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": "30015a6154bd2aef8d0848639c98385965744ef4b2fb839692a5ae2632c8bacb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3d18d12968db25682253382ebf7b0db0c5... | by
rw [← List.filterMap_eq_filter]
apply listFilterMap .id
simp only [Primrec₂, Option.guard, decide_eq_true_eq]
exact ite (hf.comp snd) (option_some_iff.mpr snd) (const none) | false | {
"chosen_tokens": 42,
"rejected_tokens": 47,
"token_jaccard": 0.857143,
"token_length_ratio": 1.119048
} | markdown_preamble_trailing | preamble | 7f88fd78b03b057bcc0d570e84d8488a24669c8097a86df580bfcef16cf6d99d | 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.Computability.Primrec.Basic
public import Mathlib.Logic.Encodable.Pi
Namespace:
Primrec
Local context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reser... | Here is the proof:
by
rw [← List.filterMap_eq_filter]
apply listFilterMap .id
simp only [Primrec₂, Option.guard, decide_eq_true_eq]
exact ite (hf.comp snd) (option_some_iff.mpr snd) (const none) | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Computability/Primrec",
"family_id": "listfilter",
"file_id": "mathlib/Mathlib/Computability/Primrec/List.lean",
"sample_id": "61faa742462a962beb76f3f1545ae04bd1db702ab9662b4c763eedf05c97a9ff"
} | train |
{
"chosen_proof_sha256": "aa83c7abdbb25c29bbdc8a05c929751eafeeab66442c5c16368b86eecdc282c9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "630ba8af62ca18a3656e76cecc9507d63e... | by
rw [← image_singleton, preimage_image_mk_eq_mul]
simp | false | {
"chosen_tokens": 9,
"rejected_tokens": 14,
"token_jaccard": 0.642857,
"token_length_ratio": 1.555556
} | markdown_preamble_trailing | trailing | 7fd556fb9d465df0ddb5d92b7d4beb52f4923831b49d8b961bbb1a1462e11f4e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Quotient
public import Mathlib.Algebra.Group.Action.Opposite
public import Mathlib.Algebra.Group.Subgroup.MulOpposite
public import Mathlib.GroupTheory... | by
rw [← image_singleton, preimage_image_mk_eq_mul]
simp
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "GroupTheory/Coset",
"family_id": "preimage_mk_one",
"file_id": "mathlib/Mathlib/GroupTheory/Coset/Defs.lean",
"sample_id": "423b546f69a419e2fd6b478d8c8a356a73c2e1fde067e823b879f61db25761a8"
} | train |
{
"chosen_proof_sha256": "ba5868d82bf97e5ff1d70a97d543dc36082af3a6a7d0e0038bdd9326c14ca5d6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp only [CompleteLattice.isCompactElement_iff_exists_le_iSup_of_le_iSup,
Finset.sup_eq_iSup] at h ⊢
intro ι s hb
replace hb : (b : α) ≤ iSup ((↑) ∘ s) := le_trans hb <| (coe_iSup s) ▸ le_refl _
obtain ⟨t, ht⟩ := h ι ((↑) ∘ s) hb
exact ⟨t, (by simpa using ht : (b : α) ≤ _)⟩ | true | {
"chosen_tokens": 84,
"rejected_tokens": 8,
"token_jaccard": 0.0625,
"token_length_ratio": 0.095238
} | unsolved_goals | unsolved_goals | 7ffcdc5e4f9f2e135eca5a9e2407f35e95c75bee1c8aeffb0030ca1ea00aa864 | 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.CompleteLatticeIntervals
public import Mathlib.Order.CompactlyGenerated.Basic
Namespace:
Set.Iic
Local context:
/-
Copyright (c) 2024 Oliver Nash. All ... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "c76f8519aac34e8686ed3330142233ec4c1d3e28a09eede6f93203b2525e1c56",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/CompactlyGenerated",
"family_id": "iscompactelement",
"file_id": "mathlib/Mathlib/Order/CompactlyGenerated/Intervals.lean",
"sample_id": "c5320067570b9bfc10d7ea602085cbc08d55ba3d758c561b8444485698919964"
} | train |
{
"chosen_proof_sha256": "a3c59538351f137c4bd7f08ab998d351346f182e72d5bba3bd7e2722c4d90e7a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
apply injective_of_mono (isoFinYonedaComponents F X).hom
simp only [Iso.inv_hom_id_apply]
ext x
rw [isoFinYonedaComponents_hom_apply]
simp only [← Functor.map_comp_apply, ← op_comp, CompHausLike.const_comp,
← isoFinYonedaComponents_hom_apply, Iso.inv_hom_id_apply, Function.comp_apply] | true | {
"chosen_tokens": 49,
"rejected_tokens": 8,
"token_jaccard": 0.028571,
"token_length_ratio": 0.163265
} | unsolved_goals | unsolved_goals | 80364c3aaa050a503407d4cd07901f98f402326d6a4e1516dcba1cb85bd9b392 | 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.Condensed.Discrete.LocallyConstant
public import Mathlib.Condensed.Equivalence
public import Mathlib.Topology.Category.LightProfinite.Extend
Namespace:
Conden... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "1f22f8636252d432d5335989443b28b5235de1f454d5ae6b2b4d00bd0037deba",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Condensed/Discrete",
"family_id": "isofinyonedacomponents_inv_comp",
"file_id": "mathlib/Mathlib/Condensed/Discrete/Colimit.lean",
"sample_id": "10fa2791fbfc644d39bf2d81c08a78f073daf00278ede35b73bb7ef9624d8ac7"
} | train |
{
"chosen_proof_sha256": "76e3cedf98327795879092160501512043429427fee2f5b0e827ab58125ee1b5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "fd12a4593741e5435daa603c9da043cf9d... | by
rw [IsSimplyConnected, simply_connected_iff_loops_nullhomotopic,
← isPathConnected_iff_pathConnectedSpace]
refine .and .rfl ⟨fun h x p hp ↦ ?_, fun h x p ↦ ?_⟩
· lift x to s using by simpa using hp 0
rcases h x {
toFun := fun t ↦ ⟨p t, hp t⟩
source' := by simp
target' := by simp
}... | true | {
"chosen_tokens": 153,
"rejected_tokens": 5,
"token_jaccard": 0.054545,
"token_length_ratio": 0.03268
} | wrong_namespace_qualification | wrong_namespace_qualification | 805c69aa145db9c04a7bf2bb1c191df87c25c463faadf4a442133214af7ed10a | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
public import Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps
public import Mathlib.Topology.H... | by
exact OProverMissing.missing_856d27b4c3ec | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "715b96cd58ba26356397f11bc1b092b98a1a6db681d4a3160f7f414cbffd6ff8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/FundamentalGroupoid",
"family_id": "issimplyconnected_iff_exists_homotopy_refl_forall_mem",
"file_id": "mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean",
"sample_id": "856d27b4c3ec825fbdd58bf9dd767cc218a09754dd1106fc2031d94545f79fef"
} | train |
{
"chosen_proof_sha256": "55204f99d585e2efb5ba03ffeb23f7cf47ecd4bbf63bb94951cbe11d0f4291b8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8a6273c23225b18d4154b0a3b8a34f1562... | by
rw [NormNum.IsInt.neg_to_eq h rfl]
simp only [ne_eq, neg_eq_zero]
apply ne_of_gt
simpa using! w | false | {
"chosen_tokens": 24,
"rejected_tokens": 29,
"token_jaccard": 0.84,
"token_length_ratio": 1.208333
} | markdown_preamble_trailing | trailing | 80b45cf81caf0fa5b6f3a975fbef18e26c5862c7c281965b2db2efd3f90da015 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Invertible
public import Mathlib.Algebra.Order.Ring.Cast
public import Mathlib.Tactic.HaveI
public import Mathlib.Tactic.NormNum.Core
Namespace:... | by
rw [NormNum.IsInt.neg_to_eq h rfl]
simp only [ne_eq, neg_eq_zero]
apply ne_of_gt
simpa using! w
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": "Tactic/Positivity",
"family_id": "nz_of_isnegnat",
"file_id": "mathlib/Mathlib/Tactic/Positivity/Core.lean",
"sample_id": "a1caf932105bba53888c1df521c9cec9b3321d4d5015df07eac43fb607350537"
} | train |
{
"chosen_proof_sha256": "aea622174bd7323409020f456ba921f86430f2be7a1ef46c09da2cf27c53cb7b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e8c807a815f30c0f1ea7687200d8d50d2f... | by
have hh : 0 < Γ.strictWidthInfty := Γ.strictWidthInfty_pos_iff.mpr Fact.out
have hΓ : Γ.strictWidthInfty ∈ Γ.strictPeriods := Γ.strictWidthInfty_mem_strictPeriods
have hanal := ModularFormClass.analyticAt_cuspFunction_zero f hh hΓ
have hper := periodic_comp_ofComplex f hΓ
simp_rw [IsZeroAtImInfty, ZeroAtFi... | true | {
"chosen_tokens": 96,
"rejected_tokens": 5,
"token_jaccard": 0.0625,
"token_length_ratio": 0.052083
} | wrong_namespace_qualification | wrong_namespace_qualification | 80c043ee395f85e0fcbaac03c01b4eefe0b5a936e4fb690089a1fb37a8431cb1 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.NumberTheory.ModularForms.QExpansion
public import Mathlib.NumberTheory.ModularForms.LevelOne.Basic
public import Mathlib.NumberTheory.ModularForms.EisensteinS... | by
exact OProverMissing.missing_79ff1c574653 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "8a404881c04003945ba301479e23ff10d15390c814fde9caf061ea27cac82de3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/ModularForms",
"family_id": "iszeroatiminfty_of_valueatinfty_eq_zero",
"file_id": "mathlib/Mathlib/NumberTheory/ModularForms/CuspFormSubmodule.lean",
"sample_id": "79ff1c5746539b95fb7b7bd18831f461eb7a59f72b537902f6b1acccf75b210c"
} | train |
{
"chosen_proof_sha256": "8debc390e3c9d99e3175632da6d0d7fd484f1961b70777e58cda8dfa8a7eb610",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8a042b931a223bd86aa1c34f5d07f04bc0... | by
let a := (lfp.comp h).lfp
refine (lfp_le _ ?_).antisymm (lfp_le _ (Eq.le ?_))
· exact lfp_le _ h.onDiag.map_lfp.le
have ha : (lfp ∘ h) a = a := (lfp.comp h).map_lfp
calc
h a a = h a (h a).lfp := congr_arg (h a) ha.symm
_ = (h a).lfp := (h a).map_lfp
_ = a := ha | false | {
"chosen_tokens": 105,
"rejected_tokens": 112,
"token_jaccard": 0.935484,
"token_length_ratio": 1.066667
} | markdown_preamble_trailing | fence | 80d0d9dd7db71d924e88033babeb58d9edcaa41785b93dcfef72da271250c033 | 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.Dynamics.FixedPoints.Basic
public import Mathlib.Order.Hom.Order
public import Mathlib.Order.BourbakiWitt
Namespace:
OrderHom
Local context:
/-
Copyright (c)... | ```lean
by
let a := (lfp.comp h).lfp
refine (lfp_le _ ?_).antisymm (lfp_le _ (Eq.le ?_))
· exact lfp_le _ h.onDiag.map_lfp.le
have ha : (lfp ∘ h) a = a := (lfp.comp h).map_lfp
calc
h a a = h a (h a).lfp := congr_arg (h a) ha.symm
_ = (h a).lfp := (h a).map_lfp
_ = a := ha
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order",
"family_id": "lfp_lfp",
"file_id": "mathlib/Mathlib/Order/FixedPoints.lean",
"sample_id": "950388e963953fd7e6eb950b29af28e1f498743a5197639d25cdf72094ea0aae"
} | train |
{
"chosen_proof_sha256": "bb1d914f2c85ca879a4ab7ab44641a41b11803eff45090adf0fe5af8fb8479b6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f45a84b72ed006d2229b6b12ac3c6213f3... | by
induction n <;> grind [AllZero, sign] | true | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.25
} | circular_dependency | circular_dependency | 811ac0e59280edbc88efe5b4908bafc2eca30cca87b7b419ca960f73b0c76b0f | 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.Real.Basic
Namespace:
Tactic.ComputeAsymptotics.UnitMonomial.AllZero
Local context:
/-
Copyright (c) 2026 Vasilii Nesterov. All rights reserved.
