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": "8f3a85b33714251a1a29f92568064949152f5e84c72c953dd09287cb0b6f1455",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [NormNum.IsNat.to_eq h rfl]
exact Nat.cast_nonneg' n | true | {
"chosen_tokens": 16,
"rejected_tokens": 8,
"token_jaccard": 0.05,
"token_length_ratio": 0.5
} | unsolved_goals | unsolved_goals | db11a052999061de3591ca82b17722b270bf5ede64bef6b5f18ad58356889b2a | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Invertible
public import Mathlib.Algebra.Order.Ring.Cast
public import Mathlib.Tactic.HaveI
public import Mathlib.Tactic.NormNum.Core
Namespace:... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "c4daf9cc9bd1142b4482128155f37d52c62741d70069740638b790eab762f2b2",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/Positivity",
"family_id": "nonneg_of_isnat",
"file_id": "mathlib/Mathlib/Tactic/Positivity/Core.lean",
"sample_id": "309747378fad33151e6310726ff0174aa0d05877efaa42bd524d8d5cf49ca2f9"
} | train |
{
"chosen_proof_sha256": "63a4845b3fb713f383d6aefe7ca814d2f2062db02f00a1f2f03bbbb49cf53c38",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
replace h (a m) : Q₁ (a ⊗ₜ m) = Q₂ (a ⊗ₜ m) := by
rw [← mul_one a, ← smul_eq_mul, ← smul_tmul', QuadraticMap.map_smul, QuadraticMap.map_smul, h]
ext x
induction x with
| tmul => simp [h]
| zero => simp
| add x y hx hy =>
have : Q₁.polarBilin = Q₂.polarBilin := by
ext
dsimp [polar]
... | true | {
"chosen_tokens": 129,
"rejected_tokens": 8,
"token_jaccard": 0.071429,
"token_length_ratio": 0.062016
} | unsolved_goals | unsolved_goals | db4a05b23ed844b81836f43fb8fbd24f75cf0303c1f87b67f90d1da7953eeedb | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.BilinearForm.TensorProduct
public import Mathlib.LinearAlgebra.QuadraticForm.Basic
public import Mathlib.Tactic.LinearCombination
Namespace:
(ro... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "8d9a5c1cf44686dc0e3b6d87703a49db13127a36fee0054aad54515bebc20d8a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/QuadraticForm",
"family_id": "basechange_ext",
"file_id": "mathlib/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean",
"sample_id": "2b78fa8807ec50bfc8ab5c93ca7eeaa86e8f4d73fc76cde03273374bc7683948"
} | train |
{
"chosen_proof_sha256": "105da6d71d1fe288f69e471a4b967a3a862bded59f3b72ae1eccbacf108cf793",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
-- Again, it is a "left inverse equals right inverse" argument in the convolution monoid.
apply left_inv_eq_right_inv
(M := Conv A (A ⊗ A))
(a := Δ[A])
· rw [Conv.mul_eq, Conv.one_eq]
simp only [comp_whiskerRight, tensor_whiskerLeft, MonObj.tensorObj.mul_def, Category.assoc,
MonObj.tensorObj.... | false | {
"chosen_tokens": 156,
"rejected_tokens": 2,
"token_jaccard": 0.018519,
"token_length_ratio": 0.012821
} | sorry_admit_sorryAx_proof_hole | admit | db8dd6781955bd68b0bb8039cde37bd6b19e6fff9fe4e80dc5b956676165a089 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Monoidal.Bimon_
public import Mathlib.CategoryTheory.Monoidal.Conv
Namespace:
CategoryTheory.HopfObj
Local context:
/-
Copyright (c) 2024 Lean... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Monoidal",
"family_id": "antipode_comul",
"file_id": "mathlib/Mathlib/CategoryTheory/Monoidal/Hopf_.lean",
"sample_id": "316e9edc7144d415a7e46a6a2a53e203a71e0c03a7d0b39706e889dc27b42fb3"
} | train |
{
"chosen_proof_sha256": "06f4331fcea43b6ff267c69e7aee0a119d3182db477558ac433ba49db7222320",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "818b600996d16659e53d7447c449dd5827... | by
let K := Algebra.adjoin F (Set.range x)
suffices #E = #K by simp [K, this, ← lift_mk_eq'.2 ⟨hx.1.aevalEquiv.toEquiv⟩]
haveI : Algebra.IsAlgebraic K E := hx.isAlgebraic
refine le_antisymm ?_ (mk_le_of_injective Subtype.val_injective)
haveI : Infinite K := hx.1.aevalEquiv.infinite_iff.1 inferInstance
simpa... | false | {
"chosen_tokens": 96,
"rejected_tokens": 101,
"token_jaccard": 0.927273,
"token_length_ratio": 1.052083
} | markdown_preamble_trailing | trailing | db9ce4472b197c521db30f011e5d720dc0be8db8e4611e2bc2a5753d5070d5e5 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.FieldTheory.IntermediateField.Adjoin.Basic
public import Mathlib.FieldTheory.MvRatFunc.Rank
public import Mathlib.RingTheory.Algebraic.Cardinality
public impor... | by
let K := Algebra.adjoin F (Set.range x)
suffices #E = #K by simp [K, this, ← lift_mk_eq'.2 ⟨hx.1.aevalEquiv.toEquiv⟩]
haveI : Algebra.IsAlgebraic K E := hx.isAlgebraic
refine le_antisymm ?_ (mk_le_of_injective Subtype.val_injective)
haveI : Infinite K := hx.1.aevalEquiv.infinite_iff.1 inferInstance
simpa... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/AlgebraicIndependent",
"family_id": "istranscendencebasis",
"file_id": "mathlib/Mathlib/RingTheory/AlgebraicIndependent/RankAndCardinality.lean",
"sample_id": "23f66eb623e6a66dd8750aa398395aca87f38f1c676ee283cf181c130d4fe931"
} | train |
{
"chosen_proof_sha256": "42771a2147cb91bca0f9e576050654f9c7f45762679c01f645ed998108481986",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5eafb8d4f210e0305decd4f2fa8d9481c3... | by
simpa using hκ.irreducible hs <| Std.lt_of_lt_of_le hsp (hφ s) | true | {
"chosen_tokens": 19,
"rejected_tokens": 5,
"token_jaccard": 0.1,
"token_length_ratio": 0.263158
} | wrong_namespace_qualification | wrong_namespace_qualification | dbba5c7c59a953d24dab2804161e961263d55484c906e11436c82d4350505efc | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Probability.Kernel.Composition.Comp
Namespace:
ProbabilityTheory.Kernel
Local context:
/-
Copyright (c) 2025 Bryan Wang. All rights reserved.
Released under ... | by
exact OProverMissing.missing_e2222f0bad75 | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "5cf2e967e2a16fa94f4a10281bcae4682f524618156a8067a2fdf0620fa527bf",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability/Kernel",
"family_id": "isirreducible_of_le_measure",
"file_id": "mathlib/Mathlib/Probability/Kernel/Irreducible.lean",
"sample_id": "e2222f0bad75d2e7deb1fd25fdb3ceb2a09b7f0046dfa22182f45b77e039767c"
} | train |
{
"chosen_proof_sha256": "ffc956ff392a03eea39dc40bd447e1d225bcdcdd1b88caef9ee391f6b7a9878c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
simp [card_of_finrank k V h] | true | {
"chosen_tokens": 8,
"rejected_tokens": 8,
"token_jaccard": 0.071429,
"token_length_ratio": 1
} | unsolved_goals | unsolved_goals | dbbaae38c6c20bfb3aafb81c546ccd10f3b5565e38b0f1cf96ba0a103f4a7d42 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Ring.GeomSum
public import Mathlib.Algebra.GroupWithZero.Units.Fintype
public import Mathlib.Data.Finite.Sum
public import Mathlib.GroupTheory.GroupAct... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "2cc64911b870965fa06b50d4633de8b8db6a00126ef1bc7a0c851098d7f2b369",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Projectivization",
"family_id": "card_of_finrank_two",
"file_id": "mathlib/Mathlib/LinearAlgebra/Projectivization/Cardinality.lean",
"sample_id": "1bdf66f02f800f9f6bc8f802f4eb7010531ea0fed644e6fb319377ffa1bd244a"
} | train |
{
"chosen_proof_sha256": "603b10c8e8d8264cdfe3ea1be9c171071aba17b8e3411dbc942a8512b4e99c56",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "119ff96b176e6f80d0102930408c2c230e... | by
haveI := Fintype.ofFinite ι
haveI := fun i => Fintype.ofFinite (κ i)
classical
refine (Basis.piTensorProduct b).dualBasis.ext (fun p ↦ ?_)
refine (Basis.piTensorProduct b).ext (fun q ↦ ?_)
simp [Finsupp.single_apply, Fintype.prod_ite_zero, ← funext_iff] | false | {
"chosen_tokens": 67,
"rejected_tokens": 72,
"token_jaccard": 0.871795,
"token_length_ratio": 1.074627
} | markdown_preamble_trailing | preamble | dc0ed27cb68b81ab0b7599fb0db7fd3b7effde970d34ae0d96f1004177b2bbff | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Dual.Basis
public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
public import Mathlib.LinearAlgebra.PiTensorProduct.Basis
Namespace:
PiTe... | Here is the proof:
by
haveI := Fintype.ofFinite ι
haveI := fun i => Fintype.ofFinite (κ i)
classical
refine (Basis.piTensorProduct b).dualBasis.ext (fun p ↦ ?_)
refine (Basis.piTensorProduct b).ext (fun q ↦ ?_)
simp [Finsupp.single_apply, Fintype.prod_ite_zero, ← funext_iff] | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/PiTensorProduct",
"family_id": "dualdistrib_dualdistribinvofbasis_left_inverse",
"file_id": "mathlib/Mathlib/LinearAlgebra/PiTensorProduct/Dual.lean",
"sample_id": "57919ee9fe831200d4f5697405ab93efe8a6295ef8776bc2c7eff0fee53e76ad"
} | train |
{
"chosen_proof_sha256": "894118c14dc5e14117bef012c677ec8f7a21a4cdd6ea74ed8d6415838166eb42",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3f8d2316a45406574dc026647834f24f20... | by
by_cases hE : CompleteSpace E; swap
· simp [integral, hE, hasSum_zero]
rw [integral_tsum (fun i ↦ (hF_int i).1)]
· exact (hF_sum.of_norm_bounded fun i ↦ norm_integral_le_integral_norm _).hasSum
have (i : ι) : ∫⁻ a, ‖F i a‖ₑ ∂μ = ‖∫ a, ‖F i a‖ ∂μ‖ₑ := by
dsimp [enorm]
rw [lintegral_coe_eq_integral _... | false | {
"chosen_tokens": 144,
"rejected_tokens": 149,
