216
} | unsolved_goals | unsolved_goals | 78f514c1615585531791afb98a734528bdd415fe113a78bdfa58360225d9460f | 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.Constructions.FiniteProductsOfBinaryProducts
public import Mathlib.CategoryTheory.Limits.FullSubcategory
public import Mathlib.CategoryTh... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "96707b9d8d8fe48a7c0d213dadfb974d362741721dcec2e406783dcae461a9f0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/ObjectProperty",
"family_id": "isclosedunderfinitecoproducts",
"file_id": "mathlib/Mathlib/CategoryTheory/ObjectProperty/FiniteProducts.lean",
"sample_id": "d26cc5be978dead2b623c3a4790cc21af0d92aeaa8715f5d06e969086e47e5d8"
} | train |
{
"chosen_proof_sha256": "801cfb0a412f7e9f521af786c135901e8a7681f3edb0eee3dd6e24550333a788",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7a185c9e616497e88188303b7a6930b622... | by
constructor
· obtain a | a := a <;> obtain b | b := b
· rw [sumLexLift, mem_map]
rintro ⟨c, hc, rfl⟩
exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩
· refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_
· rintro ⟨c, hc, rfl⟩
exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)
· rintro ⟨c... | true | {
"chosen_tokens": 287,
"rejected_tokens": 3,
"token_jaccard": 0.044444,
"token_length_ratio": 0.010453
} | circular_dependency | circular_dependency | 79f866ff335c1c0f8fbe2f6fca45c300f9404e569e6e9a780c50fe7ac0a5307a | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finset.Sum
public import Mathlib.Data.Sum.Order
public import Mathlib.Order.Interval.Finset.Defs
Namespace:
Finset
Local context:
/-
Copyright (c) 2022 ... | by
exact mem_sumLexLift | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "cd40224adc1b9edad13eda42c1703cff3c0ccae3bb0a5d1a15b040c0dbd79daf",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Sum",
"family_id": "mem_sumlexlift",
"file_id": "mathlib/Mathlib/Data/Sum/Interval.lean",
"sample_id": "fcb3cb79819283ab318174f307cb21c4367decc586f83a6b3fb969ef48ad9494"
} | train |
{
"chosen_proof_sha256": "1ce00e07b29f18d90867caf068379ad134d68969aa30ca0e3ea5aac4b2537419",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b8f5e4251a21078b41e8816acb665bda0d... | by
rw [pullbackShiftFunctorZero_inv_app]
simp only [Functor.id_obj, pullbackShiftIso, eqToIso.inv, eqToHom_app, shiftFunctorZero',
Iso.trans_inv, NatTrans.comp_app, eqToIso_refl, Iso.refl_inv, NatTrans.id_app, assoc]
erw [comp_id] | false | {
"chosen_tokens": 46,
"rejected_tokens": 50,
"token_jaccard": 0.896552,
"token_length_ratio": 1.086957
} | missing_wrong_import | import | 79fdfa8c9ed2667bcea7f52fcf14ca19654801306a9cd76345474d85c9d37ec1 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Shift.Adjunction
public import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
Namespace:
CategoryTheory
Local context:
/-
Copyright (c) 20... | import Mathlib.WrongImport
by
rw [pullbackShiftFunctorZero_inv_app]
simp only [Functor.id_obj, pullbackShiftIso, eqToIso.inv, eqToHom_app, shiftFunctorZero',
Iso.trans_inv, NatTrans.comp_app, eqToIso_refl, Iso.refl_inv, NatTrans.id_app, assoc]
erw [comp_id] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Shift",
"family_id": "pullbackshiftfunctorzero'_inv_app",
"file_id": "mathlib/Mathlib/CategoryTheory/Shift/Pullback.lean",
"sample_id": "9d858434b5715b6466fd7efd13eba53c42a1ae0530865db41d7c8c0a8af13979"
} | train |
{
"chosen_proof_sha256": "de4e82cdf62623b5f6eb51d83e7f17e091632bf91d46d2109ff7b01e77e7f85f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
have hinj : Function.Injective (algebraMap F[X] K) := by
rw [IsScalarTower.algebraMap_eq F[X] F⟮X⟯ K]
exact (algebraMap F⟮X⟯ K).injective.comp (IsFractionRing.injective F[X] F⟮X⟯)
rw [injective_iff_map_eq_zero (algebraMap F[X] (↥(ringOfIntegers F K)))]
intro p hp
rw [← Subtype.coe_inj, Subalgebra.coe... | false | {
"chosen_tokens": 111,
"rejected_tokens": 2,
"token_jaccard": 0.025641,
"token_length_ratio": 0.018018
} | sorry_admit_sorryAx_proof_hole | sorry | 7a765987aa3fd156f39bd0a76dcaf480aa8b517f1c327d4428c89bdce8fa77c7 | 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... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory",
"family_id": "algebramap_injective",
"file_id": "mathlib/Mathlib/NumberTheory/FunctionField.lean",
"sample_id": "143ab2863e5ef8203ad9cdb078d9a96faf3dccd3f496872d4f958e2b6081b962"
} | train |
{
"chosen_proof_sha256": "5d0edc0cc6cc1668b437d5481146309b64fe078c9ece4c0df2b3120bce3ca534",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "30c038fad28d31c8e5506864f2ea72186e... | by
