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": "2ec97858c3d58fdf4fe63a79d350f35bc6aeb51e4c5540c62fc1b50ae3db70dc",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e9f1b50cf06579e6c3b06af93b5a67fd43... | by
simp [toInt, (exists_translation val).choose_spec] | false | {
"chosen_tokens": 12,
"rejected_tokens": 19,
"token_jaccard": 0.857143,
"token_length_ratio": 1.583333
} | markdown_preamble_trailing | fence | ee05fc5cdb91ecf72849f9fbf2d659f99dc88d53d3a5bc55b83c7e85d43a513c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Data.List.Pairwise
public import Batteries.Tactic.GeneralizeProofs
public import Mathlib.Tactic.Order.CollectFacts
Namespace:
Mathlib.Tactic.Order.ToInt
Lo... | ```lean
by
simp [toInt, (exists_translation val).choose_spec]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Tactic/Order",
"family_id": "toint_le_toint",
"file_id": "mathlib/Mathlib/Tactic/Order/ToInt.lean",
"sample_id": "0a540970c97c25da7889f6fd2aaf6aa1ff05b98f178319841aaf0767908b62e7"
} | train |
{
"chosen_proof_sha256": "5a23d360fd69064fcfa72d36f2b8acb7fa78696675236423d19ef91035657ba8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "92bd547cc8e3c70d9fdff8bf30a6d290d3... | by
have _ := t.adj₂.isLeftAdjoint
refine ⟨fun h ↦ by rw [← map_rightToLeft_app]; exact G.map_epi _, fun h ↦ ?_⟩
rw [epi_rightToLeft_app_iff_epi_map_adj₁_unit_app]
simpa using epi_comp (t.adj₂.unit.app (H.obj X)) (H.map (t.adj₂.counit.app X ≫ t.adj₁.unit.app X)) | true | {
"chosen_tokens": 85,
"rejected_tokens": 3,
"token_jaccard": 0.046512,
"token_length_ratio": 0.035294
} | circular_dependency | circular_dependency | ee18c16159a818421101de62b1b8fc5cad2541a64f45cff8d53135b2214a5232 | 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.Adjunction.Opposites
public import Mathlib.CategoryTheory.Adjunction.Unique
public import Mathlib.CategoryTheory.Monad.Adjunction
Namespace:
Ca... | by
exact epi_rightToLeft_app_iff | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "ddc482d0d6fba2edf01bb84f2468dd9fd76e044d18a7ea81fc0872c827bf4c6c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Adjunction",
"family_id": "epi_righttoleft_app_iff",
"file_id": "mathlib/Mathlib/CategoryTheory/Adjunction/Triple.lean",
"sample_id": "45bdf565a874c8590b8b9e375ecaaa19d5c42795fb168fac5a8abcf519632618"
} | train |
{
"chosen_proof_sha256": "397dccc7f0b407dbb3cdb395970fa551240d91f8fe8d8b6664c36ca222b24937",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simp [h]
-- used in the `nlinarith` normalization steps. The `_` argument is for uniformity. | false | {
"chosen_tokens": 25,
"rejected_tokens": 2,
"token_jaccard": 0.047619,
"token_length_ratio": 0.08
} | by_exact_placeholder | by? | ee3bd8c9e361c4b26e6bd1744283cee677f4c49dda674ced0123c2f4cf4cb3a2 | 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.Ineq
public import Mathlib.Data.Nat.Cast.Order.Ring
Namespace:
Mathlib.Tactic.Linarith
Local context:
/-
Copyright (c) 2020 Robert Y. Lewis. All rights ... | 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/Linarith",
"family_id": "mul_zero_eq",
"file_id": "mathlib/Mathlib/Tactic/Linarith/Lemmas.lean",
"sample_id": "890928cea67490b486f404ea924568c9df0ae3523215f6af6df2d816e57e58f7"
} | train |
{
"chosen_proof_sha256": "82dcb0744887ee0b2b5e66ebed4e1d1db29ac81ed20eb128a4882cf8f95fc26d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "43907cb29f977ec17d763b2547778aa86b... | by
rw [aeval_def, aeval_def, ← coe_eval₂Hom, ← coe_eval₂Hom, map_eval₂Hom, ←
IsScalarTower.algebraMap_eq, Function.comp_def] | true | {
"chosen_tokens": 30,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.1
} | circular_dependency | circular_dependency | ee48d6f75702c5da7975234bcda47ce6931e015f02521c36145d3c57fd46a5ab | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Subalgebra.Tower
public import Mathlib.Algebra.MvPolynomial.Eval
Namespace:
MvPolynomial
Local context:
/-
Copyright (c) 2022 Yuyang Zhao. Al... | by
exact aeval_algebraMap_apply | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "45c36b824df0a71cc3b55591248d7a788dc004061e2e389e93371119f3b6afbb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/MvPolynomial",
"family_id": "aeval_algebramap_apply",
"file_id": "mathlib/Mathlib/RingTheory/MvPolynomial/Tower.lean",
"sample_id": "f073ff6b550aeff5c73e243e84ebf2d91d1d169b0b3b2c4fe873846d588b6590"
} | train |
{
"chosen_proof_sha256": "a1aa8b6078219bed7946cc7376ea4840b9f886018bc53ce22deefb93a518874b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
change c = (ZMod.ringEquivCongr hn) (toFun n g)
rw [← toFun_unique' n g (ZMod.ringEquivCongr hn.symm c)
(fun t ht ↦ by rw [hc t ht, ZMod.ringEquivCongr_val]), ← ZMod.ringEquivCongr_symm hn,
RingEquiv.apply_symm_apply] | false | {
"chosen_tokens": 58,
"rejected_tokens": 5,
"token_jaccard": 0.030303,
"token_length_ratio": 0.086207
} | sorry_admit_sorryAx_proof_hole | sorryAx | ee6f80b73f01e3f1281a99a35b2ad1fecb2ee59ff7dd0970735351583463d57f | 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.Aut
public import Mathlib.NumberTheory.Padics.RingHoms
public import Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity
public import Mathlib.Ring... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/Cyclotomic",
"family_id": "unique",
"file_id": "mathlib/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean",
"sample_id": "9eef900fc40300308ff1ecb2aed829f9ba2f08a85e294734879bf55ae56f16ca"
} | train |
{
"chosen_proof_sha256": "598e0d2c4be030094c0dbca69a5ab0955fb2fcd00976225f2cdbe9fd5333ec6f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0e7d9d80f9ff2362b1a3f4e14dd4db88de... | by
simp only [List.applyId]
cases h₃ : List.dlookup x (List.map Prod.toSigma (xs.zip ys)) with
| none =>
dsimp [Option.getD]
rw [h₁.mem_iff]
| some val =>
have h₂ : ys.Nodup := h₁.nodup_iff.1 h₀
replace h₁ : xs.length = ys.length := h₁.length_eq
dsimp
induction xs generalizing ys with
... | true | {
"chosen_tokens": 314,
"rejected_tokens": 2,
"token_jaccard": 0.010204,
"token_length_ratio": 0.006369
} | invalid_tactic | invalid_tactic | ee724fde98b857ec18cbd839c62dba6a0ea7a6e96c0c5256737ec4874bc8f62b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Data.MLList.Basic
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.ZeroLEOne
public import Mathlib.Data.DFinsupp.Defs
public... | by
__oprover_invalid_tactic_f81f8323cb28 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "14357a8b814e64f70e03d6fdfbf8e9aab01ee485782e15c79488bc1a423be472",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Testing/Plausible",
"family_id": "applyid_mem_iff",
"file_id": "mathlib/Mathlib/Testing/Plausible/Functions.lean",
"sample_id": "f81f8323cb2859a12bdc22124ed1f8f7177b728cd0e3476841f3fae387cfd30b"
} | train |
{
"chosen_proof_sha256": "fb278dc04d580276be7c164bc304823032754bdaf50110401d324e4d50eff42b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
simp [mem_def, or_and_right, exists_or] | false | {
"chosen_tokens": 9,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.555556
} | sorry_admit_sorryAx_proof_hole | sorryAx | eea6608762fd6737734fe12b74ea1be0ddcf199d63508ed0ebc91ba910b00eb3 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.Monoid.NatCast
public import Mathlib.Algebra.Ring.Nat
public import Mathlib.Data.Sigma.Basic
public... | 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": "SetTheory",
"family_id": "mem_cons",
"file_id": "mathlib/Mathlib/SetTheory/Lists.lean",
"sample_id": "96dbde4f4151db83b5ae77d34dc896b57800591a2988fbcdab30f0f92312396b"
} | train |
{
"chosen_proof_sha256": "e064b140ea88edff320b5c2da73ccde8cd1103d2a90b6e31a432655ee413c425",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1030995450c428ffdced06a8e3ef05db04... | by
rw [← Semigroup.mem_center_iff]
exact Iff.rfl | true | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.272727,
"token_length_ratio": 0.25
} | circular_dependency | circular_dependency | eeacf16943bd3416a414d44bdd14ada2c7d23bf6a1993811104dc683527df04f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Submonoid.Operations
public import Mathlib.GroupTheory.Subsemigroup.Center
Namespace:
Submonoid
Local context:
/-
Copyright (c) 2021 Eric Wiese... | by
exact mem_center_iff | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "c632675d0a6c71ce194c5c1b710cd95e9669b65b8359b345104489d3f8f4d9eb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "GroupTheory/Submonoid",
"family_id": "mem_center_iff",
"file_id": "mathlib/Mathlib/GroupTheory/Submonoid/Center.lean",
"sample_id": "0329a22051730a8fdf389196512cdc9c856c4238f5a4d017fdd705ece3a68b4f"
} | train |
{
"chosen_proof_sha256": "228b521f0401eea33d29434cfc850bb07b080dd681e51957a62ad89f2e88ea24",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by rwa [sign_apply, if_neg <| asymm ha, if_pos] | false | {
"chosen_tokens": 13,
"rejected_tokens": 2,
"token_jaccard": 0.076923,
"token_length_ratio": 0.153846
} | sorry_admit_sorryAx_proof_hole | sorry | eee5014b17e295abd299a5b478463c2fb8bb51e5386eba100ce74c867acf7c30 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GroupWithZero.Defs
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.Algebra.Order.Ring.Defs
public import Mathlib.Tactic.DeriveFintype --... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Sign",
"family_id": "sign_neg",
"file_id": "mathlib/Mathlib/Data/Sign/Defs.lean",
"sample_id": "3edd16c3ca0318c2e5d045f7be510639e5b1ebe6db412620dbe457e127e3ccb4"
} | train |
{
"chosen_proof_sha256": "edfba660bad11fda81e3675a3dfc917ea4d8a1ad555ebc02b9f4efe9064c3712",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "eb30e3700a643dea643a5205a242d36b20... | by
intro y hy
obtain ⟨x, (rfl : f x = y), h'⟩ := exists_ne_zero_of_finsum_mem_ne_zero hy
grind [mem_support] | false | {
"chosen_tokens": 26,
"rejected_tokens": 33,
"token_jaccard": 0.916667,
"token_length_ratio": 1.269231
} | markdown_preamble_trailing | fence | ef0ee77312338d5339f167ddcfeeff059c016179f398ea3b6b51ed2763c6cd46 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.LocallyFinsupp
public import Mathlib.Topology.Spectral.Basic
Namespace:
Function.locallyFinsupp
Local context:
/-
Copyright (c) 2026 Raphael Douglas... | ```lean
by
intro y hy
obtain ⟨x, (rfl : f x = y), h'⟩ := exists_ne_zero_of_finsum_mem_ne_zero hy
grind [mem_support]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/LocallyFinsupp",
"family_id": "support_map_subset_of_forall_mem",
"file_id": "mathlib/Mathlib/Topology/LocallyFinsupp/Pushforward.lean",
"sample_id": "3b27666f8db369265222b785fd8e88c116b6148890f43b6b80939bde09ac3038"
} | train |
{
"chosen_proof_sha256": "f13ac1fd7bb3c4deec76c60552efe11696b97aca72a742edff11d90154604aae",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
apply elim_of_ne
rintro rfl
exact hl.notMem hj | true | {
"chosen_tokens": 10,
"rejected_tokens": 8,
"token_jaccard": 0.0625,
"token_length_ratio": 0.8
} | unsolved_goals | unsolved_goals | ef159c1a3602bb1c572f6467f3c6ca1440f5f786f4d9d12dd203e1f193b3a54f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.List.Nodup
public import Mathlib.Data.Set.Prod
Namespace:
List.TProd
Local context:
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Release... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b312ad9c187435d65336cfd9c9f55003070b241ed639aad0060f412837b94a1b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "36cd5f448af1bf1ecf44142699671ba22a07e7da26e9b39c0449f7bdf315a9b7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5e5a3af9a844517e88c0337432636443b0... | by
obtain ⟨P, φ, hsurj⟩ := H
obtain ⟨g, rfl⟩ := h
obtain ⟨a, ha⟩ := hsurj (algebraMap _ _ g)
have : IsLocalization.Away (f * g) (Localization.Away (φ a)) :=
ha ▸ .mul' (Localization.Away f) _ _ _
have : IsStandardEtale R (Localization.Away a) := .of_isLocalizationAway a
exact .mk _ (IsLocalization.Away.... | true | {
"chosen_tokens": 116,
"rejected_tokens": 5,
"token_jaccard": 0.071429,
"token_length_ratio": 0.043103
} | circular_dependency | circular_dependency | ef33b2d6d5337b62730b97d7c8628f39f6a24004f44f30c771b97036d40c1335 | 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.Etale.StandardEtale
public import Mathlib.RingTheory.LocalRing.ResidueField.Instances
public import Mathlib.RingTheory.RingHom.StandardSmooth
public... | by
exact HasStandardEtaleSurjectionOn.of_dvd | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "0889cafc10672d8068b2176e3a3eee7572788cd57e50d584d8565308d88a0af7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Unramified",
"family_id": "hasstandardetalesurjectionon",
"file_id": "mathlib/Mathlib/RingTheory/Unramified/LocalStructure.lean",
"sample_id": "dfed06d389dc5d68e96360f58a96e6261e6d908c48220f135bdf99a31b4e44ba"
} | train |
{
"chosen_proof_sha256": "06f4331fcea43b6ff267c69e7aee0a119d3182db477558ac433ba49db7222320",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9ad8141b680b0c8ac6adb8c9f1a6e8450d... | by
let K := Algebra.adjoin F (Set.range x)
suffices #E = #K by simp [K, this, ← lift_mk_eq'.2 ⟨hx.1.aevalEquiv.toEquiv⟩]
haveI : Algebra.IsAlgebraic K E := hx.isAlgebraic
refine le_antisymm ?_ (mk_le_of_injective Subtype.val_injective)
haveI : Infinite K := hx.1.aevalEquiv.infinite_iff.1 inferInstance
simpa... | true | {
"chosen_tokens": 96,
"rejected_tokens": 5,
"token_jaccard": 0.037037,
"token_length_ratio": 0.052083
} | wrong_namespace_qualification | wrong_namespace_qualification | ef540060d6cc529e4a7cc4a5a7a0ee2f8113a24bcdc3a28230a8baacb9aca582 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.IntermediateField.Adjoin.Basic
public import Mathlib.FieldTheory.MvRatFunc.Rank
public import Mathlib.RingTheory.Algebraic.Cardinality
public impor... | by
exact OProverMissing.missing_23f66eb623e6 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "b8b545dda2e848be3ee8f3769b4e9cdd1a5e03889479bc7f1dec4ad4a0aa04ac",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/AlgebraicIndependent",
"family_id": "istranscendencebasis",
"file_id": "mathlib/Mathlib/RingTheory/AlgebraicIndependent/RankAndCardinality.lean",
"sample_id": "23f66eb623e6a66dd8750aa398395aca87f38f1c676ee283cf181c130d4fe931"
} | train |
{
"chosen_proof_sha256": "caf7c1208514616ff661b06fa1617859cfde2ba543e0b78c240ce9e176c8198e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simpa using h [[]] [[], []] | true | {
"chosen_tokens": 15,
"rejected_tokens": 8,
"token_jaccard": 0.076923,
"token_length_ratio": 0.533333
} | unsolved_goals | unsolved_goals | ef82e70677932199ade1f3599e4ab697891a57e01986703e2586193dc2b90756 | 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.Basic
Namespace:
InformationTheory
Local context:
/-
Copyright (c) 2026 Elazar Gershuni. All rights reserved.