"token_jaccard": 0.941176,
"token_length_ratio": 1.034722
} | markdown_preamble_trailing | trailing | dc4cc39a6d31e4ef9e7f2a32ed51faa859690c022d0202bd32c864cbc59099c0 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable
public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.Topology.Algebra.IsUni... | by
by_cases hE : CompleteSpace E; swap
· simp [integral, hE, hasSum_zero]
rw [integral_tsum (fun i ↦ (hF_int i).1)]
· exact (hF_sum.of_norm_bounded fun i ↦ norm_integral_le_integral_norm _).hasSum
have (i : ι) : ∫⁻ a, ‖F i a‖ₑ ∂μ = ‖∫ a, ‖F i a‖ ∂μ‖ₑ := by
dsimp [enorm]
rw [lintegral_coe_eq_integral _... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Integral",
"family_id": "hassum_integral_of_summable_integral_norm",
"file_id": "mathlib/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean",
"sample_id": "a4e28351bd9a0b41f4d5ba3c08dc4215d63a0d8b4b1907903fe2be3c09e51fcd"
} | train |
{
"chosen_proof_sha256": "d46e9dc9cd151de5455487d42f3fd890f31162d28298263c77fbfdfebe8161e5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
dsimp [destruct]
rcases f0 : s.1 0 with - | a' <;> intro h
· injection h with h'
rw [← h']
obtain ⟨f, al⟩ := s
apply Subtype.ext
dsimp [think, tail]
rw [← f0]
exact (Stream'.eta f).symm
· contradiction | false | {
"chosen_tokens": 65,
"rejected_tokens": 5,
"token_jaccard": 0.041667,
"token_length_ratio": 0.076923
} | sorry_admit_sorryAx_proof_hole | sorryAx | dc517ef0a9a5c9cd6925f53b5dc447ac7ed56163db9ef3b9325a5e586dbc7fe8 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Nat.Find
public import Mathlib.Data.Stream.Init
public import Mathlib.Logic.Relator
public import Mathlib.Tactic.Common
public import Batteries.Tactic.Lin... | by
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/Seq",
"family_id": "destruct_eq_think",
"file_id": "mathlib/Mathlib/Data/Seq/Computation.lean",
"sample_id": "cf0e248544ee9d6ed6e902aaa1a7849b4ce61aeb489bff40d94a5eee7d981eac"
} | train |
{
"chosen_proof_sha256": "f9aa1f2b0c3ec5667a46cb932e6917f3da85ddecbe4e4201b9fff6f973d74358",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
obtain ⟨a, b, hij⟩ | ⟨a, b, hab⟩ := h
· exact Lex.left _ _ (hr _ _ hij)
· exact Lex.right _ _ (hs _ _ _ hab) | true | {
"chosen_tokens": 46,
"rejected_tokens": 8,
"token_jaccard": 0.071429,
"token_length_ratio": 0.173913
} | unsolved_goals | unsolved_goals | dcc05359c9291761e1844ec50340d7fb715564638a7956ec09a5d91eb1ad8029 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Logic.Function.Defs
public import Mathlib.Order.Defs.Unbundled
public import Batteries.Logic
Namespace:
Sigma
Local context:
/-
Copyright (c) 2021 Yaël Dilli... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "fb5945bfef9761abaa40a4699371e72c7942f202e9d512de263d772c1281314e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Sigma",
"family_id": "lex",
"file_id": "mathlib/Mathlib/Data/Sigma/Lex.lean",
"sample_id": "ca4d4f9887ee3a892e6506dc6023e4d2417fcaf8ede1273a6e1bb19013900ef6"
} | train |
{
"chosen_proof_sha256": "dd729798355787e2f50246686d3074ecb0b708500ef8ae38cb56d8d7dbde9c7f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8657a579acb61ab7d8c3c4f5eccef630ca... | by
classical
induction s using Finset.induction with
| empty => simp
| insert _ _ _ _ => simp_all [lcm_eq_nat_lcm, Nat.factorization_lcm] | false | {
"chosen_tokens": 28,
"rejected_tokens": 35,
"token_jaccard": 0.913043,
"token_length_ratio": 1.25
} | markdown_preamble_trailing | fence | dd2584b875cfc15da3567ad48bc83d7a6e25e9251521b6467ca181fffc6d6c56 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GCDMonoid.Finset
public import Mathlib.Algebra.GCDMonoid.Nat
public import Mathlib.Data.Nat.GCD.Basic
public import Mathlib.RingTheory.Coprime.Lemmas
p... | ```lean
by
classical
induction s using Finset.induction with
| empty => simp
| insert _ _ _ _ => simp_all [lcm_eq_nat_lcm, Nat.factorization_lcm]
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/GCDMonoid",
"family_id": "factorization_lcm",
"file_id": "mathlib/Mathlib/Algebra/GCDMonoid/FinsetLemmas.lean",
"sample_id": "724a8437e52e6581c6973db3a8c052c428831eeb06b3711d37815c86ff48be5a"
} | train |
{
"chosen_proof_sha256": "a1aa8b6078219bed7946cc7376ea4840b9f886018bc53ce22deefb93a518874b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
change c = (ZMod.ringEquivCongr hn) (toFun n g)
rw [← toFun_unique' n g (ZMod.ringEquivCongr hn.symm c)
(fun t ht ↦ by rw [hc t ht, ZMod.ringEquivCongr_val]), ← ZMod.ringEquivCongr_symm hn,
RingEquiv.apply_symm_apply] | true | {
"chosen_tokens": 58,
"rejected_tokens": 8,
"token_jaccard": 0.028571,
"token_length_ratio": 0.137931
} | unsolved_goals | unsolved_goals | dd8ea58cd440957f0c7eb0093a3d20f7f5edee70c52c4597835ab67943ed0a03 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Ring.Aut
public import Mathlib.NumberTheory.Padics.RingHoms
public import Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity
public import Mathlib.Ring... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e97e24884878fe0740831dff521ece2b8373761f634b9ff757f1b8c7bf91c9e7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Cyclotomic",
"family_id": "unique",
"file_id": "mathlib/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean",
"sample_id": "9eef900fc40300308ff1ecb2aed829f9ba2f08a85e294734879bf55ae56f16ca"
} | train |
{
"chosen_proof_sha256": "5e09f885c765d1667ba798f4a88bd6a2385014bfb40b892ea398481ff81d2e8d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "87892b698b5aa0c2157518ba3c76b8916d... | by
-- The key is that φ preserves labels, so edges labelled 's' stay labelled 's'
let e : v ⟶ ι s • v := ⟨s, rfl⟩
-- φ.map e is an edge from φ.obj v, and its label is preserved
have h := (φ.map e).property
-- This says: `ι (φ.map e).val • φ.obj v = φ.obj (ι s • v)`
-- We need to show `(φ.map e).val = s`
h... | false | {
"chosen_tokens": 216,
"rejected_tokens": 221,
"token_jaccard": 0.96,
"token_length_ratio": 1.023148
} | markdown_preamble_trailing | trailing | dd975c851d378a1fee31d4ddf6ca2bccacb73e1cf08593c553fd6e4c99205214 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Action.Defs
public import Mathlib.Algebra.Group.Basic
public import Mathlib.Combinatorics.Quiver.Covering
public import Mathlib.Combinatorics.Qui... | by
-- The key is that φ preserves labels, so edges labelled 's' stay labelled 's'
let e : v ⟶ ι s • v := ⟨s, rfl⟩
-- φ.map e is an edge from φ.obj v, and its label is preserved
have h := (φ.map e).property
-- This says: `ι (φ.map e).val • φ.obj v = φ.obj (ι s • v)`
-- We need to show `(φ.map e).val = s`
h... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/Quiver",
"family_id": "map_smul_of_comp_labelling_eq",
"file_id": "mathlib/Mathlib/Combinatorics/Quiver/Schreier.lean",
"sample_id": "abc072079305390f0bef511c2c60314512d25bfb1b8f705b8e28581cf1707079"
} | train |
{
"chosen_proof_sha256": "e9b659a1781e41131bb39de2a0211b972065e3468a0c84eab21865825ba5a5a7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7f463aa725b7762c153926d14972d9db6f... | by
rw [← Rat.isSquare_intCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg (mod_cast hz),
Rat.cast_intCast] | true | {
"chosen_tokens": 19,
"rejected_tokens": 2,
"token_jaccard": 0.0625,
"token_length_ratio": 0.105263
} | invalid_tactic | invalid_tactic | ddc547b0ff98548a0b9e86d840355a1165de28e12642e09fb746fc783fc63977 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Rat
public import Mathlib.Data.Nat.Prime.Int
public import Mathlib.Data.Rat.Sqrt
public import Mathlib.Analysis.Real.Sqrt
public import Mathlib... | by
__oprover_invalid_tactic_9c259b24270f | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e70eacf945b4856ad5fc128f39c26b5629cbb0279a30afdae702c4445eef0b08",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Real",
"family_id": "irrational_sqrt_intcast_iff_of_nonneg",
"file_id": "mathlib/Mathlib/NumberTheory/Real/Irrational.lean",
"sample_id": "9c259b24270f51c0a7679a42e8272d4047f65364bf40859a29277a457689ac84"
} | train |
{
"chosen_proof_sha256": "679574657f773b5361b65d8bc217854b5dee221b7dde4fd6f07b322f738d2333",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4dbb11ad844b5c9a90a253f6b1ac76bca6... | by
rwa [← map_natCast' _ he, ← map_natCast' _ he, he'.eq_iff, Nat.cast_inj] at h | false | {
"chosen_tokens": 23,
"rejected_tokens": 30,
"token_jaccard": 0.888889,
"token_length_ratio": 1.304348
} | markdown_preamble_trailing | fence | dddc649bdfc68de2bbd52f1dcb780984ea1b5c81c079620c7c277de38638a7d4 | 0 | 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.CharZero.Defs
public import Mathlib.Algebra.Group.Equiv.Defs
public import Mathlib.Data.Nat.Cast.Basic
Namespace:
(root)
Local context:
/-
Copyright ... | ```lean
by
rwa [← map_natCast' _ he, ← map_natCast' _ he, he'.eq_iff, Nat.cast_inj] at h
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/CharZero",
"family_id": "charzero",
"file_id": "mathlib/Mathlib/Algebra/CharZero/AddMonoidHom.lean",
"sample_id": "bf23546e5597548a8a703fba0f21b06b38585a3dc1e1cc164336dbb8b96373a4"
} | train |
{