refine TopologicalSpace.ext_nhds fun x ↦ ?_
ext s
rw [mem_nhds_toTopsp, H] | true | {
"chosen_tokens": 18,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.166667
} | unknown_identifier | unknown_identifier | 7a7fe09111004f23eed4d079c9847a0a776051036109959571d3c4a0c1900823 | 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.Analysis.Filter
public import Mathlib.Topology.Bases
public import Mathlib.Topology.LocallyFinite
Namespace:
Ctop.Realizer
Local context:
/-
Copyright (... | by
exact __oprover_missing_80671f5e23cc | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "8cb8a10397e47036930b32d832cda0ffd4352bb5adfb7fd8ce5eddc29d366b5e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Analysis",
"family_id": "ext",
"file_id": "mathlib/Mathlib/Data/Analysis/Topology.lean",
"sample_id": "80671f5e23cc162f2c4d734bcae6d4b38501c2182a9dd22e1a1e199c28f65d78"
} | train |
{
"chosen_proof_sha256": "f13ac1fd7bb3c4deec76c60552efe11696b97aca72a742edff11d90154604aae",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f8994d13abe40d9fcb9cf9840e64c01799... | by
apply elim_of_ne
rintro rfl
exact hl.notMem hj | false | {
"chosen_tokens": 10,
"rejected_tokens": 15,
"token_jaccard": 0.666667,
"token_length_ratio": 1.5
} | markdown_preamble_trailing | preamble | 7ab76bfbab8411713a9a44b7afd6c60279ef39ba91a61a2003990272cba5b9b0 | 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.List.Nodup
public import Mathlib.Data.Set.Prod
Namespace:
List.TProd
Local context:
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Release... | Here is the proof:
by
apply elim_of_ne
rintro rfl
exact hl.notMem hj | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Prod",
"family_id": "elim_of_mem",
"file_id": "mathlib/Mathlib/Data/Prod/TProd.lean",
"sample_id": "1de4a2ae584ee9a9376192b0c305c1c4427057eda902e8636079376cebce05ed"
} | train |
{
"chosen_proof_sha256": "73a3ff17197caa13c9881ebe96a98ed8cb7dce96f5c0218437447afab1f809f0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
refine Finset.measurable_sup' _ ?_
simpa [Finset.mem_range] | false | {
"chosen_tokens": 14,
"rejected_tokens": 3,
"token_jaccard": 0.166667,
"token_length_ratio": 0.214286
} | by_exact_placeholder | exact? | 7ab8392dc2388d9be4a2edb5ec4deea45d8a894a4b1c0620c1e67841595e9353 | 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.Measure.AEMeasurable
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 licens... | 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": "MeasureTheory/Order",
"family_id": "finset",
"file_id": "mathlib/Mathlib/MeasureTheory/Order/Lattice.lean",
"sample_id": "2d8dea88dd355420151a905a1257de8aac829b897e12eb600a797e8418da7629"
} | train |
{
"chosen_proof_sha256": "9fa7af5d48ddb6db2529f448df0e30aa81c953ad207f9980f8c0f3df88f4e48e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
intro V
have : ∀ x : V.1, ∃ (i : ι) (r : Γ(X, V.1)) (rU : Γ(X, U i)),
X.basicOpen r = X.basicOpen rU ∧ x.1 ∈ X.basicOpen r := by
intro ⟨x, hxV⟩
obtain ⟨i, hi⟩ := TopologicalSpace.Opens.mem_iSup.mp (hU.ge (Set.mem_univ x))
exact ⟨i, exists_basicOpen_le_affine_inter V.2 (U i).2 _ ⟨hxV, hi⟩⟩
cho... | false | {
"chosen_tokens": 413,
"rejected_tokens": 3,
"token_jaccard": 0.020833,
"token_length_ratio": 0.007264
} | by_exact_placeholder | placeholder | 7ace3179a83da61a6a37593706de68e9c2a71800ef040c8a431c5e8af521e82d | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
public import Mathlib.AlgebraicGeometry.Properties
public import Mathlib.Tactic.DepRewrite
Namespace:
AlgebraicGeomet... | by
exact 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": "AlgebraicGeometry/IdealSheaf",
"family_id": "le_of_isup_eq_top",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/Basic.lean",
"sample_id": "66542818e479c89882f725bfc7d9605b37b594439204873e38030237ce086893"
} | train |
{
"chosen_proof_sha256": "426d456c7ecb6c1bdb729eb16ba1456459d7dc0d59c4a4d027a77d78ce78493f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a7eb67679c9a628af80fcc71372a23c70d... | by
simp_rw [← Nat.not_odd_iff_even]
constructor <;> intro h
· intro _ w ho
have := (w.three_le_chromaticNumber_of_odd_loop ho).trans h.chromaticNumber_le
norm_cast
· apply colorable_iff_forall_connectedComponents.2
intro c
obtain ⟨_, hv⟩ := c.nonempty_supp
use fun a ↦ Fin.ofNat 2 (c.connecte... | true | {
"chosen_tokens": 212,
"rejected_tokens": 3,
"token_jaccard": 0.027778,
"token_length_ratio": 0.014151
