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": "16b3be2598502346de2754703a4649aa758c506146fa34fc091635ed0bf9b555",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rcases isEmpty_or_nonempty X with h | h
· exact ⟨Classical.arbitrary Y, funext <| h.elim⟩
· exact ⟨f (Classical.arbitrary X), hf.eq_const _⟩ | true | {
"chosen_tokens": 39,
"rejected_tokens": 8,
"token_jaccard": 0.032258,
"token_length_ratio": 0.205128
} | unsolved_goals | unsolved_goals | 2ac185c77e31a997a73526df4ae9948823965acf12cb5c4f42000afeadb30542 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Notation.Indicator
public import Mathlib.Topology.Connected.LocallyConnected
public import Mathlib.Topology.Sets.Closeds
Namespace:
IsLocallyConstant
... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "407636fda83c5b7a83adb732b1a29fb090cebf292fcc87156769bda23304b774",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/LocallyConstant",
"family_id": "exists_eq_const",
"file_id": "mathlib/Mathlib/Topology/LocallyConstant/Basic.lean",
"sample_id": "41091d79c58a55895294fdd103ee3f3bbfd8236b3858691c14a8a3dfe9a1b2a2"
} | train |
{
"chosen_proof_sha256": "b656fc39071ddeaac02c29e4861f9ce7d1311a7b76fad806354a82acd6bcd29b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "785e48ee70dd3a551ef74441e20aad6cee... | by
cases isEmpty_or_nonempty ι
· rw [iSup_of_empty', iSup_of_empty']
· exact hdf.ciSup_le fun x ↦ hdg.le_ciSup_of_le B x (H x) | true | {
"chosen_tokens": 28,
"rejected_tokens": 5,
"token_jaccard": 0.12,
"token_length_ratio": 0.178571
} | circular_dependency | circular_dependency | 2ac81cb114284df4816758415945e26e9668a422c87ff18678889dd26f684ec4 | 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.ConditionallyCompletePartialOrder.Basic
public import Mathlib.Order.GaloisConnection.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2026 Jireh... | by
exact Directed.ciSup_mono | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "3a82591f2c4c79b5346fbecdbd75c2479a35e84e068139b0084e9adc6b23192b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Order/ConditionallyCompletePartialOrder",
"family_id": "directed",
"file_id": "mathlib/Mathlib/Order/ConditionallyCompletePartialOrder/Indexed.lean",
"sample_id": "66a9d446584881e6ae181bf60c62b6c4435815bff91f04bcf763c4007fc9aaae"
} | train |
{
"chosen_proof_sha256": "91037b5a5d4107cfc6e47498527b6da8e785a774a3e33005feebe3831fc4a2a4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "357a0e4c1d470dcb98d6df000adeb3562e... | by
rw [← tendsto_norm_atTop_iff_cobounded]
simpa [norm_intCast_eq_abs_mul_norm_one] using tendsto_intCast_atTop_atTop
|>.comp (tendsto_abs_atBot_atTop.sup tendsto_abs_atTop_atTop)
|>.atTop_mul_const (norm_pos_iff.mpr one_ne_zero) | true | {
"chosen_tokens": 32,
"rejected_tokens": 3,
"token_jaccard": 0.04,
"token_length_ratio": 0.09375
} | unknown_identifier | unknown_identifier | 2b08faf0a8bb69affd853206e48d69c1aeab22e23f2004d393261561415dc3c6 | 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.Module
public import Mathlib.Algebra.Order.Field.Power
public import Mathlib.Algebra.Polynomial.Monic
public import Mathlib.Analysis.Asymp... | by
exact __oprover_missing_2031f82d6bd5 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "e076ec07edeaefcdb4036d705e99faf54d5ddc24438a39deca0be7e543107511",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/SpecificLimits",
"family_id": "tendsto_intcast_atbot_sup_attop_cobounded",
"file_id": "mathlib/Mathlib/Analysis/SpecificLimits/Normed.lean",
"sample_id": "2031f82d6bd5b2086402ba4895d772d5452bfbf03175b7504ea5ff92425b4486"
} | train |
{
"chosen_proof_sha256": "56b4e98d034de1994cb22dfefff34d5e35c25783ae7eef0d3fe3f45cbf3ff033",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e0b2d88f43c7c482b1479e13f5c9bc4c3c... | by
simpa using mul_vle_mul_left ((vle_total 0 1).resolve_right not_vle_one_zero) x | true | {
"chosen_tokens": 15,
"rejected_tokens": 2,
"token_jaccard": 0.071429,
"token_length_ratio": 0.133333
} | invalid_tactic | invalid_tactic | 2b0bbbc67eee158c9ea0dec75cec4dd96bb7a2bb679e29d1bd2ae5a4d729aae1 | 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.Valuation.Basic
public import Mathlib.Data.NNReal.Defs
public import Mathlib.Topology.Defs.Filter
public import Mathlib.Order.Filter.Bases.Basic
Na... | by
__oprover_invalid_tactic_27226b4cef48 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "39c17893e48efecb814760c686e396dc4aeab84b04d2553f82ef8a7025bc94fc",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Valuation",
"family_id": "zero_vle",
"file_id": "mathlib/Mathlib/RingTheory/Valuation/ValuativeRel/Basic.lean",
"sample_id": "27226b4cef484b3c92a34e5c76b5413440a1f9efdabd65d7d5bb3fbb80618898"
} | train |
{
"chosen_proof_sha256": "cde0cb2c54480a9a9ba9dd0ed55105a1ff6f413d177d544464139dccab437478",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d4e1b22d41edfa6c165c88bc1ca936a47f... | by
classical
simp_rw [← AlgHom.coe_toRingHom, ← coeff_map, subProdXSubC, Polynomial.map_sub,
Polynomial.map_prod, Polynomial.map_sub, map_X, map_C, toSplittingField,
AlgHom.coe_toRingHom, MvPolynomial.aeval_X, dif_pos h,
← (finEquivRoots (Monics.splits_finsetProd h)).symm.prod_comp, Equiv.apply_symm_app... | true | {
"chosen_tokens": 165,
"rejected_tokens": 3,
"token_jaccard": 0.012987,
"token_length_ratio": 0.018182
} | circular_dependency | circular_dependency | 2b0ec5528b7a125de6b2e375a9b81c38254226fb1feaece9d8bc3df50c354a21 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.CharP.Algebra
public import Mathlib.Data.Multiset.Fintype
public import Mathlib.FieldTheory.IsAlgClosed.Basic
public import Mathlib.FieldTheory.Splitti... | by
exact toSplittingField_coeff | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "6dce43230cd4fb166894bdbd8b35b5ceb1af7ef9d3745ca0a76e68ef20ba9c47",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/IsAlgClosed",
"family_id": "tosplittingfield_coeff",
"file_id": "mathlib/Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean",
"sample_id": "dfc0baea2eabd164611e1d1c93cf31cf2ae01a14f1445c0b6a91023c66e00ae4"
} | train |
{
"chosen_proof_sha256": "055208fdff31df9d8fb76bf05bb6a68b4de72a41035c9aa9cc8d72c2b3b25d67",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "db95c7c59b1a744d161d6d13a5a1ed3269... | by
simp [Algebra.smul_def, ← nat_rawCast_2] | true | {
"chosen_tokens": 10,
"rejected_tokens": 5,
"token_jaccard": 0.153846,
"token_length_ratio": 0.5
} | wrong_namespace_qualification | wrong_namespace_qualification | 2b163f1426e145f24aa67ae233bcfc05c164db8ac921f9658ca906355d474461 | 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.Defs
public import Mathlib.Algebra.Algebra.Basic
public import Mathlib.Tactic.Ring.RingNF
Namespace:
Mathlib.Tactic.Algebra
Local context:
/-... | by
exact OProverMissing.missing_029e00c3f6a5 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1603f346fdf1ead97ce3a40f1add2bd5044bc756b503ecc10dd47aef6f566edd",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/Algebra",
"family_id": "rat_ofnat_smul",
"file_id": "mathlib/Mathlib/Tactic/Algebra/Lemmas.lean",
"sample_id": "029e00c3f6a53264f7ab33ca9285b495e3986d3f7052bd853ac23e6d759b17de"
} | train |
{
"chosen_proof_sha256": "93aa954ac038910f3cc0d692da80205ae5a7230c0f479de23b1a6ec2e526a44a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [FractionalOperation.tt, Multiset.mem_map] at rin
rw [show r = ![r 0, r 1] by simp [← List.ofFn_inj]]
apply lt_of_le_of_ne (mcf.right (r 0) (r 1)).left
intro equ
have asymm : r 0 ≠ r 1 := by
rcases (mcf.right (r 0) (r 1)).right equ with ⟨ha0, hb1⟩ | ⟨ha1, hb0⟩
· rw [ha0, hb1] at hab
exact ... | false | {
"chosen_tokens": 237,
"rejected_tokens": 3,
"token_jaccard": 0.025,
"token_length_ratio": 0.012658
} | by_exact_placeholder | exact? | 2b2ae743c980fc73f4778bfd3781b160415ea67c8f7e1bb7df768553e0447e1b | 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.Fin
public import Mathlib.Algebra.Order.BigOperators.Group.Multiset
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Linea... | 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": "Combinatorics/Optimization",
"family_id": "function",
"file_id": "mathlib/Mathlib/Combinatorics/Optimization/ValuedCSP.lean",
"sample_id": "4158bc4533e6e85b1b86e67b31a3d6eca02e3c8f701cb75f73a81c4775a53885"
} | train |
{
"chosen_proof_sha256": "4bbd86e73cbf3bb6938d81696c0fac5d64455a44efb1861863f6e712d6a46845",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "286b97964583ec10d54aee23d52750899f... | by
rw [Finset.disjoint_left]
rintro x hx hxc
obtain ⟨x, hx', rfl⟩ := mem_map.mp hxc
exact hs.compl_notMem hx' hx | true | {
"chosen_tokens": 30,
"rejected_tokens": 5,
"token_jaccard": 0.12,
"token_length_ratio": 0.166667
} | wrong_namespace_qualification | wrong_namespace_qualification | 2b3a8d0a0da7d24222fe2a0e13bf55a4eafca860be427bfd992c20f73fb0e309 | 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.Card
public import Mathlib.Order.UpperLower.Basic
Namespace:
Set
Local context:
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Release... | by
exact OProverMissing.missing_8a9a5d45eecf | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "51d63d3c47d50c05930532233183dd1da11a31ab9f6720d72c6457d77a829717",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/SetFamily",
"family_id": "intersecting",
"file_id": "mathlib/Mathlib/Combinatorics/SetFamily/Intersecting.lean",
"sample_id": "8a9a5d45eecf81f80cc6ac9acad10ad3865265d34cf144e568617dadd24602fc"
} | train |
{
"chosen_proof_sha256": "12279e85ffc4d0d8cf16eae5dea8e0e2916e98350ea6156d422f2d5856bb9f1e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [← sSup_image']
exact hd.isLUB_csSup (Hne.image _) H | false | {
"chosen_tokens": 17,
"rejected_tokens": 2,
"token_jaccard": 0.058824,
"token_length_ratio": 0.117647
} | sorry_admit_sorryAx_proof_hole | sorry | 2b5503a356257d696db89f9343f09e137f376d747ad538d568a11847ac6bafbd | 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.ConditionallyCompletePartialOrder.Basic
public import Mathlib.Order.GaloisConnection.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2026 Jireh... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/ConditionallyCompletePartialOrder",
"family_id": "directedon",
"file_id": "mathlib/Mathlib/Order/ConditionallyCompletePartialOrder/Indexed.lean",
"sample_id": "15e4f134fc4396fd42f4d434d7605aca7a909498c0d0844c2e35d252b2adb530"
} | train |
{
"chosen_proof_sha256": "6ed251e98bcc09fc840097a1704c71dc97a2385a1bbef8d0c694a48f689a68b3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ae377ebe4e81614b721fd67079ff2abc3a... | by
rw [OneSkeletonGraph_isLink, closedCell_zero_eq_singleton, closedCell_zero_eq_singleton,
singleton_union] | true | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.272727
} | circular_dependency | circular_dependency | 2ba6d52516f26a891f5a039837178ed0f3e0848640e5a25f848e4b0ab49b3026 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.CWComplex.Classical.Finite
public import Mathlib.Combinatorics.Graph.Basic
Namespace:
Topology.CWComplex.OneSkeletonGraph
Local context:
/-
Copyrigh... | by
exact isLink_iff_pair | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "47140aea32669f13703c8388ccbf1fc063152f5b01aafafc70d92b3035f0ed07",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/CWComplex",
"family_id": "islink_iff_pair",
"file_id": "mathlib/Mathlib/Topology/CWComplex/Classical/Graph.lean",
"sample_id": "0d7956b6ebdcf8c47b68d95a35dc56b9f93a664feaf120a85ef061f559b47042"
} | train |
{
"chosen_proof_sha256": "61743a744bcd9aa71d7e4c51326303887dfd1d0a7a128d08960e283447fbb42c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "739d2e77b2892cb2da9b9ac9b16f4da929... | by
rw [numLeaves_eq_numNodes_succ]
exact x.numNodes.zero_lt_succ | false | {
"chosen_tokens": 11,
"rejected_tokens": 15,
"token_jaccard": 0.769231,
"token_length_ratio": 1.363636
} | missing_wrong_import | import | 2bb4ef543f7c09e9d62a0a4b621121477d8345b9e84f3f526d24e1a86336bd93 | 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.Notation
public import Mathlib.Util.CompileInductive
import Batteries.Tactic.Alias
Namespace:
BinaryTree
Local context:
/-
Copyright (c) 2019 mathli... | import Mathlib.WrongImport
by
rw [numLeaves_eq_numNodes_succ]
exact x.numNodes.zero_lt_succ | {
"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/Tree",
"family_id": "numleaves_pos",
"file_id": "mathlib/Mathlib/Data/Tree/Basic.lean",
"sample_id": "06643b8b06bfa39bedc6e6457247fd95883e76318998fda10cf95f90e40967df"
} | train |
{
"chosen_proof_sha256": "fd3b7a2ce076a9542b33ce25f06638c50f65a2500253da07e8afbc3c349b995c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "59c573b6651dd9610dd6079b8c99d6d2fd... | by
induction m with
| zero => simpa
| succ m ih =>
simp_rw [pow_succ _ m, pow_mul]
exact ih.pow_add_one hpI m.succ_ne_zero | false | {
"chosen_tokens": 29,
"rejected_tokens": 36,
"token_jaccard": 0.916667,
"token_length_ratio": 1.241379
} | markdown_preamble_trailing | fence | 2bde5acd05e04ef6ef73cb14e5642881118639397ea895ddf44e4035b176129f | 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.Lemmas
public import Mathlib.LinearAlgebra.SModEq.Basic
public import Mathlib.RingTheory.Ideal.Operations
Namespace:
SModEq
Local context:
/-
C... | ```lean
by
induction m with
| zero => simpa
| succ m ih =>
simp_rw [pow_succ _ m, pow_mul]