"chosen_proof_sha256": "594293ce5d84984c59a5d88c876de5c8200038ce21c7d54ee21a2ba975b3d8b6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "10cc88fb02c4baabbf38e38ccad452ff78... | by rw [eq_bot_iff, sup_le_iff]; simp | true | {
"chosen_tokens": 9,
"rejected_tokens": 2,
"token_jaccard": 0.1,
"token_length_ratio": 0.222222
} | invalid_tactic | invalid_tactic | de0341c5cb1994fbaddc899d9e596a84446ab57613bf2618401f9685d46720d9 | 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.BoundedOrder.Basic
public import Mathlib.Order.Lattice
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Rele... | by
__oprover_invalid_tactic_d7b302fa097a | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "416fbce2dfd5d8e4ba5024b99605f69fc05a5ba1901b3b0ffc49248298e43714",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/BoundedOrder",
"family_id": "sup_eq_bot_iff",
"file_id": "mathlib/Mathlib/Order/BoundedOrder/Lattice.lean",
"sample_id": "d7b302fa097a85ad40628a8309be13365dc90a026d8ed805296461111bd052b1"
} | train |
{
"chosen_proof_sha256": "ce223990efe19a3952beb8c48c08eec52484b0eabebe4090119bef4f0e984746",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by simp [mk, ← ofCauchy_add] | false | {
"chosen_tokens": 8,
"rejected_tokens": 2,
"token_jaccard": 0.111111,
"token_length_ratio": 0.25
} | by_exact_placeholder | by? | de276568fab9f8b33a96d4f775d53e9d0853e082c52d3ca3ecb4b8873af38e9c | 0 | 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.CauSeq.Completion
public import Mathlib.Algebra.Order.Ring.Rat
public import Mathlib.Data.Rat.Cast.Defs
Namespace:
Real
Local context:
/-
Copyr... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Real",
"family_id": "mk_add",
"file_id": "mathlib/Mathlib/Data/Real/Basic.lean",
"sample_id": "210dd76ae67ac01eff7a888fdcf07f7b44a5b4586ca30f9a119d28a997965242"
} | train |
{
"chosen_proof_sha256": "dbc5a53bc3d6fc4141510bd9f512049166f30bf6d3c576c98bf05b032824182f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3b4adf8c6ba159c4e6085837800b011851... | by cfc_cont_tac)
(ha : p a := by cfc_tac) (ha' : ∀ i, q i (a i) := by cfc_tac) :
cfc f a = fun i => cfc f (a i) := by
ext i
let φ := Pi.evalStarAlgHom S A i
exact φ.map_cfc f a (by rwa [Pi.spectrum_eq]) (continuous_apply i) ha (ha' i) | true | {
"chosen_tokens": 79,
"rejected_tokens": 2,
"token_jaccard": 0.027778,
"token_length_ratio": 0.025316
} | invalid_tactic | invalid_tactic | de2810f02b36b8cbbe37329db682a3a19741ddb7bda1e0dc5f7b2d6a29e66e8f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique
public import Mathlib.Algebra.Algebra.Spectrum.Pi
public import Mathlib.Algebra.Star.StarAlgHom
Name... | by
__oprover_invalid_tactic_a3ef0314d959 | {
"accepted": false,
"failure_type": "unknown_tactic",
"output_sha256": "1720986c071f477950b363d28ec55e272866775b35ed16b16e36bcd7fc1c58ab",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/CStarAlgebra",
"family_id": "cfc_map_pi",
"file_id": "mathlib/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Pi.lean",
"sample_id": "a3ef0314d959f005739e2ea0a1bbd35c4cb66fa53980fc55488bf2d3ac971efc"
} | train |
{
"chosen_proof_sha256": "97f1a84bedb0ef6995acbe3d016b9163eb9ac17f32bd0890956d8e9bd5498309",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "642f434381ee52651be87c870d54141745... | by
refine (hB.hasLaw I).congr ?_
have : ∀ᵐ ω ∂P, ∀ i : I, B i ω = C i ω := ae_all_iff.2 fun _ ↦ h _
filter_upwards [this] with ω hω using funext fun i ↦ (hω i).symm | true | {
"chosen_tokens": 61,
"rejected_tokens": 3,
"token_jaccard": 0.025641,
"token_length_ratio": 0.04918
} | unknown_identifier | unknown_identifier | de4876aefdf437d3dd811f28c79bc21208d422ddac10f7b577021c0463c5ce68 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily
public import Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Def
public import Mathlib.Probab... | by
exact __oprover_missing_567f3c5c6d20 | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "3854c832d12260f8da7ecf4033b7244980b20bc0f12ef341f19858ebeac6583a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability/BrownianMotion",
"family_id": "isprebrownianreal",
"file_id": "mathlib/Mathlib/Probability/BrownianMotion/Basic.lean",
"sample_id": "567f3c5c6d2031f7fb93f23ec479080605b61b08ad68d9daa949e04606ce6590"
} | train |
{
"chosen_proof_sha256": "cb1bd661524d698c211f8037516f3bb09f8383d70d26a48c78df08d165e667a0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rcases Int.Prime.dvd_mul hp h with hp2 | hpp
· apply Or.intro_left
exact le_antisymm (Nat.le_of_dvd zero_lt_two hp2) (Nat.Prime.two_le hp)
· apply Or.intro_right
rw [sq, Int.natAbs_mul] at hpp
exact or_self_iff.mp ((Nat.Prime.dvd_mul hp).mp hpp) | false | {
"chosen_tokens": 67,
"rejected_tokens": 3,
"token_jaccard": 0.057143,
"token_length_ratio": 0.044776
} | by_exact_placeholder | exact? | de4d9b7748b2272e998ef79a3cffc51621db31412985917960e08b3df64ad832 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.EuclideanDomain.Basic
public import Mathlib.Algebra.EuclideanDomain.Int
public import Mathlib.Algebra.GCDMonoid.Nat
public import Mathlib.Data.Nat.Prim... | by
exact? | {
"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/Int",
"family_id": "prime_two_or_dvd_of_dvd_two_mul_pow_self_two",
"file_id": "mathlib/Mathlib/RingTheory/Int/Basic.lean",
"sample_id": "8c194e12f2fbe5697fa400d07d4529ac20ba6d8c4893ecec80e27968f828edce"
} | train |
{
"chosen_proof_sha256": "7605cac82936a34e51b347b493387fd67f6c60e4f590ba9df4da446fc48c38a4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
let l := hi.lift ((pullback.fst αX f) ≫ h') ((pullback.snd αX f) ≫ g)
(by simp only [Category.assoc, hh.toCommSq.w, pullback.condition_assoc, ← H.w])
use (pullback αX f), (pullback.fst αX f), l, (pullback.snd αX f)
refine ⟨IsPullback.of_hasPullback αX f, ?_, ?_⟩
· refine IsPullback.of_right' ?_ hi
rw... | false | {
"chosen_tokens": 219,
"rejected_tokens": 2,
"token_jaccard": 0.019608,
"token_length_ratio": 0.009132
} | sorry_admit_sorryAx_proof_hole | sorry | de63d39d0a76042311e71b364e950ecda63311c443e0f3d397ad31c7eb43a223 | 0 | 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.Extensive
public import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
public import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
Names... | 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/Adhesive",
"family_id": "ispushout",
"file_id": "mathlib/Mathlib/CategoryTheory/Adhesive/Basic.lean",
"sample_id": "1e9f94a5050c3f631d6ef4550fc4148994113fabe92f719d1a84b2cac8998b0b"
} | train |
{
"chosen_proof_sha256": "0d4d6dccf2c84c2f8c5355bc4fd7dec43e2d86ae85215b172de94088a2f48bcb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "81e1863881d17ff2139ffc9deef5d1a646... | by
ext x
simp only [spectrum.mem_iff, isUnit_iff_forall_ne_zero, not_forall, sub_apply,
Classical.not_not, Set.mem_range,
sub_eq_zero, @eq_comm _ (x • 1 : R) _, Set.mem_preimage, Algebra.algebraMap_eq_smul_one,
smul_apply, one_apply] | true | {
"chosen_tokens": 50,
"rejected_tokens": 5,
"token_jaccard": 0.055556,
"token_length_ratio": 0.1
} | wrong_namespace_qualification | wrong_namespace_qualification | ded8e25b58d4b16c92e30e3342eee5a9724de00ce89acb5e30a5a60000064d99 | 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.Ring.Units
public import Mathlib.Algebra.Algebra.Spectrum.Basic
public import Mathlib.Topology.ContinuousMap.Algebra
Namespace:
ContinuousMap
... | by
exact OProverMissing.missing_dd8a4792fd82 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "b384432fc14513cf7b08186780ad391a05726732f7998892c2d5e54436d88b9e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/ContinuousMap",
"family_id": "spectrum_eq_preimage_range",
"file_id": "mathlib/Mathlib/Topology/ContinuousMap/Units.lean",
"sample_id": "dd8a4792fd8238e946d75aea51e934ebc1fab1139603667bd836f2fb5b0acf87"
} | train |
{
"chosen_proof_sha256": "507db4142d79d356282c11878756b5338f2f435787c1651165c4326a0009c857",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
cases x; rw [Zsqrtd.norm, normSq]; simp | false | {
"chosen_tokens": 14,
"rejected_tokens": 2,
"token_jaccard": 0.071429,
"token_length_ratio": 0.142857
} | sorry_admit_sorryAx_proof_hole | sorry | dee95716440da9607832256ea4a1a366bba6ab62e6910449c1aa4d48ae3f8afd | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Archimedean.Real.Basic
public import Mathlib.Data.Complex.Basic
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.NumberTheory.Zsq... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/Zsqrtd",
"family_id": "intcast_complex_norm",
"file_id": "mathlib/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean",
"sample_id": "2eaa4c3a7a8979e42c8b78cd304c273792da5d766a00dd97d6cfbc8a6d8171e8"
} | train |
{
"chosen_proof_sha256": "8c5d22170da6aa84840f7344a2c31c2097b39c782de19947a86a24798df1b587",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "40a9e698de3c8772948fb6e716fb277d4c... | by
rintro ⟨f, a, hf⟩ ⟨g, b, hg⟩ rfl
congr
refine extensional_of_trichotomous_of_irrefl s fun x ↦ ?_
rw [← hf, hg] | true | {
"chosen_tokens": 33,
"rejected_tokens": 3,
"token_jaccard": 0.037037,
"token_length_ratio": 0.090909