} | unknown_identifier | unknown_identifier | 7ad6ebfa679381e0a4e9f7eab3119d7436611438ae49e5e073cac174b02d6a91 | 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.Combinatorics.SimpleGraph.Bipartite
public import Mathlib.Combinatorics.SimpleGraph.Circulant
public import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
p... | by
exact __oprover_missing_56f2b093801a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "54887d10e51425f1367cd7fc2841bf3c9ec4676eca97686e6e382e08cdd6c4f3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/SimpleGraph",
"family_id": "two_colorable_iff_forall_loop_even",
"file_id": "mathlib/Mathlib/Combinatorics/SimpleGraph/Coloring/Constructions.lean",
"sample_id": "56f2b093801a2a41875bf30e118429a5cc73ba8976a1adb97741fabeeedbed7b"
} | train |
{
"chosen_proof_sha256": "95f0a046edcc88c2f0897d1f675b7af88bcf9e64df19c8ace7d838acef7a93e4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a2dfa0842d99cd20b40e19c92c20cbe360... | by
have ⟨n, S, d, _, hd, ⟨e⟩⟩ := exists_end_algEquiv_pi_matrix_end R₀ R M
classical exact ⟨n, _, d, inferInstance, inferInstance, hd, ⟨e⟩⟩ | true | {
"chosen_tokens": 42,
"rejected_tokens": 2,
"token_jaccard": 0.05,
"token_length_ratio": 0.047619
} | invalid_tactic | invalid_tactic | 7aee4eea1a9784b2e66436e0d5fa531950b9d61c22e340009cf2b884bb79cae4 | 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.FreeModule.Finite.Basic
public import Mathlib.RingTheory.FiniteLength
public import Mathlib.RingTheory.SimpleModule.Isotypic
public import Mathli... | by
__oprover_invalid_tactic_929feb6854c6 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "19ebaad7edbd149f730fbe585a0e531111b5046f001ff44f2e087c7b0d6dbb07",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/SimpleModule",
"family_id": "exists_end_algequiv_pi_matrix_divisionring",
"file_id": "mathlib/Mathlib/RingTheory/SimpleModule/WedderburnArtin.lean",
"sample_id": "929feb6854c6bd754df0ddca63934028a11faf17c7ad84065207bb38e2efb282"
} | train |
{
"chosen_proof_sha256": "2b82d4fddfb467900b75edb5d85dcce54ace3fac2a5c050d4837c5985aa49b27",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "71a417e8c511aea0386249ac90264f5677... | by
lift s to Finset α using hs; exact s.exists_le_minimal ha
variable [Nonempty α] | true | {
"chosen_tokens": 19,
"rejected_tokens": 3,
"token_jaccard": 0.111111,
"token_length_ratio": 0.157895
} | unknown_identifier | unknown_identifier | 7afe8763db9b202c6b8bb11207f330b9572fefe4daf67b126a08003b453ca72b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Set.Finite.Basic
public import Mathlib.Order.Minimal
Namespace:
Set
Local context:
/-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released und... | by
exact __oprover_missing_08ef20f28dae | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "0329e9573368f94c00729ce33841202d50dea9e6b5cec2a19be51132c1c02bf9",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Preorder",
"family_id": "finite",
"file_id": "mathlib/Mathlib/Order/Preorder/Finite.lean",
"sample_id": "08ef20f28dae9ac8a4b11319080aa916285ab90a007cfc4fe216ca408e450b14"
} | train |
{
"chosen_proof_sha256": "88c077efecbeeb27504c4875f1630a5efd1eaa0f4e0d6e798f302270b541c3d4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c296264677213e7c36d5f393049036aa8... | by rw [norm_apply, LinearMap.det]; split_ifs <;> trivial
variable {R} | true | {
"chosen_tokens": 19,
"rejected_tokens": 2,
"token_jaccard": 0.052632,
"token_length_ratio": 0.105263
} | invalid_tactic | invalid_tactic | 7b3e9f2d62747c31c648253fb17f69c71997de6b9743edd9610bebd74176f915 | 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.Determinant
Namespace:
Algebra
Local context:
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as desc... | by
__oprover_invalid_tactic_5594b1e02762 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "f6dd3613bd478679bd2aae85ad7abecb284ef7ef70cd44660e7a1dee0d304b85",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "9346ea21709b5b4adcb76342b23a0fdfec46476e1c52d71821429c4b741b1882",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f15b94388f7d99ac88f3a9765d8d2dff99... | by
obtain ⟨f, g, h⟩ := exists_pair_ne (α → β)
rw [ne_eq, funext_iff, Classical.not_forall] at h
obtain ⟨a, h⟩ := h
exact nontrivial_of_ne _ _ h | true | {
"chosen_tokens": 41,
"rejected_tokens": 3,
"token_jaccard": 0.068966,
"token_length_ratio": 0.073171
} | unknown_identifier | unknown_identifier | 7b438a2b21f1827e8d7f8be5239ac160e3e01d5069e2cbb2fec82001e10f505c | 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.Push.Attr
Namespace:
(root)
Local context:
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as describe... | by