Release... | by
exact replicate | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "10a81969658925fb3d5e868b101ed603a8a778e798c7181dc8348d53927a6491",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/ComputeAsymptotics",
"family_id": "replicate",
"file_id": "mathlib/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Predicates.lean",
"sample_id": "26f5274475c00cabc639fcde1f5b699c28f967e875673ef858d4e214040ea235"
} | train |
{
"chosen_proof_sha256": "12279e85ffc4d0d8cf16eae5dea8e0e2916e98350ea6156d422f2d5856bb9f1e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "004b239a5e44a3546f0f75cadaa65c1812... | by
rw [← sSup_image']
exact hd.isLUB_csSup (Hne.image _) H | true | {
"chosen_tokens": 17,
"rejected_tokens": 3,
"token_jaccard": 0.117647,
"token_length_ratio": 0.176471
} | unknown_identifier | unknown_identifier | 81b7fd513fcb9efa4f0e9aa8bbde845ce8380abe6dce7748846d2dc911967017 | 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 __oprover_missing_15e4f134fc43 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c26714ee1632f16052aba31ec318325473c1f4a1a2b53c56fd9a75d48ff78df1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | 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": "fcb2f3f3485ce6a70b3afb69269f67e2beb686789a095f74d88d7d21aa424579",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9b1bc91f5f2c46f4c33f074e71ad609e2c... | by
ext S; simp only [smul_def, _root_.zero_smul, zero_def, Pi.zero_apply] | true | {
"chosen_tokens": 19,
"rejected_tokens": 5,
"token_jaccard": 0.105263,
"token_length_ratio": 0.263158
} | wrong_namespace_qualification | wrong_namespace_qualification | 81d3407b3d28547a738e18544fd20b29e55ea7d849313642f457bd6e5acd14cc | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorProduct.RightExactness
public import Mathlib.RingTheory.Congruence.Hom
public import Mathlib.RingTheory.FiniteType
public import Mathlib.Ri... | by
exact OProverMissing.missing_502371bcc584 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a09456493b6fb28b4218e93f6ee55e2f087e188c43ded7aab93ec3ad13e98b82",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/PolynomialLaw",
"family_id": "zero_smul",
"file_id": "mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean",
"sample_id": "502371bcc584859610c49145091eb701ac3e0cd90f5beb5f633781b964813dd7"
} | train |
{
"chosen_proof_sha256": "61e6f75295dcf0ee0f0edfdb232d6f4ff00a1685b5790d5e3eca26ccc61aea57",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "62a92841f7b0315f37f375dabbb5c7f5c1... | by
rw [← Module.finrank_mul_finrank ℝ ℂ E, Complex.finrank_real_complex] | true | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.0625,
"token_length_ratio": 0.2
} | unknown_identifier | unknown_identifier | 81f711ad1b8ac02184f4441743bc797007071394584bf6ec2a41d3c53b4cfea1 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Rat
public import Mathlib.Analysis.Complex.Cardinality
public import Mathlib.LinearAlgebra.Complex.Module
public import Mathlib.LinearAlgebra.F... | by
exact __oprover_missing_91a92b19287f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "4fa09e8b777917f0fc855de1a67270a8b209d731327ca48aeb7bab0287d9fa86",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Complex",
"family_id": "finrank_real_of_complex",
"file_id": "mathlib/Mathlib/LinearAlgebra/Complex/FiniteDimensional.lean",
"sample_id": "91a92b19287fd834779cff643bf9db6f489d497152bb848e415ab87fcceac41f"
} | train |
{
"chosen_proof_sha256": "9ee4d424ebdd38c7fae4d49084f0c22e4992fdbfa58721d1092ad5e07b192bfb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "bb91f322cc7379a2e09992630fffcf6b48... | by
exact div_nonneg (Finset.sum_nonneg fun _ _ => sq_nonneg _) <| sq_nonneg _ | false | {
"chosen_tokens": 18,
"rejected_tokens": 22,
"token_jaccard": 0.823529,
"token_length_ratio": 1.222222
} | missing_wrong_import | import | 824f583e5d71aaedc7228d34a02a05b2fbf8cff1e2e51a75089ed1fbc9e987a0 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Module.NatInt
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import Mathlib.Combinatorics.SimpleGraph.Density
public import Mathl... | import Mathlib.WrongImport
by
exact div_nonneg (Finset.sum_nonneg fun _ _ => sq_nonneg _) <| sq_nonneg _ | {
"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": "Combinatorics/SimpleGraph",
"family_id": "energy_nonneg",
"file_id": "mathlib/Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean",
"sample_id": "ea7f7ba297977c69513c6abcc1bfadf80c88bd3e61590c3a43e8307b939f76d2"
} | train |
{
"chosen_proof_sha256": "4644b39b47ec31a8d61ff06e763a66da7cadd0e1fb2b3309d9d1db8d535d144c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "184989b8ace5b0c3b0fc6a3e72a0630969... | by
rw [← sub_eq_zero]; exact hS.eq_zero_of_inner_left 𝕜 (by simpa [inner_sub_left, sub_eq_zero]) | true | {
"chosen_tokens": 21,
"rejected_tokens": 2,
"token_jaccard": 0.055556,
"token_length_ratio": 0.095238
} | invalid_tactic | invalid_tactic | 8258f8821e7aa337285f5fc86a9124c987d838608cb3e9715e0683c906b2705e | 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.Basic
public import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
Namespace:
(root)
Local context:
/-
Copyright (c) 2019 Zhou... | by
__oprover_invalid_tactic_6d45385fed9b | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "d4242e7a02db1ff956958cd35827bf6e4715831809d32899ca9b582a24b7d8f0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/InnerProductSpace",
"family_id": "dense",
"file_id": "mathlib/Mathlib/Analysis/InnerProductSpace/Continuous.lean",
"sample_id": "6d45385fed9b0ad68c7032cd158ad8e6ed294ff9a6d585f7ef6679ce0eea1fd4"
} | train |
{
"chosen_proof_sha256": "d20eb1ad8941b8e34a808b1966083034fd1512741d71702f610e71a800689ea8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [← single_zero_one, orderTop_single one_ne_zero, WithTop.coe_eq_zero] | false | {
"chosen_tokens": 13,
"rejected_tokens": 5,
"token_jaccard": 0.0625,
"token_length_ratio": 0.384615
} | sorry_admit_sorryAx_proof_hole | sorryAx | 825f0451a744941818f042894dece27e6912208a08a507a917d8635af3a28303 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Subalgebra.Lattice
public import Mathlib.Algebra.GroupWithZero.Regular
public import Mathlib.Algebra.Module.BigOperators
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": "RingTheory/HahnSeries",
"family_id": "ordertop_one",
"file_id": "mathlib/Mathlib/RingTheory/HahnSeries/Multiplication.lean",
"sample_id": "ac967f3faf553306a5c24145a1c9caf628fb00eae9e83bd29d882cbc48ac201d"
} | train |
{
"chosen_proof_sha256": "defdc0751f53e30ab813268ab9f2c357764562c8bfc11a4ae2b13dad9c027777",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq] | true | {
"chosen_tokens": 15,
"rejected_tokens": 8,
"token_jaccard": 0.058824,
"token_length_ratio": 0.533333
} | unsolved_goals | unsolved_goals | 82702d71d4c3a4688a47cbd3d41448c5e847cbcc737bdf179b4c4b8133b90af3 | 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.GaloisConnection.Basic
Namespace:
Heyting
Local context:
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as ... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "9ba45b74c5d26f2825bc366283e624541d904b72ae7707c39a38e5070be2b0fa",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Heyting",
"family_id": "isregular",
"file_id": "mathlib/Mathlib/Order/Heyting/Regular.lean",
"sample_id": "9a7f488b9072e1c3cd3e743c9cf1c9c590011f4701530db321fda7898e28d6e1"
} | train |
{
"chosen_proof_sha256": "2cd47198bc2f9b64bb28556c6ab7308c7324a109d2a3891b6dff94f0ce2c3121",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1c86ceea114f8b21399d14e3c8eb46ce17... | by
simp_rw [Nat.lt_add_one_iff, biSup_le_eq_sup] | false | {
"chosen_tokens": 9,
"rejected_tokens": 13,
"token_jaccard": 0.75,
"token_length_ratio": 1.444444
} | missing_wrong_import | import | 82736011456515ad527910285993062f0da327ba64fbd8a2de39792be9e207c2 | 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.Finset
public import Mathlib.Order.Interval.Finset.Nat
Namespace:
Nat
Local context:
/-
Copyright (c) 2018 Johannes Hölzl.... | import Mathlib.WrongImport
by
simp_rw [Nat.lt_add_one_iff, biSup_le_eq_sup] | {
"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": "Order/Lattice",
"family_id": "isup_lt_succ",
"file_id": "mathlib/Mathlib/Order/Lattice/Nat.lean",
"sample_id": "2b6b462ef4cebc5cf1be9ae0f138ed0a50ba4a689e61f2c8626b10e8cc24c3b7"
} | train |
{
"chosen_proof_sha256": "11fbcca75f5b8a4dbad7c2490be392157767b93ac85f1ccad1aff23708a841e9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
intro x hx
simp only [Subalgebra.mem_map, Subalgebra.mem_center_iff] at hx ⊢
obtain ⟨b, hb0, rfl⟩ := hx
intro bc
induction bc using TensorProduct.induction_on with
| zero => simp
| tmul b' c => simp [hb0]
| add _ _ _ _ => simp_all [add_mul, mul_add] | false | {
"chosen_tokens": 63,
"rejected_tokens": 3,
"token_jaccard": 0.02439,
"token_length_ratio": 0.047619
} | by_exact_placeholder | exact? | 82b32dfb15871df66c50b11ba4ad9c9ab48298c8adab941b4bb128f52eaae57b | 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.Central.Basic
public import Mathlib.RingTheory.Flat.Basic
public import Mathlib.LinearAlgebra.Basis.VectorSpace
Namespace:
(root)
Local context:
/-
C... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Central",
"family_id": "algebra",
"file_id": "mathlib/Mathlib/Algebra/Central/TensorProduct.lean",
"sample_id": "8c84f0871128efc9bcf9d29301aeb6213d25f58a7d79db843c43cab6078afc5a"
} | train |
{
"chosen_proof_sha256": "b663b020e08f48e798d48b7ce39be8da0dce1601bdd6b1231b5d9860f49a1922",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "65b86c085b15012e6279a23598357f9329... | by lia) :
t.IsGE ((t.truncGT a).obj X) b :=
t.isGE_truncGE_obj .. | false | {
"chosen_tokens": 25,
"rejected_tokens": 30,
"token_jaccard": 0.789474,
"token_length_ratio": 1.2
} | markdown_preamble_trailing | preamble | 833bbbe0bd9cbe1bb75a1000bd3c2faf5e13acf507f0e06421bca5213a2fd26a | 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.Triangulated.TStructure.TruncLTGE
Namespace:
CategoryTheory.Triangulated.TStructure
Local context:
/-
Copyright (c) 2026 Joël Riou. All rights... | Here is the proof:
by lia) :
t.IsGE ((t.truncGT a).obj X) b :=
t.isGE_truncGE_obj .. | {
"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/Triangulated",
"family_id": "isge_truncgt_obj",
"file_id": "mathlib/Mathlib/CategoryTheory/Triangulated/TStructure/TruncLEGT.lean",
"sample_id": "9464b08287d95c785301c54058651202fff8047c0ca4e3764eb7ffff3e6775f3"
} | train |
{
"chosen_proof_sha256": "05c1e20ccd52d20617d06c19bb3d65322091abd072a38ef854172815fc9654f9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
refine ⟨?_, fun h ↦ (isCyclic_of_card_le_two h).isMulCommutative⟩
classical
rintro ⟨⟨h⟩⟩
rw [← not_lt, ← Set.ncard_univ, Set.two_lt_ncard_iff]
rintro ⟨a, b, c, _, _, _, hab, hac, hbc⟩
apply hbc
simp_rw [Perm.ext_iff] at h
simpa [swap_apply_of_ne_of_ne hab hac] using h (swap a b) (swap b c) a | false | {
"chosen_tokens": 86,
"rejected_tokens": 3,
"token_jaccard": 0.02381,
"token_length_ratio": 0.034884
} | by_exact_placeholder | placeholder | 834852a1421aaa28d92fe15e578e114ec5461e05c680616ae398e9e99ed2db70 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Perm
public import Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic
public import Mathlib.SetTheory.Cardinal.Finite
Namespace:
Equiv.Perm
Local c... | by
exact 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/Finite",
"family_id": "ismulcommutative_iff_card_le_two",
"file_id": "mathlib/Mathlib/Data/Finite/Perm.lean",
"sample_id": "15e83a19ac7a4ffb734f70f90cc5cd72e116c7c5b68a279718e9f6a477143d88"
} | train |
{
"chosen_proof_sha256": "19a0b796b5a123aa0fa4da0bb1ec6d1209352b44dfd80eed24e61f22f0837fb5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3a8030466b32fb2ee6b5297792fab6b266... | by
refine measure_mono_null ?_ hs.ordConnected.null_frontier
rw [← closure_sdiff_interior, hs.interior_eq_empty, sdiff_empty]
exact subset_closure | false | {
"chosen_tokens": 23,
"rejected_tokens": 28,
"token_jaccard": 0.791667,
"token_length_ratio": 1.217391
} | markdown_preamble_trailing | preamble | 83570c25cdc8369a047445ec25ace77abf7faaa3e71a5bf9078737e40c8a9815 | 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.Order.UpperLower
public import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
public import Mathlib.Topology.Order.DenselyOrdered
Names... | Here is the proof:
by
refine measure_mono_null ?_ hs.ordConnected.null_frontier
rw [← closure_sdiff_interior, hs.interior_eq_empty, sdiff_empty]
exact subset_closure | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Order",
"family_id": "isantichain",
"file_id": "mathlib/Mathlib/MeasureTheory/Order/UpperLower.lean",
"sample_id": "0a46fac18823464095b42916006166b41806c313503a9332eebd632a234896e3"
} | train |
{
"chosen_proof_sha256": "f55e39083a696313fcfdd6bb61c3e62ddbca985973b59bf86a8381597fddf755",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
have : Nontrivial (Subobject X) := nontrivial_of_not_isZero h
obtain ⟨Y, s⟩ := (IsAtomic.eq_bot_or_exists_atom_le (⊤ : Subobject X)).resolve_left top_ne_bot
exact ⟨Y, (subobject_simple_iff_isAtom _).mpr s.1⟩
/-- Choose an arbitrary simple subobject of a non-zero Artinian object. -/
noncomputable def simpleSub... | false | {
"chosen_tokens": 145,
"rejected_tokens": 3,
"token_jaccard": 0.032258,
"token_length_ratio": 0.02069
} | by_exact_placeholder | exact? | 837d65b12a123529352d38eceeff9dd87743af17b3c21b525ab59c96392451e8 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Subobject.Lattice
public import Mathlib.CategoryTheory.ObjectProperty.ContainsZero
public import Mathlib.CategoryTheory.ObjectProperty.EpiMono
p... | 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/Subobject",
"family_id": "exists_simple_subobject",