exact ih.pow_add_one hpI m.succ_ne_zero
``` | {
"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/SModEq",
"family_id": "pow_pow_add_one",
"file_id": "mathlib/Mathlib/LinearAlgebra/SModEq/Pow.lean",
"sample_id": "f2813c1994d0014dcc0cbbe6ef4d234abd05fcb0aaba7d50c3b4450586130664"
} | train |
{
"chosen_proof_sha256": "6db23e1003596f7373b150a65cdc1843f9cceb615009a5e13fae5265b46dd136",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3dbc97992882d9b356771d88365a2201f6... | by
intro L₁ L₂ hflat
apply Subtype.ext
exact h L₁.val L₂.val L₁.prop L₂.prop hflat | true | {
"chosen_tokens": 30,
"rejected_tokens": 5,
"token_jaccard": 0.1875,
"token_length_ratio": 0.166667
} | wrong_namespace_qualification | wrong_namespace_qualification | 2be29a7419cf004992680d994be85b01b44fe0aac71190ea7afe03feb9cc8d97 | 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
exact OProverMissing.missing_70ee751da2d1 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1c3163e847cc85bf955bf7271f36d6b7faea8ea4f36440e64281cf0552daec97",
"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": "70ee751da2d1168fe9d7a11059342b3a3f15c210604e29d540ce2dd98fb9afcf"
} | train |
{
"chosen_proof_sha256": "ea6d6b98ce6bb5df330d0c3825d495329a7a811e1188283f31f2efe51dfb0168",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c37348c1479d7ebb1f5927cb69b83756ac... | by
rw [← LinearMap.comp_assoc, ← LinearMap.comp_assoc, TensorProduct.map_map_comp_assoc_eq] | true | {
"chosen_tokens": 17,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.117647
} | invalid_tactic | invalid_tactic | 2be6f4c30e76734dcb0bfc3143e3ff5890d28146213a5d70f454f00cd40a5df0 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorProduct.Tower
public import Mathlib.RingTheory.Coalgebra.Basic
import Mathlib.Tactic.Attr.Register
Namespace:
CoassocSimps
Local context:... | by
__oprover_invalid_tactic_4ea2d4d2d7f0 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "493e92c2ee8c8655b1b45bbcce545a2eaa2f55240dcbe5fd9fd93d2eb9e42712",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Coalgebra",
"family_id": "tensorproduct",
"file_id": "mathlib/Mathlib/RingTheory/Coalgebra/CoassocSimps.lean",
"sample_id": "4ea2d4d2d7f00624026e8f321443e14dc6b69d2bf90adb74245fe472882d93fb"
} | train |
{
"chosen_proof_sha256": "a18d962dbf46cb1bc44d7afa74d89d68066f49e5784c2cb2792cd31f2ea27cd6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1047399b822c6a1bbaabd56b32edd0709c... | by
rw [H.apply_zero_path, H.apply_one_path, H.evalAt_eq]
erw [H.evalAt_eq]
dsimp only [prodToProdTopI]
constructor
· slice_lhs 2 4 => rw [eqToHom_trans, eqToHom_refl] -- Porting note: this ↓ `simp` didn't do this
slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]
rfl
· slice_lhs 2 4 => rw [eq... | true | {
"chosen_tokens": 103,
"rejected_tokens": 3,
"token_jaccard": 0.025641,
"token_length_ratio": 0.029126
} | circular_dependency | circular_dependency | 2bf277499374f1270da7f25a593a292876be0b7dbfdf9bfaa696619678754124 | 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.Homotopy.Equiv
public import Mathlib.CategoryTheory.Equivalence
public import Mathlib.AlgebraicTopology.FundamentalGroupoid.Product
Namespace:
Contin... | by
exact eq_diag_path | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "fc9bf722f12c4f26646a03c9a39810208625371d727a6886a920b9d029784b0c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/FundamentalGroupoid",
"family_id": "eq_diag_path",
"file_id": "mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean",
"sample_id": "5b6616e952970d213dc2af95acd62437ab9f4ee924f0692e7518320229a74eaa"
} | train |
{
"chosen_proof_sha256": "c46ea331c2028c22d1cfe25ee9b2c63dfdbb45feb5dd1bf4ebf33f7cd796b96a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6f67bba633372c0ed9e66b5d0fc499a3e7... | by
refine ⟨fun h z => ?_, fun h n => by simpa only [zpow_natCast] using! h (n : ℤ)⟩
by_cases! hz : 0 ≤ z
· unfold Tendsto
lift z to ℕ using hz
simpa using! h z
· have : Tendsto (fun a => k a ^ z) l (𝓝 0) :=
Tendsto.comp (tendsto_zpow_atTop_zero hz) hk
have h : Tendsto f l (𝓝 0) := by simpa u... | false | {
"chosen_tokens": 110,
"rejected_tokens": 115,
"token_jaccard": 0.931034,
"token_length_ratio": 1.045455
} | markdown_preamble_trailing | preamble | 2c1978589a299ee8a8fda14b5dfaa6fcfd4b87f815e623eafe7ee1da5e3220b9 | 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.Eval.Defs
public import Mathlib.Analysis.Asymptotics.Lemmas
Namespace:
Asymptotics
Local context:
/-
Copyright (c) 2021 Devon Tuma. All ri... | Here is the proof:
by
refine ⟨fun h z => ?_, fun h n => by simpa only [zpow_natCast] using! h (n : ℤ)⟩
by_cases! hz : 0 ≤ z
· unfold Tendsto
lift z to ℕ using hz
simpa using! h z
· have : Tendsto (fun a => k a ^ z) l (𝓝 0) :=
Tendsto.comp (tendsto_zpow_atTop_zero hz) hk
have h : Tendsto f l (... | {
"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/Asymptotics",
"family_id": "superpolynomialdecay_iff_zpow_tendsto_zero",
"file_id": "mathlib/Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean",
"sample_id": "00521cab1e8d5624effd7f9080f88a2a7c2a46ecd1ff08ed8fe772073ff48814"
} | train |
{
"chosen_proof_sha256": "62c5d9390a895014879439c4bd46fd00c9783796de9ef2a06a744f2c74871aea",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d69b8dbc815ff0dff2149b097f513181f2... | by
apply_fun ((Sigma.desc fun a ↦ pullback.fst g₁ (Sigma.ι X a)) ≫ ·) using
(fun a b ↦ (cancel_epi _).mp)
ext a
simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_ι_app]
rw [← Category.assoc, pullback.condition]
simp only [Category.assoc, colimit.ι_desc, Cofan.mk_ι_app]
apply_fun ((Sigma.de... | false | {
"chosen_tokens": 358,
"rejected_tokens": 363,
"token_jaccard": 0.90566,
"token_length_ratio": 1.013966
} | markdown_preamble_trailing | preamble | 2c19b09382c1b789c100ae239eeab201d35a70c3b8be81f5c6e217c8b9ddd7dd | 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.EffectiveEpi.Basic
public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
public import Mathlib.Tactic.ApplyFun
Namespace:
Cat... | Here is the proof:
by
apply_fun ((Sigma.desc fun a ↦ pullback.fst g₁ (Sigma.ι X a)) ≫ ·) using
(fun a b ↦ (cancel_epi _).mp)
ext a
simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_ι_app]
rw [← Category.assoc, pullback.condition]
simp only [Category.assoc, colimit.ι_desc, Cofan.mk_ι_app]
a... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/EffectiveEpi",
"family_id": "effectiveepifamilystructofeffectiveepidesc_aux",
"file_id": "mathlib/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean",
"sample_id": "0f39df25f86b79636ec8efd74a8a03a810910ef6427851948542bfbbe91d6d31"
} | train |
{
"chosen_proof_sha256": "1e710a9d7e1e6f7c130e5c280474bf460f6d3246c3fb6e99473017f60576e601",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8646e1a8a8dc5383d0680a35efcb4d004e... | by cases x; rfl
nonrec def StateT.callCC {σ} [MonadCont m] {α β : Type _}
(f : Label α (StateT σ m) β → StateT σ m α) : StateT σ m α :=
StateT.mk (fun r => callCC fun f' => (f <| StateT.mkLabel f').run r) | false | {
"chosen_tokens": 71,
"rejected_tokens": 76,
"token_jaccard": 0.902439,
"token_length_ratio": 1.070423
} | markdown_preamble_trailing | preamble | 2c267673684fc6654f8ab8891c4792a44bc9f8e00802b54d61113ee8b0c0ef66 | 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.Control.Monad.Basic
public import Mathlib.Control.Monad.Writer
public import Mathlib.Control.Lawful
public import Batteries.Tactic.Congr
public import Batterie... | Here is the proof:
by cases x; rfl
nonrec def StateT.callCC {σ} [MonadCont m] {α β : Type _}
(f : Label α (StateT σ m) β → StateT σ m α) : StateT σ m α :=
StateT.mk (fun r => callCC fun f' => (f <| StateT.mkLabel f').run r) | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Control/Monad",
"family_id": "statet",
"file_id": "mathlib/Mathlib/Control/Monad/Cont.lean",
"sample_id": "8451c1001183081b0718334cf931fa3560f5cf1ca44e66d4b164528ee6a8ec57"
} | train |
{
"chosen_proof_sha256": "fda78651fadb9c3fc00301a4f0a7e0c199693b04a98a2e407d36dcf722f12255",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
let e := Fintype.equivFin ι
obtain ⟨i₀, i₁, ne, h⟩ := h.exists_approx_aux (Fintype.card ι) hε hb fun x y ↦ A x (e.symm y)
refine ⟨i₀, i₁, ne, fun k ↦ ?_⟩
convert! h (e k) <;> simp only [e.symm_apply_apply] | false | {
"chosen_tokens": 78,
"rejected_tokens": 2,
"token_jaccard": 0.022727,
"token_length_ratio": 0.025641
} | sorry_admit_sorryAx_proof_hole | admit | 2c50b277bbfb256310683bc7b0b15cb2b90ce78f3d82f90c5c5c3e5391c889d6 | 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.Real.Basic
public import Mathlib.Combinatorics.Pigeonhole
public import Mathlib.Algebra.Order.AbsoluteValue.Euclidean
Namespace:
AbsoluteValue.IsAdmissib... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/ClassNumber",
"family_id": "exists_approx",
"file_id": "mathlib/Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean",
"sample_id": "fc015e197e7e22d0acabad316890661898d1a5fa61e903e26832dbe043d9fa2c"
} | train |
{
"chosen_proof_sha256": "9d077355c8aafccdf2f765766657b2aef7013299b24845a61f82fd2eda23f5d5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "16e52315f7d70bd4a1b86cbcbc09e475d0... | by
zify
convert! Rat.num_mul_num_eq_num_mul_gcd q₁ q₂ <;> norm_cast | true | {
"chosen_tokens": 15,
"rejected_tokens": 2,
"token_jaccard": 0.066667,
"token_length_ratio": 0.133333
} | invalid_tactic | invalid_tactic | 2c8ecd531757390c20fcd24074ef03d0d120c0b4ca2517dab488f82066625f6c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Field.Rat
public import Mathlib.Algebra.Group.Indicator
public import Mathlib.Algebra.Order.Field.Rat
public import Mathlib.Data.Rat.Lemmas
public impo... | by
__oprover_invalid_tactic_471ab45e244c | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "d1751be3c73d22e7f3a85e4a351cc12fa071e04c5f752395529289df9e5ff240",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/NNRat",
"family_id": "num_mul_num_eq_num_mul_gcd",
"file_id": "mathlib/Mathlib/Data/NNRat/Lemmas.lean",
"sample_id": "471ab45e244c5046c4d6db5db33762cae30b42d81f33c3c388fbdb7ee1366b37"
} | train |
{
"chosen_proof_sha256": "da2d9310157ed9ceff8c58c2de745bed1ed2329391fedb4f8010b3557e4b56ed",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1782058706e57bee7d4d9918ffda568fb0... | by
have h : ∃ n, x ≤ t n := by
simp only [tendsto_atTop_atTop_iff_of_monotone ht_mono.monotone] at ht_tendsto
exact ht_tendsto x
have h' m := Nat.find_min h (m := m)
simp only [not_le] at h'
exact ⟨Nat.find h - 1, h' _ (by simp [hx]), by simp [Nat.find_spec h, hx]⟩ | true | {
"chosen_tokens": 76,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.065789
} | wrong_namespace_qualification | wrong_namespace_qualification | 2c9d193228e2c597c7245b340979c4e46077c99e7b447c9d227b2930b87d88a1 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecificLimits.ArithmeticGeometric
public import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
public import Mathlib.MeasureTheor... | by
exact OProverMissing.missing_9b4e3e0d6a76 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "33208c466f736312792a71e1140ed400042f42855ffd04e3f5648d58acd3b384",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability/Distributions",
"family_id": "strictmono",
"file_id": "mathlib/Mathlib/Probability/Distributions/Fernique.lean",
"sample_id": "9b4e3e0d6a7653b54818bb9ece9503e507bf5b7abbd8d2d4b99515af39aba54b"
} | train |
{
"chosen_proof_sha256": "fd4f7273c1abc3ee605f4507e24cb13680aff1584126fd18e76d89a937f7044b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
classical
exact Decidable.List.Lex.ne_iff H | false | {
"chosen_tokens": 11,
"rejected_tokens": 5,
"token_jaccard": 0.166667,
"token_length_ratio": 0.454545
} | sorry_admit_sorryAx_proof_hole | sorryAx | 2ce04ac8ec1a20af89e935349fe3b4a32d2127f97b4da240c993eda6fcc8455e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.List.Basic
public import Mathlib.Data.Nat.Basic
public import Mathlib.Order.RelClasses
Namespace:
List.Lex
Local context:
/-
Copyright (c) 2018 Mario Ca... | 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/List",
"family_id": "ne_iff",
"file_id": "mathlib/Mathlib/Data/List/Lex.lean",
"sample_id": "50332d0f41236ffc25367bcc287b5a5f97bce4b58e2df5be3cb1c6d4b44bee1a"
} | train |
{
"chosen_proof_sha256": "0ec8ef0b04cc9e360fe54d83ec9b445dee17ea05e7abf3a9b4757fa21a5fb78e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
simp [Irrational, Rat.forall, eq_comm] | false | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.272727
} | by_exact_placeholder | placeholder | 2d5d4073c53729387264afb981a4782a108c48dc9ec4ae0530c3ea2db1ba33f9 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Rat
public import Mathlib.Data.Nat.Prime.Int
public import Mathlib.Data.Rat.Sqrt
public import Mathlib.Analysis.Real.Sqrt
public import Mathlib... | 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": "NumberTheory/Real",
"family_id": "irrational_iff_ne_rational",
"file_id": "mathlib/Mathlib/NumberTheory/Real/Irrational.lean",
"sample_id": "71f1b32ec46df06988407bec9fec4c171851cda268d1134ad7fa0191f2142682"
} | train |
{
"chosen_proof_sha256": "b2349a1019905ca8cc28d0ef2d79317628410a89c7793fafcbce2e7a084a9bdd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | 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 _ _) | false | {
"chosen_tokens": 30,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.066667