exact __oprover_missing_3d58730a822f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "7e9fecea4822192c8e292de1f579931f44ecce6da6fda2db520a491a5c077287",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Logic/Nontrivial",
"family_id": "function",
"file_id": "mathlib/Mathlib/Logic/Nontrivial/Defs.lean",
"sample_id": "3d58730a822f37dbe124ac3de8532bdf9e6b3a34db752189dda5130b2510e7eb"
} | train |
{
"chosen_proof_sha256": "d56cb08de95550c2d5c8c73e4133b8fd22dbbf83f22597912cb3044a1cf97d24",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "229e61ac4950e38ed3ed9109978b62baa0... | by
simp_rw [IsTranscendenceBasis, algebraicIndependent_equiv, EquivLike.range_comp]
alias ⟨_, IsTranscendenceBasis.comp_equiv⟩ := isTranscendenceBasis_equiv | false | {
"chosen_tokens": 21,
"rejected_tokens": 25,
"token_jaccard": 0.85,
"token_length_ratio": 1.190476
} | missing_wrong_import | import | 7b59689ac4bf95b99be558066d58271b6c8fd3c08e0e5f8803dbc442a09b0628 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MvPolynomial.CommRing
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as ... | import Mathlib.WrongImport
by
simp_rw [IsTranscendenceBasis, algebraicIndependent_equiv, EquivLike.range_comp]
alias ⟨_, IsTranscendenceBasis.comp_equiv⟩ := isTranscendenceBasis_equiv | {
"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/AlgebraicIndependent",
"family_id": "istranscendencebasis_equiv",
"file_id": "mathlib/Mathlib/RingTheory/AlgebraicIndependent/Defs.lean",
"sample_id": "f15d582cb509e0a13008bd362768f09d19cdeaf2278912231d988aa0bbeaa66e"
} | train |
{
"chosen_proof_sha256": "164db8212cd6dbe524941c096782dd0230af417452ad9fde38bafe58fdc70918",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e4c90edf46e133292ddad63e3da005b52f... | by
ext
apply IsIntegralClosure.algebraMap_injective B A L
simp | false | {
"chosen_tokens": 10,
"rejected_tokens": 14,
"token_jaccard": 0.769231,
"token_length_ratio": 1.4
} | missing_wrong_import | import | 7bc69a1d15fae03f06c3dc0f18b3a0768bd7b09beff0cecc251bc76ac8eee245 | 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.IntegralClosure
public import Mathlib.RingTheory.RingHom.Finite
public import Mathlib.RingTheory.Localization.LocalizationLocalizatio... | import Mathlib.WrongImport
by
ext
apply IsIntegralClosure.algebraMap_injective B A L
simp | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/IntegralClosure",
"family_id": "galrestrict'_id",
"file_id": "mathlib/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean",
"sample_id": "470ba23482b167ed1d27d1364fdd8b69790cc7acc5d9ae4345731e59f6a235df"
} | train |
{
"chosen_proof_sha256": "e65faed5f7af3fe66504654d3728889128f1a2b5c77ec046ad6fc37f70906b6d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by simp [left_distrib, *, sub_eq_add_neg] | false | {
"chosen_tokens": 9,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.555556
} | sorry_admit_sorryAx_proof_hole | sorryAx | 7bda13087b592b3ce73fc3eeea9aed3afcdfb4a03ddb42c3424c9aa1d16c20d1 | 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.Basic
public import Mathlib.Algebra.Order.Ring.Defs
public import Mathlib.Data.Tree.Basic
public import Mathlib.Tactic.NormNum.Core
public import... | 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": "Tactic/CancelDenoms",
"family_id": "sub_subst",
"file_id": "mathlib/Mathlib/Tactic/CancelDenoms/Core.lean",
"sample_id": "104ce9cbaebdaa87f4904559e1ae0c57adfb3fa19b70fd7be28fa5278b9ac29a"
} | train |
{
"chosen_proof_sha256": "2b82d4fddfb467900b75edb5d85dcce54ace3fac2a5c050d4837c5985aa49b27",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
lift s to Finset α using hs; exact s.exists_le_minimal ha
variable [Nonempty α] | false | {
"chosen_tokens": 19,
"rejected_tokens": 3,
"token_jaccard": 0.111111,
"token_length_ratio": 0.157895
} | by_exact_placeholder | placeholder | 7bf34e0d186234ac9305e9a9c462944de2d7683321cd783b890cb2b1ef51df83 | 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.Finite.Basic
public import Mathlib.Order.Minimal
Namespace:
Set
Local context:
/-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released und... | 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": "Order/Preorder",
"family_id": "finite",
"file_id": "mathlib/Mathlib/Order/Preorder/Finite.lean",
"sample_id": "08ef20f28dae9ac8a4b11319080aa916285ab90a007cfc4fe216ca408e450b14"
} | train |
{
"chosen_proof_sha256": "1246a108b01c281615de5c0ac2e3b8d9e0aa2fea03820a7bb20910f1f044ae33",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp_rw [he.isIdempotentElem.mem_iff, he.isIdempotentElem.range_mem_invtSubmodule_iff,