Released under Apache 2.0 license as d... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "df3051af8ba2e3662905dff00b02f6a2922ea6ff62ed33928904990cd5926a2d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "InformationTheory/Coding",
"family_id": "uniquelydecodable",
"file_id": "mathlib/Mathlib/InformationTheory/Coding/UniquelyDecodable.lean",
"sample_id": "50e5e16cfcfd38d6568b5e49b24fb4dda95d8927c47ec6721aa44339e4393525"
} | train |
{
"chosen_proof_sha256": "63cd494fa84f771bdfe24b254aba32aae9ae2b28ee1e64d13bc4951087609f7b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
induction γ using Quotient.ind with | mk γ =>
simpa [← mk_trans, ← mk_symm, ← mk_refl, eq] using Homotopic.trans_symm γ | false | {
"chosen_tokens": 30,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.066667
} | sorry_admit_sorryAx_proof_hole | admit | efc83613e376964d676e178b4faeddb5ccad8e011d1756ef0af2c2bb87e8d813 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Groupoid.Grpd.Basic
public import Mathlib.Topology.Category.TopCat.Basic
public import Mathlib.Topology.Homotopy.Path
public import Mathlib.Data... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/FundamentalGroupoid",
"family_id": "trans_symm",
"file_id": "mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean",
"sample_id": "3d37610ea90fd4d8210d4ae3653f0e831fc85b4617585bafc5fe04742a37ccee"
} | train |
{
"chosen_proof_sha256": "eff3de0495382d14fb1653bae8a0cc2d9bf54b578da62db37b5ffda72bfdcbdf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "afd985109f6ef269349dc0bffcdceccc63... | by
simp [← le_bot_iff, not_le_iff_exists] | true | {
"chosen_tokens": 8,
"rejected_tokens": 2,
"token_jaccard": 0.111111,
"token_length_ratio": 0.25
} | invalid_tactic | invalid_tactic | f00ee8e61073715ee7acb2f5586bbce306ced294cc03c35a440902768205a0d0 | 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.ObjectProperty.ClosedUnderIsomorphisms
public import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
public import Mathlib.Order.CompleteL... | by
__oprover_invalid_tactic_27c82bde5cab | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "84aa0638ac85283a283ba37781cc99c69de4a07e4a8896b2cdc264de01ee694f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/ObjectProperty",
"family_id": "ne_bot_iff_exists",
"file_id": "mathlib/Mathlib/CategoryTheory/ObjectProperty/CompleteLattice.lean",
"sample_id": "27c82bde5cab40ddeb3cf4fe4ff5388002b1c6d7395440a1ff8ff9c86f617bfd"
} | train |
{
"chosen_proof_sha256": "ac66c6e721cc952e221200b69cfe94cb7c90e16c08e83be7b74b05ee3c9aa39f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by aesop
variable [Fintype ι] | false | {
"chosen_tokens": 7,
"rejected_tokens": 2,
"token_jaccard": 0.125,
"token_length_ratio": 0.285714
} | sorry_admit_sorryAx_proof_hole | sorry | f01d723b7e41cc3f22c75baf7d0baec0f5d98e5a6ff4f1c580478a5b112846c8 | 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.Sigma
public import Mathlib.Data.Fintype.OfMap
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Releas... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Fintype",
"family_id": "set",
"file_id": "mathlib/Mathlib/Data/Fintype/Sigma.lean",
"sample_id": "b7150f0dc88e25f3382430b108a0281da820a234916871dd325c0904c8b12aaf"
} | train |
{
"chosen_proof_sha256": "ae21492afcd16aacb24a3ebd74e7f36439c097260b4eee6f2b4d2294937fc244",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
cases F; rfl | true | {
"chosen_tokens": 5,
"rejected_tokens": 8,
"token_jaccard": 0.090909,
"token_length_ratio": 1.6
} | unsolved_goals | unsolved_goals | f032bb11f67502421fc0445b8fc1e819184d507f9b3c12c8c9e009842d763a79 | 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
Local context:
/-
Copyright (c) 2017 M... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e1bfc47e19243f1e07cf27fe20e5649df380f0cbefb679fcf6ef2a1db3b9a6c6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Analysis",
"family_id": "ofequiv_val",
"file_id": "mathlib/Mathlib/Data/Analysis/Topology.lean",
"sample_id": "0aeef97f4cfea55c0640efb59e6d6792baab329447e04559e5b0bd1794a70844"
} | train |
{
"chosen_proof_sha256": "8f849e0c2e5e9b058d0be7efa5644ffd7d171530c0bf0be894d1f2aff0fde505",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6214205efd64002eb318e2ef8f8fc42439... | by
simp [wInner_cWeight_eq_expect, ψ.norm_apply] | false | {
"chosen_tokens": 9,
"rejected_tokens": 13,
"token_jaccard": 0.75,
"token_length_ratio": 1.444444
} | missing_wrong_import | import | f06aa2b37201de8b83023d4bf449e683ba1ed34caa647b711dd414336217278d | 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.Expect
public import Mathlib.Algebra.Group.AddChar
public import Mathlib.Analysis.RCLike.Inner
Namespace:
AddChar
Local context:
/-
Copy... | import Mathlib.WrongImport
by
simp [wInner_cWeight_eq_expect, ψ.norm_apply] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/Fourier",
"family_id": "winner_cweight_self",
"file_id": "mathlib/Mathlib/Analysis/Fourier/FiniteAbelian/Orthogonality.lean",
"sample_id": "76d9a3f6de2535e710efca9f1d5d77672d61eb6404d0c9c6e1be8e32402d6ace"
} | train |
{
"chosen_proof_sha256": "51d70e6a078b791d67a01b436c28657296809a50b25f00afb27df7a2d7b5b11b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "262e7e5adf8e8babe0e63c45b4903dee90... | by
apply SheafOfModules.hom_ext
ext U x
exact congr($(H U.unop) x) | true | {
"chosen_tokens": 20,
"rejected_tokens": 5,
"token_jaccard": 0.176471,
"token_length_ratio": 0.25
} | wrong_namespace_qualification | wrong_namespace_qualification | f0734afcf758245bb29ec920018b429d72d44c2c868c31e2f39c330526233171 | 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.Category.ModuleCat.Sheaf.Abelian
public import Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits
public import Mathlib.Algebra.Category.ModuleCat.Sheaf... | by
exact OProverMissing.missing_2f4e8f65d6a8 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1d771dafcc0d7485c8a06e17a181d18253bf8b881f91c382dc5e0382af0f0065",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Modules",
"family_id": "hom_ext",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Modules/Sheaf.lean",
"sample_id": "2f4e8f65d6a8377847b7d2fb1bab8531851b8db89761aa4db8d2e360d21ed05e"
} | train |
{
"chosen_proof_sha256": "c3e85a55413d2bdbfb981811de0e7022d287d440d63c91a6c4ae1530e90621b9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
simpa [beta, unbeta, coprimes] using mod_eq_of_modEq
((chineseRemainderOfFinset (l[·]) (coprimes (l[·])) Finset.univ
(by simp [coprimes])
(by simpa using Set.pairwise_univ.mpr (pairwise_coprime_coprimes _))).prop i (by simp))
(coprimes_lt _ _) | false | {
"chosen_tokens": 67,
"rejected_tokens": 3,
"token_jaccard": 0.034483,
"token_length_ratio": 0.044776
} | by_exact_placeholder | exact? | f094f0308943b6ea5172b1f1a5a1d6f6bd6d6bed50b7ef1ab70a783fe8e8c2af | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.ModEq
public import Mathlib.Data.Nat.ChineseRemainder
public import Mathlib.Data.Nat.Prime.Defs
public import Mathlib.Data.Nat.Pairing
public import M... | 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": "Logic/Godel",
"family_id": "beta_unbeta_coe",
"file_id": "mathlib/Mathlib/Logic/Godel/GodelBetaFunction.lean",
"sample_id": "8dd69de99772e4b60d7d7c33b72ef5655efe8caa0668569bdbff7dc2b9a6a47c"
} | train |
{
"chosen_proof_sha256": "d93c674782daaf325f075697a98c3704ebb23b2b31510b6d8f8e1890f07f7bba",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "405da6d6559020f606063085aeff9021bd... | by
simp_rw [noZeroSMulDivisors_iff, or_iff_not_imp_left]
exact ⟨fun h r hr m eq ↦ h eq hr, fun h r m eq hr ↦ h r hr m eq⟩ | false | {
"chosen_tokens": 33,
"rejected_tokens": 40,
"token_jaccard": 0.894737,
"token_length_ratio": 1.212121
} | markdown_preamble_trailing | fence | f0d86fbefe0be20fae7af12a90cb20af9ea9cab28024fe42fb214fe9574c44ae | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Module.Torsion.Free
public import Mathlib.Tactic.Contrapose
Namespace:
(root)
Local context:
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reser... | ```lean
by
simp_rw [noZeroSMulDivisors_iff, or_iff_not_imp_left]
exact ⟨fun h r hr m eq ↦ h eq hr, fun h r m eq hr ↦ h r hr m eq⟩
``` | {
"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/NoZeroSMulDivisors",
"family_id": "nozerosmuldivisors_iff_right_eq_zero_of_smul",
"file_id": "mathlib/Mathlib/Algebra/NoZeroSMulDivisors/Defs.lean",
"sample_id": "dac74140b5028292313ba8ba24d22052ad637c133b6e08b981669e5ff7752981"
} | train |
{
"chosen_proof_sha256": "24551563a4e7d2141b3625f37d2a2f5a066ec7e2be8f2231f8bc3aa32f433342",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d99aae00c5ca2aaa8d6ff924384016fd0c... | by
rw [isUnit_iff_and, and_iff_right ha] | true | {
"chosen_tokens": 8,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.375
} | unknown_identifier | unknown_identifier | f11659d9565f9873a790ac3a5bf15cef6941f572162a9d0dc34eeab09b5fc53e | 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.Maximal
Namespace:
Submonoid
Local context:
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as descr... | by
exact __oprover_missing_8d47a007615a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c2a96c4d30f845f9980da0b0069a18d844b6bf0e11874b36c83183bc723aa902",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Ideal",
"family_id": "isunit_iff_of_ne_zero",
"file_id": "mathlib/Mathlib/RingTheory/Ideal/Nonunits.lean",
"sample_id": "8d47a007615a250bf24cd20685fe57ad901c62e3e8a8706c9506725d94de99c7"
} | train |
{
"chosen_proof_sha256": "82c27cb6e99a7d6e45601e4127587eb4a9e18558df87ff23a7cf9764fabc33f4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
simp [galRestrict', galRestrict', Subalgebra.algebraMap_eq] | false | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.1,
"token_length_ratio": 0.181818
} | sorry_admit_sorryAx_proof_hole | sorry | f1388ddceea1419c2c899e93563748b40f8df14dc285d9228d5fd1c1a96db4b9 | 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... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/IntegralClosure",
"family_id": "algebramap_galrestrict'_apply",
"file_id": "mathlib/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean",
"sample_id": "12e4ed73b2164c9034e09e737bb93b021152af848a112fdd9ed5bf5c79a07e22"
} | train |
{
"chosen_proof_sha256": "65f2254f5d37c07b087f477ca102b136468a343c041fb416c4b1c505f1281550",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [← linearIndepOn_univ_iff, ← Finset.coe_univ, show @Finset.univ (Fin 2) _ = {0,1} from rfl,
Finset.coe_insert, Finset.coe_singleton, LinearIndepOn.pair_iff _ (by trivial)]
simp | false | {
"chosen_tokens": 48,
"rejected_tokens": 2,
"token_jaccard": 0.03125,
"token_length_ratio": 0.041667
} | by_exact_placeholder | by? | f17cc3451f130505797ef4551c70e1b478ff8d612a5a0a3b81d5a334c92920da | 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.Tuple.Reflection
public import Mathlib.LinearAlgebra.Dual.Defs
public import Mathlib.LinearAlgebra.Finsupp.SumProd
public import Mathlib.LinearAlgebra... | 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/LinearIndependent",
"family_id": "linearindependent",
"file_id": "mathlib/Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean",
"sample_id": "fb692754d646fb760ef043b5d79501f41f1d28e50ad2da85d3e2605daffd725b"
} | train |
{
"chosen_proof_sha256": "4acb529aee5f1ece8076fbbdc844587e9d93366665ac8d5f6045ad6127fa4525",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
have := lift_lt.{_, max (u + 1) v}.2 (lift_lt_univ c)
rw [lift_lift, lift_univ, univ_umax.{u, v}] at this
exact this | true | {
"chosen_tokens": 40,
"rejected_tokens": 8,
"token_jaccard": 0.09375,
"token_length_ratio": 0.2
} | unsolved_goals | unsolved_goals | f1bbc1ef0c05ab80e553f6410e67f9887290073aed26bf4701718eda3b3a6977 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.SetTheory.Ordinal.Basic
Namespace:
Cardinal
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as de... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "bd943cb64286310513638bec0fd0e80fbfdbb9cffe28b8feea552363e234cf7e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "SetTheory/Ordinal",
"family_id": "lift_lt_univ",
"file_id": "mathlib/Mathlib/SetTheory/Ordinal/Univ.lean",
"sample_id": "beebaf6401da7ba6ae02cab75a18db90132912fa7666bc24027b84d9741587a1"
} | train |
{
"chosen_proof_sha256": "187dbb6cbe73173dbea1f8f380afdbd3c2e979ea26a53fbec596e63a5f1c929e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by by_cases a = 0 <;> grind [zero_le] | false | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.25
} | by_exact_placeholder | exact? | f1f91a332ac2afd361de4f1547aaf502447e01923e093e604b7813b593b064cf | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.Finite.Basic
public import Mathlib.RingTheory.Valuation.Basic
Namespace:
FiniteField
Local context:
/-
Copyright (c) 2026 María Inés de Frutos-Fe... | 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": "FieldTheory/Finite",
"family_id": "valuation_algebramap_le_one",
"file_id": "mathlib/Mathlib/FieldTheory/Finite/Valuation.lean",
"sample_id": "9f7eb7bcc0d3153a05f5019eda72791728388ea6466ec87e305f6457e18d50ee"
} | train |
{
"chosen_proof_sha256": "99f33d97a3ff079eb01e172cf32d5125fc28f1ebfb2b8c076d51536e340d9ee6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
apply tendsto_integral_filter_of_dominated_convergence _ _ _ hf.norm
· filter_upwards with v
nth_rewrite 2 [show f v = cexp (- (0 : ℝ) * ‖v‖ ^ 2) • f v by simp]
apply (Tendsto.cexp _).smul_const
exact tendsto_inv_atTop_zero.ofReal.neg.mul_const _
· filter_upwards with c using
AEStronglyMeasur... | false | {
"chosen_tokens": 144,
"rejected_tokens": 2,
"token_jaccard": 0.014286,
"token_length_ratio": 0.013889