} | sorry_admit_sorryAx_proof_hole | admit | 2db597b4238ff7e5c8ee68b5a011a6356f58ebff573573bd0ca7ebbc5c66e79e | 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.Function
public import Mathlib.Order.Interval.Set.LinearOrder
Namespace:
(root)
Local context:
/-
Copyright (c) 2020 Heather Macbeth. All rights res... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/Interval",
"family_id": "surjon_ici_of_monotone_surjective",
"file_id": "mathlib/Mathlib/Order/Interval/Set/SurjOn.lean",
"sample_id": "16580c7b1066ea68f7db2b4cc23ac07bcdd4aa4217971b759537bed8271b6fb6"
} | train |
{
"chosen_proof_sha256": "badc607969163b460b5422c403e504297c9199325e39a0ffb23b23e6ce679da2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [ncoeff_apply, neg_sub, Int.sub_neg, add_sub_cancel_left] | true | {
"chosen_tokens": 13,
"rejected_tokens": 8,
"token_jaccard": 0.058824,
"token_length_ratio": 0.615385
} | unsolved_goals | unsolved_goals | 2e30d261dca200bd42a30493920abddffedf67d94fa7edf2c803f48d58654f1f | 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.Vertex.HVertexOperator
public import Mathlib.Data.Int.Interval
Namespace:
VertexOperator
Local context:
/-
Copyright (c) 2024 Scott Carnahan. All rig... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "392a0fed2d32d7800f087ecfc18a0b4604bd4bbae3bc30b59f83e0a0e224af67",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Vertex",
"family_id": "coeff_eq_ncoeff",
"file_id": "mathlib/Mathlib/Algebra/Vertex/VertexOperator.lean",
"sample_id": "e3c482c18dafac02bf8a0fd6128bd4ffa9ee2fc18b435651f9c2883c55628c8a"
} | train |
{
"chosen_proof_sha256": "c860446eb6ebd47f56bc80449291da30bdf73e713b21c6c028e07a7ce9d17e8f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [Iso.conj_apply, Iso.eq_inv_comp]; simp [f.hom.comm] | false | {
"chosen_tokens": 20,
"rejected_tokens": 3,
"token_jaccard": 0.0625,
"token_length_ratio": 0.15
} | by_exact_placeholder | exact? | 2e7fec6cb7e34286c34ca3fdc95848ade499dfded5ed2034aa00d7c7218752a5 | 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.Grp.Basic
public import Mathlib.Algebra.Ring.PUnit
public import Mathlib.CategoryTheory.Adjunction.Limits
public import Mathlib.CategoryTheory... | 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": "CategoryTheory/Action",
"family_id": "iso",
"file_id": "mathlib/Mathlib/CategoryTheory/Action/Basic.lean",
"sample_id": "624cb99c0cbd1997bbdc0b1f1c47bd9e2e1caf360a81585dc2e8556116e02b79"
} | train |
{
"chosen_proof_sha256": "ffc956ff392a03eea39dc40bd447e1d225bcdcdd1b88caef9ee391f6b7a9878c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
simp [card_of_finrank k V h] | false | {
"chosen_tokens": 8,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.625
} | sorry_admit_sorryAx_proof_hole | sorryAx | 2e8bc6afe474cf2e998dc2d18cf565d3c2b279a2df6330a0e1b96d0d8beee73b | 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
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": "LinearAlgebra/Projectivization",
"family_id": "card_of_finrank_two",
"file_id": "mathlib/Mathlib/LinearAlgebra/Projectivization/Cardinality.lean",
"sample_id": "1bdf66f02f800f9f6bc8f802f4eb7010531ea0fed644e6fb319377ffa1bd244a"
} | train |
{
"chosen_proof_sha256": "5d0edc0cc6cc1668b437d5481146309b64fe078c9ece4c0df2b3120bce3ca534",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
refine TopologicalSpace.ext_nhds fun x ↦ ?_
ext s
rw [mem_nhds_toTopsp, H] | false | {
"chosen_tokens": 18,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.166667
} | by_exact_placeholder | placeholder | 2ea70940f63d237d36781e67c4a60feca5dccf1aef9c3e4568c636040b9cbb3b | 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.Analysis.Filter
public import Mathlib.Topology.Bases
public import Mathlib.Topology.LocallyFinite
Namespace:
Ctop.Realizer
Local context:
/-
Copyright (... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Analysis",
"family_id": "ext",
"file_id": "mathlib/Mathlib/Data/Analysis/Topology.lean",
"sample_id": "80671f5e23cc162f2c4d734bcae6d4b38501c2182a9dd22e1a1e199c28f65d78"
} | train |
{
"chosen_proof_sha256": "9a8be05ccc00ea06c742ae40a8ee2f89b7edc8fc907070b2e2dab340e52a4333",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | 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] | true | {
"chosen_tokens": 51,
"rejected_tokens": 8,
"token_jaccard": 0.032258,
"token_length_ratio": 0.156863
} | unsolved_goals | unsolved_goals | 2ee7c8362d034ee756e03dca70c3cb5f2fc9f825f7cd2a70499dd0e3ff28860b | 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.Tree.Basic
public import Mathlib.Control.Applicative
public import Mathlib.Control.Traversable.Basic
Namespace:
BinaryTree
Local context:
/-
Copyright (... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "99fc4937c1beeeedc14df853db7b4153aba39418f6309bf9998692ea9d3c7cca",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "2b5a7d59d6ab27179c1c1bc56a6c116c83ca6fccc9d560e5875bd86a9aa70480",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
simp [List.mem_iff_get, Fin.exists_iff, eq_comm (b := b)] | false | {
"chosen_tokens": 18,
"rejected_tokens": 2,
"token_jaccard": 0.0625,
"token_length_ratio": 0.111111
} | sorry_admit_sorryAx_proof_hole | admit | 2ef2e1b1e567c7c4fddabd87a0a5934b768ccac7449678de69778525daeec2b3 | 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.Data.List.Defs
Namespace:
List
Local context:
/-
Copyright (c) 2024 Miyahara Kō. All rights reserved.
Released ... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/List",
"family_id": "mem_iterate",
"file_id": "mathlib/Mathlib/Data/List/Iterate.lean",
"sample_id": "32ce66990d1ffba436ebb7dc106eea97303764f97b8b35040bc36958f838afdb"
} | train |
{
"chosen_proof_sha256": "b4230b54afe23a3c3ef25af1bbb191756411be86b06eb1ac6af58052e937afcd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c6c084351715c62d7f25bc79f783809f58... | by
simp [PresheafedSpace.congr_app (F.map_id j)] | false | {
"chosen_tokens": 13,
"rejected_tokens": 17,
"token_jaccard": 0.8,
"token_length_ratio": 1.307692
} | missing_wrong_import | import | 2f45dcf012036fe300d4224395396903214ef8dcc7c1e240e4cbe8144c6a0a95 | 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.RingedSpace.PresheafedSpace
public import Mathlib.Topology.Category.TopCat.Limits.Basic
public import Mathlib.Topology.Sheaves.Limits
public import Ma... | import Mathlib.WrongImport
by
simp [PresheafedSpace.congr_app (F.map_id j)] | {
"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/RingedSpace",
"family_id": "map_id_c_app",
"file_id": "mathlib/Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean",
"sample_id": "d0d0056d6763aa2dc8a9996175ddeb4b3a077afcfff88c7bba8165c2216d8b23"
} | train |
{
"chosen_proof_sha256": "dae1c5b0ed2fe337b0ef71d64dc3ef20222db1f81ac15117533aa09e94ca0f93",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
obtain ⟨u, u_cov, u_props⟩ := sc_st
exact
⟨u,
s_sub.trans u_cov,
fun n ↦
⟨(u_props n).1,
disjoint_of_subset (fun ⦃_⦄ a ↦ a) t_sub (u_props n).2⟩⟩ | false | {
"chosen_tokens": 50,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.06
} | by_exact_placeholder | exact? | 2f7048e42002074dbdc755ed94fe25ad80d123afa4fad662ec0e4d7ada593fa5 | 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.Continuous
public import Mathlib.Topology.NhdsSet
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Releas... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/Separation",
"family_id": "hasseparatingcover",
"file_id": "mathlib/Mathlib/Topology/Separation/SeparatedNhds.lean",
"sample_id": "f8bf5611d48800247cbe0df58105517cb729c0618719c6fcdd68a62dbe1a0152"
} | train |
{
"chosen_proof_sha256": "318c2cb9db795a2dd46d3f5a27d1f76a3d76756692909ad6bddc78fa03561671",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
constructor
rw [← trans_dualMap_symm_flip e]
exact LinearEquiv.bijective _ | false | {
"chosen_tokens": 13,
"rejected_tokens": 3,
"token_jaccard": 0.142857,
"token_length_ratio": 0.230769
} | by_exact_placeholder | exact? | 2fb98968e4727ea518c1bbf91f4fa71862e5e2dddc615ab5f111d4bde8d3d508 | 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.Dual.Lemmas
Namespace:
LinearEquiv
Local context:
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as ... | 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": "LinearAlgebra/PerfectPairing",
"family_id": "isreflexive_of_equiv_dual_of_isreflexive",
"file_id": "mathlib/Mathlib/LinearAlgebra/PerfectPairing/Basic.lean",
"sample_id": "7da1d9db371103784bbc6daf0c80df8acd3ad5e9a823b6012d8ef842a94b24f4"
} | train |
{
"chosen_proof_sha256": "38369068946cc9341941259052323d13d916cefa46cf4a0009f06b6990667aa7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0c0e4150f1cfb1e9439885a29fd7c58e25... | by
rw [basis_apply_powersetCard, exteriorPower.basis_apply, ιMulti_family_apply_coe] | true | {
"chosen_tokens": 12,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.166667
} | invalid_tactic | invalid_tactic | 2fbd283df7a1a90cb9ef6ac980f77033db10ae8177874c32cf5b3a7d53423731 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.ExteriorAlgebra.Grading
public import Mathlib.LinearAlgebra.ExteriorPower.Basis
Namespace:
ExteriorAlgebra
Local context:
/-
Copyright (c) 2026... | by
__oprover_invalid_tactic_dfb013882af3 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "30b6b073d01cbc1db05c58ce24281a7140d17b1d4c820d26467f35fe4d819d36",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/ExteriorAlgebra",
"family_id": "basis_eq_coe_basis",
"file_id": "mathlib/Mathlib/LinearAlgebra/ExteriorAlgebra/Basis.lean",
"sample_id": "dfb013882af3dab10b8a5b0e2b5f55bfe7bbceeec4f48cee972a6e94a5340486"
} | train |
{
"chosen_proof_sha256": "7c1a95bd387fa99509421bd9a6c435e484eea5c2a9222b6585afb56f5e4cfe03",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
apply tendsto_nhdsNE_of_tendsto_atTop _ _ h_neg
convert! h_pos | true | {
"chosen_tokens": 9,
"rejected_tokens": 8,
"token_jaccard": 0.071429,
"token_length_ratio": 0.888889
} | unsolved_goals | unsolved_goals | 300e31c61a6838a2a0812f94718e734eb2606ade1af500f2636490adeca83f74 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.Algebra.Order.Field
public import Mathlib.Topology.Maps.Basic
public import Mathlib.Analysis.Asymptotics.Lemmas
Namespace:
Tactic.ComputeAsymptotics
... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "5269fac6894e732c3c4e11054d038a307612f2e1d93c820690bd80916eba8453",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/ComputeAsymptotics",
"family_id": "tendsto_nhdsne_of_tendsto_attop_nhds_of_eq",
"file_id": "mathlib/Mathlib/Tactic/ComputeAsymptotics/Lemmas.lean",
"sample_id": "fb8d70162809714734e39e1803545fb5adac8b0c13fe3dded4a257dc63c28f16"
} | train |
{
"chosen_proof_sha256": "ba260adf630a9d5dff8237b3bea2f33b1eff285cacd7e89777abb74bff12da31",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3ac2f1d971950cbc9a18df02d16cfd2713... | by
cases x; rfl
nonrec def ExceptT.callCC {ε} [MonadCont m] {α β : Type _}
(f : Label α (ExceptT ε m) β → ExceptT ε m α) : ExceptT ε m α :=
ExceptT.mk (callCC fun x : Label _ m β => ExceptT.run <| f (ExceptT.mkLabel x)) | true | {
"chosen_tokens": 73,
"rejected_tokens": 5,
"token_jaccard": 0.052632,
"token_length_ratio": 0.068493
} | wrong_namespace_qualification | wrong_namespace_qualification | 30180d62520f234cfdf1320f58d1965db02564bfffdb00c5116629fe1647481c | 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.Control.Monad.Basic
public import Mathlib.Control.Monad.Writer
public import Mathlib.Control.Lawful
public import Batteries.Tactic.Congr
public import Batterie... | by
exact OProverMissing.missing_eb0438819632 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "e5260bb4f8305a400825b4f750820f978d378cb8029ca6187dc7f52884515ddc",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Control/Monad",
"family_id": "exceptt",
"file_id": "mathlib/Mathlib/Control/Monad/Cont.lean",
"sample_id": "eb0438819632308cf7faaf08d1c110925f195fcd644c416f40abd06f69c1666f"
} | train |
{
"chosen_proof_sha256": "a24cc2dfc6b516924da62f552de8528961f8853b68b3464a392b74c6594e45ef",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0f0633797d29bc379a07723d24c5fedc5d... | by
ext x
apply (IsIntegralClosure.equiv A (integralClosure A L₃) L₃ B₃).symm.injective
ext
simp [galRestrict', Subalgebra.algebraMap_eq] | false | {
"chosen_tokens": 33,
"rejected_tokens": 38,
"token_jaccard": 0.851852,
"token_length_ratio": 1.151515
} | markdown_preamble_trailing | trailing | 30699fbdeb86fbc729a3410b7942ed38f363a49a822fb7ad86928efaeddc739e | 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
ext x
apply (IsIntegralClosure.equiv A (integralClosure A L₃) L₃ B₃).symm.injective
ext
simp [galRestrict', Subalgebra.algebraMap_eq]
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/IntegralClosure",
"family_id": "galrestrict'_comp",
"file_id": "mathlib/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean",
"sample_id": "2f3292c8af1fa98f0dfb39f23e5916bd9e2de4337e8d92b7857cea3205fe336c"
} | train |
{
"chosen_proof_sha256": "a8bd42ecc90333f6177ba81b0d401c4a0d5b772d591ccefb5c358cae0906bf4e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
let e : (L ⊗[R] S) ≃ₐ[K] L ⊗[K] (K ⊗[R] S) :=
(Algebra.TensorProduct.cancelBaseChange R K K L S).symm
let e2 : L ⊗[K] (K ⊗[R] S) ≃ₐ[K] (K ⊗[R] S) ⊗[K] L := Algebra.TensorProduct.comm ..