"file_id": "mathlib/Mathlib/CategoryTheory/Subobject/ArtinianObject.lean",
"sample_id": "9044b1a9ec5361ec717fcb988a173f7c747b987f4a363348924f9ee3f41d531b"
} | train |
{
"chosen_proof_sha256": "0220babf450eb445fb686e51233d9dc7bd43814cdb2cf6da4315eb591a53bef9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [← natCast_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (Int.natCast_ne_zero.mpr hn),
Int.cast_natCast]
rfl | false | {
"chosen_tokens": 22,
"rejected_tokens": 3,
"token_jaccard": 0.05,
"token_length_ratio": 0.136364
} | by_exact_placeholder | exact? | 8388cb4255eac8c2f5327b5f0d77ac20a67ffa98e7b3b6a393a06cccaf8fe873 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Field.Basic
public import Mathlib.Algebra.Order.Group.Unbundled.Int
public import Mathlib.Algebra.Module.NatInt
public import Mathlib.GroupTheory.Quoti... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/CharZero",
"family_id": "nsmul_mem_zmultiples_iff_exists_sub_div",
"file_id": "mathlib/Mathlib/Algebra/CharZero/Quotient.lean",
"sample_id": "a86ece7891869444431a13676c31d6de05fdf4898f0e667ba12ab2a36fc2b894"
} | train |
{
"chosen_proof_sha256": "a8a91e5033815085839eaad1f6a9d8721d67997ed1799815646887dd0bdc0a37",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2287237bfe25cfcf9e2c3471e496f26861... | by
simp [AlgEquiv.ext_iff, Subtype.ext_iff, AlgEquiv.restrictNormal_apply] | true | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.2
} | unknown_identifier | unknown_identifier | 83ac361c840ae2058b083e274477c29f521dc80a428b1d7c78a7613284d2b6f6 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.Splits
public import Mathlib.FieldTheory.Galois.Notation
public import Mathlib.FieldTheory.IntermediateField.Basic
public import Mathlib.Fie... | by
exact __oprover_missing_e7bd69cfd0da | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "b721a9559137952d5e2b9a7140f0550004513700b5dc57a45125b203e13779aa",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/Normal",
"family_id": "algequiv",
"file_id": "mathlib/Mathlib/FieldTheory/Normal/Defs.lean",
"sample_id": "e7bd69cfd0dadb9fdaab62111c02ae20ae7bf73cdd0cc212f534d85bcf679dc1"
} | train |
{
"chosen_proof_sha256": "9a21e4324d10fe830df7d8256fcb74edf6972d935383079b80c071c0dccb073e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b92eb11d145737a83591d6e333398970dc... | by
simpa using toInt_le_toInt val j i | true | {
"chosen_tokens": 7,
"rejected_tokens": 5,
"token_jaccard": 0.090909,
"token_length_ratio": 0.714286
} | wrong_namespace_qualification | wrong_namespace_qualification | 848fb2f04ce053b871ba8ac16046a3e30d71cb24386781bc08ab076e160bbaea | 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.List.Pairwise
public import Batteries.Tactic.GeneralizeProofs
public import Mathlib.Tactic.Order.CollectFacts
Namespace:
Mathlib.Tactic.Order.ToInt
Lo... | by
exact OProverMissing.missing_c7eeb279213f | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "31b2b05362dc0f6eb2a89a2c2f404f4c574e1b7101c98049288b9856ed547a8c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/Order",
"family_id": "toint_nlt_toint",
"file_id": "mathlib/Mathlib/Tactic/Order/ToInt.lean",
"sample_id": "c7eeb279213fe096be67e549c7a7f69ed83061d62764c53bfce17ab42221096a"
} | train |
{
"chosen_proof_sha256": "8f3b01149f94ae9d113cd2afe1cda1c376344890d5f097114694fdbd6fd42eb6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1079d2160a585c07511be441b8c3dc8092... | by
rcases s.eq_empty_or_nonempty with (rfl | hs); · simp
rcases t.eq_empty_or_nonempty with (rfl | ht); · simp
simp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or] | true | {
"chosen_tokens": 48,
"rejected_tokens": 3,
"token_jaccard": 0.038462,
"token_length_ratio": 0.0625
} | circular_dependency | circular_dependency | 849cf0126fc02ac9cc3b7dda1ffe4168e847bad38adcf930bd2916580332e14f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.TypeTags.Basic
public import Mathlib.Topology.Bornology.Basic
Namespace:
Bornology
Local context:
/-
Copyright (c) 2022 Yury Kudryashov. All ri... | by
exact isBounded_prod | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "83b0d7c437e53ea57ece7ad820efd8174deaf3baae8e0c589578749f4643ba86",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Bornology",
"family_id": "isbounded_prod",
"file_id": "mathlib/Mathlib/Topology/Bornology/Constructions.lean",
"sample_id": "c7abf66d2d061cad451c9a404bc80bcd09d739b8aa88cc477da70de900f24a00"
} | train |
{
"chosen_proof_sha256": "35928b4ec6c4dd8e48530f64dacc44b471422b62b6f0a8ae08f8ed59d2ec96f4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
calc
_ = 𝟙 _ ⊗≫ (F.map₂ β ▷ η.app b ≫ η.naturality g) ▷ θ.app b ⊗≫
η.app a ◁ θ.naturality g ⊗≫ 𝟙 _ := by
bicategory
_ = 𝟙 _ ⊗≫ η.naturality f ▷ θ.app b ⊗≫
η.app a ◁ (G.map₂ β ▷ θ.app b ≫ θ.naturality g) ⊗≫ 𝟙 _ := by
rw [η.naturality_naturality]
bicategory
_ = _... | false | {
"chosen_tokens": 112,
"rejected_tokens": 2,
"token_jaccard": 0.03125,
"token_length_ratio": 0.017857
} | sorry_admit_sorryAx_proof_hole | sorry | 8519be5fe12fe566f5218ffa1a27e65eda9d07c8eae710fcb62b90533538e12c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Bicategory.Functor.Lax
public import Mathlib.Tactic.CategoryTheory.Bicategory.Basic
Namespace:
CategoryTheory.Lax.OplaxTrans
Local context:
/-... | 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/Bicategory",
"family_id": "vcomp_naturality_naturality",
"file_id": "mathlib/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Lax.lean",
"sample_id": "325825a927e5735338dd52a28d2dff4f12d1ad248dc0bc8ca3daba7d87f856c6"
} | train |
{
"chosen_proof_sha256": "ecfe891855dc120f1040ab8b4113f995f2ce6be82fd87d86c20a75c2f1e66b44",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
exact .tower_top (K := F⟮(X : F⟮X⟯)⟯) _
variable {y : K} | true | {
"chosen_tokens": 26,
"rejected_tokens": 8,
"token_jaccard": 0.136364,
"token_length_ratio": 0.307692
} | unsolved_goals | unsolved_goals | 851fedd9fafba0caffca0cec300529e36dde29069b3fdf0d1cbd659c86146bab | 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.Degree
public import Mathlib.RingTheory.DedekindDomain.IntegralClosure
public import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
pu... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "6fb4b0b0f18f0e288eb0e9ec48b6b6953124fdb7096f3426fcdb563cf7205245",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory",
"family_id": "algebra",
"file_id": "mathlib/Mathlib/NumberTheory/FunctionField.lean",
"sample_id": "6381270961bbd859431dc91f4de11a424555613e987c739bd7c51cfea5347942"
} | train |
{
"chosen_proof_sha256": "de8a32b5f73d36b3c5675db1bfb4e3c6a3d479eee57e39397db99ac101c60702",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "310883994a3c9ca44f3b346c085fc78418... | by
refine ⟨fun H ↦ ?_, isTopologicallyNilpotent_of_constantCoeff⟩
replace H : Tendsto (fun n ↦ constantCoeff (f ^ n)) atTop (nhds 0) :=
continuous_constantCoeff R |>.tendsto' 0 0 constantCoeff_zero |>.comp H
simpa only [map_pow] using! H | true | {
"chosen_tokens": 54,
"rejected_tokens": 2,
"token_jaccard": 0.025,
"token_length_ratio": 0.037037
} | invalid_tactic | invalid_tactic | 8541ae21b91d12147dfb7eae302420078a1af539eef3e8cf9ff8b35acedf1643 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finsupp.Interval
public import Mathlib.RingTheory.Ideal.Quotient.Defs
public import Mathlib.RingTheory.MvPowerSeries.PiTopology
public import Mathlib.Topo... | by
__oprover_invalid_tactic_460fd408bbfa | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "9a37532cc48611faa8ea55e49db29cfffe19fc3244dda83b6c6f1a9949db522d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/MvPowerSeries",
"family_id": "istopologicallynilpotent_iff_constantcoeff",
"file_id": "mathlib/Mathlib/RingTheory/MvPowerSeries/LinearTopology.lean",
"sample_id": "460fd408bbfa2c9d3696e93eaf82711398320704b30a3dd64ec1d4fbb298afc9"
} | train |
{
"chosen_proof_sha256": "6f31eac0331d8119b977de849699fd5e0d1fb2f77898bc029e261ff94821e033",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by simpa only [zero_mul, zero_add] using div_add_mod a 0 | false | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.25
} | by_exact_placeholder | placeholder | 8542cf2283119839b815e3b2167bc8c6346e8c4e46d5416a0798437c93dbcfea | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Ring.Defs
public import Mathlib.Order.RelClasses
Namespace:
EuclideanDomain
Local context:
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
R... | by
exact 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/EuclideanDomain",
"family_id": "mod_zero",
"file_id": "mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean",
"sample_id": "4793854c2e3af40b9d027f7b421519fea77582f2694be3b8462681fa81f0c2f6"
} | train |
{
"chosen_proof_sha256": "f534ed3effb9755a614b851ef33384815c516806a3924c7986894d2e4e795cd3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [← isRegular_pairing_iff, Pairing.isRegular_iff_nonempty_weakRankFunction]
exact (P.weakRankFunctionEquiv ℕ).symm.nonempty_congr | false | {
"chosen_tokens": 21,
"rejected_tokens": 2,
"token_jaccard": 0.052632,
"token_length_ratio": 0.095238
} | sorry_admit_sorryAx_proof_hole | sorry | 85691a281fc98ab670b6ffcea885b3d7e47578d51d10b62ff20358577f1bc7f3 | 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.SimplicialSet.AnodyneExtensions.Rank
public import Mathlib.Data.Finite.Sigma
Namespace:
SSet.Subcomplex.PairingCore
Local context:
/-
Copyr... | 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/SimplicialSet",
"family_id": "isregular_iff_nonempty_weakrankfunction",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/RankNat.lean",
"sample_id": "467d6115e45bfeaa364bc7d6dc84e27f5374de312ed692a0238a4a35b99ebeeb"
} | train |
{
"chosen_proof_sha256": "0b0b9e8a744c1d7e4020f039217d12486b9369dcf62ab6b9ae3cf10e259b4ee9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
simp [← factorPowSucc_comp_fontaineThetaModPPow n] | false | {
"chosen_tokens": 7,
"rejected_tokens": 3,
"token_jaccard": 0.111111,
"token_length_ratio": 0.428571
} | by_exact_placeholder | exact? | 85a31c5481f8ee992a826ad2569167e8f1a28dba5748da45f4f8064c74939f46 | 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.AdicCompletion.Functoriality
public import Mathlib.RingTheory.AdicCompletion.RingHom
public import Mathlib.RingTheory.Perfectoid.Untilt
public impor... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Perfectoid",
"family_id": "factorpowsucc_fontainethetamodppow_eq",
"file_id": "mathlib/Mathlib/RingTheory/Perfectoid/FontaineTheta.lean",
"sample_id": "aeb71f8e68a66dc07c62978a48769bca8b7ac587e71fccaeb6caa13c22e6a3db"
} | train |
{
"chosen_proof_sha256": "b33ee369bdfae6b8f7816eb5661ea638ec7dc5ec9125117e507e6aaa3923f7b0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d437f1656af8a39ef7edd102454e4e05fa... | by
intro a b h
refine le_of_lt_or_eq (or_iff_not_imp_right.2 fun hne ↦ ?_)
classical
exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),
fun j hj ↦ notMem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_gt hj,
(h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩ | true | {
"chosen_tokens": 72,
"rejected_tokens": 5,
"token_jaccard": 0.078947,
"token_length_ratio": 0.069444
} | wrong_namespace_qualification | wrong_namespace_qualification | 85b546c09616adf940e085829b82398d88ec6e4e2fbd45570c1ba38831a2e236 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Group.PiLex
public import Mathlib.Data.DFinsupp.Order
public import Mathlib.Data.DFinsupp.NeLocus
public import Mathlib.Order.WellFoundedSet
Nam... | by
exact OProverMissing.missing_cfce958e665e | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "58d2bf85a439cdef1547b2e84e32b33ea7fcd7104bfda5ecb205683046c17cc5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/DFinsupp",
"family_id": "tolex_monotone",
"file_id": "mathlib/Mathlib/Data/DFinsupp/Lex.lean",
"sample_id": "cfce958e665e03e21c1db8b838eb32bd3a0604379292b7cebef2b66be097f2db"
} | train |
{
"chosen_proof_sha256": "ab4cb4aa5cd70f70198092337097cb05b18354e67218420118c17e78a73513e0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
refine ⟨fun h ↦ ?_, fun ⟨h₁, h₂⟩ ↦ ?_⟩
· refine evalInduction h fun a h IH ↦ ?_
rcases e : f a with - | a'
· rw [Part.mem_unique h
(PFun.mem_fix_iff.2 <| Or.inl <| Part.mem_some_iff.2 <| by rw [e]; rfl)]
exact ⟨ReflTransGen.refl, e⟩
· rcases PFun.mem_fix_iff.1 h with (h | ⟨_, h, _⟩) <... | false | {
"chosen_tokens": 215,
"rejected_tokens": 3,
"token_jaccard": 0.035714,
"token_length_ratio": 0.013953
} | by_exact_placeholder | placeholder | 85d018d97b9595f3f0fba702e19c5821b8c8fb7fff7fb8d44c70afd17b05a972 | 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.PFun
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Fintype.Defs
Nam... | 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": "Computability",
"family_id": "mem_eval",
"file_id": "mathlib/Mathlib/Computability/StateTransition.lean",
"sample_id": "9fe3719b8a587cc320eadf69f2619355af22157136ae9400bafc484518741ba5"
} | train |
{
"chosen_proof_sha256": "306e987910ff023d43690d85687bf5ae9a7b00372cbbe2300a8443541f4fbbe4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8597d33dd3f22278ae75c80beaa1edceeb... | by
intro Y f hf
simp only [Presheaf.χ_app, Opposite.op_unop] at hf ⊢
choose a ha using fun Z (g : Z ⟶ Y) (hg : (Sieve.pullback f ((χ m).app X x)).arrows g) => hg
refine ⟨(hF _ hf).amalgamate a ?_, ?_⟩
· introv Y₁ h
apply (mono_iff_injective (m.app (.op Z))).mp inferInstance
simp_rw [NatTrans.naturalit... | true | {
"chosen_tokens": 165,
"rejected_tokens": 4,
"token_jaccard": 0.028169,
"token_length_ratio": 0.024242
} | circular_dependency | circular_dependency | 85d69378d0fcf3cd112215a2b5a2e27365b71dad0802d33f028338e6b64901c4 | 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.Sites.Closed
public import Mathlib.CategoryTheory.Sites.Equivalence