he.isIdempotentElem.ker_mem_invtSubmodule_iff, forall_and, and_iff_left_iff_imp, ← mul_def]
intro h x hx
simpa [he.isSelfAdjoint.star_eq] using! congr(star $(h _ (star_mem hx))) | true | {
"chosen_tokens": 55,
"rejected_tokens": 8,
"token_jaccard": 0.027027,
"token_length_ratio": 0.145455
} | unsolved_goals | unsolved_goals | 7c0b81179dfdac553c55db4c6ca378ce1fb10fe02723f57388753a9c3d832dbb | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.CStarAlgebra.Classes
public import Mathlib.Analysis.InnerProductSpace.Adjoint
Namespace:
VonNeumannAlgebra
Local context:
/-
Copyright (c) 2022 Kim ... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "ef05cc6f370400e5f1321b2b0491b694916374fa91d27049a433d689a3d749e4",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/VonNeumannAlgebra",
"family_id": "isstarprojection",
"file_id": "mathlib/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean",
"sample_id": "4180d4b77b71dc3fb2a1ccac790bfafb68b3d5a563a499f05b950658539e98e7"
} | train |
{
"chosen_proof_sha256": "5403bbdd6b8db64d0404aea07e7fcb0b3cc956807fed1c2cfdeeee9bac43ff23",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "17bf13272abd6adb07ae2877731b7c8fc4... | by
constructor
simp [h.neg_to_eq, div_nonneg] | true | {
"chosen_tokens": 10,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.3
} | unknown_identifier | unknown_identifier | 7c412effa71635470fe3834a98760c6e7e300495d485bdff8d92d21704f92aa2 | 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.Floor.Semiring
public import Mathlib.Data.NNRat.Order
public import Mathlib.Data.Rat.Floor
Namespace:
Mathlib.Meta.NormNum
Local context:
/-
Co... | by
exact __oprover_missing_a3a4139f0052 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "11387462df0846358ae2630fb5a0f5c197b3182eb5bf98c0dfa89dd14270be55",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/NNRat",
"family_id": "israt",
"file_id": "mathlib/Mathlib/Data/NNRat/Floor.lean",
"sample_id": "a3a4139f0052b0a32136e52937242d5f55eef48c36e8634d6c5292950c12886e"
} | train |
{
"chosen_proof_sha256": "e9b602b11908c749a9de1394766214188bf5ab4658bf596bbcd37cd55e75b8a2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d0879bf6448c6e4a9ff954465980f5fa1f... | by
rw [← S.toNatSubmodule_toAddSubmonoid, ← Submodule.fg_iff_addSubmonoid_fg,
Submodule.fg_iff_exists_fin_linearMap]
exact exists_congr fun n => ⟨fun ⟨f, hf⟩ => ⟨f, hf ▸ LinearMap.range_toAddSubmonoid _⟩,
fun ⟨f, hf⟩ => ⟨f.toNatLinearMap, Submodule.toAddSubmonoid_inj.mp <|
hf ▸ LinearMap.range_toAddSu... | true | {
"chosen_tokens": 68,
"rejected_tokens": 2,
"token_jaccard": 0.032258,
"token_length_ratio": 0.029412
} | invalid_tactic | invalid_tactic | 7c43df5c6766e2fb7bd5a0048520ab1e8d6a50bf147420c236206a1c902b960b | 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.Congruence.Defs
public import Mathlib.LinearAlgebra.Basis.Cardinality
public import Mathlib.LinearAlgebra.DFinsupp
public import Mathlib.LinearA... | by
__oprover_invalid_tactic_e2bfab7a5a1b | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "c1a88b654de93427400df4afd2d45901f3bf64d9e66c3db262c1946cf9d5dc4a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Finiteness",
"family_id": "addsubmonoid",
"file_id": "mathlib/Mathlib/RingTheory/Finiteness/Cardinality.lean",
"sample_id": "e2bfab7a5a1b1912f82cb1a9e10696dbb413a79fc583a7da1b4cad05b058c943"
} | train |
{
"chosen_proof_sha256": "75232f0c801e33fc6a2f2b42c9fbde78aef8e1d36674a56a7714ae3434228c02",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cbf345f5a8e8911a533324abd4e16d3741... | by
simp only [(v.filterAt_basis_closedBall x).mem_iff, ← and_imp, subset_def, mem_setOf] | true | {
"chosen_tokens": 20,
"rejected_tokens": 5,
"token_jaccard": 0.1,
"token_length_ratio": 0.25
} | wrong_namespace_qualification | wrong_namespace_qualification | 7c7b3b1b246aa21901d77c05ee1d8ae8dc7ba25edd50d3358e3b79a84d94a310 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Measure.AbsolutelyContinuous
Namespace:
VitaliFamily
Local context:
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released unde... | by
exact OProverMissing.missing_419897952885 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c94b0565bfb1a82c34506202c469b89ae8173cffa5a6f678f3dcc56b3386fc18",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Covering",
"family_id": "mem_filterat_iff",
"file_id": "mathlib/Mathlib/MeasureTheory/Covering/VitaliFamily.lean",
"sample_id": "419897952885489eef42100d7198d985d0ed456c5f3c123051bf15c5b6fac476"