have heq : Algebra.TensorProduct.map (Algebra.ofId K L) (AlgHom.id R S) =
(e.symm.toAlgHom.comp e2.symm.toAlgHom).c... | false | {
"chosen_tokens": 226,
"rejected_tokens": 3,
"token_jaccard": 0.038462,
"token_length_ratio": 0.013274
} | by_exact_placeholder | placeholder | 3072adf4bc9854c82e8b39c351a72a959821193e70443d707921a045cdea1bd8 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.PurelyInseparable.Basic
public import Mathlib.RingTheory.Flat.Basic
public import Mathlib.RingTheory.Spectrum.Prime.Topology
Namespace:
(root)
Lo... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Spectrum",
"family_id": "primespectrum",
"file_id": "mathlib/Mathlib/RingTheory/Spectrum/Prime/Homeomorph.lean",
"sample_id": "389d6b6e4066fc2cabec7aefc51cac6e21581a5a3e730cc7243010a6f145a966"
} | train |
{
"chosen_proof_sha256": "2851538f3a7c40f88dfdee204d38577a3a636017ca196f0ed156206cf7eba4b2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
induction hx using span_induction with
| mem x h => obtain ⟨i, rfl⟩ := h; cases j <;> simp [h]
| zero => simp
| add u v _ _ hu hv => simp [hu, hv]
| smul t u _ hu => simp [hu] | false | {
"chosen_tokens": 59,
"rejected_tokens": 2,
"token_jaccard": 0.027778,
"token_length_ratio": 0.033898
} | sorry_admit_sorryAx_proof_hole | sorry | 307f86bc62b7fe9e54b9cf9748a187bc89efe04481ed5a2adff9a58a8305b32f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Lie.Matrix
public import Mathlib.Algebra.Lie.OfAssociative
public import Mathlib.Algebra.Lie.Weights.Basic
public import Mathlib.LinearAlgebra.Eigenspa... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/RootSystem",
"family_id": "apply_sum_inl_eq_zero_of_mem_span_h",
"file_id": "mathlib/Mathlib/LinearAlgebra/RootSystem/GeckConstruction/Basic.lean",
"sample_id": "06f9768559f2fe8088fc3798603424c8e8c34edf39453ccd8c0d887cb067503b"
} | train |
{
"chosen_proof_sha256": "f6142b8f4680c491cc9a84d46312116399269b0b324f299edcee5c0f95b61ddc",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "58002da4ccd4d62f1b8ce3942439ad0ce3... | by
intro x y hxy
have := M.isUnit_algebraMap_end_of_le_basicOpen _ hle
rw [Module.End.isUnit_iff] at this
exact this.injective hxy | true | {
"chosen_tokens": 27,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.074074
} | invalid_tactic | invalid_tactic | 309ea5cc2b5b5afbc0a979b160d9dde441a4ddf2a815fb47ce97993a254db68c | 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.Localization
public import Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent
public import Mathlib.AlgebraicGeometry.AffineSche... | by
__oprover_invalid_tactic_ebebed96f1f5 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "4a0292f291caead584e429a3c452856bf0e9bee08474cfd2ff7b88d99fd75dc3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Modules",
"family_id": "issmulregular_of_le_basicopen",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Modules/Tilde.lean",
"sample_id": "ebebed96f1f59661bd31143f834b43b5b9941049affa74d994da26eb1385a91c"
} | train |
{
"chosen_proof_sha256": "6932312c5f36cec3a14fb1d6fd3d6a1837663ad65e9f1ee82ae3d44d58f2d03c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a5c039edafb94f9c2f4f534603a685dad7... | by
let _ : Nontrivial N := by simp [← Module.supportDim_ne_bot_iff_nontrivial R, dim]
rw [PrimeSpectrum.zeroLocus_eq_singleton]
apply le_antisymm
· intro p hp
by_contra nmem
push _ ∈ _ at nmem
have : p < ⟨maximalIdeal R, IsMaximal.isPrime' (maximalIdeal R)⟩ :=
lt_of_le_of_ne (IsLocalRing.le_ma... | true | {
"chosen_tokens": 135,
"rejected_tokens": 2,
"token_jaccard": 0.015152,
"token_length_ratio": 0.014815
} | invalid_tactic | invalid_tactic | 30e3cb0dfeec61431e5d6d5638ebcd422d7bddeaa842b3c83989a3620e4f922b | 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.KrullDimension.NonZeroDivisors
public import Mathlib.RingTheory.Spectrum.Prime.Module
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Naili... | by
__oprover_invalid_tactic_062acd11f877 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b57e9138f752240bdd5cbb3bcb4852ad3078ba6a706fe4fc7a597da81e17fcd9",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/KrullDimension",
"family_id": "support_of_supportdim_eq_zero",
"file_id": "mathlib/Mathlib/RingTheory/KrullDimension/Module.lean",
"sample_id": "062acd11f87739fb15d069f3d86b913dd7cb13fa854b155bc79d97daea43f761"
} | train |
{
"chosen_proof_sha256": "7492371e89f22fb3632f66da0f33c643e10177499de4f2d7febebab11d42e4f8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c0a7c5068f13964705a2f5697590139cfa... | by
symm
calc
0 = ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ := by
rw [(commute_lmul_lmul_sq (a + b + c)).lie_eq]
_ = _ := by rw [aux0, aux1, aux2, aux3, nsmul_add, nsmul_add] | false | {
"chosen_tokens": 70,
"rejected_tokens": 77,
"token_jaccard": 0.935484,
"token_length_ratio": 1.1
} | markdown_preamble_trailing | fence | 31178cdf59865953c167821ec1d317a24be1a43656405ecf995a397d9de1fc38 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Lie.OfAssociative
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license a... | ```lean
by
symm
calc
0 = ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ := by
rw [(commute_lmul_lmul_sq (a + b + c)).lie_eq]
_ = _ := by rw [aux0, aux1, aux2, aux3, nsmul_add, nsmul_add]
``` | {
"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/Jordan",
"family_id": "two_nsmul_lie_lmul_lmul_add_add_eq_zero",
"file_id": "mathlib/Mathlib/Algebra/Jordan/Basic.lean",
"sample_id": "31295720883c04924c41b3b7a4b8a52765977942c19e8e921c01332c5cadf5db"
} | train |
{
"chosen_proof_sha256": "4824a182947d6de23921d345ff1615bb9443926fe4268ae7e79b135cafd2e987",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rintro X ⟨hX⟩
have := hX.prop_diag_obj
simp only [isCardinalPresentable_iff] at this ⊢
exact isCardinalPresentable_of_isColimit _ hX.isColimit κ hJ | false | {
"chosen_tokens": 27,
"rejected_tokens": 2,
"token_jaccard": 0.04,
"token_length_ratio": 0.074074
} | sorry_admit_sorryAx_proof_hole | sorry | 311a07b10842cb222361fceadf3b61186e88f0fdf43ea5a2b1d6f5d623829169 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Types.Filtered
public import Mathlib.CategoryTheory.Limits.Yoneda
public import Mathlib.CategoryTheory.Presentable.Basic
public import Ma... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Presentable",
"family_id": "isclosedundercolimitsofshape_iscardinalpresentable",
"file_id": "mathlib/Mathlib/CategoryTheory/Presentable/Limits.lean",
"sample_id": "f97b91f107eb7b4f9d321215e0984f13daac29b819bde8fd09eea122778dc081"
} | train |
{
"chosen_proof_sha256": "0b0b9e8a744c1d7e4020f039217d12486b9369dcf62ab6b9ae3cf10e259b4ee9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f90fc72aaab263f72c2cf73ac9538f2e1e... | by
simp [← factorPowSucc_comp_fontaineThetaModPPow n] | true | {
"chosen_tokens": 7,
"rejected_tokens": 3,
"token_jaccard": 0.111111,
"token_length_ratio": 0.428571
} | circular_dependency | circular_dependency | 3144e61f0b57a6d0d1a4eafa252cf17bd4a6b4dece679e96abc46487a19cf89d | 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.AdicCompletion.Functoriality
public import Mathlib.RingTheory.AdicCompletion.RingHom
public import Mathlib.RingTheory.Perfectoid.Untilt
public impor... | by
exact factorPowSucc_fontaineThetaModPPow_eq | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "47390c68f04a5c4852b50a19d0e6f80a4c3c0085c36161e1fef819687422d39f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Perfectoid",
"family_id": "factorpowsucc_fontainethetamodppow_eq",
"file_id": "mathlib/Mathlib/RingTheory/Perfectoid/FontaineTheta.lean",
"sample_id": "aeb71f8e68a66dc07c62978a48769bca8b7ac587e71fccaeb6caa13c22e6a3db"
} | train |
{
"chosen_proof_sha256": "8855b3aa1fcebcd1bfc8f7c993730be08249feb4e40c04eec6356b1306bf4c78",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
cases o₁ <;> rfl | true | {
"chosen_tokens": 8,
"rejected_tokens": 8,
"token_jaccard": 0.071429,
"token_length_ratio": 1
} | unsolved_goals | unsolved_goals | 31df3b675bb684e4f22e4199c419eb2e934c7c06b9bc949812b1d50a52940e3a | 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.Ordering.Basic
public import Mathlib.Order.Defs.Unbundled
Namespace:
Ordering
Local context:
/-
Copyright (c) 2017 Microsoft Corporation. All rights res... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "d97c118a51f4d94cbaa94d739d9fc739c65058be6992d3b1f3a09e87fcba875c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Ordering",
"family_id": "dthen_eq_then",
"file_id": "mathlib/Mathlib/Data/Ordering/Lemmas.lean",
"sample_id": "ced9a8be21b100f92f5cf1865633692067c84047c80974cbc15b46518f499a9c"
} | train |
{
"chosen_proof_sha256": "f5148887f1a7dfc8799d4c5143dcee9d63a0a776aa7b9be04cad5dfcce2d500a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
refine ⟨fun {U} hU ↦ ?_⟩
exact .of_comp (a := f.s₀) (𝒱.f ·) (f.h₀ ·)
(fun _ ↦ Scheme.Hom.continuous _) (fun i ↦ funext <| by simp [← Scheme.Hom.comp_apply])
(fun _ ↦ Scheme.Hom.continuous _) U.2 (hU.isCompactOpenCovered 𝒱) | false | {
"chosen_tokens": 85,
"rejected_tokens": 2,
"token_jaccard": 0.025,
"token_length_ratio": 0.023529
} | sorry_admit_sorryAx_proof_hole | admit | 31efb9aff66a5f087738cd23eea037915a00f12d873ecc4c6a98f9cb124b8488 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.Affine
public import Mathlib.AlgebraicGeometry.Properties
public import Mathlib.AlgebraicGeometry.PullbackCarrier
public import Mat... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Cover",
"family_id": "of_hom",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Cover/QuasiCompact.lean",
"sample_id": "b53835cfef00f711bf8e92a6529c08ab65e2ebdbf08f272d243eb3a0265fab0b"
} | train |
{
"chosen_proof_sha256": "59b2a49c3792a452986527f1fd02d5e769f33e00ec8381ee8f5c72d68ad2ecd4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
simp [le_antisymm_iff, h x y, h y x]
variable [IsDomain S] [Nontrivial R] | false | {
"chosen_tokens": 22,
"rejected_tokens": 2,
"token_jaccard": 0.066667,
"token_length_ratio": 0.090909
} | sorry_admit_sorryAx_proof_hole | admit | 321f6b67827ad4d0c42945f305ba192341f6ba75030b8682c4ecdd08245d4199 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.Normed.Field.WithAbs
public import Mathlib.Analysis.SpecialFunctions.Pow.Real
Namespace:
AbsoluteValue
Local context:
/-
Copyright (c) 2025 Michael ... | 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": "Analysis/AbsoluteValue",
"family_id": "isequiv",
"file_id": "mathlib/Mathlib/Analysis/AbsoluteValue/Equivalence.lean",
"sample_id": "99f1e37e3ec119044c219636cfcc8ffdd76a19fdac55cff796f9c655385230aa"
} | train |
{
"chosen_proof_sha256": "59cc5684ec22bcbe1a47c40b86884f4ebd0253cacb27e5dc09b1539a4441b454",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ca6497bd2a5f61079765826c69fc7379d1... | by
simpa [ChainComplex.of.d] using ((inhomogeneousChains A).d_comp_d (n + 2) (n + 1) n) | true | {
"chosen_tokens": 29,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.103448
} | unknown_identifier | unknown_identifier | 322b38c853613ab119291a5ddb59093328d02f365a45e8dfea6c238874a66e71 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Homology.ConcreteCategory
public import Mathlib.RepresentationTheory.Coinvariants
public import Mathlib.RepresentationTheory.Homological.Resolution
pub... | by
exact __oprover_missing_dbf6ffda00f4 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "8e5eabca450ae3df49b2e77cdc620c2a9abbb398837513dd51c3ec8828389c1f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RepresentationTheory/Homological",
"family_id": "inhomogeneouschains",
"file_id": "mathlib/Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean",
"sample_id": "dbf6ffda00f48d6763a755f6b1baea6be9639bc7a19fe67ae570fed274326119"
} | train |
{
"chosen_proof_sha256": "9c3ded6353915dd072b1066cf2ce119a7132edcb22b959bc51e04f3927cf0477",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
have h1 (y : ℝ) : 𝐞 y = fourier 1 (y : UnitAddCircle) := by
rw [fourierChar_apply, fourier_coe_apply]
push_cast
ring_nf
simpa only [h1, Int.cast_one, ofReal_one, div_one, mul_one] using hasDerivAt_fourier 1 1 x | false | {
"chosen_tokens": 49,
"rejected_tokens": 2,
"token_jaccard": 0.029412,
"token_length_ratio": 0.040816
} | sorry_admit_sorryAx_proof_hole | admit | 3263c8990b33b625e5485e8967a750f3aa5d55dc718bd0214523bdf17345bcd0 | 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.Calculus.FDeriv.Analytic
public import Mathlib.Analysis.Fourier.AddCircle
public import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analy... | 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": "Analysis/Fourier",
"family_id": "hasderivat_fourierchar",
"file_id": "mathlib/Mathlib/Analysis/Fourier/FourierTransformDeriv.lean",
"sample_id": "5adb15e5706aefda52cadd04c16407f0f1f64e18c9972fb156247c44ba666c48"
} | train |
{
"chosen_proof_sha256": "894118c14dc5e14117bef012c677ec8f7a21a4cdd6ea74ed8d6415838166eb42",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0b5bf10ac555cb918a9c33514f288ff7c5... | by
by_cases hE : CompleteSpace E; swap
· simp [integral, hE, hasSum_zero]
rw [integral_tsum (fun i ↦ (hF_int i).1)]
· exact (hF_sum.of_norm_bounded fun i ↦ norm_integral_le_integral_norm _).hasSum
have (i : ι) : ∫⁻ a, ‖F i a‖ₑ ∂μ = ‖∫ a, ‖F i a‖ ∂μ‖ₑ := by
dsimp [enorm]
rw [lintegral_coe_eq_integral _... | true | {
"chosen_tokens": 144,
"rejected_tokens": 5,
"token_jaccard": 0.045455,
"token_length_ratio": 0.034722
} | wrong_namespace_qualification | wrong_namespace_qualification | 32b077fbda3a35ba2c9f49fb35b9c85123357eebdfeba69388e9ae7b391ae6a5 | 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.Constructions.Polish.StronglyMeasurable
public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.Topology.Algebra.IsUni... | by