} | unknown_identifier | unknown_identifier | df122cb9182694ca0715ec07d35c16e064fdab2b910354947873b9d81e5d140e | 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.Sum.Order
public import Mathlib.Order.Hom.Lex
public import Mathlib.Order.RelIso.Set
public import Mathlib.Order.UpperLower.Basic
public import Mathlib.Or... | by
exact __oprover_missing_c360ffd93195 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "920979c161884480a4bd263a7263c5c0256808166c6779734e0c4c23f9aab373",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order",
"family_id": "torelembedding_injective",
"file_id": "mathlib/Mathlib/Order/InitialSeg.lean",
"sample_id": "c360ffd931955f77da36029b2c2c6057afeb7e861c5fc96268f3b26c2470d808"
} | train |
{
"chosen_proof_sha256": "58196917b1a24b2be60c95c2b5d60e57c3e3c717c1a775b2f741e63bbfde64ca",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7fc44abc4155c5513e4bbdb5b46fb28fc6... | by
have h := lintegral_condCDF ((dirac Unit.unit).prod μ) x
simpa only [fst_prod, prod_prod, measure_univ, one_mul, lintegral_dirac] using! h | false | {
"chosen_tokens": 33,
"rejected_tokens": 40,
"token_jaccard": 0.928571,
"token_length_ratio": 1.212121
} | markdown_preamble_trailing | fence | df30f9b5050c3ba2447f63c9d64ae15bb4c8881bcf5e39df8c002ab4476969fd | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Probability.Kernel.Disintegration.CondCDF
public import Mathlib.Tactic.CrossRefAttribute
Namespace:
ProbabilityTheory
Local context:
/-
Copyright (c) 2023 Ré... | ```lean
by
have h := lintegral_condCDF ((dirac Unit.unit).prod μ) x
simpa only [fst_prod, prod_prod, measure_univ, one_mul, lintegral_dirac] using! 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": "Probability",
"family_id": "ofreal_cdf",
"file_id": "mathlib/Mathlib/Probability/CDF.lean",
"sample_id": "41dfd38896021be841aea499e4a561d7890d2e4973f5f2dd9b0701af439b41dd"
} | train |
{
"chosen_proof_sha256": "956d0d39593774896edc51480ad38b07e1edca7a7a86a5d842d9b626672e0346",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "1985f9bc8f1d271e9024a4e9b57f47f644... | by
rw [isNoetherian_iff', wellFoundedGT_iff_monotone_chain_condition]
variable [IsNoetherian R M] | false | {
"chosen_tokens": 13,
"rejected_tokens": 20,
"token_jaccard": 0.846154,
"token_length_ratio": 1.538462
} | markdown_preamble_trailing | fence | df47fef6b0c1945d58fb1f7f60ca2c1543d491bb058b2c5d8d868bcae2b07163 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.Filter.AtTopBot.Basic
public import Mathlib.RingTheory.Finiteness.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2018 Mario Carneiro, Kevin Bu... | ```lean
by
rw [isNoetherian_iff', wellFoundedGT_iff_monotone_chain_condition]
variable [IsNoetherian R M]
``` | {
"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/Noetherian",
"family_id": "monotone_stabilizes_iff_noetherian",
"file_id": "mathlib/Mathlib/RingTheory/Noetherian/Defs.lean",
"sample_id": "a21a121517cc485ec8b0fac3320d37a95ffa767d8d52dbb2b3c8baa9ed0c81a3"
} | train |
{
"chosen_proof_sha256": "1a3ab110d23e9c5a2fcf5c8f847d9eb4eca9aceac610399fa60f5ab4f28d4c1a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cc198a628839d26b84d2d45a4274cbe04c... | by
obtain ⟨c₁, hc₁, c₂, _, h⟩ := hf b hb
refine ⟨c₁, hc₁, ?_⟩
filter_upwards [h]
exact fun _ H u hu => (H u hu).1 | true | {
"chosen_tokens": 49,
"rejected_tokens": 5,
"token_jaccard": 0.09375,
"token_length_ratio": 0.102041
} | wrong_namespace_qualification | wrong_namespace_qualification | dfc398597cdb868d5ca82828afde77fa5c1bdd9ff95b99194ed2ae1329078ef6 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecialFunctions.Pow.Real
public import Mathlib.Algebra.Order.ToIntervalMod
public import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Al... | by
exact OProverMissing.missing_a2b4875ff5d1 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "3e05d5e56de7a45ff5241391ad8b77d0035c03d48f743271e1e8c99142bdd7c9",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Computability/AkraBazzi",
"family_id": "eventually_attop_ge",
"file_id": "mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean",
"sample_id": "a2b4875ff5d193c274945a397a372732cc39a1c971c7cc22bbea2e4368fe786d"
} | train |
{
"chosen_proof_sha256": "72e566f5060686e112fed795e07f0f9fff6e410e86e784e0d5fb3063e2798458",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6c38d9122f13fca6acc952741077e5353e... | by
nontriviality α using Subsingleton.elim a 0
obtain rfl | h := eq_or_ne a 0
· rw [normUnit_zero, zero_mul, normUnit_zero]
· rw [normUnit_mul h (Units.ne_zero _), normUnit_coe_units, mul_inv_eq_one] | true | {
"chosen_tokens": 42,
"rejected_tokens": 3,
"token_jaccard": 0.03125,
"token_length_ratio": 0.071429
} | unknown_identifier | unknown_identifier | dfd45f2176b4e9abc1326e7ce855ef0e386472da31f8caf31b4658befa03ee13 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Ring.Associated
Namespace:
(root)
Local context:
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as desc... | by
exact __oprover_missing_80e10db21753 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "69cc636998a3d566ccc6f782a3b839e93f4e7c2a4a4d27665ee7d86f7f14de4a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/GCDMonoid",
"family_id": "normunit_mul_normunit",
"file_id": "mathlib/Mathlib/Algebra/GCDMonoid/Basic.lean",
"sample_id": "80e10db2175365293b28a61fb2baadedb77fa1adfc56606eaf56f02d6e239f96"
} | train |
{
"chosen_proof_sha256": "5af7d1d4d55c3a73fbff3849f778662494382e96b2f978d7cb9fc32bd567bdc1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9dd28da9ecf3c55e13be992634a8a033a... | by
intro f
obtain ⟨⟨f⟩, rfl⟩ := (bijective_leftHomotopyClassToHom L X Y).2 f
exact ⟨f, rfl⟩ | true | {
"chosen_tokens": 27,
"rejected_tokens": 2,
"token_jaccard": 0.052632,
"token_length_ratio": 0.074074
} | invalid_tactic | invalid_tactic | e00c41090846da0caf24584284de55a37a12d8ef5e9ae7133d86b4305cee74ec | 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.BifibrantObjectHomotopy
Namespace:
HomotopicalAlgebra
Local context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Re... | by
__oprover_invalid_tactic_17c8f688ffbf | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b0fc25684118d3615137d7395da7edad57c073ae729c00dd9ac826d5b4c1ba03",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/ModelCategory",
"family_id": "map_surjective_of_islocalization",
"file_id": "mathlib/Mathlib/AlgebraicTopology/ModelCategory/FundamentalLemma.lean",
"sample_id": "17c8f688ffbfe3f08997e2e441397e109ef966a6e5472eeef0ec671a2f11c5b3"
} | train |
{
"chosen_proof_sha256": "8f9fde22f4f5e3c6a342c774b51f2e9a3888b3631f985c3a4bfd65b729742429",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
ext : 1
simp [derivative'] | false | {
"chosen_tokens": 8,
"rejected_tokens": 3,
"token_jaccard": 0.1,
"token_length_ratio": 0.375
} | by_exact_placeholder | placeholder | e01a3736bee50197d3da8d7b02368d6dfe0e69ecf1d6f6b6965ca04d20a1fa70 | 0 | 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
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/Polynomial",
"family_id": "mkderivation_one_eq_derivative",
"file_id": "mathlib/Mathlib/Algebra/Polynomial/Derivation.lean",
"sample_id": "bbc6b49baeedf3e55b05cb3f8d67c144875e1981fa10f0ed9325586b2eeede01"
} | train |
{
"chosen_proof_sha256": "c0bd9ef2de4bda624d2c87bea71eb5ca5fdf0a8bf96e99ba0215cfd11acebae5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "94a2116bd60b99beb1018a62dbb0caea14... | by
constructor
· intro h
constructor
rintro _ _ f (hf : epimorphisms C f)
rw [epi_iff_surjective]
rw [← h] at hf
exact hf
· intro
apply le_antisymm (surjective_le_epimorphisms C)
intro _ _ f hf
have : Epi f := hf
change Function.Surjective ((forget C).map f)
rw [← epi_iff_s... | true | {
"chosen_tokens": 68,
"rejected_tokens": 5,
"token_jaccard": 0.083333,
"token_length_ratio": 0.073529
} | wrong_namespace_qualification | wrong_namespace_qualification | e071ab865477f81173b1e02a7fc0f9132d60878de71f612a0cade8e4d7da6b84 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
public import Mathlib.CategoryTheory.Limits.Preserves.Basic
public import Mathlib.CategoryTheory.Limits.Shapes.Imag... | by
exact OProverMissing.missing_57e63da4c8d7 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "fd4b504bfacd8f24cbfc3d39c84caf099edfcbc8198923639b1a0f7edf00b2c1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/ConcreteCategory",
"family_id": "surjective_eq_epimorphisms_iff",
"file_id": "mathlib/Mathlib/CategoryTheory/ConcreteCategory/EpiMono.lean",
"sample_id": "57e63da4c8d7ec6a478c9fdfc7ae432be0adeebf4eec412381dc418d3098ca3d"
} | train |
{
"chosen_proof_sha256": "47e39958d92c203177453f8d36c36c3e273fa3ce6ffb582eb81c95f2e2f8425a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "00b449300b9da4fbf1fe53aee201d84aa2... | by
apply isTopologicalBasis_of_isOpen_of_nhds (fun U hU ↦ hU.1) <| fun x U hx hU ↦ ?_
obtain ⟨V, ⟨hoV, hcV⟩, hfx, hVf⟩ : ∃ V ∈ {V | IsOpen V ∧ IsCompact V}, f x ∈ V ∧ V ⊆ f '' U :=
(PrespectralSpace.isTopologicalBasis (X := Y)).isOpen_iff.mp
(hf.isOpen_iff_image_isOpen.mp hU) (f x) ⟨x, hx, rfl⟩
... | false | {
"chosen_tokens": 155,
"rejected_tokens": 160,
"token_jaccard": 0.940299,
"token_length_ratio": 1.032258