} | train |
{
"chosen_proof_sha256": "4a41798036ff76446dc750a37ef284895b5d38963c6bb7b381d67845e6108343",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "564b84906f8963f8e9e5daabed9de260d2... | by
have : JacobsonSpace ↥X := LocallyOfFiniteType.jacobsonSpace (f ≫ i)
refine ext_of_fromSpecResidueField_eq f g i (S ∩ closedPoints X) ?_ ?_ H'
· rwa [dense_iff_closure_eq, JacobsonSpace.closure_inter_closedPoints_eq_closure hS,
← dense_iff_closure_eq]
· intro x ⟨hxS, hx⟩
rw [← cancel_epi (Spec.map ... | true | {
"chosen_tokens": 126,
"rejected_tokens": 3,
"token_jaccard": 0.015873,
"token_length_ratio": 0.02381
} | unknown_identifier | unknown_identifier | 7cdd831064333d1d904deb43e69d1278fab84be0963d04b5a20ce8472d0ad07f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.Finite
public import Mathlib.FieldTheory.IsAlgClosed.Basic
Namespace:
AlgebraicGeometry
Local context:
/-
Copyright (c) 2026 Andr... | by
exact __oprover_missing_da422d22caf2 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "0587da8e3db98e7a7c6a5712f673ed109d6b2e894c25a4028d373e091cba4795",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/AlgClosed",
"family_id": "ext_of_apply_eq",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/AlgClosed/Basic.lean",
"sample_id": "da422d22caf26f3e5568440ff4f4ce39cf647f59520cb7416b45b8017ec51f47"
} | train |
{
"chosen_proof_sha256": "048280cfb8f07df3e6d2a0a1b3323465e6476c927989956effb4e1caeb51283e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
obtain ⟨x, w₁, w₂⟩ := x
obtain ⟨⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩, -⟩ := h
simp [Prod.toFun', Prod.invFun', h₁, h₂] | false | {
"chosen_tokens": 57,
"rejected_tokens": 2,
"token_jaccard": 0.041667,
"token_length_ratio": 0.035088
} | sorry_admit_sorryAx_proof_hole | admit | 7ce0fc1c53a078a52ab51fc2a4c4cd0fed623635f03401657ad90c211345df81 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.FiberBundle.Basic
Namespace:
Bundle.Trivialization
Local context:
/-
Copyright (c) 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache... | by
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/FiberBundle",
"family_id": "prod",
"file_id": "mathlib/Mathlib/Topology/FiberBundle/Constructions.lean",
"sample_id": "c32bfd18b1eacc9948f2069f737c203f16059abd126861ca53c4f4fff6855765"
} | train |
{
"chosen_proof_sha256": "b82467ef621f5a21aaae8eaa62dc55df65f51c9c9619a918a2bac8a08b7ed281",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
delta Irrational
contrapose!
rintro ⟨⟨rx, rfl⟩, ⟨ry, rfl⟩⟩
exact ⟨rx + ry, cast_add rx ry⟩ | true | {
"chosen_tokens": 29,
"rejected_tokens": 8,
"token_jaccard": 0.047619,
"token_length_ratio": 0.275862
} | unsolved_goals | unsolved_goals | 7d1bf5685297cc486c32cac4762f20a9fa7f8a034a7081d38e7fa26a9e789427 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Rat
public import Mathlib.Data.Nat.Prime.Int
public import Mathlib.Data.Rat.Sqrt
public import Mathlib.Analysis.Real.Sqrt
public import Mathlib... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "fe8a752cc83edb72233070545843b2739325238a18eb5dede21517407ec6ca46",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Real",
"family_id": "add_cases",
"file_id": "mathlib/Mathlib/NumberTheory/Real/Irrational.lean",
"sample_id": "d0d0f38b9e3e428b2ae38db46c5384471141fd719dfd1fd904da355d5cb234fc"
} | train |
{
"chosen_proof_sha256": "b9e374e8c54a58f587ed34013c46e429e34757a52725bc1a7e7b25ce0f218592",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
apply injective_of_mono (X.mapIso (sigmaIso f).op).hom
apply injective_of_mono (sigmaComparison X (fun a ↦ (fiber f a).1))
ext a
specialize h a
rw [← sigmaComparison_comp_sigmaIso] at h
exact h | false | {
"chosen_tokens": 48,
"rejected_tokens": 2,
"token_jaccard": 0.034483,
"token_length_ratio": 0.041667
} | by_exact_placeholder | by? | 7d650e4065a0065272be273a5187b4539fb5779b8660d31e01bd30e3137f47b6 | 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.Condensed.Discrete.Basic
public import Mathlib.Condensed.TopComparison
public import Mathlib.Topology.Category.CompHausLike.SigmaComparison
public import Mathl... | 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": "Condensed/Discrete",
"family_id": "presheaf_ext",
"file_id": "mathlib/Mathlib/Condensed/Discrete/LocallyConstant.lean",
"sample_id": "5b180b286925047b6265b1d6b6387babba0a99b025076644a75ae77bbbe0b4d6"
} | train |
{
"chosen_proof_sha256": "f994609f95dbe1be796aaafe12c37132a7b266348fd2421e6b7d6339c7e0ee00",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