exact OProverMissing.missing_a4e28351bd9a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "9c93dea653893b9177b06b4187286709785db390a7ecf150278e352d356341cb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Integral",
"family_id": "hassum_integral_of_summable_integral_norm",
"file_id": "mathlib/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean",
"sample_id": "a4e28351bd9a0b41f4d5ba3c08dc4215d63a0d8b4b1907903fe2be3c09e51fcd"
} | train |
{
"chosen_proof_sha256": "a27f59db44cea34ebdac75d20e4acc76ed165c06147e346a2eafa82d1ac01d9a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5b2ad4056a63f71ffd19b6edff0a4e71e8... | by
obtain ⟨β, _, sgn, g, hg, hβ, hf⟩ := exists_signed_sum s f
refine
⟨β ⊕ (Fin (n - ∑ i ∈ s, (f i).natAbs)), inferInstance, Sum.elim sgn 0,
Sum.elim g (Classical.arbitrary (Fin (n - Finset.sum s fun i => Int.natAbs (f i)) → α)),
?_, by simp [hβ, h], fun a ha => by simp [hf _ ha]⟩
rintro (b | b) ... | true | {
"chosen_tokens": 126,
"rejected_tokens": 3,
"token_jaccard": 0.036364,
"token_length_ratio": 0.02381
} | circular_dependency | circular_dependency | 333846360b4e1d90250b7e95fdc5a68aafd1196164bdce2babb307a2306b4945 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GroupWithZero.Units.Lemmas
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import Mathlib.Algebra.Order.Ring.Cast
public import Ma... | by
exact exists_signed_sum | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "bf4d9a66f1b62856ef84e463cf148a2ae0f481a680d5220ce0fa63cce2ed698d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Sign",
"family_id": "exists_signed_sum",
"file_id": "mathlib/Mathlib/Data/Sign/Basic.lean",
"sample_id": "3236ac1bb6f48e9d26b20fb63c3f2eb332af2bf7a5e2f318caa01e95dce53a89"
} | train |
{
"chosen_proof_sha256": "f18be9dd1c59e0ebc0b61eaff66514bafe594968c74bd37a4cddbaa1b0ea3c89",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rintro _ (rfl | rfl | rfl) _ (rfl | rfl | rfl) h <;>
simp only
[cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h ⊢
· exact ((Ring.neg_one_ne_one_of_char_ne_two hR).symm h).elim
· exact ((Ring.neg_one_ne_one_of_char_ne_two hR) h).elim | false | {
"chosen_tokens": 70,
"rejected_tokens": 3,
"token_jaccard": 0.058824,
"token_length_ratio": 0.042857
} | by_exact_placeholder | placeholder | 334182da61cfd887c965db718c91e2d37de833a6e42f5519a9ba4e93686c0b21 | 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.Defs
public import Mathlib.Algebra.Group.Fin.Basic
public import Mathlib.Algebra.Ring.ULift
public import Mathlib.Algebra.Ring.Opposite
public im... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/CharP",
"family_id": "int",
"file_id": "mathlib/Mathlib/Algebra/CharP/Basic.lean",
"sample_id": "197f653d4b36fd6849b8b6ebccef410407e7a70fbd810f7a0a75770cb14b42e2"
} | train |
{
"chosen_proof_sha256": "87c6abbbe5679e2db362b7fc92a4107a665977a26b58a1424f12d41fb84a8bc6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "03a76a6ea9df8634d131435398f0347bcd... | by
unfold logDeriv
simp [field]
ring | true | {
"chosen_tokens": 8,
"rejected_tokens": 2,
"token_jaccard": 0.111111,
"token_length_ratio": 0.25
} | invalid_tactic | invalid_tactic | 33517cf1c18f84492de57e053ec10f966db8785be8441cfe995877851451c62f | 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.Derivation.MapCoeffs
public import Mathlib.FieldTheory.PrimitiveElement
Namespace:
Differential
Local context:
/-
Copyright (c) 2024 Daniel Weber.... | by
__oprover_invalid_tactic_2e2771fc6ad1 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "57eb9d50936b1cdf31f783b24a390476808b1b3f484884b80173a10db9d6f88d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "a73fac1f1d110d78557015f3b6ec0716b6ddeed8c82d3ea7e29e1289e23f50bb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8a0eee3ca3d7371d45983d225651a1f869... | by
rw [preimage_image_mk_eq_iUnion_image, iUnion_subtype, ← image2_mul, ← iUnion_image_right]
simp only [SetLike.mem_coe] | true | {
"chosen_tokens": 20,
"rejected_tokens": 2,
"token_jaccard": 0.0625,
"token_length_ratio": 0.1
} | invalid_tactic | invalid_tactic | 33a2ca26c5e926a0375358bcd717f7c3c324828494536a2a01ed6939665bfa63 | 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.Quotient
public import Mathlib.Algebra.Group.Action.Opposite
public import Mathlib.Algebra.Group.Subgroup.MulOpposite
public import Mathlib.GroupTheory... | by
__oprover_invalid_tactic_8b9a52eec598 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "55b65fefefb991fd9bea405d330a6807b930a6aa429baf916c4d98b474149706",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "4e24363fb7ba779b1d49eb8030dc7b2f8941ee6210796f291a186ebbe93a5464",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
apply complementedLattice_of_sSup_atoms_eq_top
have : ∀ i ∈ s, ∃ t : Set α, f i = sSup t ∧ ∀ a ∈ t, IsAtom a := fun i hi ↦ by
replace h := complementedLattice_iff_isAtomistic.mp (h i hi)
obtain ⟨u, hu, hu'⟩ := eq_sSup_atoms (⊤ : Iic (f i))
refine ⟨(↑) '' u, ?_, ?_⟩
· replace hu : f i = ↑(sSup u) ... | false | {
"chosen_tokens": 244,
"rejected_tokens": 2,
"token_jaccard": 0.012195,
"token_length_ratio": 0.008197
} | sorry_admit_sorryAx_proof_hole | sorry | 33d10c34098b6d9abba0d8d2da91231f5f747dc540405e82e5b6307d724ff69e | 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.CompleteLatticeIntervals
public import Mathlib.Order.CompactlyGenerated.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2024 Oliver Nash. All r... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/CompactlyGenerated",
"family_id": "complementedlattice_of_complementedlattice_iic",
"file_id": "mathlib/Mathlib/Order/CompactlyGenerated/Intervals.lean",
"sample_id": "31d3eaad908c26db9fa8cd7a18a465f80cc33c5d4402214fcc63790030c8fb99"
} | train |
{
"chosen_proof_sha256": "f0fe75d1087a41898b65fcdcdb5116eac360500295f3e4e2a2c616668dd8129d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
nth_rw 2 [← Quiver.Path.reverse_reverse p]
apply congr_comp_reverse | true | {
"chosen_tokens": 14,
"rejected_tokens": 8,
"token_jaccard": 0.052632,
"token_length_ratio": 0.571429
} | unsolved_goals | unsolved_goals | 33d42a290b722fdf6160b5687ab7e5668fd8df731d7a97c74ce50246f0133dcd | 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.Groupoid
public import Mathlib.CategoryTheory.PathCategory.Basic
Namespace:
Quiver.FreeGroupoid
Local context:
/-
Copyright (c) 2022 Rémi Bott... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "955da69216b933be4050ed3544213ab8dfce6fdcaa40ceacb0a2eab48bc8af13",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Groupoid",
"family_id": "congr_reverse_comp",
"file_id": "mathlib/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean",
"sample_id": "5ae786a0f2c5c57019f4c1e4edc7524aa65574003fae2570dc839740f449cb98"
} | train |
{
"chosen_proof_sha256": "5e2bc200513a76ad4fb8bd2871f6348a8c6776f4fb4e2960dcaa904707705fee",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cf6ae4c397cb6879927e93c99b97bb1838... | by
-- Using `have` not `let` since it is only existence of `NormedSpace` structure that we need.
have : NormedSpace ℝ E := .restrictScalars ℝ K E
have : FiniteDimensional ℝ E := FiniteDimensional.trans ℝ K E
infer_instance
variable {E} | true | {
"chosen_tokens": 53,
"rejected_tokens": 3,
"token_jaccard": 0.030303,
"token_length_ratio": 0.056604
} | circular_dependency | circular_dependency | 3404b06a9978715f843fe94f9488cdb567cb9c52972f0346db4807a6460f66aa | 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.Module.FiniteDimension
public import Mathlib.Analysis.RCLike.Basic
public import Mathlib.Topology.Instances.RealVectorSpace
Namespace:
FiniteD... | by
exact proper_rclike | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "7e7636e7a5f2a1767bb5ef73f1ed194bf27ea95bdd052084d26bc4ec894a94ed",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/RCLike",
"family_id": "proper_rclike",
"file_id": "mathlib/Mathlib/Analysis/RCLike/Lemmas.lean",
"sample_id": "6a8e5453b7a4db469efd31b432f7b75acf3e7eec129c2d2a6b546cf031d8a71a"
} | train |
{
"chosen_proof_sha256": "851de4796bc656b1357eef23d637285576683be0e867f23ab28b2999ef31f864",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5348f324874c43ec8d969e75ffd6ef39a0... | by
letI := (Sᵀ.toNormedAddCommGroup hS.transpose)
letI := (Sᵀ.toInnerProductSpace hS.transpose.posSemidef)
ext i j
rw [LDL.lowerInv, Basis.coePiBasisFun.toMatrix_eq_transpose, coe_gramSchmidtBasis]
rfl
noncomputable instance LDL.invertibleLowerInv : Invertible (LDL.lowerInv hS) := by
rw [LDL.lowerInv_eq_gr... | true | {
"chosen_tokens": 119,
"rejected_tokens": 2,
"token_jaccard": 0.022222,
"token_length_ratio": 0.016807
} | invalid_tactic | invalid_tactic | 3428710be41746cf04a0b99b80948cd0a8912b7e114c3e49afac2cf74b8265b5 | 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.GramSchmidtOrtho
public import Mathlib.Analysis.Matrix.PosDef
Namespace:
(root)
Local context:
/-
Copyright (c) 2022 Alexander Ben... | by
__oprover_invalid_tactic_a95312207d02 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "eefca2e10e37cf2ff32fc78d18a0bacc23e8be541e733c2e09a93a61fc0765c4",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Matrix",
"family_id": "ldl",
"file_id": "mathlib/Mathlib/Analysis/Matrix/LDL.lean",
"sample_id": "a95312207d0271d9516315b3dd214e2726915b26938f24e5ee114b92566a3ab3"
} | train |
{
"chosen_proof_sha256": "b2fc7f29216c9249b22ab675eb45a0b56030097a0051e019b9740cd584e92cb3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
simp only [liftFun_iff_succ r, forall_iff_succ, cons_val_succ, cons_val_zero, ← succ_castSucc,
castSucc_zero]
variable [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α} | false | {
"chosen_tokens": 45,
"rejected_tokens": 2,
"token_jaccard": 0.032258,
"token_length_ratio": 0.044444
} | sorry_admit_sorryAx_proof_hole | sorry | 34546ee3d7f2f4b3bc277e66aa167c125e73ba6842052a6263f51a0e9da915e7 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Logic.Equiv.Fin.Basic
public import Mathlib.Order.Fin.Basic
public import Mathlib.Order.PiLex
public import Mathlib.... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/Fin",
"family_id": "liftfun_veccons",
"file_id": "mathlib/Mathlib/Order/Fin/Tuple.lean",
"sample_id": "34e8994f5ec220f3f31f1cf7f6b1b16bc75863bde20089fcb254304fbec368fa"
} | train |
{
"chosen_proof_sha256": "d4687275514802a11968138ca35e457b8f90609ca53ed1eef831fff19e4d31f5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7b2f0763bd6dff91efe83580b9c6d31cae... | by
rw [← Nat.card_prod]; exact Nat.card_congr Subgroup.groupEquivQuotientProdSubgroup | false | {
"chosen_tokens": 16,
"rejected_tokens": 20,
"token_jaccard": 0.8125,
"token_length_ratio": 1.25
} | missing_wrong_import | import | 35035681bfee8134ac926abdc2b2d203fca61f4e5038b938b6a2cf480d2f0d06 | 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 ... | import Mathlib.WrongImport
by
rw [← Nat.card_prod]; exact Nat.card_congr Subgroup.groupEquivQuotientProdSubgroup | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "GroupTheory/Coset",
"family_id": "card_eq_card_quotient_mul_card_subgroup",
"file_id": "mathlib/Mathlib/GroupTheory/Coset/Card.lean",
"sample_id": "fd18d30e8b2b216d023535e089915b87e05c113a9e9d53b5e82b2353ec29f830"
} | train |
{
"chosen_proof_sha256": "2c1ae7a6d731fe18987c9a16e015e76772289e20b04a8b5d9fc3cf3d2c6b5c61",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "048b9be495a268385d167a57cca1fa7813... | by
unfold mkFinSnoc
exact coe_mk (v := Fin.snoc (N.subtype ∘ b) y) _ _
/-- Let `b` be a basis for a submodule `N ≤ O`. If `y ∈ O` is linear independent of `N`
and `y` and `N` together span the whole of `O`, then there is a basis for `O`
whose basis vectors are given by `Fin.snoc b y`. -/
noncomputable def mkFinSno... | true | {
"chosen_tokens": 262,
"rejected_tokens": 5,
"token_jaccard": 0.033333,
"token_length_ratio": 0.019084
} | wrong_namespace_qualification | wrong_namespace_qualification | 354b70acb2054097e8e076bbd89f081d20d6cc427827733badaf7c8b4d73e3bd | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Basis.Basic
public import Mathlib.LinearAlgebra.Pi
Namespace:
Module.Basis
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights rese... | by
exact OProverMissing.missing_6eee5d2ff910 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ba4e06754d8abe64f86af9d038e888dd3053fdf0e4425f235921d49f852880e1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Basis",
"family_id": "coe_mkfinsnoc",
"file_id": "mathlib/Mathlib/LinearAlgebra/Basis/Fin.lean",
"sample_id": "6eee5d2ff91023d609bd493224edd9b87b2fa60a603a305b0c65f8936a71c40f"
} | train |
{
"chosen_proof_sha256": "10363aa97aac5a8e4a6ed13d0ebf67594ed188c89c81a730878f4693d1b889fa",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
ext x
simp only [LinearMap.comp_apply, apply, map_add, map_smul, add_assoc]
simp only [smul_add, ← mul_smul, ← add_smul, ← mul_add (f x), mul_assoc] | false | {
"chosen_tokens": 38,
"rejected_tokens": 2,
"token_jaccard": 0.04,
"token_length_ratio": 0.052632
} | sorry_admit_sorryAx_proof_hole | admit | 354c58e32a655125a2d3160becc057859b10fbad5073f127e89904e27b6c879b | 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.GroupAction.SubMulAction.OfFixingSubgroup
public import Mathlib.LinearAlgebra.Charpoly.BaseChange
public import Mathlib.LinearAlgebra.Dual.BaseChan... | 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/Transvection",
"family_id": "comp_smul_smul",
"file_id": "mathlib/Mathlib/LinearAlgebra/Transvection/Basic.lean",
"sample_id": "19c9230dd9532a6df572a3d4cf4ed8cfd2fc18850e231e933699018c9fe60632"
} | train |
{
"chosen_proof_sha256": "eb4847d4ec18aeaffa781e4a9bdcbca4898991cdee673fe5996e2fd2c98cb1bf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b5c69aaa1bed68b17e4f32a5e7a2a5a68a... | by
rw [Function.Injective] at hv
push Not at hv
rcases hv with ⟨i₁, i₂, heq, hne⟩
exact f.map_eq_zero_of_eq v heq hne
/-!