public import Mathlib.CategoryTheory.Subobject.Classifier.Defs
public import... | by
exact GrothendieckTopology. | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "403071b67998f3f9ee5a39109f0e1362913ce18d8cb8c1abeb22226054d18083",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Topos",
"family_id": "grothendiecktopology",
"file_id": "mathlib/Mathlib/CategoryTheory/Topos/Sheaf.lean",
"sample_id": "20b48f91a7f4ad693944c040d924608c454aa3ef8bafb869719f264b357fd194"
} | train |
{
"chosen_proof_sha256": "f8689006584708e8f5f2ea5a3447138e87630e033d57129a1fcb9fca25aae75e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
let h := MorphismProperty.factorizationData (trivialCofibrations C) (fibrations C)
(terminal.from X.obj)
have := isCofibrant_of_cofibration h.i
have : IsFibrant h.Z := by
rw [isFibrant_iff_of_isTerminal h.p terminalIsTerminal]
infer_instance
exact ⟨BifibrantObject.mk h.Z, homMk h.i, inferInstan... | true | {
"chosen_tokens": 180,
"rejected_tokens": 8,
"token_jaccard": 0.076923,
"token_length_ratio": 0.044444
} | unsolved_goals | unsolved_goals | 85e43e0345e51df9ddc6d8c30318b5d4950f3ca3fcf989370622d2ef1d0285f7 | 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.ModelCategory.CofibrantObjectHomotopy
public import Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy
public import Mathlib.Categ... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "3c17cf65f1141443d5fda3c6491ee0aa1e1052b0307eca31b1a84ac0eade55e9",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/ModelCategory",
"family_id": "exists_bifibrant",
"file_id": "mathlib/Mathlib/AlgebraicTopology/ModelCategory/BifibrantObjectHomotopy.lean",
"sample_id": "1b257d253a1d99e1ebaccfa098b639ce8bc9949faf2e55ad6ca74bb6e26b343e"
} | train |
{
"chosen_proof_sha256": "4bbd86e73cbf3bb6938d81696c0fac5d64455a44efb1861863f6e712d6a46845",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [Finset.disjoint_left]
rintro x hx hxc
obtain ⟨x, hx', rfl⟩ := mem_map.mp hxc
exact hs.compl_notMem hx' hx | false | {
"chosen_tokens": 30,
"rejected_tokens": 2,
"token_jaccard": 0.041667,
"token_length_ratio": 0.066667
} | by_exact_placeholder | by? | 85e4e72cfdeb240029ac27503970f1f447ef6cdfb91b0dcd9e0e1b6eab4a42a0 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Card
public import Mathlib.Order.UpperLower.Basic
Namespace:
Set
Local context:
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Release... | 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": "Combinatorics/SetFamily",
"family_id": "intersecting",
"file_id": "mathlib/Mathlib/Combinatorics/SetFamily/Intersecting.lean",
"sample_id": "8a9a5d45eecf81f80cc6ac9acad10ad3865265d34cf144e568617dadd24602fc"
} | train |
{
"chosen_proof_sha256": "bb1aa1df4a3618112e80043cd006efbcdf4906086196e285cd31ba1676129c32",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
rw [← closure_openCell_eq_closedCell, E.closed.closure_subset_iff, ← E.union]
apply subset_union_of_subset_right
exact subset_iUnion_of_subset n
(subset_iUnion (fun (j : ↑(E.I n)) ↦ openCell (C := C) n j) ⟨i, hi⟩) | false | {
"chosen_tokens": 53,
"rejected_tokens": 2,
"token_jaccard": 0.029412,
"token_length_ratio": 0.037736
} | sorry_admit_sorryAx_proof_hole | admit | 860ef14fc007eef49c917f1fe1f9b337e9f77d9b1a85bbd9af0558a67d046115 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.CWComplex.Classical.Finite
public import Mathlib.Analysis.Normed.Module.RCLike.Real
Namespace:
Topology
Local context:
/-
Copyright (c) 2025 Floris ... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/CWComplex",
"family_id": "relcwcomplex",
"file_id": "mathlib/Mathlib/Topology/CWComplex/Classical/Subcomplex.lean",
"sample_id": "26a28f745aa7034105aa52a93b588fb63e8e92ad6d8d99ffd3be3558367e64ea"
} | train |
{
"chosen_proof_sha256": "eb3d60d437f15cfc69d1d37a56f9a9b73ef1279b8f1ffb7d0327dabbc53ce4fd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7e5e76fd4da0d1114a6376ee390941064e... | by
ext
simp only [alternatingMapLinearEquiv_comp_ιMulti, LinearMap.compAlternatingMap_apply,
LinearMap.coe_comp, comp_apply, alternatingMapLinearEquiv_apply_ιMulti]
/-! Functoriality of the exterior powers. -/
variable (n) in
/-- The linear map between `n`th exterior powers induced by a linear map between the... | true | {
"chosen_tokens": 110,
"rejected_tokens": 2,
"token_jaccard": 0.018868,
"token_length_ratio": 0.018182
} | invalid_tactic | invalid_tactic | 861f04404ba855c66c6ebf4baabaedc5cd86ba903a61932406b89f608dcaadec | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Module.Presentation.Basic
public import Mathlib.LinearAlgebra.ExteriorAlgebra.OfAlternating
Namespace:
exteriorPower
Local context:
/-
Copyright (c) ... | by
__oprover_invalid_tactic_b5eed30bb368 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "fe0c2b73be9cc65a08142380ca28f736ea8ee6eac8890f031adc91dd21ffbabc",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/ExteriorPower",
"family_id": "alternatingmaplinearequiv_comp",
"file_id": "mathlib/Mathlib/LinearAlgebra/ExteriorPower/Basic.lean",
"sample_id": "b5eed30bb36819b24fa9a164628ee2a2c0338152109146fc3f9ca3ebec531d4e"
} | train |
{
"chosen_proof_sha256": "4967cf04180cb0748d25ecdf64a662a7f6a1d6bdd998e61ce6f3d7d2c6f082a5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "214074a4258bfc13547146011a31a2753a... | by
rw [← map_add]
exact m.toSyn_monotone le_self_add | true | {
"chosen_tokens": 11,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.181818
} | invalid_tactic | invalid_tactic | 8639c8abe673a0bcacd37e4e68a5bd2786c424da66d8dd4de038b3dae33a22b5 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finsupp.Lex
public import Mathlib.Data.Finsupp.WellFounded
public import Mathlib.Data.List.TFAE
Namespace:
MonomialOrder
Local context:
/-
Copyright (c)... | by
__oprover_invalid_tactic_756685a57c1c | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "ec7f85bb7233429929ffade30851ddc7543cd543ceb3fcbe6221c80ef48e51dd",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Finsupp",
"family_id": "le_add_right",
"file_id": "mathlib/Mathlib/Data/Finsupp/MonomialOrder.lean",
"sample_id": "756685a57c1c40f2d410134b5fc2580aacd3ec04b425aa2b56ef0fc77bf58bf8"
} | train |
{
"chosen_proof_sha256": "238c36aa3d5b83b0f47d94bdb4d2f15b92abc10f86ad4c7efd31a50fddc87c83",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [IsTranscendenceBasis, maximal_iff, Subtype.range_val]; rfl | false | {
"chosen_tokens": 13,
"rejected_tokens": 2,
"token_jaccard": 0.076923,
"token_length_ratio": 0.153846
} | sorry_admit_sorryAx_proof_hole | sorry | 864c38833336b66d32a5b98a1a228dd20750ab13e8a5f5c257ade35911cacaca | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MvPolynomial.CommRing
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as ... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/AlgebraicIndependent",
"family_id": "istranscendencebasis_iff_maximal",
"file_id": "mathlib/Mathlib/RingTheory/AlgebraicIndependent/Defs.lean",
"sample_id": "d594da1416489b60a7466e1d8f39b122bd55afcaa513107653ab63deea58dfe7"
} | train |
{
"chosen_proof_sha256": "f8d69be5788c6d89c35f089a74bce8cf26bd128b40b0aa5d2082d7e6f83ba04a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cad11ba3cb57fb0c810af6512d2fa884f4... | by
rw [Int.natCast_dvd]
exact Int.Prime.dvd_pow hp h | false | {
"chosen_tokens": 15,
"rejected_tokens": 22,
"token_jaccard": 0.857143,
"token_length_ratio": 1.466667
} | markdown_preamble_trailing | fence | 865ef7df37e4bca74f813a495d9a8243842d9e8b229c8e0cd9de36f56ae776d5 | 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.EuclideanDomain.Basic
public import Mathlib.Algebra.EuclideanDomain.Int
public import Mathlib.Algebra.GCDMonoid.Nat
public import Mathlib.Data.Nat.Prim... | ```lean
by
rw [Int.natCast_dvd]
exact Int.Prime.dvd_pow hp h
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Int",
"family_id": "int",
"file_id": "mathlib/Mathlib/RingTheory/Int/Basic.lean",
"sample_id": "45175a44a36d53ed6e01fe3580b4a35a13cfe7dd1d0d93468e34de78ad24b899"
} | train |
{
"chosen_proof_sha256": "ad3461d90bc1a2d6d0744349016f752c25c2128b46b806f78f2e03cb5879efbf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
have uc : (iUnion fun i => Option.elim' sᶜ u i) = univ := by
apply Subset.antisymm (subset_univ _)
· simp_rw [← compl_union_self s, Option.elim', iUnion_option]
apply union_subset_union_right sᶜ us
rcases precise_refinement (Option.elim' sᶜ u) (Option.forall.2 ⟨isOpen_compl_iff.2 hs, uo⟩)
uc ... | false | {
"chosen_tokens": 143,
"rejected_tokens": 3,
"token_jaccard": 0.044776,
"token_length_ratio": 0.020979
} | by_exact_placeholder | exact? | 867a110679ce44369a2bc513f21d2996347eff46bb1abb0b61558cb88d74dbb5 | 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.Option.Basic
public import Mathlib.Topology.Separation.Regular
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserv... | 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/Compactness",
"family_id": "precise_refinement_set",
"file_id": "mathlib/Mathlib/Topology/Compactness/Paracompact.lean",
"sample_id": "45c618e63e4bfb258f2eb5bccb2da6bd795c57ebdbec6a654acc6ef23d1bc7ee"
} | train |
{
"chosen_proof_sha256": "91dab1cd2e2e53fc5f8d17a5c93b18cccb419f8155621f1a414c77fee50249a1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
have : x < 0 := lt_of_tendsto_atBot 0 hx
intro n
simpa [abs_of_neg this, lt_neg] using! lt_of_tendsto_atBot (-n) hx | false | {
"chosen_tokens": 27,
"rejected_tokens": 5,
"token_jaccard": 0.037037,
"token_length_ratio": 0.185185
} | sorry_admit_sorryAx_proof_hole | sorryAx | 8683fd2297a82fa5a159122bd27c609e352a3b6f48f3e560b8d0eb4d3cd5434b | 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.Ring.StandardPart
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Order.Filter.FilterProduct
Namespace:
Hyperreal
Loc... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/Real",
"family_id": "archimedeanclassmk_neg_of_tendsto_atbot",
"file_id": "mathlib/Mathlib/Analysis/Real/Hyperreal.lean",
"sample_id": "9f0aba5e8c958a1721ba7a16d006099909f296bcaa86d09dd7b80102ebf2e9d4"
} | train |
{
"chosen_proof_sha256": "1791f6bfaef40a4ae8b1dcdf06c489870e6b8a0fe97f09d60381a63c54a541cd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [irreducibleComponents_eq_singleton, Set.mem_singleton_iff] at hZ
simp [irreducibleComponentOpen, irreducibleComponents_eq_singleton, hZ] | false | {
"chosen_tokens": 19,
"rejected_tokens": 2,
"token_jaccard": 0.071429,
"token_length_ratio": 0.105263
} | by_exact_placeholder | by? | 86b7c1647e0109de7c408a53b5a8250876a1ef731a5a8c2f6e78be93135cef8c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
public import Mathlib.AlgebraicGeometry.Noetherian
Namespace:
AlgebraicGeometry.Scheme
Local context:
/-
Copyrigh... | 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": "AlgebraicGeometry/IdealSheaf",
"family_id": "irreduciblecomponentopen_eq_top",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/IrreducibleComponent.lean",
"sample_id": "9b3f45e527412ab937dbda0aad67341cd01387028790afa9d9ec5b4d5e084b9c"
} | train |
{
"chosen_proof_sha256": "3fc07c2c4acb95dfe246db6ffb6277493987fabf1c2b630f887c08fb23242d47",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.toShHom
g.toShHom)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← TopCat.coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
rw [imageBasicOp... | false | {
"chosen_tokens": 75,
"rejected_tokens": 2,
"token_jaccard": 0.027027,
"token_length_ratio": 0.026667
} | sorry_admit_sorryAx_proof_hole | sorry | 86d035f7ed3d1bfdbfd379070140b91720be9d2ff1ba1d017ca01a4b5cad7ced | 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.LocallyRingedSpace
public import Mathlib.Algebra.Category.Ring.Constructions
public import Mathlib.Geometry.RingedSpace.OpenImmersion
publ... | 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": "Geometry/RingedSpace",
"family_id": "imagebasicopen_image_open",
"file_id": "mathlib/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean",
"sample_id": "879654c433cff299cfe89ef259733d4899e296266c7758cbde162533660dcf70"
} | train |
{
"chosen_proof_sha256": "5b8c55401ccfdd0326db1a35194aa3affc02ada3be7e2075a756a2a2bba77cb4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [isCoskeletal_iff]
exact isRightKanExtension_iff_isIso ((coskAdj n).unit.app X)
((coskAdj n).counit.app _) (𝟙 _) ((coskAdj n).left_triangle_components X) | false | {
"chosen_tokens": 42,
"rejected_tokens": 2,
"token_jaccard": 0.05,
"token_length_ratio": 0.047619
} | by_exact_placeholder | by? | 86f84706a1673fda3d7710499e69ff590ca87fb180d35386e48ccc93cb68dcf9 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicTopology.SimplicialObject.Basic
public import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction
public import Mathlib.CategoryTheory.Functor.KanE... | 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": "AlgebraicTopology/SimplicialObject",
"family_id": "iscoskeletal_iff_isiso",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplicialObject/Coskeletal.lean",
"sample_id": "fad9f1ea8d6654621705b02879012779787dbbf5d7900dc540f712fce41505c9"
} | train |
{
"chosen_proof_sha256": "5b4476fdc1c0c0f51888b9b0f5b1dfa03fd37018bf6998a42fadc18efa75285e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "eee74c135583607225b77544c2b5eec1a0... | by
rw [quasicategory_iff_hasLiftingProperty.{u} _ terminalIsTerminal, innerFibration_iff]
exact ⟨fun h _ _ _ ⟨i, h0, hn⟩ ↦ h h0 hn,
fun h _ _ h0 hn ↦ h _ (horn_ι_mem_innerHornInclusions h0 hn)⟩ | true | {
"chosen_tokens": 49,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.061224
} | circular_dependency | circular_dependency | 8734960e11c18e84e901f8ead8eaac0301bd01aef6200970255c5419ec97ef97 | 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.Quasicategory.Basic
Namespace:
SSet
Local context:
/-
Copyright (c) 2026 Jack McKoen. All rights reserved.