} | sorry_admit_sorryAx_proof_hole | sorry | f24cbde0b979366f70a0bc4794768059d7d873df5d9ce207ea02c4c31ade800a | 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.Integral.PeakFunction
public import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
Namespace:
Real
Local context:
/-
Copyright (c)... | 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": "Analysis/Fourier",
"family_id": "tendsto_integral_cexp_sq_smul",
"file_id": "mathlib/Mathlib/Analysis/Fourier/Inversion.lean",
"sample_id": "dce31b4f3667c3830b421cb76a3fefb37b0de149d8e70ed049af7ee6287ae930"
} | train |
{
"chosen_proof_sha256": "887cd45eb0ad0270c148f43d5c6fb5160031574c8c6c116689e7fb4bdff4ffd8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rcases finite_or_infinite X with _ | _
· have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_right S X)
rwa [lift_umax.{v, u}, lift_id'] at this
have key (x : X) (s s' : S) (h : s • x = s' • x) (hc : Commute s s') : x /ₒ s = x /ₒ s' := by
rw [oreDiv_eq_iff]
refine ⟨s, s'.1, h,... | true | {
"chosen_tokens": 190,
"rejected_tokens": 8,
"token_jaccard": 0.052632,
"token_length_ratio": 0.042105
} | unsolved_goals | unsolved_goals | f288b1217155fde1b29741560b09e8035d82ac315bdd14a0cccdc44f3fc01fd7 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Pigeonhole
public import Mathlib.GroupTheory.OreLocalization.Basic
public import Mathlib.SetTheory.Cardinal.Arithmetic
Namespace:
OreLocalization... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "9e5fa086937ddbc936aca1e5073aa97e5282eed6598ca162b5cb0943fe6d74c6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "GroupTheory/OreLocalization",
"family_id": "cardinalmk_le_lift_cardinalmk_of_commute",
"file_id": "mathlib/Mathlib/GroupTheory/OreLocalization/Cardinality.lean",
"sample_id": "00174d518458fef9a309864e49f69be9bd090110f25e5905b3ad9090d83909e4"
} | train |
{
"chosen_proof_sha256": "cd750d60c8401fd18ff81d5b091050a985f85f02e4e58110a29f7aecd6f9f0ac",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "68d3eb64f1e33f838236f5814f4593cf22... | by
simp [card_eq_card_quotient_mul_card_subgroup s, @dvd_mul_right ℕ]
variable {H : Type*} [Group H] | false | {
"chosen_tokens": 21,
"rejected_tokens": 26,
"token_jaccard": 0.818182,
"token_length_ratio": 1.238095
} | markdown_preamble_trailing | preamble | f2e1555249406de1da330958165d290725c296f9313c7f4953d084731fce1ca5 | 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.GroupTheory.Coset.Basic
public import Mathlib.SetTheory.Cardinal.Finite
Namespace:
Subgroup
Local context:
/-
Copyright (c) 2018 Mitchell Rowett. All rights ... | Here is the proof:
by
simp [card_eq_card_quotient_mul_card_subgroup s, @dvd_mul_right ℕ]
variable {H : Type*} [Group H] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "GroupTheory/Coset",
"family_id": "card_quotient_dvd_card",
"file_id": "mathlib/Mathlib/GroupTheory/Coset/Card.lean",
"sample_id": "f168373cfd1f31b36e20d360cfbb18b1a6084a7395ef31c4c3eb93947132babd"
} | train |
{
"chosen_proof_sha256": "6d7fe2ba7791e82259c7c7dac10b1de23d8b4e925a2fa271084af7b3f600bee5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
by_cases hf2 : f = 0; · rw [hf2]; exact dvd_zero _
rw [factor, dif_pos (WfDvdMonoid.exists_irreducible_factor hf1 hf2)]
exact (Classical.choose_spec <| WfDvdMonoid.exists_irreducible_factor hf1 hf2).2 | false | {
"chosen_tokens": 45,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.066667
} | by_exact_placeholder | exact? | f35a8e7db0623e1e93c90dd1632daf4551e9dceb04b874670bc5a2d449d43c8a | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.CharP.Algebra
public import Mathlib.FieldTheory.SplittingField.IsSplittingField
public import Mathlib.RingTheory.Algebraic.Basic
Namespace:
Polynomial... | by
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": "FieldTheory/SplittingField",
"family_id": "factor_dvd_of_not_isunit",
"file_id": "mathlib/Mathlib/FieldTheory/SplittingField/Construction.lean",
"sample_id": "3b9bf8a1b9ae9ed64027deecfa9b2b28510490bbc0204132c8e750b77e3c634e"
} | train |
{
"chosen_proof_sha256": "197871194b11ccd2f7b68c20098508f4fb8d58251817640d3a8e7c9a33767c9c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7d23803bcb7a0b31219450a0f71c8fc39a... | by
refine ⟨fun h _ _ ↦ by rw [h], fun h ↦ ?_⟩
rw [DFunLike.ext'_iff]
apply Function.Surjective.injective_comp_right mkAlgHom_surjective
simpa [← AlgHom.coe_comp] using MvPolynomial.algHom_ext fun ⟨n, m⟩ ↦ h n m | true | {
"chosen_tokens": 54,
"rejected_tokens": 3,
"token_jaccard": 0.03125,
"token_length_ratio": 0.055556
} | circular_dependency | circular_dependency | f371a38e650cd99b2a7379d936de257bacd66cb57805b5348dcc452b7706fc8e | 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.MvPolynomial.Eval
public import Mathlib.Algebra.RingQuot
public import Mathlib.RingTheory.DividedPowers.Basic
Namespace:
DividedPowerAlgebra
Local co... | by
exact algHom_ext_iff | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "4b41f2a593c49deb962ebe4b78ce3bd32d961a187f21987ee4c90767fe7bc73f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/DividedPowerAlgebra",
"family_id": "alghom_ext_iff",
"file_id": "mathlib/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean",
"sample_id": "41bafb9f17936ff03795282fab01c1edc2ec5df90a2dc068b3d884811a5b8b02"
} | train |
{
"chosen_proof_sha256": "97f1a84bedb0ef6995acbe3d016b9163eb9ac17f32bd0890956d8e9bd5498309",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
refine (hB.hasLaw I).congr ?_
have : ∀ᵐ ω ∂P, ∀ i : I, B i ω = C i ω := ae_all_iff.2 fun _ ↦ h _
filter_upwards [this] with ω hω using funext fun i ↦ (hω i).symm | false | {
"chosen_tokens": 61,
"rejected_tokens": 3,
"token_jaccard": 0.025641,
"token_length_ratio": 0.04918
} | by_exact_placeholder | placeholder | f40309344a510224d9770d3ea2213bfcb54fa2857bc5d2ab522fb6fcb66ee5ed | 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.BrownianMotion.GaussianProjectiveFamily
public import Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Def
public import Mathlib.Probab... | 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/BrownianMotion",
"family_id": "isprebrownianreal",
"file_id": "mathlib/Mathlib/Probability/BrownianMotion/Basic.lean",
"sample_id": "567f3c5c6d2031f7fb93f23ec479080605b61b08ad68d9daa949e04606ce6590"
} | train |
{
"chosen_proof_sha256": "5914fc3ed55a874cb691c1f7b7c08b38f9e056106005fb4efe3ad9109dba7f6a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "94baa657612564dcd8266b8f23b88fc4f5... | by
apply (Ideal.homogeneousCore 𝒜 I).isHomogeneous.isPrime_of_homogeneous_mem_or_mem
· exact ne_top_of_le_ne_top h.ne_top (Ideal.toIdeal_homogeneousCore_le 𝒜 I)
rintro x y hx hy hxy
have H := h.mem_or_mem (Ideal.toIdeal_homogeneousCore_le 𝒜 I hxy)
refine H.imp ?_ ?_
· exact Ideal.mem_homogeneousCore_of_h... | false | {
"chosen_tokens": 66,
"rejected_tokens": 71,
"token_jaccard": 0.888889,
"token_length_ratio": 1.075758
} | markdown_preamble_trailing | trailing | f403dd22f2d145b528e5e9f653595d4e75bac7c3a6ad468bfbc1e02f4caa6242 | 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.GradedAlgebra.Homogeneous.Ideal
Namespace:
(root)
Local context:
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.... | by
apply (Ideal.homogeneousCore 𝒜 I).isHomogeneous.isPrime_of_homogeneous_mem_or_mem
· exact ne_top_of_le_ne_top h.ne_top (Ideal.toIdeal_homogeneousCore_le 𝒜 I)
rintro x y hx hy hxy
have H := h.mem_or_mem (Ideal.toIdeal_homogeneousCore_le 𝒜 I hxy)
refine H.imp ?_ ?_
· exact Ideal.mem_homogeneousCore_of_h... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/GradedAlgebra",
"family_id": "ideal",
"file_id": "mathlib/Mathlib/RingTheory/GradedAlgebra/Radical.lean",
"sample_id": "636539d6c0febb4c83b00bdc22993462b52f9a53f319750f1f167f122ce30681"
} | train |
{
"chosen_proof_sha256": "3db3fcbd4e98076f3d0533d22872c1a4a27f5605675d9e65936bcaf6671c864f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f1ecebbbb93ce9395668a94b0ca6950988... | by
simp_rw [← scottContinuousOn_univ] at ⊢
exact .fromProd (fun a ↦ (h₁ a).scottContinuousOn) (fun b ↦ (h₂ b).scottContinuousOn)
(fun a ↦ (h₁ a).monotone) (fun b ↦ (h₂ b).monotone) | true | {
"chosen_tokens": 59,
"rejected_tokens": 3,
"token_jaccard": 0.086957,
"token_length_ratio": 0.050847
} | unknown_identifier | unknown_identifier | f404585a26f6d125c21efbb18736bf413e4c51bd0e08e6b83622d37dbf533078 | 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.ScottContinuity
public import Mathlib.Order.Bounds.Lattice
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Christopher Hoskin. All rights reserv... | by
exact __oprover_missing_5f10fa5c3e8b | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "49d62ccef480c865a9ee8a98cb5d10270d13820caec9dc015e12c84e6a648408",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/ScottContinuity",
"family_id": "scottcontinuous",
"file_id": "mathlib/Mathlib/Order/ScottContinuity/Prod.lean",
"sample_id": "5f10fa5c3e8b99e44987fd6f1820e919de6c733d5031ca673f738ff243d9f672"
} | train |
{
"chosen_proof_sha256": "6695cef3757a420cfb80a318d6ef88b082eb4eef77453a026e398af8fe7ac543",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9f2e69ab8abb2e6cb6d83b900a61296628... | by
rw [iSup, Bool.range_eq, sSup_pair, sup_comm] | true | {
"chosen_tokens": 13,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.230769
} | circular_dependency | circular_dependency | f408bc396de89130f3c3f895044a48a5b933e57f1583ef4773ae5898578c7fb7 | 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.Bool.Set
public import Mathlib.Data.Nat.Set
public import Mathlib.Order.CompleteLattice.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Joh... | by
exact iSup_bool_eq | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "5fedb2df1b8cb8b6969b85a9360e6f03449667df4c301a59b87eb28996a68fdb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Order/CompleteLattice",
"family_id": "isup_bool_eq",
"file_id": "mathlib/Mathlib/Order/CompleteLattice/Lemmas.lean",
"sample_id": "37f235bb48e86794e4b09884c23d8d2d7e5709381d2c53948b9b73b1480e3fa6"
} | train |
{
"chosen_proof_sha256": "eaff7377a9ad125a2190653d93c4626decbfd214d6bde989ce3ba7f1259b5118",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b2e47c015eac842014bd0418a98dd81b89... | by
cases nonempty_fintype α
simpa using exists_max_image univ f univ_nonempty | true | {
"chosen_tokens": 10,
"rejected_tokens": 5,
"token_jaccard": 0.071429,
"token_length_ratio": 0.5
} | circular_dependency | circular_dependency | f444fb714b082981328cc58890c8a7a3bb5fe75c9f285f4954682165ce3256f2 | 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.Max
public import Mathlib.Data.Fintype.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released... | by
exact Finite.exists_max | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "cff0f934a9303e1bb8f27b908850b70b29b9035458347987071d64fa56a8e6dd",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "87c6abbbe5679e2db362b7fc92a4107a665977a26b58a1424f12d41fb84a8bc6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "247835032373bede62a267228c5e4b2505... | by
unfold logDeriv
simp [field]
ring | false | {
"chosen_tokens": 8,
"rejected_tokens": 15,
"token_jaccard": 0.8,
"token_length_ratio": 1.875
} | markdown_preamble_trailing | fence | f487d09facef28f5e8d6ef8378dae74c803863fcb31b5a9cd769649b7419a61e | 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.Derivation.MapCoeffs
public import Mathlib.FieldTheory.PrimitiveElement
Namespace:
Differential
Local context:
/-
Copyright (c) 2024 Daniel Weber.... | ```lean
by
unfold logDeriv
simp [field]
ring
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "FieldTheory/Differential",
"family_id": "logderiv_mul",
"file_id": "mathlib/Mathlib/FieldTheory/Differential/Basic.lean",
"sample_id": "2e2771fc6ad14eff3a9b98719630e14505321f0c2d2dbcd9d1df71f5b966db93"
} | train |
{
"chosen_proof_sha256": "a2f7906395a8fceee2188be53f90d4ed089166e4d7b933b3ba61e10fb185f1d5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
split_ifs with h
exacts [hs h, ht h] | false | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.25
} | by_exact_placeholder | exact? | f500c030a1d861d97ff842fe13223759f23266b7cb59c888378b79826ba51fed | 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.Countable
public import Mathlib.Order.ConditionallyCompleteLattice.Basic
public import Mathlib.Tactic.CrossRefAttribute
public import Mathlib.Tactic.F... | 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/MeasurableSpace",
"family_id": "measurableset",
"file_id": "mathlib/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean",
"sample_id": "aa6161864a0c5f5bc6339fe001dfcdf0b66c0b5bdb3cf1678dcb5f82be308aa0"
} | train |
{
"chosen_proof_sha256": "958795982b90c7f479780dff2a7eb96698a38b055a8fc856a14fd71baa43ed6a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c7aee18ddb84fda2619825e8c317978cac... | by rw [IsLUB.sSup_eq (h d₁ d₂ (isLUB_iff_sSup_eq.mpr rfl))]
mpr h _ d₁ d₂ _ hda := by rw [isLUB_iff_sSup_eq, ← (h d₁ d₂), IsLUB.sSup_eq hda]
alias ⟨ScottContinuous.map_sSup, ScottContinuous.of_map_sSup⟩ :=
scottContinuous_iff_map_sSup | true | {
"chosen_tokens": 61,
"rejected_tokens": 2,
"token_jaccard": 0.034483,
"token_length_ratio": 0.032787
} | invalid_tactic | invalid_tactic | f576b260c4266e6b0f58023991dff75b58e2813bd47e839418b76429e542a29d | 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.ScottContinuity.Prod
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license ... | by
__oprover_invalid_tactic_7885120c783d | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "fe51d2f00845dfceba088f47e1cfe4c166c8d1a699d09446097765153ebc774a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/ScottContinuity",
"family_id": "scottcontinuous_iff_map_ssup",
"file_id": "mathlib/Mathlib/Order/ScottContinuity/Complete.lean",