simpa using! (H.toChainHomotopy).homologyMap_eq n | false | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.25
} | by_exact_placeholder | exact? | 7d65d3c3cf51d0cdebd30857be9621ec84caa31c0bee3d12af219887f2fbc423 | 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.SimplicialObject.Homotopy
public import Mathlib.AlgebraicTopology.AlternatingFaceMapComplex
public import Mathlib.Algebra.Homology.Homotopy
... | 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": "AlgebraicTopology/SimplicialObject",
"family_id": "map_homology_eq",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplicialObject/ChainHomotopy.lean",
"sample_id": "184e16fbe9e5eec732502dd0a1ed9c46036bcaecb5ded625d98d24b0ac54477f"
} | train |
{
"chosen_proof_sha256": "d7c3832f7227fb403f537f073b6c62458e059e39149445706fd5cbe0c34cde70",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simpa only [tendsto_iff_comap] using le_iff_ultrafilter | true | {
"chosen_tokens": 8,
"rejected_tokens": 8,
"token_jaccard": 0.071429,
"token_length_ratio": 1
} | unsolved_goals | unsolved_goals | 7da3e9a37f1065c110270f3c6d015594699a0f0028b79b729328e2d0c07dd17d | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.Filter.Ultrafilter.Defs
public import Mathlib.Order.Filter.Cofinite
public import Mathlib.Order.ZornAtoms
Namespace:
Filter
Local context:
/-
Copyright... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e422fccc2758c173c5740b3634c733b26082230bfcb6f40aabeeb831ee6bb135",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Filter",
"family_id": "tendsto_iff_ultrafilter",
"file_id": "mathlib/Mathlib/Order/Filter/Ultrafilter/Basic.lean",
"sample_id": "4a94ca7d62a9fac81c4789f5e6cb34fc62a67f792fb4fbe276c40abbf6855644"
} | train |
{
"chosen_proof_sha256": "37b0177ca46b549523222694ca3ff53bf01def907bba860633ebbd21d6d3186d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
ext x
obtain ⟨I, rfl⟩ := ClassGroup.mk0_surjective x
rw [extendedHom_mk0, MonoidHom.one_apply]
exact (ClassGroup.mk0_eq_one_iff (extendedIdeal A B I).2).mpr (by simpa using h I) | false | {
"chosen_tokens": 45,
"rejected_tokens": 2,
"token_jaccard": 0.03125,
"token_length_ratio": 0.044444
} | by_exact_placeholder | by? | 7dd1945e6d25e702d82772c0d74968a8fdd7b09534a732810731a72101ca0d3b | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.FractionalIdeal.Extended
public import Mathlib.RingTheory.ClassGroup.Basic
Namespace:
ClassGroup
Local context:
/-
Copyright (c) 2026 Riccardo Bra... | 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/ClassGroup",
"family_id": "extendedhom_eq_one_of_forall_isprincipal",
"file_id": "mathlib/Mathlib/RingTheory/ClassGroup/ExtendedHom.lean",
"sample_id": "6e5dde6af19e66e06a4ae4317b8eb4e8eb4862f32bca8c9663051c5849e4b141"
} | train |
{
"chosen_proof_sha256": "30f83bf0d483bdb24895df71acc1365551186d762bcd20d14e7f1ac4b162ed7c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
obtain ⟨s, -, hscard, hs⟩ := addRothNumber_spec (univ : Finset α)
haveI := noAccidental hs
rw [← hscard, ← card_triangleIndices, ← card_triangles]
exact (locallyLinear hs).le_ruzsaSzemerediNumber | true | {
"chosen_tokens": 41,
"rejected_tokens": 8,
"token_jaccard": 0.090909,
"token_length_ratio": 0.195122
} | unsolved_goals | unsolved_goals | 7df2068fdd647fbbc9300618148aa09b4c3f1d0391e48be4e3f75ccd4059e47c | 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.Combinatorics.Additive.AP.Three.Behrend
public import Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite
public import Mathlib.Tactic.Rify
public import Mat... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e220252dbe14e1cd3c9e09821ebb0f2d54f4d3217f44dfff02eee9ac1553012f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/Extremal",
"family_id": "addrothnumber_le_ruzsaszemeredinumber",
"file_id": "mathlib/Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean",
"sample_id": "862a0c729b5cd3749b546c796ee66cf2c56a927378f8d912626bfe6707f1fa0a"
} | train |
{
"chosen_proof_sha256": "42771a2147cb91bca0f9e576050654f9c7f45762679c01f645ed998108481986",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simpa using hκ.irreducible hs <| Std.lt_of_lt_of_le hsp (hφ s) | false | {
"chosen_tokens": 19,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.105263
} | by_exact_placeholder | by? | 7e805872ff2776d4eba6c7d08fa5e7d04e415d972a0d7e7d770e708fd4d2f147 | 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.Kernel.Composition.Comp
Namespace:
ProbabilityTheory.Kernel
Local context:
/-
Copyright (c) 2025 Bryan Wang. All rights reserved.