Released under Apache 2.0 licens... | by
exact quasicategory_iff_innerFibration | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "660abf679bef5d3627f03de3886ddb25eb20fb82b18df0e3e679a54bfa950ee8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/Quasicategory",
"family_id": "quasicategory_iff_innerfibration",
"file_id": "mathlib/Mathlib/AlgebraicTopology/Quasicategory/InnerFibration.lean",
"sample_id": "3681767d9aba4e16656d0211cddaff6efe2e454378f2ebc6b91e0ff3858dab3f"
} | train |
{
"chosen_proof_sha256": "fa7ef57b668670c0961ba6cf31ace864279e791239ac9d2e2fff89a3656780bf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id] | false | {
"chosen_tokens": 18,
"rejected_tokens": 2,
"token_jaccard": 0.071429,
"token_length_ratio": 0.111111
} | sorry_admit_sorryAx_proof_hole | sorry | 873e10cdbc89dae4fa9f90d08aa80a42d79fac437736f6bdb0b7aeb24be1f4f3 | 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.Products.Basic
Namespace:
CategoryTheory.Bifunctor
Local context:
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apac... | 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/Products",
"family_id": "diagonal",
"file_id": "mathlib/Mathlib/CategoryTheory/Products/Bifunctor.lean",
"sample_id": "f4231e8dda39dee1bbddb1e4ffb21d16d698adcaa0cf76876d45ddbd2a042b9f"
} | train |
{
"chosen_proof_sha256": "73a3ff17197caa13c9881ebe96a98ed8cb7dce96f5c0218437447afab1f809f0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5f156dd91316fb432ab42faa62bdffc2ee... | by
refine Finset.measurable_sup' _ ?_
simpa [Finset.mem_range] | true | {
"chosen_tokens": 14,
"rejected_tokens": 5,
"token_jaccard": 0.230769,
"token_length_ratio": 0.357143
} | circular_dependency | circular_dependency | 873f14f3f7947175f8af1db0fe704907bb19351becb93343e19007a26929a973 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Measure.AEMeasurable
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 licens... | by
exact Finset.measurable_range_sup | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "8dd5f00dd49941eface0a084621b7a30d227ef1957b3e3a428fa2a857c379b66",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Order",
"family_id": "finset",
"file_id": "mathlib/Mathlib/MeasureTheory/Order/Lattice.lean",
"sample_id": "2d8dea88dd355420151a905a1257de8aac829b897e12eb600a797e8418da7629"
} | train |
{
"chosen_proof_sha256": "6ef68f198f9c3ddb8b2e403677d5bebc09f01883ade66189972cc6ad03aa6c3a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3b80855cb929e86418fc523ac9ee1b0eb5... | by
let f := (lmul R S).toLinearMap.prodMap (lmul R T).toLinearMap
have : (lmul R (S × T)).toLinearMap = (prodMapLinear R S T S T R).comp f :=
LinearMap.ext₂ Prod.mul_def
simp_rw [trace, this]
exact trace_prodMap' _ _ | true | {
"chosen_tokens": 64,
"rejected_tokens": 5,
"token_jaccard": 0.085714,
"token_length_ratio": 0.078125
} | wrong_namespace_qualification | wrong_namespace_qualification | 875627d90e3c53e50284bae13a69a6d892700a2436ccefeb02a6a4739805a54c | 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.FiniteDimensional.Lemmas
public import Mathlib.LinearAlgebra.Matrix.BilinearForm
public import Mathlib.LinearAlgebra.Trace
Namespace:
Algebra
L... | by
exact OProverMissing.missing_5cac71067938 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "8e8679bdeed7ea13ff2a32ba7670c633a64d5bad144fbb9581b064a7ff435d6e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Trace",
"family_id": "trace_prod_apply",
"file_id": "mathlib/Mathlib/RingTheory/Trace/Defs.lean",
"sample_id": "5cac71067938ac4bef77108bfac10db01fab6c38979ad732ebaa5982c97b9231"
} | train |
{
"chosen_proof_sha256": "bcf496a337f71ad6a5e3548ca780931dad5f52b6127e92076ca6f453d0ce3736",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a4baaccefd765a94402a3364d09d4e062... | by
refine le_trans (measure_iUnion_le _) (le_trans
(ENNReal.tsum_le_tsum <| notConvergentSeqLTIndex_spec (half_pos hε) hf hsm hs hfg) ?_)
simp_rw [ENNReal.ofReal_mul (half_pos hε).le]
rw [ENNReal.tsum_mul_left, ← ENNReal.ofReal_tsum_of_nonneg, inv_eq_one_div, tsum_geometric_two,
← ENNReal.ofReal_mul (half... | true | {
"chosen_tokens": 86,
"rejected_tokens": 3,
"token_jaccard": 0.045455,
"token_length_ratio": 0.034884
} | unknown_identifier | unknown_identifier | 875e4c8fc4e9bde14b4fa45ddc1b7519936c4af9f6fdb067008dcd04e3d384cb | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
Namespace:
MeasureTheory.Egorov
Local context:
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Releas... | by
exact __oprover_missing_444126766a6d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "042283deed19845832ef4975582f82762a28e3551c82a6b4088f02f15fb26016",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Function",
"family_id": "measure_iunionnotconvergentseq",
"file_id": "mathlib/Mathlib/MeasureTheory/Function/Egorov.lean",
"sample_id": "444126766a6d3867417667c40dd37f00d68870fb70c1031516cd452dd27d2227"
} | train |
{
"chosen_proof_sha256": "a2aaf585a4a9909c497f3f34f264608d1efd5c7508490b5e6cf98acb7351c7d2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
constructor
ext x'
simp_rw [LieIdeal.mem_killingCompl, LieModule.traceForm_comm]
refine ⟨fun hx' ↦ ?_, fun hx y _ ↦ hx ▸ LinearMap.map_zero₂ (killingForm R L') y⟩
suffices e.symm x' ∈ LinearMap.ker (killingForm R L) by
rw [IsKilling.ker_killingForm_eq_bot] at this
simpa [map_zero] using (e : L ≃ₗ[R... | false | {
"chosen_tokens": 130,
"rejected_tokens": 3,
"token_jaccard": 0.029851,
"token_length_ratio": 0.023077
} | by_exact_placeholder | exact? | 8798cd3cd389a2a9689d5042e6452c921a933fa3f97e7e4f093d31d91b7ea500 | 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.InvariantForm
public import Mathlib.Algebra.Lie.Semisimple.Basic
public import Mathlib.Algebra.Lie.TraceForm
Namespace:
LieAlgebra
Local context:... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Lie",
"family_id": "iskilling_of_equiv",
"file_id": "mathlib/Mathlib/Algebra/Lie/Killing.lean",
"sample_id": "e0c5330d2514224d882fef475861bb09dfae3c5924cf8a420d81d8d903a10d54"
} | train |
{
"chosen_proof_sha256": "31a79f6998a821c6e1751ba23882d5c421febc08424475fbe1cf68e517fa33e6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "51957d7dd68893e414fd4283c1aee53565... | by
cases L <;> grind | false | {
"chosen_tokens": 7,
"rejected_tokens": 11,
"token_jaccard": 0.636364,
"token_length_ratio": 1.571429
} | missing_wrong_import | import | 87a1d22ecb84c0459be5625941b2b527076649799dd5536d564fc1290f5586a4 | 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.SimplexCategory.GeneratorsRelations.EpiMono
Namespace:
SimplexCategoryGenRel
Local context:
/-
Copyright (c) 2025 Robin Carlier. All rights... | import Mathlib.WrongImport
by
cases L <;> grind | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/SimplexCategory",
"family_id": "isadmissible_cons_iff",
"file_id": "mathlib/Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/NormalForms.lean",
"sample_id": "6c68bdabc3643e303bfd886e5b6ece2de0d41819d4c4fb2cea020fa9d7f7d7c8"
} | train |
{
"chosen_proof_sha256": "fcfc00391c408d2d6a402d4b95c161c6a253181ebc839f73404ed0063d916643",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
refine ⟨fun m hm ↦ ?_⟩
obtain ⟨m', hm'⟩ := h ⟨m, hm.isPrime⟩
have : m = Ideal.comap (algebraMap A B) m'.asIdeal := by
rw [← PrimeSpectrum.comap_asIdeal (algebraMap A B) m', hm']
rw [Ideal.smul_top_eq_map, this]
exact (Submodule.restrictScalars_eq_top_iff _ _ _).ne.mpr
fun top ↦ m'.isPrime.ne_top <|... | false | {
"chosen_tokens": 98,
"rejected_tokens": 3,
"token_jaccard": 0.039216,
"token_length_ratio": 0.030612
} | by_exact_placeholder | placeholder | 87d5dc0d0e6ffd3a8508426afc710d2fb51b8f53a4e84f33b63cf1fd5755e54d | 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.Flat.FaithfullyFlat.Basic
public import Mathlib.RingTheory.Ideal.GoingUp
public import Mathlib.RingTheory.Spectrum.Prime.RingHom
Namespace:
(root)
... | 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/Flat",
"family_id": "module",
"file_id": "mathlib/Mathlib/RingTheory/Flat/FaithfullyFlat/Algebra.lean",
"sample_id": "662af12bfb4b038d7217e684e813afa0288e30d1adb398ad43bc110a38cae151"
} | train |
{
"chosen_proof_sha256": "2c01520ab51bc25fbe7980ca7811c6f4fd2614f49e05e36aaa18fc8557eeaddd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "19062733be5acb02f6c35df456458662dd... | by
rw [modEq_comm]
simp_rw [modEq_iff_eq_add_zsmul, QuotientAddGroup.eq_iff_sub_mem, AddSubgroup.mem_zmultiples_iff,
eq_sub_iff_add_eq', eq_comm] | false | {
"chosen_tokens": 21,
"rejected_tokens": 28,
"token_jaccard": 0.882353,
"token_length_ratio": 1.333333
} | markdown_preamble_trailing | fence | 88242f0ca172f0c6e3b63b8465baeb6b1ce4b73627b9cda529177aaf51922138 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.ModEq
public import Mathlib.GroupTheory.QuotientGroup.Defs
public import Mathlib.Algebra.Group.Subgroup.ZPowers.Basic
Namespace:
AddCommGroup
L... | ```lean
by
rw [modEq_comm]
simp_rw [modEq_iff_eq_add_zsmul, QuotientAddGroup.eq_iff_sub_mem, AddSubgroup.mem_zmultiples_iff,
eq_sub_iff_add_eq', eq_comm]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "GroupTheory/QuotientGroup",
"family_id": "modeq_iff_eq_mod_zmultiples",
"file_id": "mathlib/Mathlib/GroupTheory/QuotientGroup/ModEq.lean",
"sample_id": "f57b40da55b42c3f0a2ab8a4512907a91408e6b8713cf272d9f3041b94ceb710"
} | train |
{
"chosen_proof_sha256": "5bc8e347ea35f5e54b983043b84f723d4bb439695fcffd6e054bbe0bc3c13d4f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
classical
constructor
· simp only [basis, TwoSidedIdeal.le_iff, TwoSidedIdeal.coe_mk', setOf_subset_setOf]
intro h
constructor
· intro x hx
have (d' : _) : coeff d' (C (σ := σ) x) ∈ J := by
rw [coeff_C]; split_ifs <;> [exact hx; exact J.zero_mem]
simpa using h (C x) (fun _ _ ↦ t... | false | {
"chosen_tokens": 185,
"rejected_tokens": 5,
"token_jaccard": 0.037975,
"token_length_ratio": 0.027027
} | sorry_admit_sorryAx_proof_hole | sorryAx | 8895a94106f673784bd151c69008c029eae1eb9492a55ca89c5fc9d639718115 | 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.Finsupp.Interval
public import Mathlib.RingTheory.Ideal.Quotient.Defs
public import Mathlib.RingTheory.MvPowerSeries.PiTopology
public import Mathlib.Topo... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/MvPowerSeries",
"family_id": "basis_le_iff",
"file_id": "mathlib/Mathlib/RingTheory/MvPowerSeries/LinearTopology.lean",
"sample_id": "0ccc7aefbf796c9bc413bd34f90036d969463ddb4ebea0b7afe1f034f8ff048d"
} | train |
{
"chosen_proof_sha256": "cc4265fa4a32e1dda64a7dec7b903dcc9cceebb2f55f6e8ee131b8ddefaf2f73",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
have : Ideal.span ({a} : Set α) ≠ ⊤ := by
intro H
rw [Ideal.span_singleton_eq_top] at H
contradiction
rcases Ideal.exists_le_maximal _ this with ⟨I, Imax, H⟩
use I, Imax
apply H
apply Ideal.subset_span
exact Set.mem_singleton a | false | {
"chosen_tokens": 58,
"rejected_tokens": 5,
"token_jaccard": 0.073171,
"token_length_ratio": 0.086207
} | sorry_admit_sorryAx_proof_hole | sorryAx | 88b4242280a4e3ecff2ef868c4f87eecc65c3991810eb6fef38c6f3944b8251f | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Ideal.Maximal
Namespace:
(root)
Local context:
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as describe... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Ideal",
"family_id": "exists_max_ideal_of_mem_nonunits",
"file_id": "mathlib/Mathlib/RingTheory/Ideal/Nonunits.lean",
"sample_id": "55440f62ede7bbdc77ead5e942a883e7750663cc7506c8d69af849e8b683ebab"
} | train |
{
"chosen_proof_sha256": "fcfc00391c408d2d6a402d4b95c161c6a253181ebc839f73404ed0063d916643",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0c9c2d914957c16bba37e08631ce2d43c8... | by
refine ⟨fun m hm ↦ ?_⟩
obtain ⟨m', hm'⟩ := h ⟨m, hm.isPrime⟩
have : m = Ideal.comap (algebraMap A B) m'.asIdeal := by
rw [← PrimeSpectrum.comap_asIdeal (algebraMap A B) m', hm']
rw [Ideal.smul_top_eq_map, this]
exact (Submodule.restrictScalars_eq_top_iff _ _ _).ne.mpr
fun top ↦ m'.isPrime.ne_top <|... | true | {
"chosen_tokens": 98,
"rejected_tokens": 3,
"token_jaccard": 0.039216,
"token_length_ratio": 0.030612
} | unknown_identifier | unknown_identifier | 88bdbbce4eac1ec873fc556a2bda2fbdb8015b7460d5c9c7a17def4af444bf11 | 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.Flat.FaithfullyFlat.Basic
public import Mathlib.RingTheory.Ideal.GoingUp
public import Mathlib.RingTheory.Spectrum.Prime.RingHom
Namespace:
(root)
... | by
exact __oprover_missing_662af12bfb4b | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "2b9090b64041f4789e1239499a299e6c1e18735db440e142237a7840a9af5716",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Flat",
"family_id": "module",
"file_id": "mathlib/Mathlib/RingTheory/Flat/FaithfullyFlat/Algebra.lean",
"sample_id": "662af12bfb4b038d7217e684e813afa0288e30d1adb398ad43bc110a38cae151"
} | train |
{
"chosen_proof_sha256": "107f7ce7855f94c8cc4f2d6c868596a255522f4e82b0b2ecfe00fd289a468b21",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b8e10a38198c67a1563aa7f4d5d8b714b3... | by
classical exact continuous_piecewise hs hf continuousOn_const | false | {
"chosen_tokens": 7,
"rejected_tokens": 11,
"token_jaccard": 0.636364,
"token_length_ratio": 1.571429
} | missing_wrong_import | import | 891664579de121a6b1e6568a783e33f758946bef5ed5177f721e2e7ae78c4625 | 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.Piecewise
public import Mathlib.Topology.Clopen
Namespace:
(root)
Local context:
/-
Copyright (c) 2... | import Mathlib.WrongImport
by
classical exact continuous_piecewise hs hf continuousOn_const | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/Algebra",
"family_id": "continuous_mulindicator",
"file_id": "mathlib/Mathlib/Topology/Algebra/Indicator.lean",
"sample_id": "a1e841521763274b65199f7d26a8e6592c777a58ce061b317bd17dcc7a139bd0"
} | train |
{
"chosen_proof_sha256": "8dd1af80a36345fc761ec289040691b52374d545c9e7b00957a6cfdf28435160",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "550986a1cd495105b0fc378dff54d8c33e... | 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... | false | {
"chosen_tokens": 102,
"rejected_tokens": 109,
"token_jaccard": 0.9375,
"token_length_ratio": 1.068627
} | markdown_preamble_trailing | fence | 891beb6b22aaad6840193c24a4a6097472ed3d684c2712ba41fec48db9f09a38 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.RatFunc.AsPolynomial
public import Mathlib.RingTheory.EuclideanDomain
public import Mathlib.RingTheory.Localization.FractionRing