"sample_id": "7885120c783d836f7907adab08dd0290a06a8a8cf35af5f387b7e22d740bb408"
} | train |
{
"chosen_proof_sha256": "aea622174bd7323409020f456ba921f86430f2be7a1ef46c09da2cf27c53cb7b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
have hh : 0 < Γ.strictWidthInfty := Γ.strictWidthInfty_pos_iff.mpr Fact.out
have hΓ : Γ.strictWidthInfty ∈ Γ.strictPeriods := Γ.strictWidthInfty_mem_strictPeriods
have hanal := ModularFormClass.analyticAt_cuspFunction_zero f hh hΓ
have hper := periodic_comp_ofComplex f hΓ
simp_rw [IsZeroAtImInfty, ZeroAtFi... | false | {
"chosen_tokens": 96,
"rejected_tokens": 2,
"token_jaccard": 0.021277,
"token_length_ratio": 0.020833
} | by_exact_placeholder | by? | f5b2298cc151431196248c8a08e93525403792c1ed139b53c12093a8e9b79682 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.NumberTheory.ModularForms.QExpansion
public import Mathlib.NumberTheory.ModularForms.LevelOne.Basic
public import Mathlib.NumberTheory.ModularForms.EisensteinS... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/ModularForms",
"family_id": "iszeroatiminfty_of_valueatinfty_eq_zero",
"file_id": "mathlib/Mathlib/NumberTheory/ModularForms/CuspFormSubmodule.lean",
"sample_id": "79ff1c5746539b95fb7b7bd18831f461eb7a59f72b537902f6b1acccf75b210c"
} | train |
{
"chosen_proof_sha256": "ef657d6d5fc4b7d8887ac9017ba9ce06e5bcee28dc51338cd8d6ad2502a8eccb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
simp_rw [← union, disjoint_union_right, disjoint_iUnion_right]
exact ⟨disjointBase n i , fun _ _ ↦ disjoint_openCell_of_ne (by lia)⟩ | false | {
"chosen_tokens": 26,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.076923
} | sorry_admit_sorryAx_proof_hole | sorry | f5e05730d55da25be7402f3dd0ddb95d1b4c02281377512c0749332cda48217f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.CWComplex.Classical.Finite
public import Mathlib.Analysis.Normed.Module.RCLike.Real
Namespace:
Topology
Local context:
/-
Copyright (c) 2025 Floris ... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/CWComplex",
"family_id": "relcwcomplex",
"file_id": "mathlib/Mathlib/Topology/CWComplex/Classical/Subcomplex.lean",
"sample_id": "4b6a657545abaaf7eebda573d07dd91e0d6097e9312e9cce6c12048585c3e4e7"
} | train |
{
"chosen_proof_sha256": "54c464d6ce26bdef8908aa10edaaa4627d275c464a5704aa3ef40ec7a4c12203",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rintro ⟨m, hm⟩
have hm' : ∀ a, f a ≤ m := fun a ↦ hm <| Set.mem_range_self _
obtain ⟨a₀⟩ := ‹Nonempty α›
suffices ∀ b, f a₀ ≤ b → ∃ a, b < f a by
obtain ⟨a, ha⟩ : ∃ a, m < f a := this m (hm' a₀)
exact ha.not_ge (hm' a)
have h : ∀ a, ∃ a', f a < f a' := fun a ↦ (exists_gt a).imp (fun a' h ↦ hf h)
... | false | {
"chosen_tokens": 153,
"rejected_tokens": 3,
"token_jaccard": 0.038462,
"token_length_ratio": 0.019608
} | by_exact_placeholder | exact? | f5f4402442cd70dd6207fa4eb9bfc671f1d5a766e4ae24cb223939b9f3d96c29 | 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.SuccPred.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/SuccPred",
"family_id": "strictmono",
"file_id": "mathlib/Mathlib/Order/SuccPred/Archimedean.lean",
"sample_id": "d0ac2882ec427829126b8598a0819730b07105bce20b1228031035f70c0b76c9"
} | train |
{
"chosen_proof_sha256": "9eee49c5561e216892ae126cb44a6a373a2be5f508e4256a2d5f533053f0d013",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "21c394e35574aad91bc8d293cafaa5383d... | by
rw [(ofSet s hs).trans_ofSet hs']
ext <;> simp [hs'.interior_eq] | true | {
"chosen_tokens": 22,
"rejected_tokens": 5,
"token_jaccard": 0.095238,
"token_length_ratio": 0.227273
} | wrong_namespace_qualification | wrong_namespace_qualification | f5ff602d72fc6513f309e21650df2caab9ea091788674fedcb4d0c450b53094b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.OpenPartialHomeomorph.IsImage
Namespace:
OpenPartialHomeomorph
Local context:
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released... | by
exact OProverMissing.missing_3e916153cf2d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1dc5df246dc2fe1cb8c23f9d4a3f03f50a685894a0142280c5bdeaefeb92baf7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/OpenPartialHomeomorph",
"family_id": "ofset_trans_ofset",
"file_id": "mathlib/Mathlib/Topology/OpenPartialHomeomorph/Composition.lean",
"sample_id": "3e916153cf2d028f0fe2211091c80661d43ef75952801649ac6582ce0158842e"
} | train |
{
"chosen_proof_sha256": "9a8be05ccc00ea06c742ae40a8ee2f89b7edc8fc907070b2e2dab340e52a4333",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e7fe54825ee652c35445a03a95e094b290... | by
induction t with
| nil => rw [traverse, traverse, η.preserves_pure]
| node v l r hl hr =>
rw [traverse, traverse, η.preserves_seq, η.preserves_seq, η.preserves_map, hl, hr,
Function.comp_apply] | false | {
"chosen_tokens": 51,
"rejected_tokens": 56,
"token_jaccard": 0.833333,
"token_length_ratio": 1.098039
} | markdown_preamble_trailing | preamble | f630852fb6dbd3b820dd1cac396ae2dbfabb1fc909edd1102d20332890df7e5b | 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.Tree.Basic
public import Mathlib.Control.Applicative
public import Mathlib.Control.Traversable.Basic
Namespace:
BinaryTree
Local context:
/-
Copyright (... | Here is the proof:
by
induction t with
| nil => rw [traverse, traverse, η.preserves_pure]
| node v l r hl hr =>
rw [traverse, traverse, η.preserves_seq, η.preserves_seq, η.preserves_map, hl, hr,
Function.comp_apply] | {
"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/Tree",
"family_id": "naturality",
"file_id": "mathlib/Mathlib/Data/Tree/Traversable.lean",
"sample_id": "7f88709361e31e7cfe4a2d2240e22f16a88fdfa93df42a71071e525d8a5017f9"
} | train |
{
"chosen_proof_sha256": "c8b268100380170538174b78ca7f2045ab5992f487a935451a91e1cbc0d99b37",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "00ef4f5a3e97159007721ad98486c2afdf... | by
have h₁ : a i < supOfSeq a :=
Nat.lt_add_one_iff.mpr (le_max_of_le_right <| Finset.le_sup (by simp))
have h₂ : supOfSeq a ≤ (i + 1) * (supOfSeq a)! + 1 :=
le_trans (self_le_factorial _) (le_trans (Nat.le_mul_of_pos_left (supOfSeq a)! (succ_pos i))
(le_add_right _ _))
simpa only [coprimes] using l... | true | {
"chosen_tokens": 87,
"rejected_tokens": 5,
"token_jaccard": 0.046512,
"token_length_ratio": 0.057471
} | wrong_namespace_qualification | wrong_namespace_qualification | f63b09ce645bcd94d895f6c1d0e21514ea1db435057cef1188b364df666d428b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.ModEq
public import Mathlib.Data.Nat.ChineseRemainder
public import Mathlib.Data.Nat.Prime.Defs
public import Mathlib.Data.Nat.Pairing
public import M... | by
exact OProverMissing.missing_81a0159374a3 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "be7acb484b09252e54bfb7c47367a9747a31079f3f308761d7ea6b5fefd7a7c8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Logic/Godel",
"family_id": "coprimes_lt",
"file_id": "mathlib/Mathlib/Logic/Godel/GodelBetaFunction.lean",
"sample_id": "81a0159374a335acfc3a3be57440b1fb6c4f215c8d52062ffeefc8554afc5a68"
} | train |
{
"chosen_proof_sha256": "5cfc8f87ea330fcf416830673402289862153b1294b56a5e4d961dfa30463860",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
refine ⟨fun h => .of_comp (f := ({·} : β → Compacts β)) ?_, map_injective hf⟩
simp_rw [Function.comp_def, ← map_singleton hf]
exact h.comp singleton_injective | true | {
"chosen_tokens": 43,
"rejected_tokens": 8,
"token_jaccard": 0.076923,
"token_length_ratio": 0.186047
} | unsolved_goals | unsolved_goals | f647bd28ea8d91ee9a617020d27e1c858ce9e61c3023e66a40a4855e44fa2fae | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.Sets.Closeds
public import Mathlib.Topology.QuasiSeparated
Namespace:
TopologicalSpace.Compacts
Local context:
/-
Copyright (c) 2020 Floris van Door... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "5020a6c2a642297b0a53c5cde7e8f712789a337cdccc08855de7b987ca248699",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Sets",
"family_id": "map_injective_iff",
"file_id": "mathlib/Mathlib/Topology/Sets/Compacts.lean",
"sample_id": "a931f0b3f033fc32c017444ea0eb1a5592318ecdbcdc093591ad2758f58c4528"
} | train |
{
"chosen_proof_sha256": "b9120dc0c9841f5cc0966a2e218d4adcc087ef91568585fb5a852de6e1d93e10",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "21dca1957ea463f842223ca3af36fd706d... | by
apply Quotient.lift_unique
apply Paths.lift_unique
fapply @Quiver.Symmetrify.lift_unique _ _ _ _ _ _ _ _ _
· rw [← Functor.toPrefunctor_comp]
exact hΦ
· rintro X Y f
simp only [← Functor.toPrefunctor_comp, Prefunctor.comp_map, Paths.of_map]
change Φ.map (Groupoid.inv ((Quotient.functor redStep)... | false | {
"chosen_tokens": 139,
"rejected_tokens": 146,
"token_jaccard": 0.962963,
"token_length_ratio": 1.05036
} | markdown_preamble_trailing | fence | f64d8856c2db234d6983bc981b971cde755f38221b23d795ded5b2f4b1e145d0 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Groupoid
public import Mathlib.CategoryTheory.PathCategory.Basic
Namespace:
Quiver.FreeGroupoid
Local context:
/-
Copyright (c) 2022 Rémi Bott... | ```lean
by
apply Quotient.lift_unique
apply Paths.lift_unique
fapply @Quiver.Symmetrify.lift_unique _ _ _ _ _ _ _ _ _
· rw [← Functor.toPrefunctor_comp]
exact hΦ
· rintro X Y f
simp only [← Functor.toPrefunctor_comp, Prefunctor.comp_map, Paths.of_map]
change Φ.map (Groupoid.inv ((Quotient.functor ... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Groupoid",
"family_id": "lift_unique",
"file_id": "mathlib/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean",
"sample_id": "3e937a169766b398011457da21839b47e8a3e949c8dec6560a9e40da808f1d10"
} | train |
{
"chosen_proof_sha256": "76fa0ee83574746f588fc7899220fecd9e66129f439898c7d8ec00fbf1b68c95",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8032f9d6a97228a94984a08de206cf6fcb... | by
obtain ⟨i', hi'_lub⟩ : ∃ i', IsLUB (Set.Iio i) i' := exists_lub_Iio i
rcases lub_Iio_eq_self_or_Iio_eq_Iic i hi'_lub with hi'_eq_i | h_Iio_eq_Iic
· rw [← hi'_eq_i] at hi'_lub ⊢
exact hτ.measurableSet_lt_of_isLUB i' hi'_lub
· have h_lt_eq_preimage : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iio i := rfl
have h_Ii... | true | {
"chosen_tokens": 128,
"rejected_tokens": 3,
"token_jaccard": 0.033898,
"token_length_ratio": 0.023438
} | unknown_identifier | unknown_identifier | f67ce98cbdd36221db6089ccf00e575f9144e988a9d64e1f3d0a377f4bdb031c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Probability.Process.Adapted
public import Mathlib.MeasureTheory.Constructions.BorelSpace.WithTop
public import Mathlib.Data.ENat.Lattice
Namespace:
MeasureThe... | by
exact __oprover_missing_bfdf8a5735fb | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "32897803ca433a0d9beec2943b2ed277e1b3d2c1a186fd501a0dcb0483dede80",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability/Process",
"family_id": "isstoppingtime",
"file_id": "mathlib/Mathlib/Probability/Process/Stopping.lean",
"sample_id": "bfdf8a5735fb83291498154fe55688e7c5eabb3111989c0465481333ca9088fd"
} | train |
{
"chosen_proof_sha256": "b4cbd01d6c035dc37c5d6b8fd96656834089520d56a98eb2574c117f82f08ca7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ff462f64730661be30ba8210dab7fdd5b5... | by
simp [dens, card_biUnion h, sum_div] | true | {
"chosen_tokens": 10,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.5
} | wrong_namespace_qualification | wrong_namespace_qualification | f6b354b68e1294c29aca8a9bed2ee65c21c785e11708c699b977094fb51f70f2 | 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.Field.Defs
public import Mathlib.Data.Finset.Density
Namespace:
Finset
Local context:
/-
Copyr... | by
exact OProverMissing.missing_2e83f293bd34 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "58782bd5f256f976a5b1fc56262c1a3d1c9bea23d0c5866366b912b74bed32fd",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "362e79b2088809d4572239c27914a8ae94ced25dc58902293b9373aadce248ad",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ac907c8dc3bdd145394ac850e70ea47494... | by
haveI : Fintype ι := Fintype.ofFinite ι
rw [fromDirectSumEquiv, ← fromDFinsuppEquiv_single]
convert! rfl | true | {
"chosen_tokens": 20,
"rejected_tokens": 5,
"token_jaccard": 0.095238,
"token_length_ratio": 0.25
} | wrong_namespace_qualification | wrong_namespace_qualification | f6f70f9eae1c4405bb11626f2f870afa1905b5920027a873b2f5002de15dc05e | 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.DirectSum.Module
public import Mathlib.LinearAlgebra.Multilinear.DFinsupp
Namespace:
MultilinearMap
Local context:
/-
Copyright (c) 2024 Sophie Morel... | by
exact OProverMissing.missing_215bc10c1721 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "276859439ea45b61dd158f27a79ea1aac44893dafad095803d5112223ef1b8c5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Multilinear",
"family_id": "fromdirectsumequiv_lof",
"file_id": "mathlib/Mathlib/LinearAlgebra/Multilinear/DirectSum.lean",
"sample_id": "215bc10c17214a80cdf028f691cd9e0cb3d40b6838bce81a23b452ba150e84be"
} | train |
{
"chosen_proof_sha256": "55204f99d585e2efb5ba03ffeb23f7cf47ecd4bbf63bb94951cbe11d0f4291b8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "af47faf185b783519da8aa97ae8dad4e28... | by
rw [NormNum.IsInt.neg_to_eq h rfl]
simp only [ne_eq, neg_eq_zero]
apply ne_of_gt
simpa using! w | true | {
"chosen_tokens": 24,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.208333
} | wrong_namespace_qualification | wrong_namespace_qualification | f7086b2e5dd2f6b88d37c654ae5234053d2595d72bfb179565e082705b726884 | 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.Invertible
public import Mathlib.Algebra.Order.Ring.Cast
public import Mathlib.Tactic.HaveI
public import Mathlib.Tactic.NormNum.Core
Namespace:... | by