### Algebraic structure inherited from `MultilinearMap`
`AlternatingMap` carries the same `AddCommMonoid`, `AddCommGroup`, and `Module` structure
as `MultilinearMap`
-/ | true | {
"chosen_tokens": 72,
"rejected_tokens": 2,
"token_jaccard": 0.022222,
"token_length_ratio": 0.027778
} | invalid_tactic | invalid_tactic | 354ef4ba11542a9a99316901bb57688217619e2a7828de2ec5ac511b146c0d26 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.GroupTheory.Perm.Sign
public import Mathlib.LinearAlgebra.LinearIndependent.Defs
public import Mathlib.LinearAlgebra.Multilinear.Basis
Namespace:
AlternatingM... | by
__oprover_invalid_tactic_cb8c4de23317 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "04ff01998f8740cf4c1e0c58f8f437b3dd163e599eed72c322726a88eac06b4a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Alternating",
"family_id": "map_eq_zero_of_not_injective",
"file_id": "mathlib/Mathlib/LinearAlgebra/Alternating/Basic.lean",
"sample_id": "cb8c4de23317146f19f32df2485e1f047d44025176348b35088a93dfd2c0286c"
} | train |
{
"chosen_proof_sha256": "fd406c6900b20879ac4c2cbd6762020da2098ae8a94437f8bd328ff516fb751d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ed1245fdebd56f8a9139f5759dda744d8f... | by
have := Adjunction.rightAdjointUniq_hom_counit (mapPullbackAdj (𝟙 X)) (id X).mapPullbackAdj
rw [pullbackId, Adjunction.rightAdjointUniq_hom_counit] | true | {
"chosen_tokens": 27,
"rejected_tokens": 3,
"token_jaccard": 0.052632,
"token_length_ratio": 0.111111
} | unknown_identifier | unknown_identifier | 3550c406481c2c9774a40659175f7ec68fe597fd70b5c89aa70e32bcefe9f7f0 | 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.Comma.Over.Pullback
public import Mathlib.CategoryTheory.Monoidal.Cartesian.Basic
public import Mathlib.CategoryTheory.Adjunction.Unique
public ... | by
exact __oprover_missing_9b1bbd8b0b0a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "9b11e0022818b96a686a3e3f7c92484bae136688ad924ac338ecf6673e03b745",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/LocallyCartesianClosed",
"family_id": "pullbackid_hom_counit",
"file_id": "mathlib/Mathlib/CategoryTheory/LocallyCartesianClosed/ChosenPullbacksAlong.lean",
"sample_id": "9b1bbd8b0b0a1747cccd2009098a56d3ba588503f823228d885fa9ccd0f30c49"
} | train |
{
"chosen_proof_sha256": "aedbd85f784d006a3b8e0a1c284ce353d4c9554e1c007c5246c151cde67cde07",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3d5cbd94823c1da9fd124543545cda0f81... | by
exact ⟨fun h ↦ ⟨⟨@effectiveEpiFamilyStructOfEffectiveEpiDesc _ _ _ _ X π _ h _ _ (fun g ↦
(FinitaryPreExtensive.isIso_sigmaDesc_fst (fun a ↦ Sigma.ι X a) g inferInstance).epi_of_iso)⟩⟩,
fun _ ↦ inferInstance⟩
variable {D : Type*} [Category* D] [FinitaryPreExtensive D]
variable (F : C ⥤ D) [PreservesFinite... | true | {
"chosen_tokens": 80,
"rejected_tokens": 5,
"token_jaccard": 0.075,
"token_length_ratio": 0.0625
} | wrong_namespace_qualification | wrong_namespace_qualification | 355b1460fee3cbc6766e60cf44b6e229c86c0d18a4919fb0f51dd99e412010b5 | 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.EffectiveEpi.Preserves
public import Mathlib.CategoryTheory.EffectiveEpi.Coproduct
public import Mathlib.CategoryTheory.Extensive
public import ... | by
exact OProverMissing.missing_a3caeca3aca5 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f7987da539a33e873744d2af1896ff01b2868af55242809ccae7e41468d7fdb0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/EffectiveEpi",
"family_id": "effectiveepi_desc_iff_effectiveepifamily",
"file_id": "mathlib/Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean",
"sample_id": "a3caeca3aca5af45210b187259a35456fb858a420f6704c94186f1b14ec00a25"
} | train |
{
"chosen_proof_sha256": "7bfddf9976579c9b33ef1adb03a12367ce2324df54f03888b73e637b017ab565",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [← disjoint_iff]
exact disjoint_sdiff_iff_le hz hx
-- cf. `IsCompl.left_le_iff` and `IsCompl.right_le_iff` | true | {
"chosen_tokens": 25,
"rejected_tokens": 8,
"token_jaccard": 0.041667,
"token_length_ratio": 0.32
} | unsolved_goals | unsolved_goals | 3572f35d3a1f468f7ac6c404447b178fd422425b30981250e1fcc32346c4a46a | 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.BooleanAlgebra.Defs
public import Mathlib.Tactic.GRewrite
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
R... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "4199460063626449aa34c8e36e46a0a6b0d0f3c678ba42e730f3296f9faee3a0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "29e959508ac284606b1ea79d1f9c44fa587eb78cdc7fa0d4ba1b2b14a0490f11",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "13de1f24cd4205e4f85cb45f304f4be680... | by
simp only [Shrink.rec, eqRec_eq_cast, cast_eq_iff_heq]
rw [Equiv.symm_apply_apply] | true | {
"chosen_tokens": 18,
"rejected_tokens": 5,
"token_jaccard": 0.117647,
"token_length_ratio": 0.277778
} | wrong_namespace_qualification | wrong_namespace_qualification | 359ff4cd179e8ed7a2eba7d84f983d6230213a7bd8ecc1b1f10a2cc40c07580f | 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.Logic.Equiv.Defs
public import Mathlib.Tactic.MkIffOfInductiveProp
public import Mathlib.Tactic.PPWithUniv
Namespace:
(root)
Local context:
/-
Copyright (c) ... | by
exact OProverMissing.missing_e5f7ea238c90 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c8583cdd28a0406dbadd3db22a26137d6515acf344009f797c96ca71ce54ab19",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Logic/Small",
"family_id": "shrink",
"file_id": "mathlib/Mathlib/Logic/Small/Defs.lean",
"sample_id": "e5f7ea238c907be9131815ee500db761a2036d4f0c717eb8bb8747df40a1767e"
} | train |
{
"chosen_proof_sha256": "ee1a0ff1d116d0272d387712ee44ccfa46821cca224349b1934d9628b210bb5f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rcases isEmpty_or_nonempty ι
· simp
rcases isEmpty_or_nonempty ι'
· simp
refine ciSup_le fun j ↦ ?_
grw [hv.apply_sum_le]
simp only [map_mul]
have (f : ι → ℝ) : ⨆ i : ↥Finset.univ, f i.val = ⨆ i, f i :=
Function.Surjective.iSup_comp (fun i ↦ ⟨⟨i, Finset.mem_univ i⟩, rfl⟩) f
rw [this fun i ↦ v (... | false | {
"chosen_tokens": 136,
"rejected_tokens": 5,
"token_jaccard": 0.05,
"token_length_ratio": 0.036765
} | sorry_admit_sorryAx_proof_hole | sorryAx | 35a5a247925ca0a7a9efa829477d70646ee300ff31aed3d1c3659abc86188e50 | 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 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/Height",
"family_id": "isnonarchimedean",
"file_id": "mathlib/Mathlib/NumberTheory/Height/MvPolynomial.lean",
"sample_id": "f97002eb6f5349cba0b0cba353ad0fc3306516a27d9c160813ec6a889f48cb83"
} | train |
{
"chosen_proof_sha256": "20d35803748fd7d6978af67d3ca14fe96ffdbfdea01cb8bdd6440cb3a20e70bb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "83f4c0050682df846c30f760830154dbb6... | by
let nop := @neg_one_pow_div_two_of_three_mod_four
rw [quadratic_reciprocity', pow_mul, nop hp, nop hq, neg_one_mul] <;>
rwa [← Prime.mod_two_eq_one_iff_ne_two Fact.out, odd_of_mod_four_eq_three] | false | {
"chosen_tokens": 35,
"rejected_tokens": 39,
"token_jaccard": 0.896552,
"token_length_ratio": 1.114286
} | missing_wrong_import | import | 35b8f9d00ad0947d1f95405b553de862111d1980aa5db086e1eff558b750bd4e | 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.LegendreSymbol.Basic
public import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
Namespace:
legendreSym
Local context:
/-
Copyright... | import Mathlib.WrongImport
by
let nop := @neg_one_pow_div_two_of_three_mod_four
rw [quadratic_reciprocity', pow_mul, nop hp, nop hq, neg_one_mul] <;>
rwa [← Prime.mod_two_eq_one_iff_ne_two Fact.out, odd_of_mod_four_eq_three] | {
"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/LegendreSymbol",
"family_id": "quadratic_reciprocity_three_mod_four",
"file_id": "mathlib/Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean",
"sample_id": "87698f733f1f8800ecece55747b606be522d5d9ae2965cfc2fd1ae226fa9c49c"
} | train |
{
"chosen_proof_sha256": "7f6de06b7c519dc56f72cfe0e561560f9ccb6c57f8383a711dc7fbaa5be562ae",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c6fd5b3bbb6c2272c694e26c6809305221... | by
rw [absNorm_eq' 1 I₀ (by rw [one_smul]; rfl), OneMemClass.coe_one, map_one, abs_one,
Int.cast_one, _root_.div_one] | true | {
"chosen_tokens": 33,
"rejected_tokens": 3,
"token_jaccard": 0.04,
"token_length_ratio": 0.090909
} | unknown_identifier | unknown_identifier | 35d3ef8b373df58615d0d549084fb23dd8d55bd87d4de30b31d920652036aea4 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.FractionalIdeal.Basic
public import Mathlib.RingTheory.Ideal.Norm.AbsNorm
public import Mathlib.RingTheory.Localization.NormTrace
Namespace:
Fracti... | by
exact __oprover_missing_04175e8f6983 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "0869978178e7c3420128e96c6381d97c5637f090367ea97d3c137f1faa732ccd",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/FractionalIdeal",
"family_id": "coeideal_absnorm",
"file_id": "mathlib/Mathlib/RingTheory/FractionalIdeal/Norm.lean",
"sample_id": "04175e8f69832579c88e53e7defc7abb906d1749bebae6932b35a3cc3ac66b47"
} | train |
{
"chosen_proof_sha256": "bc537854a12e226bc8123bdf85ce0d801d68aa77f9f581d84fafad195b8a7d94",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ca1391130a23e3d68ef153258875ebd71c... | by
obtain ⟨n, f, hf, P, rfl⟩ := P.exists_fin_rename
rw [IsNilpotent.map_iff (rename_injective _ hf), MvPolynomial.isNilpotent_iff_of_fintype]
lift f to Fin n ↪ σ using hf
refine ⟨fun H i ↦ ?_, fun H i ↦ by simpa using H (i.embDomain f)⟩
by_cases H : i ∈ Set.range (Finsupp.embDomain f)
· aesop
· rw [coeff_... | true | {
"chosen_tokens": 97,
"rejected_tokens": 2,
"token_jaccard": 0.018519,
"token_length_ratio": 0.020619
} | invalid_tactic | invalid_tactic | 361932539e02e5f0fb82ab0899499dd4514051803753a869c6c3ebcff7834d9c | 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.MvPolynomial.Homogeneous
public import Mathlib.RingTheory.Polynomial.Nilpotent
Namespace:
MvPolynomial
Local context:
/-
Copyright (c) 2025 Andrew... | by
__oprover_invalid_tactic_7f4e05d647d8 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "7513c53ea4da56aaee9f30770e9f196004a07f06d2f45b58ff91eac6762a8b80",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/MvPolynomial",
"family_id": "isnilpotent_iff",
"file_id": "mathlib/Mathlib/Algebra/MvPolynomial/Nilpotent.lean",
"sample_id": "7f4e05d647d8234808511f2d65c328365b205678685e9a78a56123a55f2d6582"
} | train |
{
"chosen_proof_sha256": "cfefd25aa03c8c5b75a26d1fb3bac847d72b8789c719fd94bddb3d949b0e73ab",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "fa88b07de990b63c90a15ac530da9aeac1... | by
refine ⟨fun hn x ↦ lt_of_le_of_lt (norm_coe_le_norm f x) hn, ?_⟩
· obtain (he | he) := isEmpty_or_nonempty α
· simpa
· obtain ⟨x, hx⟩ := exist_norm_eq h
exact fun h ↦ hx ▸ h x | true | {
"chosen_tokens": 48,
"rejected_tokens": 2,
"token_jaccard": 0.033333,
"token_length_ratio": 0.041667
} | invalid_tactic | invalid_tactic | 3633032d32bfd4e2dd33ff66dfccaf834d2efa205c6ae97b7e8f2f7d2788b9c6 | 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.ContinuousMap.Bounded.Normed
Namespace:
(root)
Local context:
/-
Copyright (c) 2024 Yoh Tanimoto. All rights reserved.
Released under Apache 2.0 lic... | by
__oprover_invalid_tactic_3872ad2e2e46 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "f381f1a70338f1234b5e7244fde4200b9bdd28c8a9ee1e3032b21c1552b0fdb0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/ContinuousMap",
"family_id": "norm_lt_iff_of_compactlysupported",
"file_id": "mathlib/Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean",
"sample_id": "3872ad2e2e46d2587f8e296dc30252bf359d8b645c5d7c7851970aa025ddb54e"
} | train |
{
"chosen_proof_sha256": "eba6e30ed08f1c8872a40c988023ca98c14ec7e63590064e2ee0d7b74b74430f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
refine
Computation.eq_of_bisim
(fun c1 c2 =>
∃ (l : List α) (s : WSeq α),
c1 = Computation.corec (fun ⟨n, s⟩ =>
match Seq.destruct s with
| none => Sum.inl n
| some (none, s') => Sum.inr (n, s')
| some (some _, s') => Sum.inr (n + 1, s')) ... | true | {
"chosen_tokens": 263,
"rejected_tokens": 8,
"token_jaccard": 0.028986,
"token_length_ratio": 0.030418
} | unsolved_goals | unsolved_goals | 367aaa573632060435fc2f1fbc6b6b90fc7718af178f2ea2857903c102b08e18 | 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.DList.Basic
public import Mathlib.Data.WSeq.Basic
Namespace:
Stream'.WSeq
Local context:
/-
Copyright (c) 2017 Microsoft Corporation. All rights reser... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "8168c5fc8d4ce8751edbf96033ee0bbc2c6a6cb93799b867bddf32abe0cbab1d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/WSeq",
"family_id": "length_eq_map",
"file_id": "mathlib/Mathlib/Data/WSeq/Defs.lean",
"sample_id": "382d9c391f850ba4635e3d83c4c92384a1fa17b247df5a67f6099368f1582fb9"
} | train |
{
"chosen_proof_sha256": "feace96534351d67a98372e9c910b5669a41f603a705ba51814a94d00a6c6ea8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cb73e3847e70012451e31d1a504621a0a7... | by
intro _ ha _ hb hne
have := hrc ha hb (fun h ↦ hne (hf h))
grind | true | {
"chosen_tokens": 23,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.130435
} | unknown_identifier | unknown_identifier | 36b7e16998f9fa3db57dd194d1fd49b44db17c969d46126a46fddbb6c1b7c888 | 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.Notation
public import Mathlib.Data.Set.Pairwise.Basic
public import Mathlib.Data.SetLike.Basic
public import Mathlib.Order.Directed
public import Mat... | by
exact __oprover_missing_5bb17713869d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d66fbae3cb8bd0f12fa4601d9020527b7f02aa02daf44dc2259329e18605600e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Preorder",
"family_id": "ischain",
"file_id": "mathlib/Mathlib/Order/Preorder/Chain.lean",
"sample_id": "5bb17713869ded8211fecb07cd342899842c304348ff9eddfbbe96f8cb169aa9"
} | train |
{
"chosen_proof_sha256": "f187198a893dd0403a98bdeb7c5bab4c0cada1bb485a8d70e6ce244cdd6c05b3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | 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 | false | {
"chosen_tokens": 30,
"rejected_tokens": 5,
"token_jaccard": 0.04,
"token_length_ratio": 0.166667
} | sorry_admit_sorryAx_proof_hole | sorryAx | 36dda3f7ca76f7f356892aac32d90c779d07bf95015f573ced8a2680847a87f2 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.Normed.Group.Defs
public import Mathlib.MeasureTheory.Measure.Stieltjes
public import Mathlib.MeasureTheory.VectorMeasure.Basic
public import Mathlib.... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "MeasureTheory/VectorMeasure",
"family_id": "vectormeasure_icc",
"file_id": "mathlib/Mathlib/MeasureTheory/VectorMeasure/BoundedVariation.lean",
"sample_id": "10754817056ac7fe2728b7d3c57dd409b9884a07365c423b530430a8a51d4d7e"
} | train |
{
"chosen_proof_sha256": "37b0177ca46b549523222694ca3ff53bf01def907bba860633ebbd21d6d3186d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d7dad36c4dd4b782ac303e0c16fb450acd... | by
ext x
obtain ⟨I, rfl⟩ := ClassGroup.mk0_surjective x
rw [extendedHom_mk0, MonoidHom.one_apply]
exact (ClassGroup.mk0_eq_one_iff (extendedIdeal A B I).2).mpr (by simpa using h I) | true | {
"chosen_tokens": 45,
"rejected_tokens": 5,
"token_jaccard": 0.090909,
"token_length_ratio": 0.111111
} | wrong_namespace_qualification | wrong_namespace_qualification | 376259a361952449d7f02ec7e995816b2f078c596e2245e23effd7db12ad61c0 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.FractionalIdeal.Extended