} | markdown_preamble_trailing | preamble | e0be8bd239483da4aed00df8ece624f1be100f52c53f62f9ca053eed3daf8321 | 0 | 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.Ideal
public import Mathlib.Topology.Sets.Compacts
public import Mathlib.Topology.Sets.OpenCover
public import Mathlib.Topology.Spectral.Hom
Namespace:
... | Here is the proof:
by
apply isTopologicalBasis_of_isOpen_of_nhds (fun U hU ↦ hU.1) <| fun x U hx hU ↦ ?_
obtain ⟨V, ⟨hoV, hcV⟩, hfx, hVf⟩ : ∃ V ∈ {V | IsOpen V ∧ IsCompact V}, f x ∈ V ∧ V ⊆ f '' U :=
(PrespectralSpace.isTopologicalBasis (X := Y)).isOpen_iff.mp
(hf.isOpen_iff_image_isOpen.mp hU) (f... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Topology/Spectral",
"family_id": "topology",
"file_id": "mathlib/Mathlib/Topology/Spectral/Prespectral.lean",
"sample_id": "33918059ac643ca50e9d2e3d7ccb2d32565782d0b03a2650b0af67e33602c28c"
} | train |
{
"chosen_proof_sha256": "156fa55c2e6ebaab51efb61e1b6415d519f8bca6eaddaf4eff8b3327fd1b34c8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rcases le_total x 0 with (neg | pos)
· exact .inr <| isSquare_of_nonneg (neg_nonneg_of_nonpos neg)
· exact .inl <| isSquare_of_nonneg pos
exists_isRoot_of_odd_natDegree := exists_isRoot_of_odd_natDegree
variable [IsRealClosed R] | false | {
"chosen_tokens": 38,
"rejected_tokens": 2,
"token_jaccard": 0.037037,
"token_length_ratio": 0.052632
} | sorry_admit_sorryAx_proof_hole | sorry | e12d3e799ab9bab09a38aedfa11d3af2c09c7f06dd96e2c42c1ff2bbcfea4b8c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.Degree.Domain
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.Algebra.Ring.Semireal.Defs
public import Mathlib.Tact... | by
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": "FieldTheory/IsRealClosed",
"family_id": "of_linearorderedfield",
"file_id": "mathlib/Mathlib/FieldTheory/IsRealClosed/Basic.lean",
"sample_id": "b9d038f0b31926725a67cabc50ad8dfeb3e914d41076f9b7c6d575a332e9b255"
} | train |
{
"chosen_proof_sha256": "dc1780cb3fb0b15d1b902233abc1a076e6c3e1b7b64ffb6ddfc98b9761298fce",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "34da45620930cb8bcc150c1331c6a664b3... | by
refine .fromProd (fun a d hd₁ hd₂ c hdc ↦ ?_) (fun b d hd₁ hd₂ c hdc ↦ ?_)
· have e1 : (fun b ↦ (f a, g b)) '' d = {f a} ×ˢ (g '' d) := by aesop
simp_rw [Prod.map_apply, e1]
exact .prod (singleton_nonempty _) (hd₁.image _) isLUB_singleton (hg hd₁ hd₂ hdc)
· have e2 : ((fun a ↦ (f a, g b)) '' d) = (f ''... | false | {
"chosen_tokens": 166,
"rejected_tokens": 171,
"token_jaccard": 0.92,
"token_length_ratio": 1.03012
} | markdown_preamble_trailing | preamble | e14b0bb81c7ff2020373790c9097fb3dc33ee23b5e9c4c020e650f09436622c3 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Order.ScottContinuity
public import Mathlib.Order.Bounds.Lattice
Namespace:
(root)
Local context:
/-
Copyright (c) 2025 Christopher Hoskin. All rights reserv... | Here is the proof:
by
refine .fromProd (fun a d hd₁ hd₂ c hdc ↦ ?_) (fun b d hd₁ hd₂ c hdc ↦ ?_)
· have e1 : (fun b ↦ (f a, g b)) '' d = {f a} ×ˢ (g '' d) := by aesop
simp_rw [Prod.map_apply, e1]
exact .prod (singleton_nonempty _) (hd₁.image _) isLUB_singleton (hg hd₁ hd₂ hdc)
· have e2 : ((fun a ↦ (f a, ... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/ScottContinuity",
"family_id": "scottcontinuous",
"file_id": "mathlib/Mathlib/Order/ScottContinuity/Prod.lean",
"sample_id": "1a6f779a26d6933a1ab93cb5daf06274698653e526a8b75426906873d50dc3c8"
} | train |
{
"chosen_proof_sha256": "bb1d914f2c85ca879a4ab7ab44641a41b11803eff45090adf0fe5af8fb8479b6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "87b680f48e655bff7c8ab451ec2a00022c... | by
induction n <;> grind [AllZero, sign] | false | {
"chosen_tokens": 12,
"rejected_tokens": 16,
"token_jaccard": 0.75,
"token_length_ratio": 1.333333
} | missing_wrong_import | import | e15ce64fab7abc90c13f02e5624b160ceeaadae59a44a9ad9572cf9237f2e961 | 0 | 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... | import Mathlib.WrongImport
by
induction n <;> grind [AllZero, sign] | {
"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": "Tactic/ComputeAsymptotics",
"family_id": "replicate",
"file_id": "mathlib/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Predicates.lean",
"sample_id": "26f5274475c00cabc639fcde1f5b699c28f967e875673ef858d4e214040ea235"
} | train |
{
"chosen_proof_sha256": "81a9d05bf0ed54d4fa50cc7311fe12bd2332adf9f758867f731c0d0d23db9024",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ba8c41a3c9a3ed4626d2c2866e09bd88dd... | by
ext i
by_cases h : i = i₀
· subst h
simp
· rw [subtypeNeLift_of_neq _ _ _ _ h] | true | {
"chosen_tokens": 24,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.125
} | circular_dependency | circular_dependency | e26a0adc8bd3349d4c3d99cc10637ce74d8bfe19c4b177af8e509a377a51f5ad | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Logic.Equiv.Option
Namespace:
Function
Local context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in... | by
exact subtypeNeLift_restriction | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "b577b45f916f3ad4d4835b3899fedd07a946a1eba1f6f3d5d5b6a538eb7cefc5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data",
"family_id": "subtypenelift_restriction",
"file_id": "mathlib/Mathlib/Data/SubtypeNeLift.lean",
"sample_id": "15f253266f8e1f550df58301525f21097e11cc98ed777b7a946120509154235e"
} | train |
{
"chosen_proof_sha256": "49bdfcf01bbb1a8c4c2fb91482006ee7e27f5a1dbdbae3ae54b463c762ee66e9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6af328522e0ae1497586eaa9d33047feb1... | by
ext X; simp [Equivalence.unit] | true | {
"chosen_tokens": 10,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.3
} | unknown_identifier | unknown_identifier | e2e58748dd0e4b5eef5db9c3700472a24b005ba0a194fa7e61b4d7bb49bb0fbf | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Adjunction.Basic
public import Mathlib.CategoryTheory.Yoneda
public import Mathlib.CategoryTheory.Opposites
Namespace:
CategoryTheory.Adjunctio... | by
exact __oprover_missing_65e0c46ab808 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "bf35ea83ee7e208ca9b5f67529c1890e859879ccf20d1ef5f554c2a0c6abba26",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Adjunction",
"family_id": "rightop_eq",
"file_id": "mathlib/Mathlib/CategoryTheory/Adjunction/Opposites.lean",
"sample_id": "65e0c46ab80804e8a5cd071bf84cbfc579f19ac01cda5e58e506b75975df165e"
} | train |
{
"chosen_proof_sha256": "8f849e0c2e5e9b058d0be7efa5644ffd7d171530c0bf0be894d1f2aff0fde505",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "18167008b708b906a466e4eee73d2d19d1... | by
simp [wInner_cWeight_eq_expect, ψ.norm_apply] | true | {
"chosen_tokens": 9,
"rejected_tokens": 3,
"token_jaccard": 0.090909,
"token_length_ratio": 0.333333
} | circular_dependency | circular_dependency | e30fc623542e5281c71a8fd5476ba50b33642b4886d1f2a8424bdd07ef8fe9d8 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Expect
public import Mathlib.Algebra.Group.AddChar
public import Mathlib.Analysis.RCLike.Inner
Namespace:
AddChar
Local context:
/-
Copy... | by
exact wInner_cWeight_self | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "1209904076d2721414bd2f88050e40d914c83da391e0e4c96722dac7cc9455ce",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Fourier",
"family_id": "winner_cweight_self",
"file_id": "mathlib/Mathlib/Analysis/Fourier/FiniteAbelian/Orthogonality.lean",
"sample_id": "76d9a3f6de2535e710efca9f1d5d77672d61eb6404d0c9c6e1be8e32402d6ace"
} | train |
{
"chosen_proof_sha256": "42680d60d5e78b64557581d1ab549dd5b4e6ea0d21a689bb03eb5cdec7a02fe8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
unfold logDeriv
simp [deriv_algebraMap] | false | {
"chosen_tokens": 7,
"rejected_tokens": 3,
"token_jaccard": 0.111111,
"token_length_ratio": 0.428571
} | by_exact_placeholder | placeholder | e345837be9274fe2df5bab4d0e93a7238089b978ddaadb48545e48e8ce95282b | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Derivation.MapCoeffs
public import Mathlib.FieldTheory.PrimitiveElement
Namespace:
Differential
Local context:
/-
Copyright (c) 2024 Daniel Weber.... | by
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": "FieldTheory/Differential",
"family_id": "logderiv_algebramap",
"file_id": "mathlib/Mathlib/FieldTheory/Differential/Basic.lean",
"sample_id": "026c42c1a18ace69185b5b2383ea579a45b7bd0313b181e0953d0847129aa197"
} | train |
{
"chosen_proof_sha256": "bdd71d8bba5ab6bc0aadc26f2ebfd16f9c6d0a1ea732936b96c085c594ae8d5f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "08990ef995bd3c79af527f9a2b0631a1d0... | by
simp [pullSub, List.take_of_length_le hl, List.drop_eq_nil_iff.mpr hl] | true | {
"chosen_tokens": 17,
"rejected_tokens": 5,
"token_jaccard": 0.133333,
"token_length_ratio": 0.294118
} | wrong_namespace_qualification | wrong_namespace_qualification | e3768cd0cb7530009ac746a7429e4255936c69c78baff3cbb08a710a7d696ef2 | 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.CompleteLattice.SetLike
Namespace:
Descriptive.Tree
Local context:
/-
Copyright (c) 2024 Sven Manthe. All rights reserved.