exact OProverMissing.missing_a1caf932105b | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "162755ecab5b0686c9310caa07662a0758e51d8868fe59bc3d64ba85932447e1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/Positivity",
"family_id": "nz_of_isnegnat",
"file_id": "mathlib/Mathlib/Tactic/Positivity/Core.lean",
"sample_id": "a1caf932105bba53888c1df521c9cec9b3321d4d5015df07eac43fb607350537"
} | train |
{
"chosen_proof_sha256": "b1b2588f46af06b1f0c76c618d2cd12402ece06fbea01ebed4a3cb2260d14513",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "698ba750f435c2d8477d0231c1f5e3c795... | by
rw [mapRange_def]
refine (Finset.prod_subset support_mk_subset ?_).trans ?_
· intro i h1 h2
simp only [mem_support_toFun, ne_eq] at h1
simp only [Finset.coe_sort_coe, mem_support_toFun, mk_apply, ne_eq, h1, not_false_iff,
dite_eq_ite, ite_true, not_not] at h2
simp [h2, h0]
· refine Finset.p... | false | {
"chosen_tokens": 87,
"rejected_tokens": 91,
"token_jaccard": 0.921053,
"token_length_ratio": 1.045977
} | missing_wrong_import | import | f71b769ab34dcb653e577d30813f04560c9cc692b359c17fe772a4d72ccc78d8 | 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.GroupWithZero.Action
public import Mathlib.Data.DFinsupp.Ext
public import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
Namespace:
DFi... | import Mathlib.WrongImport
by
rw [mapRange_def]
refine (Finset.prod_subset support_mk_subset ?_).trans ?_
· intro i h1 h2
simp only [mem_support_toFun, ne_eq] at h1
simp only [Finset.coe_sort_coe, mem_support_toFun, mk_apply, ne_eq, h1, not_false_iff,
dite_eq_ite, ite_true, not_not] at h2
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": "Data/DFinsupp",
"family_id": "prod_maprange_index",
"file_id": "mathlib/Mathlib/Data/DFinsupp/BigOperators.lean",
"sample_id": "345ae0fc64c4223b685a8043b5b363bbd84f8c383d47f5ccfa20bde13d5436d1"
} | train |
{
"chosen_proof_sha256": "51d70e6a078b791d67a01b436c28657296809a50b25f00afb27df7a2d7b5b11b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1288fc6a5cfa445715d48101a0256f0dac... | by
apply SheafOfModules.hom_ext
ext U x
exact congr($(H U.unop) x) | false | {
"chosen_tokens": 20,
"rejected_tokens": 25,
"token_jaccard": 0.789474,
"token_length_ratio": 1.25
} | markdown_preamble_trailing | trailing | f728e0161850a867694e274da513091deecd760ed626de3fa45e1c94120b2f9b | 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.Category.ModuleCat.Sheaf.Abelian
public import Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits
public import Mathlib.Algebra.Category.ModuleCat.Sheaf... | by
apply SheafOfModules.hom_ext
ext U x
exact congr($(H U.unop) x)
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": "AlgebraicGeometry/Modules",
"family_id": "hom_ext",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Modules/Sheaf.lean",
"sample_id": "2f4e8f65d6a8377847b7d2fb1bab8531851b8db89761aa4db8d2e360d21ed05e"
} | train |
{
"chosen_proof_sha256": "a73fac1f1d110d78557015f3b6ec0716b6ddeed8c82d3ea7e29e1289e23f50bb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b648013aeb700becb315f9a7957067c070... | by
rw [preimage_image_mk_eq_iUnion_image, iUnion_subtype, ← image2_mul, ← iUnion_image_right]
simp only [SetLike.mem_coe] | false | {
"chosen_tokens": 20,
"rejected_tokens": 27,
"token_jaccard": 0.882353,
"token_length_ratio": 1.35
} | markdown_preamble_trailing | fence | f72c0006a3a432226cec6afe7b30edfbe71f1e3422c3f63ed8f82f301a5ce327 | 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.Quotient
public import Mathlib.Algebra.Group.Action.Opposite
public import Mathlib.Algebra.Group.Subgroup.MulOpposite
public import Mathlib.GroupTheory... | ```lean
by
rw [preimage_image_mk_eq_iUnion_image, iUnion_subtype, ← image2_mul, ← iUnion_image_right]
simp only [SetLike.mem_coe]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "GroupTheory/Coset",
"family_id": "preimage_image_mk_eq_mul",
"file_id": "mathlib/Mathlib/GroupTheory/Coset/Defs.lean",
"sample_id": "8b9a52eec598c7910eb23c645754ae85116421ba3d320b896639db65e6cbf421"
} | train |
{
"chosen_proof_sha256": "4fff200b3969b2099ac39b5df9f12ed101d4778e55341e30c66e404ab2007cba",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a775a92a4cceb4ec9087e1badba45ff9b0... | by
obtain ⟨k, rfl⟩ := hn
rw [pow_add, pow_one, pow_mul, sq]
cases s <;> simp | false | {
"chosen_tokens": 25,
"rejected_tokens": 30,
"token_jaccard": 0.814815,
"token_length_ratio": 1.2
} | markdown_preamble_trailing | trailing | f72cd6146f1f97818067312ca140822391c175e045472f54d8a22647dd903a49 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GroupWithZero.Units.Lemmas
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import Mathlib.Algebra.Order.Ring.Cast
public import Ma... | by
obtain ⟨k, rfl⟩ := hn
rw [pow_add, pow_one, pow_mul, sq]
cases s <;> simp
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Sign",
"family_id": "pow_odd",
"file_id": "mathlib/Mathlib/Data/Sign/Basic.lean",
"sample_id": "68d51948bcb20f9102720ae7d20bb860c7d2a1e39fd0c3d403bf97713844dd99"
} | train |
{
"chosen_proof_sha256": "be03b33260f8a567b6bd346d2bc5d2f454ac795779b31a7801a8bdb4b9414537",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6f7dc276bda9e16d09662fbe3919eda2fa... | by
dsimp [Functor.HasPointwiseLeftKanExtension]
simp only [hasPointwiseLeftKanExtensionAt_iff w]
refine ⟨fun h X₄ ↦ ?_, fun h _ ↦ h _⟩
rw [← Functor.hasPointwiseLeftKanExtensionAt_iff_of_iso _ _ (B.objObjPreimageIso X₄)]
apply h | false | {
"chosen_tokens": 48,
"rejected_tokens": 52,
"token_jaccard": 0.909091,
"token_length_ratio": 1.083333
} | missing_wrong_import | import | f7a96330f9aab66e7a585adc7be5b00b96de55c8199b015937beb6e15fd88bff | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.GuitartExact.Basic
public import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction
Namespace:
CategoryTheory.TwoSquare
Local context:
/-
... | import Mathlib.WrongImport
by
dsimp [Functor.HasPointwiseLeftKanExtension]
simp only [hasPointwiseLeftKanExtensionAt_iff w]
refine ⟨fun h X₄ ↦ ?_, fun h _ ↦ h _⟩
rw [← Functor.hasPointwiseLeftKanExtensionAt_iff_of_iso _ _ (B.objObjPreimageIso X₄)]
apply h | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/GuitartExact",
"family_id": "haspointwiseleftkanextension_iff",
"file_id": "mathlib/Mathlib/CategoryTheory/GuitartExact/KanExtension.lean",
"sample_id": "ca893de6923284d5337d25661aaf5c8d1221b14895d8ede2751221d59a9b92ee"
} | train |
{
"chosen_proof_sha256": "bd9e89042700066e34db2eae2a5bdca2cc788a55ff318687cbca751f9272e192",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b21d5b016f524bd98133a7e41c46044068... | by
simp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]
split_ifs <;> rfl | false | {
"chosen_tokens": 27,
"rejected_tokens": 32,
"token_jaccard": 0.791667,
"token_length_ratio": 1.185185
} | markdown_preamble_trailing | preamble | f7b621d6b3eb4f87d3912540dc3f342ad7dae1bd0cac641b35726d6b2420935e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Data.MLList.Basic
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.ZeroLEOne
public import Mathlib.Data.DFinsupp.Defs
public... | Here is the proof:
by
simp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]
split_ifs <;> 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": "Testing/Plausible",
"family_id": "list",
"file_id": "mathlib/Mathlib/Testing/Plausible/Functions.lean",
"sample_id": "25e33e2ad73b99a239389fcab2e1bb17f4373e01742868a9ff04d6e9bc1274a4"
} | train |
{
"chosen_proof_sha256": "8debc390e3c9d99e3175632da6d0d7fd484f1961b70777e58cda8dfa8a7eb610",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5f997efe8cebc3ae55d046a4d771e677d5... | by
let a := (lfp.comp h).lfp
refine (lfp_le _ ?_).antisymm (lfp_le _ (Eq.le ?_))
· exact lfp_le _ h.onDiag.map_lfp.le
have ha : (lfp ∘ h) a = a := (lfp.comp h).map_lfp
calc
h a a = h a (h a).lfp := congr_arg (h a) ha.symm
_ = (h a).lfp := (h a).map_lfp
_ = a := ha | true | {
"chosen_tokens": 105,
"rejected_tokens": 2,
"token_jaccard": 0.033333,
"token_length_ratio": 0.019048
} | invalid_tactic | invalid_tactic | f7c1c590285c84f1f83335c16dbe766f0f45022fb2ef6433999fd9a51eb7bda8 | 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.Dynamics.FixedPoints.Basic
public import Mathlib.Order.Hom.Order
public import Mathlib.Order.BourbakiWitt
Namespace:
OrderHom
Local context:
/-
Copyright (c)... | by
__oprover_invalid_tactic_950388e96395 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "4a322d19e4348a86c1a5a35ebda42ac55a661467b647f2033221587fc25f27fb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order",
"family_id": "lfp_lfp",
"file_id": "mathlib/Mathlib/Order/FixedPoints.lean",
"sample_id": "950388e963953fd7e6eb950b29af28e1f498743a5197639d25cdf72094ea0aae"
} | train |
{
"chosen_proof_sha256": "7d2f77b8c1bb155392db6b9e50f050fefd652d8cabc59188758f6cc33ac80619",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by convert! aeval_algebraMap_apply A x p | false | {
"chosen_tokens": 7,
"rejected_tokens": 2,
"token_jaccard": 0.125,
"token_length_ratio": 0.285714
} | by_exact_placeholder | by? | f7c43f8ec5bcea5f9b6cc654659693f57368d634d9a67c3a9dd47baa7bc0bea5 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Subalgebra.Tower
public import Mathlib.Algebra.MvPolynomial.Eval
Namespace:
Subalgebra
Local context:
/-
Copyright (c) 2022 Yuyang Zhao. All ... | 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/MvPolynomial",
"family_id": "mvpolynomial_aeval_coe",
"file_id": "mathlib/Mathlib/RingTheory/MvPolynomial/Tower.lean",
"sample_id": "bb469c3613f92d8cabf0255e6c012999e3bdcfb6d61f1c8b6b5d3a6dc8c383d5"
} | train |
{
"chosen_proof_sha256": "4477bb12e5cc2a166f61746299c4ae29af03f6b7f73cd9a6d3ff697daf2ff447",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G
rw [← hS1, ← Fintype.card_coe, ← Nat.card_eq_fintype_card, ← Finset.coe_sort_coe, ← Nat.card_fun]
exact Finite.card_le_of_embedding (quotientCenterEmbedding hS2) | true | {
"chosen_tokens": 47,
"rejected_tokens": 8,
"token_jaccard": 0.057143,
"token_length_ratio": 0.170213
} | unsolved_goals | unsolved_goals | f7e105969fa9d56a65268298a4d23e405deb59bb39ba29d8ae63d2b998dbe0ba | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Subgroup.Finite
public import Mathlib.GroupTheory.Commutator.Basic
public import Mathlib.GroupTheory.Rank
public import Mathlib.GroupTheory.Index... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "98fa8cf9cafe448e3703a1d232dbeda941d8919bea439d47bf06547064ed1e1d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "GroupTheory/Commutator",
"family_id": "index_center_le_pow",
"file_id": "mathlib/Mathlib/GroupTheory/Commutator/Finite.lean",
"sample_id": "865c2117838a5f9b6bdb0bfba027decbe6b709de52a194b8d1a351aa165f6f08"
} | train |
{
"chosen_proof_sha256": "586f1604059baac7ae4d22c8ed3b6c941cba1beaead4cb818f84462ea38efbce",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5e149fff8a99e4451822a4357dbc08d6be... | by
simp only [nonempty_iff_ne_empty, Ne, sumLexLift_eq_empty, not_and_or, exists_prop, not_forall] | false | {
"chosen_tokens": 16,
"rejected_tokens": 21,
"token_jaccard": 0.705882,
"token_length_ratio": 1.3125
} | markdown_preamble_trailing | trailing | f80340f087667bdd43d67352e1717d4154923556a67335964dbc76814002d0f6 | 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 ... | by
simp only [nonempty_iff_ne_empty, Ne, sumLexLift_eq_empty, not_and_or, exists_prop, not_forall]
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Sum",
"family_id": "sumlexlift_nonempty",
"file_id": "mathlib/Mathlib/Data/Sum/Interval.lean",
"sample_id": "dce4fa550c062aaa1e0e5d299822fa3b4591439eeb6e8fb744a8c9d9b1a2391e"
} | train |
{
"chosen_proof_sha256": "124731cfc6276098aa30620798da6c884c01af47d833522f931a5e1fb2b79484",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_algEquiv_matrix_end_mulOpposite R₀ R
have := Module.Finite.equiv e.toLinearEquiv
classical exact ⟨n, hn, _, _, _, .of_surjective
(Matrix.entryLinearMap R₀ _ (0 : Fin n) (0 : Fin n)) fun f ↦ ⟨fun _ _ ↦ f, rfl⟩, ⟨e⟩⟩ | false | {
"chosen_tokens": 82,
"rejected_tokens": 5,
"token_jaccard": 0.085714,
"token_length_ratio": 0.060976
} | sorry_admit_sorryAx_proof_hole | sorryAx | f815cab88a19f118c9634447de5186f39605f1536ea4e446c458e07f4c650198 | 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.FreeModule.Finite.Basic
public import Mathlib.RingTheory.FiniteLength
public import Mathlib.RingTheory.SimpleModule.Isotypic
public import Mathli... | 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/SimpleModule",
"family_id": "exists_algequiv_matrix_divisionring_finite",
"file_id": "mathlib/Mathlib/RingTheory/SimpleModule/WedderburnArtin.lean",
"sample_id": "45556f8bd9370bd1415c420e560973ba88e4c44babfc2defddcf5c8f39aa00bb"
} | train |
{
"chosen_proof_sha256": "443eea2a8757775ee8a442c763e06d326c2fd7f65ba73927f6537e2b3fa75567",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [finite_iff_subgroup_quotient]; constructor <;> assumption | false | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.090909,
"token_length_ratio": 0.181818
} | sorry_admit_sorryAx_proof_hole | admit | f82250c17f89948d17caf20341ed0704ec3639cb2aa7420fd5082651d3394538 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Subgroup.Finite
public import Mathlib.Data.Finite.Prod
public import Mathlib.GroupTheory.QuotientGroup.Basic
Namespace:
(root)
Local context:
/... | 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": "GroupTheory/QuotientGroup",
"family_id": "finite",
"file_id": "mathlib/Mathlib/GroupTheory/QuotientGroup/Finite.lean",
"sample_id": "36013d3b030a4beac323ba03c7b15bdae8a250c801f46d872bcc359cdf27248b"
} | train |
{
"chosen_proof_sha256": "a139748a86235708cf34b19e320c41795f248777d1dc57cebe0b73665a686a07",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "39ec33afe785bb2145068683a69a98c599... | by