public import Mathlib.RingTheory.ClassGroup.Basic
Namespace:
ClassGroup
Local context:
/-
Copyright (c) 2026 Riccardo Bra... | by
exact OProverMissing.missing_6e5dde6af19e | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "e996cae37284de7f25034b2334ab08e50ba7d6655f0a2622813c221c50635d65",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/ClassGroup",
"family_id": "extendedhom_eq_one_of_forall_isprincipal",
"file_id": "mathlib/Mathlib/RingTheory/ClassGroup/ExtendedHom.lean",
"sample_id": "6e5dde6af19e66e06a4ae4317b8eb4e8eb4862f32bca8c9663051c5849e4b141"
} | train |
{
"chosen_proof_sha256": "471c71a8aa6e81210c7b7ea449c4bb32d4af4dfb79ecb5a4425fcd21336ff9fe",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "acf989bf4bb4722560bd130b7474c5acc0... | by
simp only [ncoeff, HVertexOperator.coeff, LinearMap.coe_mk, AddHom.coe_mk]
exact HahnSeries.coeff_eq_zero_of_lt_order h | true | {
"chosen_tokens": 23,
"rejected_tokens": 5,
"token_jaccard": 0.157895,
"token_length_ratio": 0.217391
} | wrong_namespace_qualification | wrong_namespace_qualification | 3858d7ef88881d29bc2475ddbae59ecdf6871d9d2c1f75f977b1629cd8491902 | 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.Vertex.HVertexOperator
public import Mathlib.Data.Int.Interval
Namespace:
VertexOperator
Local context:
/-
Copyright (c) 2024 Scott Carnahan. All rig... | by
exact OProverMissing.missing_a0a40b53d145 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "48301007c1b8d53a738af50f821372f8bb24ae9827b904c9e8c0084f2eacbef4",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Vertex",
"family_id": "ncoeff_eq_zero_of_lt_order",
"file_id": "mathlib/Mathlib/Algebra/Vertex/VertexOperator.lean",
"sample_id": "a0a40b53d14535de5a165c9e14107454e6cc6b4afceef9775bdaaf0c31600cd0"
} | train |
{
"chosen_proof_sha256": "9c3779b7663f368283b7a9eee3adc6092b2e62b6c24d85d914985cdd33c9ca16",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
refine ⟨fun H => H.van_kampen' c₀ hc₀, fun H => ?_⟩
constructor
simp_rw [BinaryCofan.isVanKampen_iff] at H ⊢
intro X Y c hc X' Y' c' αX αY f hX hY
obtain ⟨d, hd, hd'⟩ :=
Limits.BinaryCofan.IsColimit.desc' hc (HT.from _ ≫ c₀.inl) (HT.from _ ≫ c₀.inr)
rw [H c' (αX ≫ HT.from _) (αY ≫ HT.from _) (f ≫ d) ... | false | {
"chosen_tokens": 195,
"rejected_tokens": 2,
"token_jaccard": 0.016667,
"token_length_ratio": 0.010256
} | sorry_admit_sorryAx_proof_hole | admit | 3871df2da27da115e888c4cbd6568e5e7e1e0973f029f278eee1d39b110d90af | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Types.Coproducts
public import Mathlib.CategoryTheory.Limits.Types.Products
public import Mathlib.CategoryTheory.Limits.Types.Pullbacks
p... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory",
"family_id": "finitaryextensive_iff_of_isterminal",
"file_id": "mathlib/Mathlib/CategoryTheory/Extensive.lean",
"sample_id": "b7454ed33b7067c1c405e0b58237396fb00e395f6ff19576d9916ffdee3bc4d9"
} | train |
{
"chosen_proof_sha256": "f994609f95dbe1be796aaafe12c37132a7b266348fd2421e6b7d6339c7e0ee00",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "92a52bbb327e83cdf5240725d5efdd23b4... | by
simpa using! (H.toChainHomotopy).homologyMap_eq n | true | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.25
} | circular_dependency | circular_dependency | 38743134bb4a1adde6fff4899380853797dc96253099eb40e9181f50887d9237 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicTopology.SimplicialObject.Homotopy
public import Mathlib.AlgebraicTopology.AlternatingFaceMapComplex
public import Mathlib.Algebra.Homology.Homotopy
... | by
exact map_homology_eq | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "2bfb5d937c1385a985e75f8506d07e9796deb0b660381a52ac172e0f2beb8a92",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/SimplicialObject",
"family_id": "map_homology_eq",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplicialObject/ChainHomotopy.lean",
"sample_id": "184e16fbe9e5eec732502dd0a1ed9c46036bcaecb5ded625d98d24b0ac54477f"
} | train |
{
"chosen_proof_sha256": "19091c655ecaa51563cbf7438e93b5ad161e207efc740ddfb20c17af665fcd74",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "028d6d8509b98611e3f56cc2c5f547656c... | 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... | false | {
"chosen_tokens": 215,
"rejected_tokens": 220,
"token_jaccard": 0.938462,
"token_length_ratio": 1.023256
} | markdown_preamble_trailing | trailing | 387f03c2b097f6e2d3e2ea34e628249cba148a80923c64e0a21ac9ad1e4a1a0b | 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... | 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... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | 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": "e0b9bed3a6acbad85454a981928799bf6c6adb0fb6d0013326ad4cea47a0b244",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "af7418511b50f18257ec80fc628fa03b92... | by
simp only [r_eq_r' S, r', Con.rel_mk] | false | {
"chosen_tokens": 13,
"rejected_tokens": 18,
"token_jaccard": 0.705882,
"token_length_ratio": 1.384615
} | markdown_preamble_trailing | preamble | 38825fc80997bced68503ed46f379384a0dff70237207ddf81f3043faa691812 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
public import Mathlib.Algebra.Group.Submonoid.Operations
public import Mathlib.Algebra.Regular.Basic
public import Math... | Here is the proof:
by
simp only [r_eq_r' S, r', Con.rel_mk] | {
"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/MonoidLocalization",
"family_id": "r_iff_exists",
"file_id": "mathlib/Mathlib/GroupTheory/MonoidLocalization/Basic.lean",
"sample_id": "194f0f4367719ea1e6824ec86d27af7e10162a98d826fa13df412b7a41e3d577"
} | train |
{
"chosen_proof_sha256": "dd729798355787e2f50246686d3074ecb0b708500ef8ae38cb56d8d7dbde9c7f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "613d32e5c711b87b51ff7b2dba0bc6c8e1... | by
classical
induction s using Finset.induction with
| empty => simp
| insert _ _ _ _ => simp_all [lcm_eq_nat_lcm, Nat.factorization_lcm] | true | {
"chosen_tokens": 28,
"rejected_tokens": 2,
"token_jaccard": 0.045455,
"token_length_ratio": 0.071429
} | invalid_tactic | invalid_tactic | 388bdb13cb4f1eaf25410a3dc865794ac0eeaf68456dbdaf1f6e3e6060f6df69 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GCDMonoid.Finset
public import Mathlib.Algebra.GCDMonoid.Nat
public import Mathlib.Data.Nat.GCD.Basic
public import Mathlib.RingTheory.Coprime.Lemmas
p... | by
__oprover_invalid_tactic_724a8437e52e | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "d92808b2d035bf16b2922a4c326d86cd275cb144aaa64c2aab393b1613255977",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/GCDMonoid",
"family_id": "factorization_lcm",
"file_id": "mathlib/Mathlib/Algebra/GCDMonoid/FinsetLemmas.lean",
"sample_id": "724a8437e52e6581c6973db3a8c052c428831eeb06b3711d37815c86ff48be5a"
} | train |
{
"chosen_proof_sha256": "e1fced99958acc84dd8fa2ad1ee6184c3708ee54a7e8d9105ea3938d1bcfadce",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
classical
refine ⟨fun h₂ ↦ by simp [isBigO_of_le' (c := 0), h₂], ?_⟩
contrapose
intro h₁
obtain ⟨e, he⟩ := exists_single_le_pos (lt_of_le_of_ne h (h₁ ·.symm))
rw [isBigO_iff'']
push Not
intro a ha
simp only [Pi.one_apply, norm_eq_abs, frequently_atTop, abs_one]
intro b
obtain ⟨c, hc⟩ := eventua... | false | {
"chosen_tokens": 264,
"rejected_tokens": 3,
"token_jaccard": 0.02381,
"token_length_ratio": 0.011364
} | by_exact_placeholder | exact? | 389a5dfeb9a6b6a183afa9e93ee57bbe820fdd85a3dfc495faed3ab952c2f26e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic
Namespace:
Function.locallyFinsuppWithin
Local context:
/-
Copyright (c) 2026 Stefan Kebekus. All rights... | 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": "Analysis/Complex",
"family_id": "zero_iff_logcounting_bounded",
"file_id": "mathlib/Mathlib/Analysis/Complex/ValueDistribution/LogCounting/Asymptotic.lean",
"sample_id": "8ccb2339aba41c643ce9ab6237c697b1faecc53decbb2a2025ef71895c9c0e9f"
} | train |
{
"chosen_proof_sha256": "dab19792c72b6e7680af848aa1b358000086832bd7758b76eb7e5a5172d70aec",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "486f6d77bda3f3ad1fcdbdbc2929bf3ae5... | by
grw [Sum.dist, ← le_add_of_nonneg_right dist_nonneg, ← le_add_of_nonneg_left dist_nonneg] | true | {
"chosen_tokens": 15,
"rejected_tokens": 5,
"token_jaccard": 0.133333,
"token_length_ratio": 0.333333
} | wrong_namespace_qualification | wrong_namespace_qualification | 38abc8c992b37abff93e816b2dffb12764189361fa90f853fa18bdb1118100ac | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.ConditionallyCompleteLattice.Group
public import Mathlib.Topology.MetricSpace.Isometry
Namespace:
Metric
Local context:
/-
Copyright (c) 2019 Sébastien... | by
exact OProverMissing.missing_21c2606d6758 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "b15f3432909b95de75e4231a4ad93e083d4d8ae9d515f5f3f73c93226805f347",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/MetricSpace",
"family_id": "sum",
"file_id": "mathlib/Mathlib/Topology/MetricSpace/Gluing.lean",
"sample_id": "21c2606d6758e396ea3dedfc301c790b246ae8c817f34e167e84ed4e781597b5"
} | train |
{
"chosen_proof_sha256": "1c8bfaba84928466d869f23a6dc4de48708f2d6673fcc79573e01b4de03f6a29",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4f39866c14e5ed6348379b32458fb4fbb1... | by
rw [normal_iff] at h ⊢
intro x; specialize h (f.symm x)
rw [← f.apply_symm_apply x, minpoly.algEquiv_eq, ← f.toAlgHom.comp_algebraMap, ← map_map]
exact ⟨h.1.map f, h.2.map _⟩ | false | {
"chosen_tokens": 57,
"rejected_tokens": 61,
"token_jaccard": 0.914286,
"token_length_ratio": 1.070175
} | missing_wrong_import | import | 38d18d4dbc52298f7b0484bf3499a8e4168b74b0e74aac0fae48edc87eea15ac | 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.Splits
public import Mathlib.FieldTheory.Galois.Notation
public import Mathlib.FieldTheory.IntermediateField.Basic
public import Mathlib.Fie... | import Mathlib.WrongImport
by
rw [normal_iff] at h ⊢
intro x; specialize h (f.symm x)
rw [← f.apply_symm_apply x, minpoly.algEquiv_eq, ← f.toAlgHom.comp_algebraMap, ← map_map]
exact ⟨h.1.map f, h.2.map _⟩ | {
"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": "FieldTheory/Normal",
"family_id": "normal",
"file_id": "mathlib/Mathlib/FieldTheory/Normal/Defs.lean",
"sample_id": "30e2dcca73b76516afafa57bf58fa1f528e0a923278fa6441a9972d75d07dd33"
} | train |
{
"chosen_proof_sha256": "5a49665e0c5d2151828a184678146c90b5369415677dd342dff1e4392eda48b8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "fdd6631a5a7abaea2efb2699b9a4116dd2... | by
simp [*] | true | {
"chosen_tokens": 5,
"rejected_tokens": 3,
"token_jaccard": 0.142857,
"token_length_ratio": 0.6
} | circular_dependency | circular_dependency | 38e72cf81d72d0b042e17ff6e770311e3ad2a4bb67bff51ae2c9b5c3344b4acc | 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.Ineq
public import Mathlib.Data.Nat.Cast.Order.Ring
Namespace:
Mathlib.Tactic.Linarith
Local context:
/-
Copyright (c) 2020 Robert Y. Lewis. All rights ... | by
exact eq_of_eq_of_eq | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "d15d5276ebe1989abbe921486f8971b3b92fbe64adcb62db6da4da2163362988",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/Linarith",
"family_id": "eq_of_eq_of_eq",
"file_id": "mathlib/Mathlib/Tactic/Linarith/Lemmas.lean",
"sample_id": "83ec59d8b8358fb8a455e8f01681be42384d6a03b957fc77c05e2d58d2ad540d"
} | train |
{
"chosen_proof_sha256": "8bb9f6292b5b7cdafeb1af9846ff9f4bfcbbf84b87fb669f9431c4e7c51bed33",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0532fa68927b4e9d297bcdb5210b2b8108... | by
simp only [iSup, hf.map_sSup', ← range_comp]
rfl | false | {
"chosen_tokens": 14,
"rejected_tokens": 19,
"token_jaccard": 0.722222,
"token_length_ratio": 1.357143
} | markdown_preamble_trailing | preamble | 38f7e4af5d4868840f3c0d810a6d8539234efb15d720ce203c92f010d9e827cb | 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.ConditionallyCompleteLattice.Basic
public import Mathlib.Order.RelIso.Basic
Namespace:
LeftOrdContinuous
Local context:
/-
Copyright (c) 2020 Yury Kudr... | Here is the proof:
by
simp only [iSup, hf.map_sSup', ← range_comp]
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": "Order",
"family_id": "map_isup",
"file_id": "mathlib/Mathlib/Order/OrdContinuous.lean",
"sample_id": "8b95081cd7f651acb1a47e7a53ca218a713ae886e12ac7935b03bee7470a950f"
} | train |
{
"chosen_proof_sha256": "e934aed5577882b0e1846e82a16eff1d5d3efe027757ba1c1868dea83273eb47",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2a1c4d5761edc1d34d895e3b69bd6ea2c7... | by
obtain ⟨T, hT₁, hT₂⟩ := has_injective_coseparator (op G) ((isSeparator_op_iff _).2 hG)
exact ⟨unop T, inferInstance, (isSeparator_unop_iff _).2 hT₂⟩ | true | {
"chosen_tokens": 42,
"rejected_tokens": 3,
"token_jaccard": 0.08,
"token_length_ratio": 0.071429
} | circular_dependency | circular_dependency | 397a2eb4305972dbfa1c04e564ffcecf32d4b57ad5d2d4665ea2e74e6789cc48 | 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.Abelian.Subobject
public import Mathlib.CategoryTheory.Limits.EssentiallySmall
public import Mathlib.CategoryTheory.Preadditive.Injective.Basic
... | by
exact has_projective_separator | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "f54969bc658347d968513a8f995eae649f443a7b61aa863a63cef6e5c8684dd2",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Generator",
"family_id": "has_projective_separator",
"file_id": "mathlib/Mathlib/CategoryTheory/Generator/Abelian.lean",
"sample_id": "a0442559c44ff5e1900294674d91b30edd8758751f47d7b1327b13d6af184bd6"
} | train |
{
"chosen_proof_sha256": "c198f493b5c0a079bb4ea7d468b89dbbd0202be048c196e631696690e8762509",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
obtain rfl | hμ := eq_zero_or_isProbabilityMeasure μ
· simp
simpa [Pi.mul_def, integral_of_ae_eq_zero_or_one, integral_one_sub_of_ae_eq_zero_or_one, mul_comm,
*] using condVar_of_ae_eq_zero_or_one bot_le hXmeas hX | true | {
"chosen_tokens": 30,
"rejected_tokens": 8,
"token_jaccard": 0.064516,
"token_length_ratio": 0.266667
} | unsolved_goals | unsolved_goals | 398bc786a96037f71cecf272ce5b090bfcbf3eff13b1ea92d677b68cfe449367 | 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.Integral.Bochner.Basic
public import Mathlib.Probability.CondVar
import Mathlib.Probability.Notation
Namespace:
ProbabilityTheory
Local context... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "2639225f5522d206f6e0dbe67fcf8efda531c80cbb5887e690e8a3255d373117",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability/Distributions",
"family_id": "variance_of_ae_eq_zero_or_one",
"file_id": "mathlib/Mathlib/Probability/Distributions/TwoValued.lean",
"sample_id": "f83dd1b97c9a129dc210da5c52474019b307010c9a4d0c19de7a0ae4adc2ce13"
} | train |
{
"chosen_proof_sha256": "3bdeb5d69c9d68024e63a03b42978b4ed5db36efc7c46e5dd1c026762b0bff70",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
ext
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· obtain @⟨n, i, h0, hn⟩ := h
simp only [iSup_iff, ofHoms_iff, Subtype.exists, exists_prop]
use n, i
· simp only [iSup_iff, ofHoms_iff] at h
obtain ⟨n, ⟨i, h0, hn⟩, _, _⟩ := h
exact horn_ι_mem_innerHornInclusions h0 hn | false | {
"chosen_tokens": 81,
"rejected_tokens": 2,
"token_jaccard": 0.027778,
"token_length_ratio": 0.024691
} | sorry_admit_sorryAx_proof_hole | sorry | 39bb1153b9c3ca38d62c94322fa2f5ad691a7dd49dd29b13a83a2be7090f6e3a | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicTopology.Quasicategory.Basic
Namespace:
SSet
Local context:
/-
Copyright (c) 2026 Jack McKoen. All rights reserved.