public import Math... | ```lean
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),
← P... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "FieldTheory/RatFunc",
"family_id": "intdegree_mul",
"file_id": "mathlib/Mathlib/FieldTheory/RatFunc/Degree.lean",
"sample_id": "48750c221d7b3b3a511e093001b0678d1d19b0dff2bd9c9bd5031532b6945ec8"
} | train |
{
"chosen_proof_sha256": "3431d3593784b843a6d2cba92338ea3a61fb0adeecb9190810dad7f00a815cd9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
induction N with
| zero => simp
| succ N ih => calc
_ ⊆ Ico (a 0) (a N) ∪ Ico (a N) (a (N + 1)) := Ico_subset_Ico_union_Ico
_ ⊆ _ := by simpa [Finset.range_add_one] using
union_subset_union_right (Ico (a N) (a (N + 1))) ih | true | {
"chosen_tokens": 70,
"rejected_tokens": 8,
"token_jaccard": 0.055556,
"token_length_ratio": 0.114286
} | unsolved_goals | unsolved_goals | 89c012e812267083ef624b42bdd20ee6b7dd5ff90df0e63a0ddb196564580a9c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finset.Range
public import Mathlib.Data.Set.Lattice
public import Mathlib.Order.Interval.Set.LinearOrder
Namespace:
(root)
Local context:
/-
Copyright (... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "a4c3ce6ba60bf3a9d3f93399aedb672d5db3ed2f4faa6dd553b16f09485a3e39",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Interval",
"family_id": "ico_subset_biunion_ico",
"file_id": "mathlib/Mathlib/Order/Interval/Set/Union.lean",
"sample_id": "80e66e4358b5a047cdbe0254803df37a3bda8284661c9db0a5c755ad758dbabf"
} | train |
{
"chosen_proof_sha256": "86252bc5f575ff25cd53bd8cf62a6e1c8e93248c81bafebff2879ba682468c22",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
apply toFinsupp_injective
ext x
simp_rw [toFinsupp_equivMapDomain, Finsupp.equivMapDomain_apply, toFinsupp_mapDomain,
Finsupp.mapDomain_equiv_apply] | false | {
"chosen_tokens": 19,
"rejected_tokens": 3,
"token_jaccard": 0.058824,
"token_length_ratio": 0.157895
} | by_exact_placeholder | placeholder | 89d006c93a0e419660c54d6276de49753c9d08567ec42c62f76eea206af3c562 | 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.SkewMonoidAlgebra.Basic
public import Mathlib.Algebra.Module.BigOperators
public import Mathlib.Algebra.Algebra.Equiv
Namespace:
SkewMonoidAlgebra
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": "Algebra/SkewMonoidAlgebra",
"family_id": "equivmapdomain_eq_mapdomain",
"file_id": "mathlib/Mathlib/Algebra/SkewMonoidAlgebra/Lift.lean",
"sample_id": "24bec75d4c990cfaa562f6c11f1e1ed4950885ceeb40f6408a0260da05969bf6"
} | train |
{
"chosen_proof_sha256": "732ed3f932374a610e819e8e805afcfdf95b627321f8ae3cc0670b9127888807",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "fdbdc32f2293a3472c9b736355b763db02... | by
obtain ⟨n, rfl⟩ | ⟨n, rfl⟩ := Int.even_or_odd n
· simp [zpow_add, ← mul_zpow, Int.negOnePow_even _ (Even.add_self n)]
· rw [Int.negOnePow_odd _ (by exact odd_two_mul_add_one n)]
simp [Units.smul_def, zpow_add, Int.two_mul, ← mul_zpow] | true | {
"chosen_tokens": 66,
"rejected_tokens": 3,
"token_jaccard": 0.060606,
"token_length_ratio": 0.045455
} | unknown_identifier | unknown_identifier | 89daae060675c49c88f4af38955d2d9f9516f6dd8a3d81cbb91ce8051de098c4 | 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.Center.Preadditive
public import Mathlib.Algebra.Ring.NegOnePow
Namespace:
CategoryTheory.CatCenter
Local context:
/-
Copyright (c) 2025 Joël ... | by
exact __oprover_missing_5b7bc29de380 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ceb0564fd9b4e3e71593d18a55f0c9d098707d0a92004dcfee2a187859e3feb2",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Center",
"family_id": "app_neg_one_zpow",
"file_id": "mathlib/Mathlib/CategoryTheory/Center/NegOnePow.lean",
"sample_id": "5b7bc29de3804ac7bf45079e461f320c398aef01effbb7aab20e4f7fde575a9c"
} | train |
{
"chosen_proof_sha256": "389a8937a3f2af4740351819b93e71736e399c56154cd81ab2e2e646a0bfda28",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c471d34833437ae28a7e54bc2a541395cc... | by
rw [eq_comm, prod_eq_prod_of_nonempty h, eq_comm, @eq_comm _ eY'] | true | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.166667,
"token_length_ratio": 0.2
} | circular_dependency | circular_dependency | 89eaf3d120f1e250a0eddbad69fb7548a43725362bc820609cbf420a6cbb0313 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.OpenPartialHomeomorph.Composition
Namespace:
OpenPartialHomeomorph
Local context:
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Rele... | by
exact prod_eq_prod_of_nonempty | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "bd5c8c099274d8353551f9dcafe15d7e0a8126aa044c206f751796988aaa77cd",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/OpenPartialHomeomorph",
"family_id": "prod_eq_prod_of_nonempty",
"file_id": "mathlib/Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean",
"sample_id": "b0b58afdb4cef1d362a08061526d43713fac33d2fd2c151db38d446b2e4f58cd"
} | train |
{
"chosen_proof_sha256": "d4554f84e387bbee7a732a6c3f4886d4878a68fc65f7eb3a496b2c61095b621d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "190a0c5110351f1f4b9f147746258119b2... | by
simp [-objD, -obj_X] | true | {
"chosen_tokens": 9,
"rejected_tokens": 2,
"token_jaccard": 0.111111,
"token_length_ratio": 0.222222
} | invalid_tactic | invalid_tactic | 8a09dcdccf5304c3479c0286a781a7159ac02df764e9ef7957985fbf2bcb1d97 | 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.HomologicalComplex
public import Mathlib.AlgebraicTopology.SimplicialObject.Basic
public import Mathlib.CategoryTheory.Abelian.Basic
Namespac... | by
__oprover_invalid_tactic_95cc52a04da2 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "f04dfaf7362ad040bed028810a6c1634fb4f4fcd15f0c11645811b7670ec516a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology",
"family_id": "normalizedmoorecomplex_objd",
"file_id": "mathlib/Mathlib/AlgebraicTopology/MooreComplex.lean",
"sample_id": "95cc52a04da20a5f5c49fe48e2d7c757cb686f4efaddac32474d2f88abd5ceb5"
} | train |
{
"chosen_proof_sha256": "24379af29dd6a4b6bd06c4a471ab0fb06ebf44b4f629deaf2e40a3cfa29440d2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c3ce2be1cb1418609db10af95ac35411db... | by
simp_rw [← card_univ, ← card_powersetCard]
refine (card_le_card fun s hs ↦ mem_biUnion.2 ⟨#s, ?_⟩).trans card_biUnion_le
exact ⟨mem_Iic.2 (mem_shatterer.1 hs).card_le_vcDim, mem_powersetCard_univ.2 rfl⟩ | true | {
"chosen_tokens": 49,
"rejected_tokens": 5,
"token_jaccard": 0.085714,
"token_length_ratio": 0.102041
} | wrong_namespace_qualification | wrong_namespace_qualification | 8a22826dde8043386b093afb4528c86d63d6d40dd8e350f40d4428e07f7685d5 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Combinatorics.SetFamily.Compression.Down
public import Mathlib.Data.Fintype.Powerset
public import Mathlib.Order.Interval.Finset.Nat
public import Mathlib.Alge... | by
exact OProverMissing.missing_c9a34ccd888b | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c61ee19905c3e382b9084cd22061d10972bf819cdea3262d9da34cd7b04b7458",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/SetFamily",
"family_id": "card_shatterer_le_sum_vcdim",
"file_id": "mathlib/Mathlib/Combinatorics/SetFamily/Shatter.lean",
"sample_id": "c9a34ccd888b6cf8e9fb2f8c6ffab0caeeb794e571e1779f7f95f329d2895ac6"
} | train |
{
"chosen_proof_sha256": "badc607969163b460b5422c403e504297c9199325e39a0ffb23b23e6ce679da2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [ncoeff_apply, neg_sub, Int.sub_neg, add_sub_cancel_left] | false | {
"chosen_tokens": 13,
"rejected_tokens": 5,
"token_jaccard": 0.066667,
"token_length_ratio": 0.384615
} | sorry_admit_sorryAx_proof_hole | sorryAx | 8a3859f44eb7fd9803b827128c5877ff2c70585c213b125790fc7e67415fd881 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Vertex.HVertexOperator
public import Mathlib.Data.Int.Interval
Namespace:
VertexOperator
Local context:
/-
Copyright (c) 2024 Scott Carnahan. All rig... | by
exact 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": "Algebra/Vertex",
"family_id": "coeff_eq_ncoeff",
"file_id": "mathlib/Mathlib/Algebra/Vertex/VertexOperator.lean",
"sample_id": "e3c482c18dafac02bf8a0fd6128bd4ffa9ee2fc18b435651f9c2883c55628c8a"
} | train |
{
"chosen_proof_sha256": "65f2254f5d37c07b087f477ca102b136468a343c041fb416c4b1c505f1281550",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "561e76fe0eca757e1f6084da3b5fc3cffc... | by
rw [← linearIndepOn_univ_iff, ← Finset.coe_univ, show @Finset.univ (Fin 2) _ = {0,1} from rfl,
Finset.coe_insert, Finset.coe_singleton, LinearIndepOn.pair_iff _ (by trivial)]
simp | true | {
"chosen_tokens": 48,
"rejected_tokens": 5,
"token_jaccard": 0.058824,
"token_length_ratio": 0.104167
} | wrong_namespace_qualification | wrong_namespace_qualification | 8aa8dc8f9ef2593b0e8bfb2ce23b4b8dc18bb951c1627dca517c5f8d351cbb24 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fin.Tuple.Reflection
public import Mathlib.LinearAlgebra.Dual.Defs
public import Mathlib.LinearAlgebra.Finsupp.SumProd
public import Mathlib.LinearAlgebra... | by
exact OProverMissing.missing_fb692754d646 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "6f2c1da2ad6f74fe22e4613ebb4e0abb7459c858831fe188a01ec8b7102900cb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/LinearIndependent",
"family_id": "linearindependent",
"file_id": "mathlib/Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean",
"sample_id": "fb692754d646fb760ef043b5d79501f41f1d28e50ad2da85d3e2605daffd725b"
} | train |
{
"chosen_proof_sha256": "e1906f3243bcdc4427e534120589c5393cc550f48188ef4c67cb950df5b01321",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "dfc66e404d868f74aa4cfd0272a599ee1a... | by
have h : whiskerLeft F (add F s i a b).hom = _ :=
((whiskeringLeft C D D).obj F).map_preimage _
exact (NatTrans.congr_app h X).trans (by simp) | true | {
"chosen_tokens": 47,
"rejected_tokens": 2,
"token_jaccard": 0.033333,
"token_length_ratio": 0.042553
} | invalid_tactic | invalid_tactic | 8adcc2261a2b57dadc2a1fc442926a1e528501db6aa9f1f3bcc49dfb46f1bbf6 | 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.CommShift
Namespace:
CategoryTheory.HasShift.Induced
Local context:
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under... | by
__oprover_invalid_tactic_2640774717b5 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b2cff663fa0a790bae7ee563db3fad18f8e8ab0c202fd9a6dea0d8738ab1f407",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Shift",
"family_id": "add_hom_app_obj",
"file_id": "mathlib/Mathlib/CategoryTheory/Shift/Induced.lean",
"sample_id": "2640774717b5cc5cf0106fb4ef5d3f69a7c6539d5084db036f9ef862ac01288e"
} | train |
{
"chosen_proof_sha256": "8855b3aa1fcebcd1bfc8f7c993730be08249feb4e40c04eec6356b1306bf4c78",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
cases o₁ <;> rfl | false | {
"chosen_tokens": 8,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.625
} | sorry_admit_sorryAx_proof_hole | sorryAx | 8b8406d66747866baaa5e51558aa08f4676774e0d17e90214cbc8acec45ef18d | 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.Ordering.Basic
public import Mathlib.Order.Defs.Unbundled
Namespace:
Ordering
Local context:
/-
Copyright (c) 2017 Microsoft Corporation. All rights res... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Ordering",
"family_id": "dthen_eq_then",
"file_id": "mathlib/Mathlib/Data/Ordering/Lemmas.lean",
"sample_id": "ced9a8be21b100f92f5cf1865633692067c84047c80974cbc15b46518f499a9c"
} | train |
{
"chosen_proof_sha256": "c3e85a55413d2bdbfb981811de0e7022d287d440d63c91a6c4ae1530e90621b9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e09c9698121c193ab550d6a3eaeef59bc9... | by
simpa [beta, unbeta, coprimes] using mod_eq_of_modEq
((chineseRemainderOfFinset (l[·]) (coprimes (l[·])) Finset.univ
(by simp [coprimes])
(by simpa using Set.pairwise_univ.mpr (pairwise_coprime_coprimes _))).prop i (by simp))
(coprimes_lt _ _) | true | {
"chosen_tokens": 67,
"rejected_tokens": 3,
"token_jaccard": 0.034483,
"token_length_ratio": 0.044776
} | circular_dependency | circular_dependency | 8b944cd418996cd9b3ccadd6de74d1e5f6fe9bc56f5050979dd752ef8ee4e425 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.ModEq
public import Mathlib.Data.Nat.ChineseRemainder
public import Mathlib.Data.Nat.Prime.Defs
public import Mathlib.Data.Nat.Pairing
public import M... | by
exact beta_unbeta_coe | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "6c209f2942e1f9bc05708f42d396566b4571827d54ae1ff7daeee1fbeb68846a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Logic/Godel",
"family_id": "beta_unbeta_coe",
"file_id": "mathlib/Mathlib/Logic/Godel/GodelBetaFunction.lean",
"sample_id": "8dd69de99772e4b60d7d7c33b72ef5655efe8caa0668569bdbff7dc2b9a6a47c"
} | train |
{
"chosen_proof_sha256": "11fbcca75f5b8a4dbad7c2490be392157767b93ac85f1ccad1aff23708a841e9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9d02c26fa4752bc8511f5fa4db9b7d94bf... | by
intro x hx
simp only [Subalgebra.mem_map, Subalgebra.mem_center_iff] at hx ⊢
obtain ⟨b, hb0, rfl⟩ := hx
intro bc
induction bc using TensorProduct.induction_on with
| zero => simp
| tmul b' c => simp [hb0]
| add _ _ _ _ => simp_all [add_mul, mul_add] | true | {
"chosen_tokens": 63,
"rejected_tokens": 7,
"token_jaccard": 0.071429,
"token_length_ratio": 0.111111
} | circular_dependency | circular_dependency | 8c01ab0fce227ae148c346fdedff0f02cfd1761370d03dd7a48f2b4b25984245 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Central.Basic
public import Mathlib.RingTheory.Flat.Basic
public import Mathlib.LinearAlgebra.Basis.VectorSpace
Namespace:
(root)
Local context:
/-
C... | by
exact Algebra.TensorProduct.includeLeft_map_center_le | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "f0b6b6cb9aa1c41f55eac8bd950cdef6b42e1091b3b4e18ade0942d044c2b6eb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Central",
"family_id": "algebra",
"file_id": "mathlib/Mathlib/Algebra/Central/TensorProduct.lean",
"sample_id": "8c84f0871128efc9bcf9d29301aeb6213d25f58a7d79db843c43cab6078afc5a"
} | train |
{
"chosen_proof_sha256": "a14f9a077bdd5b6f8dbacb2f648d28154c7ace639c1cada2afcd1dc3728bb20d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "934a7df138e5d7d9c932ccbe25af684836... | by
constructor
intro Z ⟨g, gc⟩ ⟨h, hc⟩ e
apply_fun InducedCategory.Hom.hom at e
obtain rfl : g = h := ConcreteCategory.hom_ext _ _ fun y ↦ by
rw [← (h₁ y).choose_spec]
simpa using congr(($e).base.hom (h₁ y).choose)
refine SheafedSpace.hom_stalk_ext ⟨g, gc⟩ ⟨g, hc⟩ rfl fun y ↦ ?_