constructor
· exact HallMarriageTheorem.hall_hard_inductive
· rintro ⟨f, hf₁, hf₂⟩ s
rw [← card_image_of_injective s hf₁]
apply card_le_card
intro
rw [mem_image, mem_biUnion]
rintro ⟨x, hx, rfl⟩
exact ⟨x, hx, hf₂ x⟩ | false | {
"chosen_tokens": 54,
"rejected_tokens": 61,
"token_jaccard": 0.935484,
"token_length_ratio": 1.12963
} | markdown_preamble_trailing | fence | f86219efb5cb4f802277f4645c3e4b06de51196cbb103a37732db46c7ad96573 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Basic
public import Mathlib.Data.Fintype.Powerset
public import Mathlib.Data.Set.Finite.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) ... | ```lean
by
constructor
· exact HallMarriageTheorem.hall_hard_inductive
· rintro ⟨f, hf₁, hf₂⟩ s
rw [← card_image_of_injective s hf₁]
apply card_le_card
intro
rw [mem_image, mem_biUnion]
rintro ⟨x, hx, rfl⟩
exact ⟨x, hx, hf₂ x⟩
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/Hall",
"family_id": "finset",
"file_id": "mathlib/Mathlib/Combinatorics/Hall/Finite.lean",
"sample_id": "9844a70ca9aa095c0c679f810bb6057aadb712cb3e2289c8ec909fdd876bd5a3"
} | train |
{
"chosen_proof_sha256": "b663b020e08f48e798d48b7ce39be8da0dce1601bdd6b1231b5d9860f49a1922",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by lia) :
t.IsGE ((t.truncGT a).obj X) b :=
t.isGE_truncGE_obj .. | true | {
"chosen_tokens": 25,
"rejected_tokens": 8,
"token_jaccard": 0.157895,
"token_length_ratio": 0.32
} | unsolved_goals | unsolved_goals | f879a15aa1fd44809244ddacb0c5a2137fe57cafd36dc9e1ce36b967a4b6d1e3 | 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.Triangulated.TStructure.TruncLTGE
Namespace:
CategoryTheory.Triangulated.TStructure
Local context:
/-
Copyright (c) 2026 Joël Riou. All rights... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "9c6a0dc3a4f168e3df602a1d8fa72c34ca98e1f1b7ae8523235e26ce7a412097",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Triangulated",
"family_id": "isge_truncgt_obj",
"file_id": "mathlib/Mathlib/CategoryTheory/Triangulated/TStructure/TruncLEGT.lean",
"sample_id": "9464b08287d95c785301c54058651202fff8047c0ca4e3764eb7ffff3e6775f3"
} | train |
{
"chosen_proof_sha256": "12e284d615394feaae0bf91b6d2469b44129b24d28aa683dddb8dd6ad90c1ad2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
grw [hv.apply_sum_le, ← cbiSup_eq_of_forall (by grind)]
simp | false | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.0625,
"token_length_ratio": 0.2
} | by_exact_placeholder | exact? | f89146c860e936c5e0a0e744a34533baf0266ca5b6a3bd117d28ae143a87c11d | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.Homogenize
public import Mathlib.NumberTheory.Height.Basic
import Mathlib.Algebra.Order.Ring.IsNonarchimedean
import Mathlib.Data.Fintype.Or... | 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": "NumberTheory/Height",
"family_id": "apply_sum_univ_le",
"file_id": "mathlib/Mathlib/NumberTheory/Height/MvPolynomial.lean",
"sample_id": "141f039ee25caf569389190297e8e170fe2b5f957ffbed666e13fe2928910cf7"
} | train |
{
"chosen_proof_sha256": "f187198a893dd0403a98bdeb7c5bab4c0cada1bb485a8d70e6ce244cdd6c05b3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [← Icc_union_Ioc_eq_Icc le_rfl h, VectorMeasure.of_union (by simp)
measurableSet_Icc measurableSet_Ioc, Icc_self, hf.vectorMeasure_singleton,
hf.vectorMeasure_Ioc h]
simp | true | {
"chosen_tokens": 30,
"rejected_tokens": 8,
"token_jaccard": 0.037037,
"token_length_ratio": 0.266667
} | unsolved_goals | unsolved_goals | f8d6d9b9b9181fcf10e6e196868d71b213e17a119fd74afa6ff28879397af82e | 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.Normed.Group.Defs
public import Mathlib.MeasureTheory.Measure.Stieltjes
public import Mathlib.MeasureTheory.VectorMeasure.Basic
public import Mathlib.... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "df73b8bbb0e4a4d0b09eb285144a2fa3a4a6171e552c5bfa210247f2dc6df676",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/VectorMeasure",
"family_id": "vectormeasure_icc",
"file_id": "mathlib/Mathlib/MeasureTheory/VectorMeasure/BoundedVariation.lean",
"sample_id": "10754817056ac7fe2728b7d3c57dd409b9884a07365c423b530430a8a51d4d7e"
} | train |
{
"chosen_proof_sha256": "3486bfbcb341bbc6602e3e4c5334d8af2187c1a15d03540967ecccbb4983efe2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "72a5ec820d1616bed6e09b9298c3d6ba25... | by
rw [← cancel_mono (mapBifunctorLeftUnitor F X e p hp Y').inv, assoc, assoc, Iso.hom_inv_id,
comp_id, mapBifunctorLeftUnitor_inv_naturality, Iso.hom_inv_id_assoc] | true | {
"chosen_tokens": 33,
"rejected_tokens": 3,
"token_jaccard": 0.038462,
"token_length_ratio": 0.090909
} | circular_dependency | circular_dependency | f8f44157683393ee5042bf128f54df081332a78f92c001cb0f2544184424efac | 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.GradedObject.Associator
public import Mathlib.CategoryTheory.GradedObject.Single
Namespace:
CategoryTheory.GradedObject
Local context:
/-
Copy... | by
exact mapBifunctorLeftUnitor_naturality | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "33d3809e37dd3a56fc6ca3d07e6c90b503d3768752892ce82eb43fc0bb79dd1c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/GradedObject",
"family_id": "mapbifunctorleftunitor_naturality",
"file_id": "mathlib/Mathlib/CategoryTheory/GradedObject/Unitor.lean",
"sample_id": "9547247b6d603996c4b3fb961825dcb69ba77ace42e69a17a558d84380f3ec34"
} | train |
{
"chosen_proof_sha256": "8690ef82dc2fef54f22a4e762da7bbf0fe4e6499a977029b3fad0c51781aa987",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "18212381e9e2d5955dd328bc6ac67fa9f7... | by
classical
rw [ofDirectSumEquiv]
convert! ofDFinsuppEquiv_symm_single_tprod p x | false | {
"chosen_tokens": 11,
"rejected_tokens": 18,
"token_jaccard": 0.846154,
"token_length_ratio": 1.636364
} | markdown_preamble_trailing | fence | f91f5235496e941fe9cfd5258fb9aa7c4921cff1c1cdc81164fb61f3f0bfc997 | 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.PiTensorProduct.Basic
public import Mathlib.LinearAlgebra.PiTensorProduct.DFinsupp
public import Mathlib.Algebra.DirectSum.Module
Namespace:
PiT... | ```lean
by
classical
rw [ofDirectSumEquiv]
convert! ofDFinsuppEquiv_symm_single_tprod p x
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/PiTensorProduct",
"family_id": "ofdirectsumequiv_symm_lof_tprod",
"file_id": "mathlib/Mathlib/LinearAlgebra/PiTensorProduct/DirectSum.lean",
"sample_id": "29db24e141a0e3e320d003b92675922c58df588b3780d126668224314302b43a"
} | train |
{
"chosen_proof_sha256": "bb5993596ab6f0322c25fd89be3f755c85b89f6e4d2f8ad57eb17900f2915564",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
ext; simp only [add_def, smul_def, _root_.add_smul] | false | {
"chosen_tokens": 14,
"rejected_tokens": 2,
"token_jaccard": 0.071429,
"token_length_ratio": 0.142857
} | sorry_admit_sorryAx_proof_hole | sorry | f950f96c8666dc6791b0d4b648f84ceed4268882edc1bf156aa7b80594cd5bdd | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorProduct.RightExactness
public import Mathlib.RingTheory.Congruence.Hom
public import Mathlib.RingTheory.FiniteType
public import Mathlib.Ri... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/PolynomialLaw",
"family_id": "add_smul",
"file_id": "mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean",
"sample_id": "4aa461ab228ed2d6d40d0634d68fa0428d1672b0b417d2b773b3a8831f202fa2"
} | train |
{
"chosen_proof_sha256": "80d415d211305782b1e1301d4d0be21abb30589a9327984a8f2cfe8e37267cf8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cc0e6de2bb15058021ed449d45b1ad5b54... | by
have : ∀ᵐ x ∂μ, (f - f') x = 0 := by
apply ae_eq_zero_of_integral_contMDiff_smul_eq_zero I (hf.sub hf')
intro g g_diff g_supp
simp only [Pi.sub_apply, smul_sub]
rw [integral_sub, sub_eq_zero]
· exact h g g_diff g_supp
· exact hf.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_... | false | {
"chosen_tokens": 84,
"rejected_tokens": 89,
"token_jaccard": 0.924528,
"token_length_ratio": 1.059524
} | markdown_preamble_trailing | trailing | f9556318272271ba6b4a30c43408227d4440d15b079b2ac43d1c2181fb010b96 | 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.Geometry.Manifold.PartitionOfUnity
public import Mathlib.MeasureTheory.Function.AEEqOfIntegral
Namespace:
(root)
Local context:
/-
Copyright (c) 2023 Sébasti... | by
have : ∀ᵐ x ∂μ, (f - f') x = 0 := by
apply ae_eq_zero_of_integral_contMDiff_smul_eq_zero I (hf.sub hf')
intro g g_diff g_supp
simp only [Pi.sub_apply, smul_sub]
rw [integral_sub, sub_eq_zero]
· exact h g g_diff g_supp
· exact hf.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/Distribution",
"family_id": "ae_eq_of_integral_contmdiff_smul_eq",
"file_id": "mathlib/Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean",
"sample_id": "d55812070999d9cd69897ef7878e60eedf847e5cca72e88cad19720567719cd3"
} | train |
{
"chosen_proof_sha256": "7bfddf9976579c9b33ef1adb03a12367ce2324df54f03888b73e637b017ab565",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [← disjoint_iff]
exact disjoint_sdiff_iff_le hz hx
-- cf. `IsCompl.left_le_iff` and `IsCompl.right_le_iff` | false | {
"chosen_tokens": 25,
"rejected_tokens": 5,
"token_jaccard": 0.095238,
"token_length_ratio": 0.2
} | sorry_admit_sorryAx_proof_hole | sorryAx | f95d36b7806ce5579355ff7f76edd92a24e172206c21302c20059f9be6ca05fa | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.BooleanAlgebra.Defs
public import Mathlib.Tactic.GRewrite
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
R... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/BooleanAlgebra",
"family_id": "inf_sdiff_eq_bot_iff",
"file_id": "mathlib/Mathlib/Order/BooleanAlgebra/Basic.lean",
"sample_id": "6ef6ae9c0f9613c7c3706f6b6b19cc3ed2c14562e6ea2548c5e346b1a3fa96cb"
} | train |
{
"chosen_proof_sha256": "f77254043cd1457d306845776290c0242267d27c173b195a06e4b8478f5ec277",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [Set.eq_univ_iff_forall]
intro x
rw [Set.mem_iUnion]
exact 𝒰.exists_eq x | false | {
"chosen_tokens": 20,
"rejected_tokens": 5,
"token_jaccard": 0.125,
"token_length_ratio": 0.25
} | sorry_admit_sorryAx_proof_hole | sorryAx | f96ab8eeb0849aeb3bcbe3a89eb152030998157212ece81f89771367f1412bb7 | 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.Sites.MorphismProperty
public import Mathlib.CategoryTheory.MorphismProperty.Limits
Namespace:
AlgebraicGeometry.Scheme
Local context:
/-
C... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Cover",
"family_id": "cover",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Cover/MorphismProperty.lean",
"sample_id": "3d805ac3a07f605169cd17d09107c30e012f4d0f06e8571fe5f44b26cb0ba912"
} | train |
{
"chosen_proof_sha256": "63ce66eb5d2d85e6c3aa93e37342e57afc36c526d7b717542f12e151e6f9e034",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
let := Localization.AtPrime.algebraOfLiesOver (p.under R) p
rw [isUnramifiedAt_iff_map_eq R (p.under R) p, and_iff_right,
Ideal.IsDedekindDomain.ramificationIdx_eq_one_iff hp Ideal.map_comap_le]
have : Finite (R ⧸ p.under R) :=
Ideal.finiteQuotientOfFreeOfNeBot _ (mt Ideal.eq_bot_of_comap_eq_bot hp)
... | true | {
"chosen_tokens": 92,
"rejected_tokens": 8,
"token_jaccard": 0.102564,
"token_length_ratio": 0.086957
} | unsolved_goals | unsolved_goals | f9d383b5c0fd528f0b5d3274e723ff5bd68c5d117d88d485e7a529456f6f1e58 | 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.RamificationInertia.Basic
public import Mathlib.RingTheory.LocalRing.ResidueField.Instances
public import Mathlib.RingTheory.Unramified.LocalRing
... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e505ecb6a0cc32d2b092dfe6c26910892cfff5faf1971a88c446a47acb1fecdf",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/RamificationInertia",
"family_id": "algebra",
"file_id": "mathlib/Mathlib/NumberTheory/RamificationInertia/Unramified.lean",
"sample_id": "ed0bc1a8156a91f3c77b4c93e0e33c2312cab82d14a2bd0d77876ca7788349af"
} | train |
{
"chosen_proof_sha256": "6986b7de3b811139639af9103eba6dc70b8f13e73607506a5de302cee24db96a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7ad080fabb1c3deb7fe535e2a848f748f1... | by
rw [BilinForm.toMatrixAux, Matrix.toBilin'Aux, toLinearMap₂'Aux_toMatrix₂Aux] | true | {
"chosen_tokens": 18,
"rejected_tokens": 3,
"token_jaccard": 0.058824,
"token_length_ratio": 0.166667
} | unknown_identifier | unknown_identifier | f9f83d25656b783f0465073ef91e37afcaea7fec031baca77c629bde35485bcc | 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.BilinearForm.Properties
public import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
Namespace:
(root)
Local context:
/-
Copyright (c) 2020 Anne... | by
exact __oprover_missing_1e74a857d727 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f2da8552e2a46b5625efd7bed00ae51038cf7b37b991293cf847b66f56ead8f3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Matrix",
"family_id": "linearmap",
"file_id": "mathlib/Mathlib/LinearAlgebra/Matrix/BilinearForm.lean",
"sample_id": "1e74a857d727dcb19bacc65e1dd39ea3f06dc3533cc251ea326cf338c51e0a18"
} | train |
{
"chosen_proof_sha256": "8e7840e8c83d8f557e8647b534950399050e6f0e066df7ccc73d874d601c7f9a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rcases exists_isRoot_of_odd_natDegree (f := X ^ n - C x) (by simp [hn]) with ⟨r, hr⟩
exact ⟨r, by linear_combination - (by simpa using hr : r ^ n - x = 0)⟩ | false | {
"chosen_tokens": 48,
"rejected_tokens": 5,
"token_jaccard": 0.060606,
"token_length_ratio": 0.104167
} | sorry_admit_sorryAx_proof_hole | sorryAx | fa04b9c59ef329e4ac9db395c0705f98ebf0ff96a8ec0c463a36aa63ce23fe63 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.Degree.Domain
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.Algebra.Ring.Semireal.Defs