Released under Apache 2.0 licens... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/Quasicategory",
"family_id": "innerhorninclusions_eq_isup",
"file_id": "mathlib/Mathlib/AlgebraicTopology/Quasicategory/InnerFibration.lean",
"sample_id": "0499f9ddadeadf4353cd9529cac7c5ef8cb81587f40f09d0d81cd202b497e41b"
} | train |
{
"chosen_proof_sha256": "68b7446878a3dfdb301fca861ee4b86aba230785df09d24821618a061945374b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0cf725ee07a3eb842b12fe329854e9d4ad... | by
classical
let ⟨f⟩ := quotient_prod_linearEquiv p
exact rank_prod'.symm.trans f.rank_eq | true | {
"chosen_tokens": 18,
"rejected_tokens": 5,
"token_jaccard": 0.176471,
"token_length_ratio": 0.277778
} | wrong_namespace_qualification | wrong_namespace_qualification | 39d221fe035b23dabb00f03e2add60e036cd183cc5a454fd3597b92d4a0b90e7 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Basis.VectorSpace
public import Mathlib.LinearAlgebra.Dimension.Finite
public import Mathlib.LinearAlgebra.Dimension.RankNullity
Namespace:
(roo... | by
exact OProverMissing.missing_a8f96640d4ba | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "9cacb6448c156cbdc75eca2929b9bdd3b402b57e83aa832a50fcce2c960ba969",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Dimension",
"family_id": "rank_quotient_add_rank_of_divisionring",
"file_id": "mathlib/Mathlib/LinearAlgebra/Dimension/DivisionRing.lean",
"sample_id": "a8f96640d4ba327fc7e9e6bd4ae65eab83d12cf3037d5abef2ad29e199cb314c"
} | train |
{
"chosen_proof_sha256": "7492371e89f22fb3632f66da0f33c643e10177499de4f2d7febebab11d42e4f8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4fa103ab04285b9702895808bf833a23ff... | by
symm
calc
0 = ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ := by
rw [(commute_lmul_lmul_sq (a + b + c)).lie_eq]
_ = _ := by rw [aux0, aux1, aux2, aux3, nsmul_add, nsmul_add] | true | {
"chosen_tokens": 70,
"rejected_tokens": 2,
"token_jaccard": 0.033333,
"token_length_ratio": 0.028571
} | invalid_tactic | invalid_tactic | 39d6e22a1b8531dc38257f0d792244995aea973f34fa11533a96fa01b0f74248 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Lie.OfAssociative
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license a... | by
__oprover_invalid_tactic_31295720883c | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "c81f4ca586ac14043b7a60e123d96d0230ead2144925f26f43d82eaa5b335743",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Jordan",
"family_id": "two_nsmul_lie_lmul_lmul_add_add_eq_zero",
"file_id": "mathlib/Mathlib/Algebra/Jordan/Basic.lean",
"sample_id": "31295720883c04924c41b3b7a4b8a52765977942c19e8e921c01332c5cadf5db"
} | train |
{
"chosen_proof_sha256": "b8dd807b486dcb43a0cfd46155c68422d627f5cb03d289ff5154b4c959da2515",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
simpa [← coe_subset_coe] using! Set.not_subset | false | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.181818
} | sorry_admit_sorryAx_proof_hole | admit | 3a06c731d9c2783a1be222ebacbcd02a3d1c160019702b6a851e9760fc69f86e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Tactic.Monotonicity.Attr
public import Mathlib.Tactic.SetLike
public import Mathlib.Data.Set.Basic
Namespace:
SetLike
Local context:
/-
Copyright (c) 2021 Er... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/SetLike",
"family_id": "not_le_iff_exists",
"file_id": "mathlib/Mathlib/Data/SetLike/Basic.lean",
"sample_id": "70091202abe0a1d76505ded2eda1922a6e55f006910256559adb12050fa4d2a2"
} | train |
{
"chosen_proof_sha256": "73ba32450f7eaa32ac31b9569cb16af401e9b0fc7b4daddcdd761838e6b70efb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
letI : TopologicalSpace Y := ⊥; haveI := discreteTopology_bot Y
exact (isCompact_range hf.continuous).finite_of_discrete | false | {
"chosen_tokens": 21,
"rejected_tokens": 2,
"token_jaccard": 0.052632,
"token_length_ratio": 0.095238
} | by_exact_placeholder | by? | 3a07fbb92b7fcdf273d60be8f269cfe6b472d207b4f840aaa78c2528d2cb7bb9 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Notation.Indicator
public import Mathlib.Topology.Connected.LocallyConnected
public import Mathlib.Topology.Sets.Closeds
Namespace:
IsLocallyConstant
... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/LocallyConstant",
"family_id": "range_finite",
"file_id": "mathlib/Mathlib/Topology/LocallyConstant/Basic.lean",
"sample_id": "8138a062ed7aa5e0b5f610f4aeff3a70100ca75f6fdd0cf8c052c2cf58487e47"
} | train |
{
"chosen_proof_sha256": "fc005df75a037e9e437ea57ce191d036c65586027b4016497e6ed6b62c40e9f3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "05a262d7393581017bce712b907edb4384... | by
dsimp only [LiftP, LiftP']
apply exists_iff_exists_of_mono F (f _ n α) (g _ n α)
· ext i ⟨x, _⟩
cases i <;> rfl
· intros
rw [MvFunctor.map_map]
dsimp +unfoldPartialApp [(· ⊚ ·)]
suffices (fun i => Subtype.val) = (fun i x => (MvFunctor.f P n α i x).val) by rw [this]
ext i ⟨x, _⟩
cases ... | false | {
"chosen_tokens": 111,
"rejected_tokens": 116,
"token_jaccard": 0.927273,
"token_length_ratio": 1.045045
} | markdown_preamble_trailing | trailing | 3a4a4a438ba5b6ac6006e54e5014ce6261a493e9e2fb0bb07c6668ba380ae03a | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fin.Fin2
public import Mathlib.Data.TypeVec
public import Mathlib.Logic.Equiv.Defs
Namespace:
MvFunctor
Local context:
/-
Copyright (c) 2018 Jeremy Avig... | by
dsimp only [LiftP, LiftP']
apply exists_iff_exists_of_mono F (f _ n α) (g _ n α)
· ext i ⟨x, _⟩
cases i <;> rfl
· intros
rw [MvFunctor.map_map]
dsimp +unfoldPartialApp [(· ⊚ ·)]
suffices (fun i => Subtype.val) = (fun i x => (MvFunctor.f P n α i x).val) by rw [this]
ext i ⟨x, _⟩
cases ... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Control/Functor",
"family_id": "liftp_predlast_iff",
"file_id": "mathlib/Mathlib/Control/Functor/Multivariate.lean",
"sample_id": "9f7fcfe93c14fb87931e79553027be989fd1cf1216f6e4592f76a5cfd75311b1"
} | train |
{
"chosen_proof_sha256": "8dd1af80a36345fc761ec289040691b52374d545c9e7b00957a6cfdf28435160",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1a652632662ffd2c69af3248efeb0f3501... | by
simp only [intDegree, add_sub, sub_add, sub_sub_eq_add_sub, sub_sub, sub_eq_sub_iff_add_eq_add]
norm_cast
rw [← Polynomial.natDegree_mul x.denom_ne_zero y.denom_ne_zero, ←
Polynomial.natDegree_mul (RatFunc.num_ne_zero (mul_ne_zero hx hy))
(mul_ne_zero x.denom_ne_zero y.denom_ne_zero),
← Polynomia... | true | {
"chosen_tokens": 102,
"rejected_tokens": 2,
"token_jaccard": 0.032258,
"token_length_ratio": 0.019608
} | invalid_tactic | invalid_tactic | 3a5a101eec821896472f0b33c586b7bf9380083e1b65d17d00d49c592d32e2a5 | 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.RatFunc.AsPolynomial
public import Mathlib.RingTheory.EuclideanDomain
public import Mathlib.RingTheory.Localization.FractionRing
public import Math... | by
__oprover_invalid_tactic_48750c221d7b | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "3b576b6b84b139a92e301c39735c8b03b468a4037c2b3acd023ba7b45d98b88c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/RatFunc",
"family_id": "intdegree_mul",
"file_id": "mathlib/Mathlib/FieldTheory/RatFunc/Degree.lean",
"sample_id": "48750c221d7b3b3a511e093001b0678d1d19b0dff2bd9c9bd5031532b6945ec8"
} | train |
{
"chosen_proof_sha256": "e849d95b5db02f9c868ba8b0e417961eeea08fc774f074fe05a938dadf630d6e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d15e4ade9e9b79b5272408bd0853223ef4... | by
by_contra!
exact hA (hP.subobject_eq_top this) | true | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.166667,
"token_length_ratio": 0.272727
} | unknown_identifier | unknown_identifier | 3af692074f3d66c71d252186b387fa6e55bdf72b605049d20700002c9de61aff | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.ExtremalEpi
public import Mathlib.CategoryTheory.Generator.Basic
public import Mathlib.CategoryTheory.Limits.Presentation
Namespace:
CategoryTh... | by
exact __oprover_missing_bf6d1403eb44 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "12fd2466f71c4f41864681a45b9aed05fb29563b40aa89f30ef2163c0b7d6374",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Generator",
"family_id": "exists_of_subobject_ne_top",
"file_id": "mathlib/Mathlib/CategoryTheory/Generator/StrongGenerator.lean",
"sample_id": "bf6d1403eb4460ab7069e5f743251c32cc0683a5c043812b0396ebf8e760f9d6"
} | train |
{
"chosen_proof_sha256": "c785440b2e793b1a2afd030ccba707286c555ec39b3feca32e7d614cb913da7e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
let ⟨z, hz⟩ := (F.map (algebraMap R ℤ_[p])).evalSubFactor x y
simp only [Polynomial.eval_map_algebraMap] at hz
calc
‖F.aeval x - F.aeval y‖ = ‖z‖ * ‖x - y‖ := by simp [hz]
_ ≤ 1 * ‖x - y‖ := by gcongr; apply PadicInt.norm_le_one
_ = ‖x - y‖ := by simp | false | {
"chosen_tokens": 90,
"rejected_tokens": 2,
"token_jaccard": 0.02381,
"token_length_ratio": 0.022222
} | sorry_admit_sorryAx_proof_hole | admit | 3b38342a69e6bae52493abf42ed745b63648ef60756ce74e01bb424663256091 | 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.Identities
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.NumberTheory.Padics.PadicIntegers
public import Mathlib... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/Padics",
"family_id": "padic_polynomial_dist",
"file_id": "mathlib/Mathlib/NumberTheory/Padics/Hensel.lean",
"sample_id": "7734458b70658c76e232f731e838552dc282d9587d72ef9af109c7d77dc4c28f"
} | train |
{
"chosen_proof_sha256": "1ce00e07b29f18d90867caf068379ad134d68969aa30ca0e3ea5aac4b2537419",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0ec603376cff234c3462a3aa6c0a2d3215... | by
rw [pullbackShiftFunctorZero_inv_app]
simp only [Functor.id_obj, pullbackShiftIso, eqToIso.inv, eqToHom_app, shiftFunctorZero',
Iso.trans_inv, NatTrans.comp_app, eqToIso_refl, Iso.refl_inv, NatTrans.id_app, assoc]
erw [comp_id] | true | {
"chosen_tokens": 46,
"rejected_tokens": 3,
"token_jaccard": 0.035714,
"token_length_ratio": 0.065217
} | circular_dependency | circular_dependency | 3b4d48befed62bf0f1226229b22f74380c118d044223f7ce9d875ed9d154902e | 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.Shift.Adjunction
public import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
Namespace:
CategoryTheory
Local context:
/-
Copyright (c) 20... | by
exact pullbackShiftFunctorZero'_inv_app | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "e62659ff4a83b8ae5b72ed688538e06a93578e7bade6a16c0ff9a87a6897ba1d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Shift",
"family_id": "pullbackshiftfunctorzero'_inv_app",
"file_id": "mathlib/Mathlib/CategoryTheory/Shift/Pullback.lean",
"sample_id": "9d858434b5715b6466fd7efd13eba53c42a1ae0530865db41d7c8c0a8af13979"
} | train |
{
"chosen_proof_sha256": "d409d608f76c2016efc5fdd42930cab75c91e811c1cdb05b5a33c8b551b1384f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "326f40f774f4b7c77d4400168187e23827... | by
intro i
have h_eq_iUnion : {ω | τ ω ≤ i} = ⋃ k ≤ i, {ω | τ ω = k} := by
ext ω
simp only [Set.mem_setOf_eq, Set.mem_iUnion, exists_prop]
cases τ ω with
| top => simp
| coe a => norm_cast; simp
rw [h_eq_iUnion]
refine MeasurableSet.biUnion (Set.to_countable _) fun k hk => ?_
exact f.mono ... | true | {
"chosen_tokens": 91,
"rejected_tokens": 3,
"token_jaccard": 0.039216,
"token_length_ratio": 0.032967
} | circular_dependency | circular_dependency | 3b68759fae54cb38b8d8ee08eb72bfc3f98facf04d55052ddbe64f3d9c882365 | 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 isStoppingTime_of_measurableSet_eq | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "144990c5d793be30b9b90f94ecb488925dcf459e714ad2816a4dd8892b3f5831",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability/Process",
"family_id": "isstoppingtime_of_measurableset_eq",
"file_id": "mathlib/Mathlib/Probability/Process/Stopping.lean",
"sample_id": "63d4c6f7bdfaad78a31356f5390bcb2b6c5c4366f0f8ad12785fcd4685b1c7e7"
} | train |
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