Released under Apache 2.0 li... | by
exact OProverMissing.missing_4fba185756ba | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "eadec439be49bd25fc97ff1d3e8135532a1a5972b123b66e8c35c33e5133e522",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "SetTheory/Descriptive",
"family_id": "mem_pullsub_short",
"file_id": "mathlib/Mathlib/SetTheory/Descriptive/Tree.lean",
"sample_id": "4fba185756badbaa3d579461622f7431189517e5fb7071a627d47ad8323f43b0"
} | train |
{
"chosen_proof_sha256": "3a259627f5a5b32fa98a969454c4436a807645c0fdfa0bcf717382246cd616fd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
apply Stream'.ext; intro n
cases n <;> rfl | true | {
"chosen_tokens": 14,
"rejected_tokens": 8,
"token_jaccard": 0.055556,
"token_length_ratio": 0.571429
} | unsolved_goals | unsolved_goals | e391990692053e4022446fd8dddf4246db6ca9e99bcaa1dc1f6f7ddb755ffe8f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Stream.Defs
public import Mathlib.Logic.Function.Basic
public import Mathlib.Data.Nat.Basic
public import Mathlib.Tactic.Common
Namespace:
Stream'
Local... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "875b042262b31d7859253425e5195e587b69724726f762af3d3cd6db1de2cc95",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Stream",
"family_id": "const_eq",
"file_id": "mathlib/Mathlib/Data/Stream/Init.lean",
"sample_id": "89ecc5dec26210b39010cde2e335b6d4cf0a4c1bf7eb4d3bd4ba691ff88b6aba"
} | train |
{
"chosen_proof_sha256": "a90544dc513a05353e26becaa7c94b77ef02a2abe517eb6e4b51662aba7c3cd3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6c4b77c678413a1803601330d163141e89... | by
simpa [← Basis.coord_apply] using LinearMap.congr_fun (basis_coord R n b s) x | false | {
"chosen_tokens": 20,
"rejected_tokens": 24,
"token_jaccard": 0.863636,
"token_length_ratio": 1.2
} | missing_wrong_import | import | e3a52afda624cbcdce425c40ce5ac9fa0c1a62233b604e7709c525156be9b06e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.ExteriorPower.Basic
public import Mathlib.LinearAlgebra.ExteriorPower.Pairing
public import Mathlib.RingTheory.Finiteness.Subalgebra
public impor... | import Mathlib.WrongImport
by
simpa [← Basis.coord_apply] using LinearMap.congr_fun (basis_coord R n b s) x | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/ExteriorPower",
"family_id": "basis_repr_apply",
"file_id": "mathlib/Mathlib/LinearAlgebra/ExteriorPower/Basis.lean",
"sample_id": "5c25dc9430d8645a6917c9b8ba0f572387c246c6713ecceebd925b758bc2e637"
} | train |
{
"chosen_proof_sha256": "55394e3b94e4002c95f3216c1764d8bfdefb9347ecc72f11001d5f75ebad2dcb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "fbcf882d1eaeaa3d8c9140c5df42fa2a0c... | by
rintro x ⟨hf, hg⟩
rw [mem_fixedPoints_iff] at *
rw [comp_apply, hg, hf] | true | {
"chosen_tokens": 22,
"rejected_tokens": 3,
"token_jaccard": 0.058824,
"token_length_ratio": 0.136364
} | circular_dependency | circular_dependency | e3c3630c76505294850432fb40bc577b775b580b4872a75922a8777ec74c3f2e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Set.Operations
Namespace:
Function
Local context:
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as descr... | by
exact inter_subset_fixedPoints_comp | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "b987ab2babce512194b884122cc8f0fee577a5ea3abbc5c07d67cb59a8ff7971",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Dynamics/FixedPoints",
"family_id": "inter_subset_fixedpoints_comp",
"file_id": "mathlib/Mathlib/Dynamics/FixedPoints/Defs.lean",
"sample_id": "65f5358ecacae8480f414a0ec307a37a8131f8d46fb796e1817fa646fac75533"
} | train |
{
"chosen_proof_sha256": "186eee1641eb21a1d1c5d6e7242727e956490464bf744e56de75eb2d692d16ff",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4a080b9895c1efd55f50ec53f39f68ef54... | by
rw [← singleton_vadd]; exact (h₁ x).add h₂ | false | {
"chosen_tokens": 17,
"rejected_tokens": 22,
"token_jaccard": 0.8,
"token_length_ratio": 1.294118
} | markdown_preamble_trailing | trailing | e3d42cc7997015f95bb78a442e2c41ed2372b82eaf3c09e2a1d53ae4430b0ccf | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
public import Mathlib.Algebra.Ring.Action.Pointwise.Set
public import Mathlib.Topology.Bornology.Basic
Namespace:
A... | by
rw [← singleton_vadd]; exact (h₁ x).add h₂
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": "Topology/Bornology",
"family_id": "vadd_absorbs",
"file_id": "mathlib/Mathlib/Topology/Bornology/Absorbs.lean",
"sample_id": "88a199b64f05b0ccb332981cf02b5c961ab80a822fd1dae1175ba359f20a3e82"
} | train |
{
"chosen_proof_sha256": "7308aa6374fea597c48250db2901872d2efda7ec2b018c03749527c542943e3a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
obtain ⟨n, hn⟩ := exists_rat_eq_convergent h
exact ⟨n, hn.symm ▸ convs_eq_convergent ξ n⟩ | false | {
"chosen_tokens": 22,
"rejected_tokens": 2,
"token_jaccard": 0.058824,
"token_length_ratio": 0.090909
} | sorry_admit_sorryAx_proof_hole | sorry | e4181044c816113275b6b82e51973c7f4d366de6209e1dacd4b7dd7ecb98cdd1 | 0 | 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.ContinuedFractions.Computation.ApproximationCorollaries
public import Mathlib.Algebra.ContinuedFractions.Computation.Translations
public import Mathlib... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/DiophantineApproximation",
"family_id": "exists_convs_eq_rat",
"file_id": "mathlib/Mathlib/NumberTheory/DiophantineApproximation/ContinuedFractions.lean",
"sample_id": "88dc792ee34d82bdbe14dbea6a9bf9efe530e0a2ea1640a11d02fb687a45c35f"
} | train |
{
"chosen_proof_sha256": "598e0d2c4be030094c0dbca69a5ab0955fb2fcd00976225f2cdbe9fd5333ec6f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c6ea5e3ef49586d24e451298b05a23e2d4... | by
simp only [List.applyId]
cases h₃ : List.dlookup x (List.map Prod.toSigma (xs.zip ys)) with
| none =>
dsimp [Option.getD]
rw [h₁.mem_iff]
| some val =>
have h₂ : ys.Nodup := h₁.nodup_iff.1 h₀
replace h₁ : xs.length = ys.length := h₁.length_eq
dsimp
induction xs generalizing ys with
... | false | {
"chosen_tokens": 314,
"rejected_tokens": 321,
"token_jaccard": 0.979798,
"token_length_ratio": 1.022293
} | markdown_preamble_trailing | fence | e4630180ab38f365d85471fbc2a7edc17dd870324c6a71664cac901689e57e57 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Data.MLList.Basic
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.ZeroLEOne
public import Mathlib.Data.DFinsupp.Defs
public... | ```lean
by
simp only [List.applyId]
cases h₃ : List.dlookup x (List.map Prod.toSigma (xs.zip ys)) with
| none =>
dsimp [Option.getD]
rw [h₁.mem_iff]
| some val =>
have h₂ : ys.Nodup := h₁.nodup_iff.1 h₀
replace h₁ : xs.length = ys.length := h₁.length_eq
dsimp
induction xs generalizing ys... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Testing/Plausible",
"family_id": "applyid_mem_iff",
"file_id": "mathlib/Mathlib/Testing/Plausible/Functions.lean",
"sample_id": "f81f8323cb2859a12bdc22124ed1f8f7177b728cd0e3476841f3fae387cfd30b"
} | train |
{
"chosen_proof_sha256": "dbc5a53bc3d6fc4141510bd9f512049166f30bf6d3c576c98bf05b032824182f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9ae8a4625029f9ac2796e24d13902444fe... | by cfc_cont_tac)
(ha : p a := by cfc_tac) (ha' : ∀ i, q i (a i) := by cfc_tac) :
cfc f a = fun i => cfc f (a i) := by
ext i
let φ := Pi.evalStarAlgHom S A i
exact φ.map_cfc f a (by rwa [Pi.spectrum_eq]) (continuous_apply i) ha (ha' i) | false | {
"chosen_tokens": 79,
"rejected_tokens": 86,
"token_jaccard": 0.945946,
"token_length_ratio": 1.088608
} | markdown_preamble_trailing | fence | e46432044006992c37b35a13124082ee01ca2c2f48e20b314e4fad7c262be8d1 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique
public import Mathlib.Algebra.Algebra.Spectrum.Pi
public import Mathlib.Algebra.Star.StarAlgHom
Name... | ```lean
by cfc_cont_tac)
(ha : p a := by cfc_tac) (ha' : ∀ i, q i (a i) := by cfc_tac) :
cfc f a = fun i => cfc f (a i) := by
ext i
let φ := Pi.evalStarAlgHom S A i
exact φ.map_cfc f a (by rwa [Pi.spectrum_eq]) (continuous_apply i) ha (ha' i)
``` | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/CStarAlgebra",
"family_id": "cfc_map_pi",
"file_id": "mathlib/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Pi.lean",
"sample_id": "a3ef0314d959f005739e2ea0a1bbd35c4cb66fa53980fc55488bf2d3ac971efc"
} | train |
{
"chosen_proof_sha256": "363f3ee37f17de1a17c56efabc2dfd6a0b7c412da4a7733db969356b0041dbda",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
apply SetLike.coe_injective
simp only [coe_centralizer, Set.centralizer_centralizer_centralizer]
variable {M} in | false | {
"chosen_tokens": 19,
"rejected_tokens": 2,
"token_jaccard": 0.052632,