Released under ... | 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/Kernel",
"family_id": "isirreducible_of_le_measure",
"file_id": "mathlib/Mathlib/Probability/Kernel/Irreducible.lean",
"sample_id": "e2222f0bad75d2e7deb1fd25fdb3ceb2a09b7f0046dfa22182f45b77e039767c"
} | train |
{
"chosen_proof_sha256": "8b5027e289eea500a3bcfb155a81791b1a51bc31d201ba8ef228cee84d9c2f28",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1328c6946c14123810833c46f6825053fa... | by
simp [IsIdempotentElem, ← star_mul]
alias ⟨_, IsIdempotentElem.star⟩ := IsIdempotentElem.star_iff | true | {
"chosen_tokens": 20,
"rejected_tokens": 5,
"token_jaccard": 0.105263,
"token_length_ratio": 0.25
} | wrong_namespace_qualification | wrong_namespace_qualification | 7e97c6bc4b773534d536b29be342f92ad773170c983893bd34f22ba1642aeb73 | 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.Star.SelfAdjoint
public import Mathlib.Algebra.Module.Basic
public import Mathlib.Algebra.Module.Equiv.Defs
public import Mathlib.Algebra.Module.Linear... | by
exact OProverMissing.missing_d2e80bfd09f7 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ab23365fd8632d286b08ca615458823568e27c5f2c70594b84fff7963333f482",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Star",
"family_id": "isidempotentelem",
"file_id": "mathlib/Mathlib/Algebra/Star/Module.lean",
"sample_id": "d2e80bfd09f7c8ffd630b3233e0baf00abf1e879c638b96895f9ef4ded614180"
} | train |
{
"chosen_proof_sha256": "b925b94bb1cfb3ff34ceacbe250abd23143fbe7540078964bfa526bb642e3c9f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
classical
simpa [dotProduct_comm] using! (isSymmetric_toEuclideanLin_iff.mpr hA).im_inner_self_apply _ | false | {
"chosen_tokens": 17,
"rejected_tokens": 5,
"token_jaccard": 0.105263,
"token_length_ratio": 0.294118
} | sorry_admit_sorryAx_proof_hole | sorryAx | 7f0a291701f1a7784e69e2424170e04b16d28b4a90eb53af80e67cdab8aa14d6 | 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.InnerProductSpace.PiL2
public import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.Analysis.InnerProductSpace.Adjoint
Namespace:
Matrix
Loca... | 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": "Analysis/Matrix",
"family_id": "ishermitian",
"file_id": "mathlib/Mathlib/Analysis/Matrix/Hermitian.lean",
"sample_id": "f7bbb727f1ea351360c7052b952d4fc3113423ef399aac4de4fe2b1d680daf36"
} | train |
{
"chosen_proof_sha256": "a90544dc513a05353e26becaa7c94b77ef02a2abe517eb6e4b51662aba7c3cd3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "74398be95deeaf66769f719ac90f40d249... | by
simpa [← Basis.coord_apply] using LinearMap.congr_fun (basis_coord R n b s) x | true | {
"chosen_tokens": 20,
"rejected_tokens": 3,
"token_jaccard": 0.047619,
"token_length_ratio": 0.15
} | circular_dependency | circular_dependency | 7f182e9f1ef28adea082b090eccd216911e16e2af053a3301ac9f68e6308d1c8 | 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.ExteriorPower.Basic
public import Mathlib.LinearAlgebra.ExteriorPower.Pairing
public import Mathlib.RingTheory.Finiteness.Subalgebra
public impor... | by
exact basis_repr_apply | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "1fba34c09993a37d176b36897273bd81ec922867e74ae405f6efd33a8f8fd1d5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/ExteriorPower",
"family_id": "basis_repr_apply",
"file_id": "mathlib/Mathlib/LinearAlgebra/ExteriorPower/Basis.lean",
"sample_id": "5c25dc9430d8645a6917c9b8ba0f572387c246c6713ecceebd925b758bc2e637"
} | train |
{
"chosen_proof_sha256": "de3f7865b87871f2b8374f7e15ee4575771052e51846b14aedf16b2bb0b0042a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
ext ⟨a, b⟩
simp only [mem_smulAntidiagonal, mem_singleton, Prod.ext_iff]
constructor
· rintro ⟨has, hat, hst⟩
obtain rfl :=
(hs.min_le hns has).eq_of_not_lt fun hlt =>
(SMul.smul_lt_smul_of_lt_of_le hlt <| ht.min_le hnt hat).ne' hst
exact ⟨rfl, IsCancelSMul.left_cancel _ _ _ hst⟩
· ri... | false | {
"chosen_tokens": 92,
"rejected_tokens": 3,
"token_jaccard": 0.042553,
"token_length_ratio": 0.032609
} | by_exact_placeholder | placeholder | 7f6b2ef97c2fd4f7c9c55e9f8886229ba6f7eb65e68003e24550f0aa58309bed | 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.Pointwise.Set.Scalar
public import Mathlib.Data.Set.SMulAntidiagonal
Namespace:
Finset
Local context:
/-
Copyright (c) 2024 Scott Carnahan. All... | 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/Finset",
"family_id": "smulantidiagonal_min_smul_min",
"file_id": "mathlib/Mathlib/Data/Finset/SMulAntidiagonal.lean",
"sample_id": "986207edb34e921fae14ac5de27f5721ed1936c5cf23f19974ccfb40895a7398"
} | train |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.