rw [← (h₁ y).choose_... | true | {
"chosen_tokens": 158,
"rejected_tokens": 2,
"token_jaccard": 0.017241,
"token_length_ratio": 0.012658
} | invalid_tactic | invalid_tactic | 8c2e8344bf1a0380244746b8c6fa37aa3f588797df8bed056b9bd6e225a54259 | 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.Geometry.RingedSpace.PresheafedSpace.HasColimits
public import Mathlib.Geometry.RingedSpace.Stalks
public import Mathlib.Topology.Sheaves.Functors
Namespace:
... | by
__oprover_invalid_tactic_36f6292b2bea | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b689190bd4e840c05368912875a34f6d5ff7ae6f3b456f4fe6cde9653bc2d625",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Geometry/RingedSpace",
"family_id": "epi_of_base_surjective_of_stalk_mono",
"file_id": "mathlib/Mathlib/Geometry/RingedSpace/SheafedSpace.lean",
"sample_id": "36f6292b2bea01ba2949cab99df0942afa3d01db01b2c1e30655e18e62dad252"
} | train |
{
"chosen_proof_sha256": "31b27ce5f99a3bf78bf18a3e524b1460389b6e7ac4458283ab4e0eee4a7b4805",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "def358f035864fb5fca9334490b9dd7233... | by
unfold unfolds; rw [corec_eq] | false | {
"chosen_tokens": 8,
"rejected_tokens": 13,
"token_jaccard": 0.615385,
"token_length_ratio": 1.625
} | markdown_preamble_trailing | trailing | 8c671c3dfeac12d7f8141e13a69eb10674a2bc90ac5145510ca7145f9fa044b7 | 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.Stream.Defs
public import Mathlib.Logic.Function.Basic
public import Mathlib.Data.Nat.Basic
public import Mathlib.Tactic.Common
Namespace:
Stream'
Local... | by
unfold unfolds; rw [corec_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": "Data/Stream",
"family_id": "unfolds_eq",
"file_id": "mathlib/Mathlib/Data/Stream/Init.lean",
"sample_id": "767863c4f273ffd20ab08a8048315c1504cb395ceb5b970fd18d5025077b58cc"
} | train |
{
"chosen_proof_sha256": "53c62f34c0746917df7ca2e7b0151f3d1cc2f5a2e7dd8a4133dc4a6cf7d81680",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
intro f g hfg
have : f.toAlgHom = g.toAlgHom := by
apply (hμ.powerBasis K).algHom_ext
rw [AlgEquiv.coe_algHom, AlgEquiv.coe_algHom, powerBasis_gen,
← autToPow_spec K hμ g, ← autToPow_spec K hμ f, hfg]
exact AlgEquiv.coe_algHom_injective this | false | {
"chosen_tokens": 59,
"rejected_tokens": 2,
"token_jaccard": 0.03125,
"token_length_ratio": 0.033898
} | sorry_admit_sorryAx_proof_hole | admit | 8c87b79e6f7a98f4cd2e16b9be5cf4a47374f051214cdebb0452b49782d7433e | 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.Cyclotomic.PrimitiveRoots
public import Mathlib.FieldTheory.PolynomialGaloisGroup
Namespace:
IsPrimitiveRoot
Local context:
/-
Copyright (c) 202... | 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/Cyclotomic",
"family_id": "auttopow_injective",
"file_id": "mathlib/Mathlib/NumberTheory/Cyclotomic/Gal.lean",
"sample_id": "21df6f68f8686f6d7b4ab378fa3f7d2564d582425a9b504a6d726062d4362c74"
} | train |
{
"chosen_proof_sha256": "e6c5467020ffc4f18638257b26b5e8e286b407b28f63974096a991066b7eb6a4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6c168c8eafbed5003e492ac3a2d6e9416d... | by rw [← div_eq_mul_inv, ← h3, mul_div_cancel_right₀ _ h2] | true | {
"chosen_tokens": 14,
"rejected_tokens": 5,
"token_jaccard": 0.0625,
"token_length_ratio": 0.357143
} | wrong_namespace_qualification | wrong_namespace_qualification | 8c986f7e48b70d93115000d38d4dc4d8bc21850cfd23fccd4c99d72400d450a8 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Field.Basic
public import Mathlib.Algebra.Order.Ring.Defs
public import Mathlib.Data.Tree.Basic
public import Mathlib.Tactic.NormNum.Core
public import... | by
exact OProverMissing.missing_6be9545aedd7 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "16c8ed511401576fde6d5955bda2452974a0ec0c08a14a799e3d7590d19f5fab",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/CancelDenoms",
"family_id": "inv_subst",
"file_id": "mathlib/Mathlib/Tactic/CancelDenoms/Core.lean",
"sample_id": "6be9545aedd7e6b94e2cba861b00ad8b06f63ac214c443b7f533604392c9fc8e"
} | train |
{
"chosen_proof_sha256": "5914fc3ed55a874cb691c1f7b7c08b38f9e056106005fb4efe3ad9109dba7f6a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "911fa94098dd26004ff50169dbf98303f1... | by
apply (Ideal.homogeneousCore 𝒜 I).isHomogeneous.isPrime_of_homogeneous_mem_or_mem
· exact ne_top_of_le_ne_top h.ne_top (Ideal.toIdeal_homogeneousCore_le 𝒜 I)
rintro x y hx hy hxy
have H := h.mem_or_mem (Ideal.toIdeal_homogeneousCore_le 𝒜 I hxy)
refine H.imp ?_ ?_
· exact Ideal.mem_homogeneousCore_of_h... | true | {
"chosen_tokens": 66,
"rejected_tokens": 5,
"token_jaccard": 0.088235,
"token_length_ratio": 0.075758
} | wrong_namespace_qualification | wrong_namespace_qualification | 8cd3e6b23c491d3580dc0cacb0eee01ad14116d848301c335c7d6e9ce1ad6f50 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
Namespace:
(root)
Local context:
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.... | by
exact OProverMissing.missing_636539d6c0fe | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "1064428c677591b347891e8362c2b706c0bf93a9a75cdac46c739b114c5cfa38",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/GradedAlgebra",
"family_id": "ideal",
"file_id": "mathlib/Mathlib/RingTheory/GradedAlgebra/Radical.lean",
"sample_id": "636539d6c0febb4c83b00bdc22993462b52f9a53f319750f1f167f122ce30681"
} | train |
{
"chosen_proof_sha256": "0520ccfd6af7e55e37364ebbda9a263098edd7a0ec6ad9af960d91cce2f0e2d9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [← ClassGroup.mk_mk0 (FractionRing A), extendedHom_mk]
variable [IsDedekindDomain B] | false | {
"chosen_tokens": 19,
"rejected_tokens": 3,
"token_jaccard": 0.052632,
"token_length_ratio": 0.157895
} | by_exact_placeholder | placeholder | 8d36f7bf3cc6649e2f83ad2ce23db280d0e98e6a968e00d5505b0f89f38f47ba | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.FractionalIdeal.Extended
public import Mathlib.RingTheory.ClassGroup.Basic
Namespace:
ClassGroup
Local context:
/-
Copyright (c) 2026 Riccardo Bra... | by
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/ClassGroup",
"family_id": "extendedhom_mk",
"file_id": "mathlib/Mathlib/RingTheory/ClassGroup/ExtendedHom.lean",
"sample_id": "b16015c2c8cb752b26e08d47468536e58a1c1a2c7cc95ef8b2185f3dd54359d8"
} | train |
{
"chosen_proof_sha256": "c975c088fa8efb7c0cc15ea499939f263cd071ef49bfcc3a4f706c833af7cab0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
tfae_have 1 → 2 := fun _ ↦ inferInstance
tfae_have 2 → 3 := fun _ ↦ IsAtomic.exists_atom _
tfae_have 3 → 1 := fun ⟨I, hI⟩ ↦ by
have ⟨_, h⟩ := isSimpleRing_iff_isTwoSided_imp.mp ‹IsSimpleRing R›
simp_rw [← isFullyInvariant_iff_isTwoSided] at h
have := isSimpleModule_iff_isAtom.mpr hI
obtain eq |... | false | {
"chosen_tokens": 127,
"rejected_tokens": 2,
"token_jaccard": 0.017241,
"token_length_ratio": 0.015748
} | sorry_admit_sorryAx_proof_hole | sorry | 8d3c4dae441a9770b09caa9f889a65bafc5d6c877c079cada1fe2d4a603cb18e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
public import Mathlib.RingTheory.FiniteLength
public import Mathlib.RingTheory.SimpleModule.Isotypic
public import Mathli... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/SimpleModule",
"family_id": "issimplering",
"file_id": "mathlib/Mathlib/RingTheory/SimpleModule/WedderburnArtin.lean",
"sample_id": "e7de025d50f83819819386b2f880ed49a8eebee597e932160cded70716ac05cb"
} | train |
{
"chosen_proof_sha256": "eba6e30ed08f1c8872a40c988023ca98c14ec7e63590064e2ee0d7b74b74430f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c18ef6777010a60698f317842829f54e2c... | 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')) ... | false | {
"chosen_tokens": 263,
"rejected_tokens": 268,
"token_jaccard": 0.941176,
"token_length_ratio": 1.019011
} | markdown_preamble_trailing | preamble | 8d46d352b0acc02d5fd7931b9809cc9af7987e9fb76bbc152105915aa41d29b1 | 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.DList.Basic
public import Mathlib.Data.WSeq.Basic
Namespace:
Stream'.WSeq
Local context:
/-
Copyright (c) 2017 Microsoft Corporation. All rights reser... | Here is the proof:
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') => Su... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/WSeq",
"family_id": "length_eq_map",
"file_id": "mathlib/Mathlib/Data/WSeq/Defs.lean",
"sample_id": "382d9c391f850ba4635e3d83c4c92384a1fa17b247df5a67f6099368f1582fb9"
} | train |
{
"chosen_proof_sha256": "097944a1a41dd54c4d08892020794d36ebd2632227e02e3aff759ac09d3dd6c6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
haveI : Nonempty ι := Fintype.card_pos_iff.mp (hn.symm ▸ Nat.succ_pos _)
haveI := Classical.decEq ι
-- Choose an arbitrary element `x : ι` and `y : t x`.
let x := Classical.arbitrary ι
have tx_ne : (t x).Nonempty := by
rw [← Finset.card_pos]
calc
0 < 1 := Nat.one_pos
_ ≤ #(.biUnion {x} ... | false | {
"chosen_tokens": 367,
"rejected_tokens": 2,
"token_jaccard": 0.008197,
"token_length_ratio": 0.00545
} | sorry_admit_sorryAx_proof_hole | admit | 8d515ec863331c8a96979a98637ec693b7094aa681b6362d5a27056376a0be92 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Basic
public import Mathlib.Data.Fintype.Powerset
public import Mathlib.Data.Set.Finite.Basic
Namespace:
HallMarriageTheorem
Local context:
/-
C... | 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": "Combinatorics/Hall",
"family_id": "hall_hard_inductive_step_a",
"file_id": "mathlib/Mathlib/Combinatorics/Hall/Finite.lean",
"sample_id": "b15d779c153730a548af1bcb1242b64da3cdb5f65c04ae8beeb21d386329dcf8"
} | train |
{
"chosen_proof_sha256": "886ef528de3066c5206bdbcb8b42b2d7a0a7e9340445f6b246cca5e2a1bdcafc",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
rw [smul_Spec_def, smul_Spec_def]
simp_rw [smul_restrictAppIso_hom_apply, ← ConcreteCategory.comp_apply, Category.assoc]
have :
f ≫ (ΓSpecIso S).inv ≫ (Spec S).presheaf.map U.leTop.op ≫ (Hom.appIso (Spec.map f) U).inv =
(ΓSpecIso R).inv ≫ (Spec R).presheaf.map (Spec.map f ''ᵁ U).leTop.op := by
... | false | {
"chosen_tokens": 108,
"rejected_tokens": 5,
"token_jaccard": 0.021277,
"token_length_ratio": 0.046296
} | sorry_admit_sorryAx_proof_hole | sorryAx | 8d689755e613a7e5178eb46b6d8643d68c07df78ecf69a3bbd5905335e069127 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Category.ModuleCat.Localization
public import Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent
public import Mathlib.AlgebraicGeometry.AffineSche... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry/Modules",
"family_id": "restrictappiso_smul_spec",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/Modules/Tilde.lean",
"sample_id": "cfdd53edfe1acc3d2b51c43a1c120daa3d309e493c2f938aa9d41f5faeec220c"
} | train |
{
"chosen_proof_sha256": "e1fc953f2e811d1c999db981871450abfdfcde88bd7349db65e45507d796d374",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
let ℬ := {s | s ∈ 𝒜} ∪ {s | sᶜ ∈ 𝒜}
have count_ℬ : ℬ.Countable := by
apply h.union
have : compl '' 𝒜 = {s | sᶜ ∈ 𝒜} := by
ext s
simpa using ⟨fun ⟨x, x_mem, hx⟩ ↦ by simp [← hx, x_mem], fun hs ↦ ⟨sᶜ, hs, by simp⟩⟩
exact this ▸ h.image compl
let f : Set (Set (Set α)) → Set α := fun A ... | true | {
"chosen_tokens": 246,
"rejected_tokens": 8,
"token_jaccard": 0.051948,
"token_length_ratio": 0.03252
} | unsolved_goals | unsolved_goals | 8d73cfb0505474da0386e5518697b1a4a0db5091c9bdd21d27ee3ef02bef68ae | 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.Finite.Prod
public import Mathlib.MeasureTheory.SetSemiring
Namespace:
MeasureTheory
Local context:
/-
Copyright (c) 2024 Etienne Marion. All rights res... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "9e97f920a60e2743317f63040a29640c46c1dce68696b1498e54be6c66ee1018",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory",
"family_id": "countable_generatesetalgebra",
"file_id": "mathlib/Mathlib/MeasureTheory/SetAlgebra.lean",
"sample_id": "c9b478f786784f9ac4583d69de5a7f68ad959e59303de1658982ea618e1c423e"
} | train |
{
"chosen_proof_sha256": "38369068946cc9341941259052323d13d916cefa46cf4a0009f06b6990667aa7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "73a3cbacd26aabe1c9fcecf97006df24ef... | by
rw [basis_apply_powersetCard, exteriorPower.basis_apply, ιMulti_family_apply_coe] | false | {
"chosen_tokens": 12,
"rejected_tokens": 19,
"token_jaccard": 0.846154,
"token_length_ratio": 1.583333
} | markdown_preamble_trailing | fence | 8deff741aa8c675df5047b3166efd52dedc66aea6e82d8e13d7059f74a8f3ed4 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.ExteriorAlgebra.Grading
public import Mathlib.LinearAlgebra.ExteriorPower.Basis
Namespace:
ExteriorAlgebra
Local context:
/-
Copyright (c) 2026... | ```lean
by
rw [basis_apply_powersetCard, exteriorPower.basis_apply, ιMulti_family_apply_coe]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/ExteriorAlgebra",
"family_id": "basis_eq_coe_basis",
"file_id": "mathlib/Mathlib/LinearAlgebra/ExteriorAlgebra/Basis.lean",
"sample_id": "dfb013882af3dab10b8a5b0e2b5f55bfe7bbceeec4f48cee972a6e94a5340486"
} | train |
{
"chosen_proof_sha256": "8269b551fa1e3baafeb590db545d184ccf387498c0689c5533fadd958fae33b7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
let adj := mapPullbackAdj g
let U : Over Z := Over.mk (φ₁ ≫ snd f g)
let φ₁' : U ⟶ (pullback g).obj (Over.mk f) := Over.homMk φ₁
let φ₂' : U ⟶ (pullback g).obj (Over.mk f) := Over.homMk φ₂ (by simpa using! h₂.symm)
have : φ₁' = φ₂' := by
apply (adj.homEquiv U _).symm.injective
apply (Over.forget X)... | false | {
"chosen_tokens": 126,
"rejected_tokens": 3,
"token_jaccard": 0.044444,
"token_length_ratio": 0.02381
} | by_exact_placeholder | exact? | 8df32fd3bb374f85976cd0159e1cf56fe77b4f01aa3709430427890e65319927 | 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.Comma.Over.Pullback
public import Mathlib.CategoryTheory.Monoidal.Cartesian.Basic
public import Mathlib.CategoryTheory.Adjunction.Unique
public ... | 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/LocallyCartesianClosed",
"family_id": "hom_ext",
"file_id": "mathlib/Mathlib/CategoryTheory/LocallyCartesianClosed/ChosenPullbacksAlong.lean",
"sample_id": "56dfb71d32b180653fe5db01d6101ad74566b666462b27c7889ed2d552e18dc1"
} | train |
{
"chosen_proof_sha256": "d56cb08de95550c2d5c8c73e4133b8fd22dbbf83f22597912cb3044a1cf97d24",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1afe6ea91860f1c986d85a12f7c2ff36c9... | by
simp_rw [IsTranscendenceBasis, algebraicIndependent_equiv, EquivLike.range_comp]