"token_length_ratio": 0.105263
} | by_exact_placeholder | by? | e46a4a9ccb7d77c5e92d504a7eaa438d7516f165f88e415be977cc1fece7045c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.GroupTheory.Subsemigroup.Centralizer
public import Mathlib.GroupTheory.Submonoid.Center
Namespace:
Submonoid
Local context:
/-
Copyright (c) 2021 Thomas Brow... | 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": "GroupTheory/Submonoid",
"family_id": "centralizer_centralizer_centralizer",
"file_id": "mathlib/Mathlib/GroupTheory/Submonoid/Centralizer.lean",
"sample_id": "a921660930bcd0f4c9bd0e614c1768bb72ffc642b6565ee11d368d38df7c680d"
} | train |
{
"chosen_proof_sha256": "ff8a2edf1706af29211010fc997e519b058214e2acf8d5e15ccbda28fa02aa4d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [Submonoid.eq_top_iff']
intro x
induction x with | _ a
induction a with
| one => exact Submonoid.one_mem _
| of x => exact subset_closure <| by simp [range, of]
| mul x y hx hy => exact Submonoid.mul_mem _ hx hy | false | {
"chosen_tokens": 55,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.054545
} | by_exact_placeholder | exact? | e488dc514a8a3997590112b8da35690178bf8b36f3d4a45abb0db6317c67be6f | 0 | 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.FreeMonoid.Basic
public import Mathlib.Algebra.Group.Submonoid.Operations
public import Mathlib.GroupTheory.Congruence.Hom
Namespace:
PresentedMonoid
... | 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/PresentedMonoid",
"family_id": "closure_range_of",
"file_id": "mathlib/Mathlib/Algebra/PresentedMonoid/Basic.lean",
"sample_id": "6fc83af12926d67f82e7ea31db06b0a746a421286c20a9485cd6854580deafe5"
} | train |
{
"chosen_proof_sha256": "018854a18bc4ddad72fb2634603cc3211f19d9c9b481d90eec379823ad5a6440",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8d1c313ba94921d07ea1a0040f878f4467... | by
simp only [← zsmul_eq_mul, ofAdd_zsmul, map_zpow] | true | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.272727
} | unknown_identifier | unknown_identifier | e494692203a4f53c5f513a358ebbee1784d8af94eeadd1fb53bc0832b527095c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Order.SemiconjSup
Namespace:
CircleDeg1Lift
Local context:
/-
Copyright (c) 2020 Yury Kudryashov. All rig... | by
exact __oprover_missing_8e6d776a112c | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d867d33b3eddfdee5b1f3d8cc8f0d4f6eb7c6edfce4ee68dec2eb02bf0e91ebb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Dynamics/Circle",
"family_id": "translate_zpow",
"file_id": "mathlib/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean",
"sample_id": "8e6d776a112cd89de7aa1f1a126b48dfead5089fd7abc542320a1969d23d73fc"
} | train |
{
"chosen_proof_sha256": "a577b07c0d33a825fd2224d959c4010ad6df89972c66448b93e986876b718d2f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2b642afd7ce1ed1e1837b4bb707a053310... | by
calc
Var[X; μ | m]
_ = μ[X ^ 2 - 2 * X * μ[X | m] + μ[X | m] ^ 2 | m] := by rw [condVar, sub_sq]
_ =ᵐ[μ] μ[X ^ 2 | m] - 2 * μ[X | m] ^ 2 + μ[X | m] ^ 2 := by
have aux₀ : Integrable (X ^ 2) μ := hX.integrable_sq
have aux₁ : Integrable (2 * X * μ[X | m]) μ := by
rw [mul_assoc]
... | true | {
"chosen_tokens": 288,
"rejected_tokens": 3,
"token_jaccard": 0.031746,
"token_length_ratio": 0.010417
} | unknown_identifier | unknown_identifier | e4d870a0abe6553c1f65d53d619e4619b8c74758a9de10c4b964080427e3f3fb | 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.ConditionalExpectation.PullOut
public import Mathlib.MeasureTheory.Integral.Average
public import Mathlib.Probability.Moments.Variance
... | by
exact __oprover_missing_56bb01dbdf12 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "b06209a8391beb4631a67bbdd6ee34087b65f550caeebe12c5e1346ac0141193",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability",
"family_id": "condvar_ae_eq_condexp_sq_sub_sq_condexp",
"file_id": "mathlib/Mathlib/Probability/CondVar.lean",
"sample_id": "56bb01dbdf1213a5571dfdae532b08e414ed78cc621e447a3cc9e4531d26ca79"
} | train |
{
"chosen_proof_sha256": "bdd86c88ef7416db2f30c34d0d939f5c26ce704f80d0007fe2f340c238437f0f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e1a889ef8d34f3c24811588b71ee3a8a19... | by
refine IsPullback.of_forall_isPullback_app fun X => ?_
rw [Types.isPullback_iff]
refine ⟨congr(($(comp_χ_eq m)).app X), ?_, ?_⟩
· simpa using! (mono_iff_injective (m.app X)).mp (inferInstance)
· simp only [Functor.const_obj_obj, Functor.sieves_obj, χ_app, Opposite.op_unop,
TypeCat.hom_ofHom, TypeCat.... | true | {
"chosen_tokens": 123,
"rejected_tokens": 3,
"token_jaccard": 0.016949,
"token_length_ratio": 0.02439
} | unknown_identifier | unknown_identifier | e4f858d19f4af3774d53a0b1649b4583fad48b7bb78c54f5f26a499051cc8c79 | 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 __oprover_missing_e8c9dc5f379d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "56d8e37a1910b600afedc60e4ba1afc1830c8bac9ac37b2689b5dfcfc2773de7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Topos",
"family_id": "presheaf",
"file_id": "mathlib/Mathlib/CategoryTheory/Topos/Sheaf.lean",
"sample_id": "e8c9dc5f379d2760fb24d04c1fd1b200b12c9f621f4607d9bb2924f7901f72b2"
} | train |
{
"chosen_proof_sha256": "407de8aea2c8c096a4b6c72b70c997e3bb00015ef6148a6a35ed2aaaed552283",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "fe26dba9dee59cdbe8ccc661ca4aeb3988... | by
obtain ⟨T', e, h'⟩ := hF T hT
exact (ShortComplex.exact_iff_of_iso
(F.mapShortComplex.mapIso ((shortComplexOfDistTriangleIsoOfIso e hT)))).2 h' | false | {
"chosen_tokens": 36,
"rejected_tokens": 41,
"token_jaccard": 0.851852,
"token_length_ratio": 1.138889
} | markdown_preamble_trailing | trailing | e51e995438167d7e6e6ca3840514af9d351377186e954ddd7eb889fb65a45e7a | 0 | 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.ShortComplex.Exact
public import Mathlib.CategoryTheory.Shift.ShiftSequence
public import Mathlib.CategoryTheory.Triangulated.Functor
public i... | by
obtain ⟨T', e, h'⟩ := hF T hT
exact (ShortComplex.exact_iff_of_iso
(F.mapShortComplex.mapIso ((shortComplexOfDistTriangleIsoOfIso e hT)))).2 h'
This completes the proof. | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Triangulated",
"family_id": "ishomological",
"file_id": "mathlib/Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean",
"sample_id": "99a463e8b538cc1129c05262ca34214f41e74d8600f9e3aa84ec10439902b1f8"
} | train |
{
"chosen_proof_sha256": "fd734d0795994b2df39d22bec1fc6dc26c9a299bef650b0d07eb3f85b3d6c929",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "572db0c3d1eca879ca2f4b84f5b5bc898e... | by
rw [logHeight_eq_log_mulHeight]
exact log_nonneg <| x.one_le_mulHeight | true | {
"chosen_tokens": 12,
"rejected_tokens": 5,
"token_jaccard": 0.214286,
"token_length_ratio": 0.416667
} | wrong_namespace_qualification | wrong_namespace_qualification | e5aad54417fb656db3478b2193d4e9aee9a51f8bfb67aa4e3e934540e8578dd6 | 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.Projectivization.Basic
public import Mathlib.NumberTheory.Height.Basic
Namespace:
Projectivization
Local context:
/-
Copyright (c) 2026 Michael... | by
exact OProverMissing.missing_5bf0e0681c55 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ec4d7488e3ed492df0db21140eac51abfc3c7f2a2ea21d1a87907f1536a7e2d1",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Height",
"family_id": "logheight_nonneg",
"file_id": "mathlib/Mathlib/NumberTheory/Height/Projectivization.lean",
"sample_id": "5bf0e0681c55eec8bc5b1c759d43192dd7b331a9558e5d94174d45141766d782"
} | train |
{
"chosen_proof_sha256": "a923c894044509b83369abcd9431b69d970849ba97f75437eb89f230df9823e5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "35834d0e98d574f6e1ed88656df05a2d1e... | by
rw [IsHausdorff.eq_iff_smodEq (I := 𝔭)]
simp only [smul_eq_mul, mul_top]
intro n
cases n
· simp
· simp [SModEq, mk_pow_fontaineTheta] | true | {
"chosen_tokens": 32,
"rejected_tokens": 5,
"token_jaccard": 0.076923,
"token_length_ratio": 0.15625
} | wrong_namespace_qualification | wrong_namespace_qualification | e5e378d24a4b511073c55ca348aa6d93dfca0de859222172ab8a6744db594bef | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.AdicCompletion.Functoriality
public import Mathlib.RingTheory.AdicCompletion.RingHom
public import Mathlib.RingTheory.Perfectoid.Untilt
public impor... | by
exact OProverMissing.missing_0884596d7746 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "224c0d630c8fcf8814d5ee96e1555f8189450f8d4fe89f89103d87906420ddef",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Perfectoid",
"family_id": "fontainetheta_teichmuller",