public import Mathlib.Tact... | by
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": "FieldTheory/IsRealClosed",
"family_id": "exists_eq_pow_of_odd",
"file_id": "mathlib/Mathlib/FieldTheory/IsRealClosed/Basic.lean",
"sample_id": "4791ed0fbefe25e9f05c1dec233b6ddc5e09c117965a9cb9936eb95639f9e525"
} | train |
{
"chosen_proof_sha256": "74ef8f872938daf0468322013395261e7267b7edbfb838bf7741225d397b7c55",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [← card k V]
have : Nat.card V > 0 := Nat.card_pos
lia | false | {
"chosen_tokens": 21,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.095238
} | sorry_admit_sorryAx_proof_hole | admit | fa0e82f69be5bf39dcb91769e347f49b9b4144ae4b0fc279abd7fa3a7d1cc1a0 | 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.GeomSum
public import Mathlib.Algebra.GroupWithZero.Units.Fintype
public import Mathlib.Data.Finite.Sum
public import Mathlib.GroupTheory.GroupAct... | 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": "LinearAlgebra/Projectivization",
"family_id": "card",
"file_id": "mathlib/Mathlib/LinearAlgebra/Projectivization/Cardinality.lean",
"sample_id": "de57e8cad85d8b44648cf10f3d51ce2288da48a2805529ae83b12ceae714db81"
} | train |
{
"chosen_proof_sha256": "cb1bd661524d698c211f8037516f3bb09f8383d70d26a48c78df08d165e667a0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6989f734ba078bd696f50a35920dafc641... | by
rcases Int.Prime.dvd_mul hp h with hp2 | hpp
· apply Or.intro_left
exact le_antisymm (Nat.le_of_dvd zero_lt_two hp2) (Nat.Prime.two_le hp)
· apply Or.intro_right
rw [sq, Int.natAbs_mul] at hpp
exact or_self_iff.mp ((Nat.Prime.dvd_mul hp).mp hpp) | true | {
"chosen_tokens": 67,
"rejected_tokens": 3,
"token_jaccard": 0.057143,
"token_length_ratio": 0.044776
} | circular_dependency | circular_dependency | fa34a54be47df85918d4d4bb180a89b30bad3e3e2e2dc1112fe459bf66c982ba | 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.EuclideanDomain.Basic
public import Mathlib.Algebra.EuclideanDomain.Int
public import Mathlib.Algebra.GCDMonoid.Nat
public import Mathlib.Data.Nat.Prim... | by
exact prime_two_or_dvd_of_dvd_two_mul_pow_self_two | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "fb19c0294d8772bd586deef5b6b35c6681d085d091a92a77f20d63fe4777abfe",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Int",
"family_id": "prime_two_or_dvd_of_dvd_two_mul_pow_self_two",
"file_id": "mathlib/Mathlib/RingTheory/Int/Basic.lean",
"sample_id": "8c194e12f2fbe5697fa400d07d4529ac20ba6d8c4893ecec80e27968f828edce"
} | train |
{
"chosen_proof_sha256": "0c3061002fd6080e1b0c2c6f335b3c8a50d21cd02337f3272c2365c7c29242da",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6b2eb75d0e9cee43259678b2a625c571bc... | by
obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (NeZero.ne n)
intro i
simpa using (h i).injective.ne (by decide : (0 : Fin 3) ≠ 1) | false | {
"chosen_tokens": 42,
"rejected_tokens": 46,
"token_jaccard": 0.90625,
"token_length_ratio": 1.095238
} | missing_wrong_import | import | fa473beaef49c74d79e1c1230952f4e790b1b0851b8963da5c4b2d06b8633658 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.Convex.Between
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.Tactic.Continuity
Namespace:
Polygon
Local context:
/-
Copyright (c) 20... | import Mathlib.WrongImport
by
obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (NeZero.ne n)
intro i
simpa using (h i).injective.ne (by decide : (0 : Fin 3) ≠ 1) | {
"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": "Geometry/Polygon",
"family_id": "hasnondegeneratevertices",
"file_id": "mathlib/Mathlib/Geometry/Polygon/Basic.lean",
"sample_id": "1a320ce7ba94d641553130a5996d892f469c6c053d11bc8e9c32a512ac9b6215"
} | train |
{
"chosen_proof_sha256": "d988ca19062348f5a91c0b5ac55ee8cf0c8bfcea69c1997f85ea9823ef01e86a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
refine ⟨fun h => Set.infinite_prod.1 ?_, ?_⟩
· rw [← image_uncurry_prod] at h
exact h.of_image _
· rintro (⟨hs, b, hb⟩ | ⟨ht, a, ha⟩)
· exact hs.image2_left hb (hfs _ hb)
· exact ht.image2_right ha (hft _ ha) | false | {
"chosen_tokens": 71,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.070423
} | sorry_admit_sorryAx_proof_hole | sorryAx | fa6ca32851f464013e1652313b9c0014aa86a090c204d59e88d22da24284d297 | 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.Data.Fintype.Prod
public import Mathlib.Data.Fintype.Pi
public import Mathlib.Algebra.Order.Group.Multiset
public i... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Finite",
"family_id": "infinite_image",
"file_id": "mathlib/Mathlib/Data/Finite/Prod.lean",
"sample_id": "21544d2f51f369dc7014292be7e61bf52982b39514987f4c5a1ef51ecf1523b5"
} | train |
{
"chosen_proof_sha256": "9427a4321d73c521b7283942cf3246fe882d0cae4ae971903987493e7ed9579b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "09f801cfefe7ea06610ee0102da6822633... | by
rw [isUnit_iff_exists]
exact ⟨n, inv_mul_of_unit _ h, mul_inv_of_unit _ h⟩ | true | {
"chosen_tokens": 17,
"rejected_tokens": 3,
"token_jaccard": 0.133333,
"token_length_ratio": 0.176471
} | unknown_identifier | unknown_identifier | fa74dd762057b0b0d341e9b48021931b8229ae30377b6655d52f4d2b686ee826 | 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.Associated
public import Mathlib.Data.ZMod.Basic
public import Mathlib.RingTheory.Coprime.Lemmas
Namespace:
ZMod
Local context:
/-
Copyr... | by
exact __oprover_missing_7c3c6fabe63d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "730c95b19a4e55853480c1cfa4f0282f9ea8ae2205ac7e70ab0404f5ac258b17",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/ZMod",
"family_id": "isunit_inv",
"file_id": "mathlib/Mathlib/Data/ZMod/Units.lean",
"sample_id": "7c3c6fabe63da232039bfcfad352f1cc6f001b2ce6fe12a8d6099283072984c1"
} | train |
{
"chosen_proof_sha256": "7e80ec87feb47441a23ea0489b04ea9d8954472b614c08ce067659d50bc27e69",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [hasBasis_self, (hasBasis_nhdsSet _).forall_iff (by grind)]
intro U ⟨hU, h_KU⟩
obtain ⟨L, hL, hKL, hLU⟩ := exists_compact_between hK hU h_KU
exact ⟨L, by rwa [← subset_interior_iff_mem_nhdsSet], hL, hLU⟩ | false | {
"chosen_tokens": 53,
"rejected_tokens": 2,
"token_jaccard": 0.03125,
"token_length_ratio": 0.037736
} | by_exact_placeholder | by? | fa974d87bd4888292b6b3209e08ac3e2f96c7b2b8f753d1fa553ba2110ed6c5d | 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.Compactness.Compact
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as... | 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": "Topology/Compactness",
"family_id": "iscompact",
"file_id": "mathlib/Mathlib/Topology/Compactness/LocallyCompact.lean",
"sample_id": "ec0e8191257eb538c96b568fc34a540eaf43a4e996b7fbafb697f19a5d8137e8"
} | train |
{
"chosen_proof_sha256": "105da6d71d1fe288f69e471a4b967a3a862bded59f3b72ae1eccbacf108cf793",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "274b4f9cd10909114b550d1e9e2d114135... | by
-- Again, it is a "left inverse equals right inverse" argument in the convolution monoid.
apply left_inv_eq_right_inv
(M := Conv A (A ⊗ A))
(a := Δ[A])
· rw [Conv.mul_eq, Conv.one_eq]
simp only [comp_whiskerRight, tensor_whiskerLeft, MonObj.tensorObj.mul_def, Category.assoc,
MonObj.tensorObj.... | true | {
"chosen_tokens": 156,
"rejected_tokens": 2,
"token_jaccard": 0.018519,
"token_length_ratio": 0.012821
} | invalid_tactic | invalid_tactic | faefdb0c1a4dfb16e565cd8401ff605d23b532bd779df295657a8ef2735dac88 | 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.Monoidal.Bimon_
public import Mathlib.CategoryTheory.Monoidal.Conv
Namespace:
CategoryTheory.HopfObj
Local context:
/-
Copyright (c) 2024 Lean... | by
__oprover_invalid_tactic_316e9edc7144 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "16903bab18cca94395c17353e82074094cf1cafb49b76cb066438de74df85702",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Monoidal",
"family_id": "antipode_comul",
"file_id": "mathlib/Mathlib/CategoryTheory/Monoidal/Hopf_.lean",
"sample_id": "316e9edc7144d415a7e46a6a2a53e203a71e0c03a7d0b39706e889dc27b42fb3"
} | train |
{
"chosen_proof_sha256": "9342ddf1a3670b64a91f411236ca705a4f712a410fef1d38cc662b4389a061cb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7eb57228a163703ead56c3c31e0edfa6a8... | by
rw [← image_commutatorSet_closureCommutatorRepresentatives G]
exact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _)) | true | {
"chosen_tokens": 24,
"rejected_tokens": 3,
"token_jaccard": 0.105263,
"token_length_ratio": 0.125
} | circular_dependency | circular_dependency | fb9fa7aeef1da5dc41f91f04f721b32151067b936f9319513ba7add25384a86b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Subgroup.Finite
public import Mathlib.GroupTheory.Commutator.Basic
public import Mathlib.GroupTheory.Rank
public import Mathlib.GroupTheory.Index... | by
exact card_commutatorSet_closureCommutatorRepresentatives | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "a492ca3d778cae007620ac29501c34872e3f5b01f8fb7718b2f0f59efe97e960",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "GroupTheory/Commutator",
"family_id": "card_commutatorset_closurecommutatorrepresentatives",
"file_id": "mathlib/Mathlib/GroupTheory/Commutator/Finite.lean",
"sample_id": "83c5bc19a8475efdecefba11a168fc6bb576240b8897c80b4ded959f3ff7248c"
} | train |
{
"chosen_proof_sha256": "1e7bb9669789dabc1d0b4a44d7bd38a7c58136fdfc801571fd7b5e8b4e1c7606",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a4eb052d860f24edc427522755e82d280... | by
conv_lhs => rw [← (mateEquiv adj₁ adj₂).right_inv α]
exact (mateEquiv_counit adj₁ adj₂ ((mateEquiv adj₁ adj₂).symm α) d) | true | {
"chosen_tokens": 38,
"rejected_tokens": 5,
"token_jaccard": 0.136364,
"token_length_ratio": 0.131579
} | wrong_namespace_qualification | wrong_namespace_qualification | fc07bc3e3b9a592cff36ec0921c3ff297fc31b4c4fe7b6002c4086fbf020a8c2 | 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.Adjunction.Basic
public import Mathlib.CategoryTheory.Functor.TwoSquare
public import Mathlib.CategoryTheory.HomCongr
Namespace:
CategoryTheory... | by
exact OProverMissing.missing_4c02a391b7ac | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "5a4fab0c29198f803ecee62039477850c92cd9e568d5eb3d98f72e4e757ad36c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Adjunction",
"family_id": "mateequiv_counit_symm",
"file_id": "mathlib/Mathlib/CategoryTheory/Adjunction/Mates.lean",
"sample_id": "4c02a391b7ac277a192dd766770fc91e8893ee8cde8243ce667618594d9c0e76"
} | train |
{
"chosen_proof_sha256": "19091c655ecaa51563cbf7438e93b5ad161e207efc740ddfb20c17af665fcd74",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d3ed95d2114a8fd9d1ee96ff835de25fc3... | by
intro x y h
by_cases hx : x ∈ xs <;> by_cases hy : y ∈ xs
· rw [List.mem_iff_getElem?] at hx hy
obtain ⟨i, hx⟩ := hx
obtain ⟨j, hy⟩ := hy
suffices some x = some y by injection this
have h₂ := h₁.length_eq
rw [List.applyId_zip_eq h₀ h₂ _ _ _ hx] at h
rw [← hx, ← hy]; congr
apply (Lis... | true | {
"chosen_tokens": 215,
"rejected_tokens": 5,
"token_jaccard": 0.047619,
"token_length_ratio": 0.023256
} | wrong_namespace_qualification | wrong_namespace_qualification | fc4c2b5ab9a129497a5ad09940749ee763eec3aa784deb5d1df78d23c9c5e290 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Data.MLList.Basic
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.ZeroLEOne
public import Mathlib.Data.DFinsupp.Defs
public... | by
exact OProverMissing.missing_1d731b6f1a0a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "0d94b5475912bb5aa03ee546cb4afa1233159c376e0c56e4a9e9a37b39be414c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Testing/Plausible",
"family_id": "applyid_injective",
"file_id": "mathlib/Mathlib/Testing/Plausible/Functions.lean",
"sample_id": "1d731b6f1a0aa28042e1c006480d5c7f9dc42d3f0a8f8f5b52bbb265d67e141d"
} | train |
{
"chosen_proof_sha256": "e08724fba84126ab1ada747c46865f50f539e3028cb4c27da440863f309c14d1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "76ec52606c6847d51d5dec2714d2a3c51c... | by
borelize G
rw [modularCharacterFun_eq_haarScalarFactor MeasureTheory.Measure.haar g]
exact haarScalarFactor_pos_of_isHaarMeasure _ _ | true | {
"chosen_tokens": 17,
"rejected_tokens": 5,
"token_jaccard": 0.176471,
"token_length_ratio": 0.294118
} | wrong_namespace_qualification | wrong_namespace_qualification | fc5b70ce8cac9b3505c98f5a706d3419f21b03b28a57c907f93fd39d4b0110fb | 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.Function.LocallyIntegrable
public import Mathlib.MeasureTheory.Group.Integral
public import Mathlib.MeasureTheory.Group.Measure
public import Mat... | by
exact OProverMissing.missing_5d6d44820ec4 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "6d2e42bdb4e1b04aade046cc6168629faec230f54b8cfd10b70b6b2a6771f3b5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Group",
"family_id": "modularcharacterfun_pos",
"file_id": "mathlib/Mathlib/MeasureTheory/Group/ModularCharacter.lean",
"sample_id": "5d6d44820ec47fa0236df113ccc6391b9ab3172c1aa71f5359c1564641b4264c"
} | train |
{
"chosen_proof_sha256": "d216e83441f49f61f8d591a6008f2029d6a48f6f63ee5482956becb053185e63",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6235791abd41c06ce11aa0d46e38fd7ac2... | by
rw [commute_iff_eq, hp.le_iff_mul_eq_right hq |>.mp h, hp.le_iff_mul_eq_left hq |>.mp h] | false | {
"chosen_tokens": 25,
"rejected_tokens": 30,
"token_jaccard": 0.75,
"token_length_ratio": 1.2
} | markdown_preamble_trailing | preamble | fcf3c2aec84165d9c63bc7475349dfc1ae9077d41f52e556095f10ed35be4064 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order
public import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric
import Mat... | Here is the proof:
by