alias ⟨_, IsTranscendenceBasis.comp_equiv⟩ := isTranscendenceBasis_equiv | true | {
"chosen_tokens": 21,
"rejected_tokens": 3,
"token_jaccard": 0.111111,
"token_length_ratio": 0.142857
} | circular_dependency | circular_dependency | 8ea0c3702e806018da88c6a9f44d0d9352283c826d2b73924e83c4f8e50cd862 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MvPolynomial.CommRing
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as ... | by
exact isTranscendenceBasis_equiv | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "349c187611a38322513dd8291f4544125dc67d61a1c0a92c78b256e97edc90b3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/AlgebraicIndependent",
"family_id": "istranscendencebasis_equiv",
"file_id": "mathlib/Mathlib/RingTheory/AlgebraicIndependent/Defs.lean",
"sample_id": "f15d582cb509e0a13008bd362768f09d19cdeaf2278912231d988aa0bbeaa66e"
} | train |
{
"chosen_proof_sha256": "4d69ce6ddd118e205dc838354a84f5c72260ded77b2d93b7242fcdda08dda204",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e904f2b2bc2150eaeb5f082549806fb4c5... | by
rwa [← hasPointwiseRightDerivedFunctorAt_iff F L W] | true | {
"chosen_tokens": 9,
"rejected_tokens": 2,
"token_jaccard": 0.1,
"token_length_ratio": 0.222222
} | invalid_tactic | invalid_tactic | 8ecb61d91a32633b256de9adaf691ff844d6ae55a7079a39dc61fb75b60575a9 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Functor.Derived.RightDerived
public import Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
public import Mathlib.CategoryTheory.Localizati... | by
__oprover_invalid_tactic_039e1da8660a | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "580e8f348462902d745bc6e1c9348bd01aa0463216dab73ed3236bdfde086779",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Functor",
"family_id": "haspointwiserightderivedfunctorat",
"file_id": "mathlib/Mathlib/CategoryTheory/Functor/Derived/PointwiseRightDerived.lean",
"sample_id": "039e1da8660af549f0b84de8a8d3a1a21828f8928f6164e5d53b92bab06d1d90"
} | train |
{
"chosen_proof_sha256": "9c3779b7663f368283b7a9eee3adc6092b2e62b6c24d85d914985cdd33c9ca16",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "68abd3e690faef28c6d313fc42ae55322e... | 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) ... | true | {
"chosen_tokens": 195,
"rejected_tokens": 2,
"token_jaccard": 0.016667,
"token_length_ratio": 0.010256
} | invalid_tactic | invalid_tactic | 8ee48d16076ce0efbb35cc9651a3611904e557ea2382155c61a34f3275f07f91 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Types.Coproducts
public import Mathlib.CategoryTheory.Limits.Types.Products
public import Mathlib.CategoryTheory.Limits.Types.Pullbacks
p... | by
__oprover_invalid_tactic_b7454ed33b70 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "08f75b99c3c4c91ed068afee8bd658ec64932b7b66848b1fb33a3c1e788bb012",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "d0f64a8fe631e03ccedfddafbaffe13122e5547990701f69b60f4ed018241b86",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
ext f : 1
apply (Matrix.stdBasis _ _ _).ext
intro ⟨i, j⟩
simp only [LinearMap.coe_comp, LinearMap.coe_mk, AddHom.coe_mk, Function.comp_apply, map_sum,
map_smul, stdBasis_eq_single, LinearMap.coe_sum, Finset.sum_apply,
LinearMap.smul_apply, LinearMap.id_coe, id_eq]
ext k l
simp [sum_apply, Matrix.... | false | {
"chosen_tokens": 84,
"rejected_tokens": 2,
"token_jaccard": 0.022222,
"token_length_ratio": 0.02381
} | by_exact_placeholder | by? | 8f5d730d1adf0f3a3e57d7755282e4455eb1c30aafabc0acc72e5afc29499a07 | 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.Azumaya.Defs
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Yunzhou Xie. All righ... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Azumaya",
"family_id": "alghom",
"file_id": "mathlib/Mathlib/Algebra/Azumaya/Matrix.lean",
"sample_id": "fbfa2ca2104fb09021e16e7f3542762a6ce83fb6ca653dc82a4df2bcb75e0375"
} | train |
{
"chosen_proof_sha256": "bb5993596ab6f0322c25fd89be3f755c85b89f6e4d2f8ad57eb17900f2915564",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "28000438c5c8b56a4e4bd3586216319d20... | by
ext; simp only [add_def, smul_def, _root_.add_smul] | true | {
"chosen_tokens": 14,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.214286
} | unknown_identifier | unknown_identifier | 8f5e88ad3a6c2cdd8f8cb9e68a3aafaa3f9c7e9367a418a9866dde032e626ba5 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorProduct.RightExactness
public import Mathlib.RingTheory.Congruence.Hom
public import Mathlib.RingTheory.FiniteType
public import Mathlib.Ri... | by
exact __oprover_missing_4aa461ab228e | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "75f60bd80f07d9be411f0e03023ac57fd7a647f3128712d968d0a447d00c9ca9",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/PolynomialLaw",
"family_id": "add_smul",
"file_id": "mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean",
"sample_id": "4aa461ab228ed2d6d40d0634d68fa0428d1672b0b417d2b773b3a8831f202fa2"
} | train |
{
"chosen_proof_sha256": "19a0b796b5a123aa0fa4da0bb1ec6d1209352b44dfd80eed24e61f22f0837fb5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
refine measure_mono_null ?_ hs.ordConnected.null_frontier
rw [← closure_sdiff_interior, hs.interior_eq_empty, sdiff_empty]
exact subset_closure | true | {
"chosen_tokens": 23,
"rejected_tokens": 8,
"token_jaccard": 0.04,
"token_length_ratio": 0.347826
} | unsolved_goals | unsolved_goals | 8fc52e7a2c60288ccc21487b9ec9b76463104f940224ac42d05a03ab78a23f83 | 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.Order.UpperLower
public import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
public import Mathlib.Topology.Order.DenselyOrdered
Names... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "6584f9190fc28c49db534da1daea2b8245c9b0b6d90bde339b735a88af7d1b63",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Order",
"family_id": "isantichain",
"file_id": "mathlib/Mathlib/MeasureTheory/Order/UpperLower.lean",
"sample_id": "0a46fac18823464095b42916006166b41806c313503a9332eebd632a234896e3"
} | train |
{
"chosen_proof_sha256": "ea6d6b98ce6bb5df330d0c3825d495329a7a811e1188283f31f2efe51dfb0168",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f06c65161176fa9df19e9c48097892fb38... | by
rw [← LinearMap.comp_assoc, ← LinearMap.comp_assoc, TensorProduct.map_map_comp_assoc_eq] | false | {
"chosen_tokens": 17,
"rejected_tokens": 24,
"token_jaccard": 0.846154,
"token_length_ratio": 1.411765
} | markdown_preamble_trailing | fence | 9028b0f52f1a3867e2d5f02aa6188dab55c13c079d81060736539aec15aca578 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.TensorProduct.Tower
public import Mathlib.RingTheory.Coalgebra.Basic
import Mathlib.Tactic.Attr.Register
Namespace:
CoassocSimps
Local context:... | ```lean
by
rw [← LinearMap.comp_assoc, ← LinearMap.comp_assoc, TensorProduct.map_map_comp_assoc_eq]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Coalgebra",
"family_id": "tensorproduct",
"file_id": "mathlib/Mathlib/RingTheory/Coalgebra/CoassocSimps.lean",
"sample_id": "4ea2d4d2d7f00624026e8f321443e14dc6b69d2bf90adb74245fe472882d93fb"
} | train |
{
"chosen_proof_sha256": "2f2eac7accbcc17b05097bd1c9316c1f6c3d52ca67232fbfd3c0c187a0e1b260",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [← le_div_iff₀ pi_div_two_pos, div_eq_inv_mul (W k) _]
rw [W_eq_integral_sin_pow_div_integral_sin_pow, le_div_iff₀ (integral_sin_pow_pos _)]
convert! integral_sin_pow_succ_le (2 * k + 1)
rw [integral_sin_pow (2 * k)]
simp | false | {
"chosen_tokens": 46,
"rejected_tokens": 2,
"token_jaccard": 0.037037,
"token_length_ratio": 0.043478
} | sorry_admit_sorryAx_proof_hole | sorry | 90cadcbf3987c56f87e9ee007189a10be8229725aad39bf7b70bf8bdb495d4d6 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
Namespace:
Real.Wallis
Local context:
/-
Copyright (c) 2021 Hanting Zhang. All rights reserved.
Released under Apac... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/Real",
"family_id": "le_w",
"file_id": "mathlib/Mathlib/Analysis/Real/Pi/Wallis.lean",
"sample_id": "01224b14a9dc704c2fd71fd678b713d03a65ff9d6ee15bc743185909d5552045"
} | train |
{
"chosen_proof_sha256": "bb5b4eccf3c9035f221ecef92ea8d7966625741540da04e726a173d5b6907beb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3033f816b6132c5bc0a106b5e710753aa1... | by
apply Functor.ext; rotate_left
· intro X
ext i
specialize h i
have := congr_obj h X
simpa
· intro X Y g
funext i
specialize h i
have := congr_hom h g
simpa | true | {
"chosen_tokens": 37,
"rejected_tokens": 3,
"token_jaccard": 0.043478,
"token_length_ratio": 0.081081
} | unknown_identifier | unknown_identifier | 90ddf0958de3b822d1ab6500bc962212a8dde106118f9d93fbe41afe145f71e9 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.EqToHom
public import Mathlib.CategoryTheory.NatIso
public import Mathlib.CategoryTheory.Products.Basic
Namespace:
Functor
Local context:
/-
C... | by
exact __oprover_missing_f1e96e086d10 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "fc9d3852ad7e151c810a9e75948272c407eba1e7140266dd2fa683583bdb5011",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Pi",
"family_id": "pi_ext",
"file_id": "mathlib/Mathlib/CategoryTheory/Pi/Basic.lean",
"sample_id": "f1e96e086d107cbb628f9333e3a0d90e5f01c368a40a55b24a768acd38a2bd9b"
} | train |
{
"chosen_proof_sha256": "343a166fea2e90fed4c3043781c3d8f443aa011a6cdb85e131c437406e24ed78",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6e4084086462a436ac3eef56faeb185418... | by
have : C a • D f = D (C a * f) := by simp
rw [this, C_mul', D.map_smul] | true | {
"chosen_tokens": 29,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.068966
} | invalid_tactic | invalid_tactic | 919c954efc4d20c6c697ccfb4d207fe70728f71ae645df32f1ab67982423d4cd | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.AlgebraMap
public import Mathlib.Algebra.Polynomial.Derivative
public import Mathlib.Algebra.Polynomial.Module.AEval
public import Mathlib.R... | by
__oprover_invalid_tactic_f98433b4daad | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "c8ef849e945657333aeef2466182d2d216545329efc3a0a42a5425090fbe6a45",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Polynomial",
"family_id": "c_smul_derivation_apply",
"file_id": "mathlib/Mathlib/Algebra/Polynomial/Derivation.lean",
"sample_id": "f98433b4daad566e797079ce130b71338bc97d14ae011a049aadb8ae222adbed"
} | train |
{
"chosen_proof_sha256": "9ed2efcc5bc8272d2a70d89cfb0ba3dc939ae6d8c6a39b0504302e5c2afc9b8d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
obtain ⟨x, v₁, v₂⟩ := x
obtain ⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩ := h
simp [Prod.toFun', Prod.invFun', h₁, h₂] | true | {
"chosen_tokens": 53,
"rejected_tokens": 8,
"token_jaccard": 0.115385,
"token_length_ratio": 0.150943
} | unsolved_goals | unsolved_goals | 91d3837e7475eb285102d03d528dcfbdd9784e43a76e6ccb4f83c38f698f1659 | 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.FiberBundle.Basic
Namespace:
Bundle.Trivialization
Local context:
/-
Copyright (c) 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b337a49e0b685a15b078a4f54b9e32486cb59f04899bfda1ec89e61d8da7837d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/FiberBundle",
"family_id": "prod",
"file_id": "mathlib/Mathlib/Topology/FiberBundle/Constructions.lean",
"sample_id": "b765d1892775d5e6e6a62ecd85051d3ac26c69fedc8cb19fe0955539acfca9f3"
} | train |
{
"chosen_proof_sha256": "cc860a95c409479a8d192fe8d26a650d014f625436d7b17247ecda1e72c0ab68",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
induction ψ using Quotient.inductionOn
induction θ using Quotient.inductionOn
simp_rw [← QuotientAddGroup.mk_zsmul, ← QuotientAddGroup.mk_add,
QuotientAddGroup.eq_iff_sub_mem, ← smul_sub, ← sub_sub]
exact AddSubgroup.zsmul_mem_zmultiples_iff_exists_sub_div hz | false | {
"chosen_tokens": 40,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.075
} | by_exact_placeholder | placeholder | 91e894c7c61d4403480af9a1cbbb184b6badf43ec7a37c996284e51aa0e6924a | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Field.Basic
public import Mathlib.Algebra.Order.Group.Unbundled.Int
public import Mathlib.Algebra.Module.NatInt
public import Mathlib.GroupTheory.Quoti... | 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/CharZero",
"family_id": "zmultiples_zsmul_eq_zsmul_iff",
"file_id": "mathlib/Mathlib/Algebra/CharZero/Quotient.lean",
"sample_id": "c6688be02d1db8888f8f73bfb3dc49d8f7954b075b4191fe5618031c2b9977e0"
} | train |
{
"chosen_proof_sha256": "a76aedca09d06687f23aabb71a976948a3f5261d4be2092b4354b20a250ef4d4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d52a3705d55c120cb4dee3283877b5e3a... | by
simp [h]
/- ExProd.equateZero -/ | true | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.25
} | unknown_identifier | unknown_identifier | 92015ca04c744068c1ebbc8e79cb35a3ef189d8c38fedfefedbfc0c6032ab1e2 | 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 __oprover_missing_c124e4499748 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ae447a9876d057381754ebdd1fb19ee91c512b403b660778146a79cec1765eba",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/Algebra",
"family_id": "smul_one_eq_zero",
"file_id": "mathlib/Mathlib/Tactic/Algebra/Lemmas.lean",
"sample_id": "c124e4499748820a17df65d5a170a2889e6ded7f6f676ef140d015398e461d38"
} | train |
{
"chosen_proof_sha256": "b77e7e870da0ed2ed6f8b7ac1969e927be1488555d4e8afb15051655e1aaced2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a5285c68d16fa2a2346f4fd34255ffe7bd... | by
simp [← Function.comp_def, Tendsto, ← map_map, map_inv_atTop] | true | {
"chosen_tokens": 15,
"rejected_tokens": 3,
"token_jaccard": 0.071429,
"token_length_ratio": 0.2
} | unknown_identifier | unknown_identifier | 92515cc23625d70d50c68f9519769e85bca419ac3ec1235b245cdf8e5619d855 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Group.MinMax
public import Mathlib.Order.Filter.AtTopBot.Basic
public import Mathlib.Order.Filter.AtTopBot.Map
public import Mathlib.Order.Filter... | by
exact __oprover_missing_105c96e643ef | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "6640fdd6c5b666a8f20d623043460993fb36c90bd44a5169b972f1d8d873c286",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Order/Filter",
"family_id": "tendsto_comp_inv_attop_iff",
"file_id": "mathlib/Mathlib/Order/Filter/AtTopBot/Group.lean",
"sample_id": "105c96e643efa1f749e66cefffdea4f338def0d05aec0167af11a6685aed3ac3"
} | train |
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