rw [commute_iff_eq, hp.le_iff_mul_eq_right hq |>.mp h, hp.le_iff_mul_eq_left hq |>.mp h] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/CStarAlgebra",
"family_id": "commute_of_le",
"file_id": "mathlib/Mathlib/Analysis/CStarAlgebra/Projection.lean",
"sample_id": "9c77b43c796b9a66a9282957c2b67d46b27f3beda4c44634fa70becc056cd2ca"
} | train |
{
"chosen_proof_sha256": "b310f14e4f1b5c03181254cbcff43ceec8a366fb3cce3389a0bcba30d3ef880d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "da9f340528cf03713b6735daff262145f2... | by
by_cases h0 : p = 0
· simp [h0]
by_cases h_top : p = ∞
· simp only [h_top, eLpNorm, eLpNormEssSup, ENNReal.top_ne_zero, if_true,
if_false]
apply essSup_eq
exact h.comp (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)
simp only [eLpNorm_eq_eLpNorm' h0 h_top, eLpNorm', one_div]
congr 1
... | false | {
"chosen_tokens": 81,
"rejected_tokens": 86,
"token_jaccard": 0.909091,
"token_length_ratio": 1.061728
} | markdown_preamble_trailing | preamble | fd32fa56b05702cc6d93cd99fa80b39ffb29cc39d57c28cc677cc53f4bb63998 | 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.HasLaw
public import Mathlib.Probability.Moments.Variance
public import Mathlib.MeasureTheory.Function.UniformIntegrable
Namespace:
ProbabilityThe... | Here is the proof:
by
by_cases h0 : p = 0
· simp [h0]
by_cases h_top : p = ∞
· simp only [h_top, eLpNorm, eLpNormEssSup, ENNReal.top_ne_zero, if_true,
if_false]
apply essSup_eq
exact h.comp (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)
simp only [eLpNorm_eq_eLpNorm' h0 h_top, eLpNorm', ... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Probability",
"family_id": "elpnorm_eq",
"file_id": "mathlib/Mathlib/Probability/IdentDistrib.lean",
"sample_id": "9a2df3c5de7a292cbf62d67b3b00525a929d8c851243d6ce0af4fd45360425d2"
} | train |
{
"chosen_proof_sha256": "0e69e1ffdf18797894f9d6a7bd81718318be487366506848fac73b20c0a23f03",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "229982fa3e3692bb975dd7914f9e2dfa6c... | by
rw [internallyProjective_iff_tensor_condition]
refine ⟨fun h A B e he S g ↦ ?_, fun h A B e he S g ↦ ?_⟩
· specialize h e S ((β_ _ _).hom ≫ g)
obtain ⟨S', π, hπ, g', hh⟩ := h
refine ⟨S', π, hπ, (β_ _ _).inv ≫ g', ?_⟩
simp [← hh]
· specialize h e S ((β_ _ _).inv ≫ g)
obtain ⟨S', π, hπ, g', hh⟩... | false | {
"chosen_tokens": 151,
"rejected_tokens": 155,
"token_jaccard": 0.925,
"token_length_ratio": 1.02649
} | missing_wrong_import | import | fd68541312e92487163b18e8a1c37e82393465edc3e32d3541774dc8031b5ef2 | 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.Preadditive.Projective.Internal
public import Mathlib.Condensed.Light.Epi
public import Mathlib.Condensed.Light.Functors
public import Mathlib.C... | import Mathlib.WrongImport
by
rw [internallyProjective_iff_tensor_condition]
refine ⟨fun h A B e he S g ↦ ?_, fun h A B e he S g ↦ ?_⟩
· specialize h e S ((β_ _ _).hom ≫ g)
obtain ⟨S', π, hπ, g', hh⟩ := h
refine ⟨S', π, hπ, (β_ _ _).inv ≫ g', ?_⟩
simp [← hh]
· specialize h e S ((β_ _ _).inv ≫ g)
... | {
"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": "Condensed/Light",
"family_id": "internallyprojective_iff_tensor_condition",
"file_id": "mathlib/Mathlib/Condensed/Light/InternallyProjective.lean",
"sample_id": "e1cca0717031d58aaaf848a8d08234c2dff2f8f85f108568fd9abbcb33e5e463"
} | train |
{
"chosen_proof_sha256": "c628935d9750505c8adb2f5d44da9ce697c92da40b87a39fbb8e2a552d65d0b4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
ext; exact geometricallyReduced_iff _ | false | {
"chosen_tokens": 6,
"rejected_tokens": 5,
"token_jaccard": 0.375,
"token_length_ratio": 0.833333
} | sorry_admit_sorryAx_proof_hole | sorryAx | fd6d204074abcf13128165747e074d7a0749e58f853353eb9dc71697df24d484 | 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.Artinian
public import Mathlib.AlgebraicGeometry.Geometrically.Basic
public import Mathlib.AlgebraicGeometry.Morphisms.SchemeTheoreticallyDom... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Geometrically",
"family_id": "geometricallyreduced",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Geometrically/Reduced.lean",
"sample_id": "8fa0d3b17dbb462bc253c216aedf5600d245ea36ea5fef5dc975ff5afcf7c43e"
} | train |
{
"chosen_proof_sha256": "b2349a1019905ca8cc28d0ef2d79317628410a89c7793fafcbce2e7a084a9bdd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "12e7ffd87a87551cb23a2885c902209a7a... | by
rw [← Ioi_union_left, ← Ioi_union_left]
exact
(surjOn_Ioi_of_monotone_surjective h_mono h_surj a).union_union
(@image_singleton _ _ f a ▸ surjOn_image _ _) | true | {
"chosen_tokens": 30,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.066667
} | invalid_tactic | invalid_tactic | fe1f79d77fd6aed889086f3d6d975d142ca2816bfb1e9510f64c8f9cff8c87a7 | 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.Function
public import Mathlib.Order.Interval.Set.LinearOrder
Namespace:
(root)
Local context:
/-
Copyright (c) 2020 Heather Macbeth. All rights res... | by
__oprover_invalid_tactic_16580c7b1066 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b8ad03a9377f68f1de44062eea13597ed3a30f54cd3f9d2ecd60049637c36c7d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Interval",
"family_id": "surjon_ici_of_monotone_surjective",
"file_id": "mathlib/Mathlib/Order/Interval/Set/SurjOn.lean",
"sample_id": "16580c7b1066ea68f7db2b4cc23ac07bcdd4aa4217971b759537bed8271b6fb6"
} | train |
{
"chosen_proof_sha256": "5668706c687646eb1a543fe99da9ca45de8dce6aa203fed4f6e41809be74067a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "587a3a6ebe4fb7f9e8b2bca4769899b66e... | by
obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos
have hk₀' : (0 : ℝ) < k := Int.cast_pos.mpr hk₀
have hden : ((j / k : ℚ).den : ℤ) ≤ k := by
convert! le_of_dvd hk₀ (Rat.den_dvd j k)
exact Rat.intCast_div_eq_divInt _ _
refine ⟨j / k, ?_, Nat.cast_le.mp (hden.trans hk₁)⟩
rw [← div_... | false | {
"chosen_tokens": 192,
"rejected_tokens": 196,
"token_jaccard": 0.956522,
"token_length_ratio": 1.020833
} | missing_wrong_import | import | fea5e445f0ecf531c447510cf4a6d9a8a18de16ebf60d8a1e12148fd46792c65 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.NumberTheory.Real.Irrational
public import Mathlib.RingTheory.Coprime.Lemmas
public import Mathlib.RingTheory.Int.Basic
public import Mathlib.Tactic.Basic
Nam... | import Mathlib.WrongImport
by
obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos
have hk₀' : (0 : ℝ) < k := Int.cast_pos.mpr hk₀
have hden : ((j / k : ℚ).den : ℤ) ≤ k := by
convert! le_of_dvd hk₀ (Rat.den_dvd j k)
exact Rat.intCast_div_eq_divInt _ _
refine ⟨j / k, ?_, Nat.cast_le.mp (hd... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/DiophantineApproximation",
"family_id": "exists_rat_abs_sub_le_and_den_le",
"file_id": "mathlib/Mathlib/NumberTheory/DiophantineApproximation/Basic.lean",
"sample_id": "f7c063a4e5099ef9d392d9d1440be2d9799586ee41a1ffac00a85cda728bea8f"
} | train |
{
"chosen_proof_sha256": "ff5b125214a3c976ea99dae2474b0a2f292cf08626af894b2e62bc6e985e11b4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "81c79c52eecec98e2493470604cad60abe... | by
apply freeYonedaHomEquiv.injective
simp only [freeYonedaHomEquiv_comp, Equiv.apply_symm_apply]
variable (C) | false | {
"chosen_tokens": 18,
"rejected_tokens": 23,
"token_jaccard": 0.809524,
"token_length_ratio": 1.277778
} | markdown_preamble_trailing | trailing | fec7b034d69a1dc3d34be425dc3d2ca348dc1f03cb5763bee58dbcebf9fcded5 | 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.Generator.Basic
public import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
Namespace:
CategoryTheory.Presheaf
Local context:
/-
Copyrig... | by
apply freeYonedaHomEquiv.injective
simp only [freeYonedaHomEquiv_comp, Equiv.apply_symm_apply]
variable (C)
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/Generator",
"family_id": "freeyonedahomequiv_symm_comp",
"file_id": "mathlib/Mathlib/CategoryTheory/Generator/Presheaf.lean",
"sample_id": "2c8f4c46b465c3b61a9d9f12832774c7822ae516e1e560bc50ecc34f4e2ef998"
} | train |
{
"chosen_proof_sha256": "156fa55c2e6ebaab51efb61e1b6415d519f8bca6eaddaf4eff8b3327fd1b34c8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7023cf0fd56d90184e0f93e416074e4ba1... | by
rcases le_total x 0 with (neg | pos)
· exact .inr <| isSquare_of_nonneg (neg_nonneg_of_nonpos neg)
· exact .inl <| isSquare_of_nonneg pos
exists_isRoot_of_odd_natDegree := exists_isRoot_of_odd_natDegree
variable [IsRealClosed R] | true | {
"chosen_tokens": 38,
"rejected_tokens": 3,
"token_jaccard": 0.074074,
"token_length_ratio": 0.078947
} | unknown_identifier | unknown_identifier | fece8a02b77f73127ee44b266ecdbee6ca4989dc4e3ff87d1583b4cf58727cc1 | 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.Domain
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.Algebra.Ring.Semireal.Defs
public import Mathlib.Tact... | by
exact __oprover_missing_b9d038f0b319 | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "b0e35e417854898447c4fafe8f7055e9110b898bd9f9b856d255df8dd4875df9",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/IsRealClosed",
"family_id": "of_linearorderedfield",
"file_id": "mathlib/Mathlib/FieldTheory/IsRealClosed/Basic.lean",
"sample_id": "b9d038f0b31926725a67cabc50ad8dfeb3e914d41076f9b7c6d575a332e9b255"
} | train |
{
"chosen_proof_sha256": "7d031507a20582e22ca4729c3a20ae04f876f5f252c02c99897df94087378755",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "eefa6d8a0659afef7ba5bbcd19f7c9f76e... | by
ext j
rw [assoc, assoc, enrichedComp_π, id_comp, tensorHom_def', assoc,
← MonoidalCategory.whiskerLeft_comp_assoc, enrichedId_π,
whisker_exchange_assoc, MonoidalCategory.whiskerRight_id, assoc, assoc,
Iso.inv_hom_id_assoc]
dsimp
rw [e_comp_id, comp_id] | true | {
"chosen_tokens": 47,
"rejected_tokens": 5,
"token_jaccard": 0.074074,
"token_length_ratio": 0.106383
} | wrong_namespace_qualification | wrong_namespace_qualification | fedb638c5be7d8eb5573f67caa2aa6e798f500d18a8f84ace0958ef16d5b661c | 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.Monoidal.FunctorCategory
public import Mathlib.CategoryTheory.Enriched.Ordinary.Basic
public import Mathlib.CategoryTheory.Functor.Category
publ... | by
exact OProverMissing.missing_623cf51d346b | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d6bf0ee5819e999d5abbba29e9aedcb278bf30ca9caaae706151f9b794ff6e6d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Enriched",
"family_id": "enriched_comp_id",
"file_id": "mathlib/Mathlib/CategoryTheory/Enriched/FunctorCategory.lean",
"sample_id": "623cf51d346b229ca13fed1da60cd6df5ba3c3809b92103ef6f67007cec38024"
} | train |
{
"chosen_proof_sha256": "c311ad4b581caa0a3b895b5e4dcbcd3cc30063d829252ea41d678e32db406435",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
tfae_have 1 ↔ 2 := Sheaf.isConstant_iff_isIso_counit_app _ _ _
tfae_have 1 ↔ 3 := ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩
tfae_have 1 ↔ 4 := Sheaf.isConstant_iff_mem_essImage _ LightProfinite.isTerminalPUnit adjunction X
tfae_have 1 ↔ 5 :=
have : functor.Faithful := inferInstance
have : functor.Full := inferInsta... | false | {
"chosen_tokens": 149,
"rejected_tokens": 3,
"token_jaccard": 0.016667,
"token_length_ratio": 0.020134
} | by_exact_placeholder | exact? | ff0c30ecd470d8af41854c5ade7a55215e2ddc6f58bb84cf5a9e2546d4c4f5d2 | 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.Colimit
public import Mathlib.Condensed.Discrete.Module
Namespace:
LightCondSet
Local context:
/-
Copyright (c) 2024 Dagur Asgeirsson. All... | 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": "Condensed/Discrete",
"family_id": "isdiscrete_tfae",
"file_id": "mathlib/Mathlib/Condensed/Discrete/Characterization.lean",
"sample_id": "4edfa6f1dfc9284bddb186d5cd60c15b919148603ac348edfd11fc15fed97087"
} | train |
{
"chosen_proof_sha256": "3f320a505bb6b269ad39068f91895095fa32692c32719fd89a78bd89e6e3c623",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | 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) | true | {
"chosen_tokens": 64,
"rejected_tokens": 8,
"token_jaccard": 0.046512,
"token_length_ratio": 0.125
} | unsolved_goals | unsolved_goals | ff36e99145c397b086e6dcc40beee2c688d1383b4ff6365cf6d05c579689caad | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.InnerProductSpace.PiL2
public import Mathlib.Analysis.InnerProductSpace.Spectrum
public import Mathlib.LinearAlgebra.Eigenspace.Charpoly
Namespace:
L... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "aaa10d9e2e18985a11b384b2b89e78e9eb706e3ec6647e65a0f70c4870acdb74",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "956d0d39593774896edc51480ad38b07e1edca7a7a86a5d842d9b626672e0346",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "06ed38f6debd03daf214edd7588ee3722c... | by
rw [isNoetherian_iff', wellFoundedGT_iff_monotone_chain_condition]
variable [IsNoetherian R M] | true | {
"chosen_tokens": 13,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.153846
} | invalid_tactic | invalid_tactic | ffa3fe7d3fdde395aa3cbd116fee881dfc77f13916abfbdf23ab6e53f7a1044b | 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.AtTopBot.Basic
public import Mathlib.RingTheory.Finiteness.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2018 Mario Carneiro, Kevin Bu... | by
__oprover_invalid_tactic_a21a121517cc | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "fde5879b0007d8a93a1b9953ea86fc4366956237ea64b4d16897ba203885227d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Noetherian",
"family_id": "monotone_stabilizes_iff_noetherian",
"file_id": "mathlib/Mathlib/RingTheory/Noetherian/Defs.lean",
"sample_id": "a21a121517cc485ec8b0fac3320d37a95ffa767d8d52dbb2b3c8baa9ed0c81a3"
} | train |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.