"file_id": "mathlib/Mathlib/RingTheory/Perfectoid/FontaineTheta.lean",
"sample_id": "0884596d77461ad97de009c6f8b8d539833c319659cef7be0f8610e17da4299a"
} | train |
{
"chosen_proof_sha256": "42680d60d5e78b64557581d1ab549dd5b4e6ea0d21a689bb03eb5cdec7a02fe8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d015ef1b5c637ff972dc63934850928ed1... | by
unfold logDeriv
simp [deriv_algebraMap] | true | {
"chosen_tokens": 7,
"rejected_tokens": 3,
"token_jaccard": 0.111111,
"token_length_ratio": 0.428571
} | unknown_identifier | unknown_identifier | e5fb78cabcd71bc97c01d07e35922281aee6d077397bee38dbed8e8d8415ddce | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Derivation.MapCoeffs
public import Mathlib.FieldTheory.PrimitiveElement
Namespace:
Differential
Local context:
/-
Copyright (c) 2024 Daniel Weber.... | by
exact __oprover_missing_026c42c1a18a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "acc443b90d6ae3ebd9c8f957836d90a83a2a8c0e38400fd43a2209b58605cf24",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/Differential",
"family_id": "logderiv_algebramap",
"file_id": "mathlib/Mathlib/FieldTheory/Differential/Basic.lean",
"sample_id": "026c42c1a18ace69185b5b2383ea579a45b7bd0313b181e0953d0847129aa197"
} | train |
{
"chosen_proof_sha256": "efb528cee40199b8a7f10c4af81ecc5b602aa05a4991b3b60a5968591ef7ad2c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
dsimp [section_]
rw [assoc, dsimp% X.hom.naturality, comp_id]
exact ed.s'_comp_ε | true | {
"chosen_tokens": 24,
"rejected_tokens": 8,
"token_jaccard": 0.041667,
"token_length_ratio": 0.333333
} | unsolved_goals | unsolved_goals | e607631f47e054ca315a6b6f1e741508f29c9d36923edc13ebcb03447e960826 | 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.Homotopy
public import Mathlib.AlgebraicTopology.AlternatingFaceMapComplex
public import Mathlib.AlgebraicTopology.CechNerve
public import Mat... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "2858a7b4be4e3b02b0efa758201e23ddbddf9aedef914abfc0289048fd745d15",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology",
"family_id": "section_app_comp_hom_app",
"file_id": "mathlib/Mathlib/AlgebraicTopology/ExtraDegeneracy.lean",
"sample_id": "2c31379e40cbecc4b70db87b9684b86d81f7fb6fcff5b69f3c8ba705b4e20f03"
} | train |
{
"chosen_proof_sha256": "e26bc9c73cc42d866d4dadb94c4f9430e4046c553d052939bf6c8d1e66df3de3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0a6f4bb20aaaaf779ecf54289d8b160260... | by
refine hK.induction_on ⟨∅, by simp⟩ (fun s t hst ⟨W, Wc, Wo, hKW, hWU⟩ ↦ ?_) ?_ ?_
· use W, Wc, Wo, subset_trans hst hKW, hWU
· intro s t ⟨W₁, Wc₁, Wo₁, hKW₁, hWU₁⟩ ⟨W₂, Wc₂, Wo₂, hKW₂, hWU₂⟩
exact ⟨W₁ ∪ W₂, Wc₁.union Wc₂, Wo₁.union Wo₂, Set.union_subset_union hKW₁ hKW₂,
Set.union_subset hWU₁ hWU₂⟩
... | true | {
"chosen_tokens": 182,
"rejected_tokens": 2,
"token_jaccard": 0.017857,
"token_length_ratio": 0.010989
} | invalid_tactic | invalid_tactic | e61edf5dfe93d0b8cfd1256929925a82f00b345dfb406a20942a30039c21e23c | 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.Ideal
public import Mathlib.Topology.Sets.Compacts
public import Mathlib.Topology.Sets.OpenCover
public import Mathlib.Topology.Spectral.Hom
Namespace:
... | by
__oprover_invalid_tactic_bad13452563a | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "dc4cfc1cd759f5e14e7891398413460ff0d0ad5db91dd302647a141b9a62a36c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Spectral",
"family_id": "prespectralspace",
"file_id": "mathlib/Mathlib/Topology/Spectral/Prespectral.lean",
"sample_id": "bad13452563a2e4433cf2964ac3661b1745a1d26c8fd4dce023319c5196a9b1e"
} | train |
{
"chosen_proof_sha256": "3f00e9105f1d5704619d01594b14b1a2c0b0c68fe25f9bbf22b6c6f972b81c9f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "877aceedde815db9287812ae1efc9e2b30... | by
rw [moduleCat_exact_iff_function_exact] at hC ⊢
exact rTensor_exact M hC | true | {
"chosen_tokens": 12,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.166667
} | invalid_tactic | invalid_tactic | e6e2af2afe02914c6d4fb820267857235e4158a0953ed64c9da5f163d9056227 | 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.Basic
public import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
public import Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed
Namespace... | by
__oprover_invalid_tactic_a26f956f637c | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "ee40da114424fa8d7e761d3980499cd260f0e36b905ac8390918f01ef71cdaf7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Flat",
"family_id": "rtensor_shortcomplex_exact",
"file_id": "mathlib/Mathlib/RingTheory/Flat/CategoryTheory.lean",
"sample_id": "a26f956f637c764b2f59cef3d4d51fd447c50773705cdd84c865b39ccbebb619"
} | train |
{
"chosen_proof_sha256": "db758460f3feb84a60fe6730f2e7a0ca8f4614d60686027e4a4aa0bdb93eeac5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "762cc97dcff582c8c7abd91dcd8f80b24e... | by
by_cases hμ : SFinite μ
swap; · rw [compProd_of_not_sfinite _ _ hμ]; simp
by_cases hν : SFinite ν
swap; · rw [compProd_of_not_sfinite _ _ hν]; simp
let s := κ.mutuallySingularSet η
have hs : MeasurableSet s := Kernel.measurableSet_mutuallySingularSet κ η
symm
refine ⟨s, hs, ?_⟩
rw [compProd_apply h... | true | {
"chosen_tokens": 184,
"rejected_tokens": 5,
"token_jaccard": 0.029851,
"token_length_ratio": 0.027174
} | circular_dependency | circular_dependency | e715233f7e1b1c656f2acadee5ac2e9ce43cf45b1e6d578e347df0df854e6ad7 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Probability.Kernel.Composition.MeasureCompProd
public import Mathlib.Probability.Kernel.RadonNikodym
Namespace:
MeasureTheory.Measure
Local context:
/-
Copyr... | by
exact MutuallySingular.compProd_of_right | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "fcc8f52fe37e395077694ac40e4079267326a6f17df55f31deeb90789c060839",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability/Kernel",
"family_id": "mutuallysingular",
"file_id": "mathlib/Mathlib/Probability/Kernel/Composition/AbsolutelyContinuous.lean",
"sample_id": "56824650124ef37b3559baee31b1277720d3a9f018ce7db07f8712e4b6c90072"
} | train |
{
"chosen_proof_sha256": "760a46c45735c2286ac991019ea42b32b81c6e8ac4991d80d094da0aa71c32a8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "95d16f8d87604e31af296cb85c57b5f4bc... | by
dsimp [destruct]
cases f0 : s.1 0 <;> intro h
· contradiction
· apply Subtype.ext
funext n
induction n with
| zero => injection h with h'; rwa [h'] at f0
| succ n IH => exact s.2 IH | true | {
"chosen_tokens": 53,
"rejected_tokens": 2,
"token_jaccard": 0.026316,
"token_length_ratio": 0.037736
} | invalid_tactic | invalid_tactic | e782a773dcd5fc20a642f86e98bc3a424fa41bf6bf0859d6a4a1f5495e1ba3d1 | 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.Find
public import Mathlib.Data.Stream.Init
public import Mathlib.Logic.Relator
public import Mathlib.Tactic.Common
public import Batteries.Tactic.Lin... | by
__oprover_invalid_tactic_5ec0b0e55183 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "0156e296883168cf0f8c430a64bac182d96a8ea134a52814107376b65614f5e6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Seq",
"family_id": "destruct_eq_pure",
"file_id": "mathlib/Mathlib/Data/Seq/Computation.lean",
"sample_id": "5ec0b0e55183ff152ce79ab5d2c02cf74a8634036c497bc83a0db9d263f074b7"
} | train |
{
"chosen_proof_sha256": "8ea2eb314aa329962f79f410c370fc03bc8d9d4238b02a2baace82f999396bf2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by simp [left_distrib, *] | false | {
"chosen_tokens": 7,
"rejected_tokens": 2,
"token_jaccard": 0.125,
"token_length_ratio": 0.285714
} | sorry_admit_sorryAx_proof_hole | admit | e790e5919324c47a89e2bf661643ba35f690ac97d93ea74dc72dd30ad283563f | 0 | 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
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": "Tactic/CancelDenoms",
"family_id": "add_subst",
"file_id": "mathlib/Mathlib/Tactic/CancelDenoms/Core.lean",
"sample_id": "7b5b722d1bc16ae3899d329eba58a7b2d1599243b0ea9acc8a97cda675c28cc8"
} | train |
{
"chosen_proof_sha256": "3dd577e7ce2be81d92ba051dd678c134f2a6bcae0a357014a7b7228cadb929c6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
refine strictMonoOn_of_lt_add_one ⟨?_⟩ fun n _ _ hn ↦
pow_ssubset_pow_succ_of_pow_ne_closure hX₁ hX hn
rintro - - n hn m ⟨-, hmn⟩ hm
apply hn
obtain rfl | hm₀ := m.eq_zero_or_pos
· simp [eq_comm (a := (1 : Set _)), coe_set_eq_one, -Set.subset_singleton_iff,
hX.coe.not_subset_singleton] at hm
· ... | false | {
"chosen_tokens": 138,
"rejected_tokens": 2,
"token_jaccard": 0.016393,
"token_length_ratio": 0.014493
} | sorry_admit_sorryAx_proof_hole | sorry | e818a51a5a29fc97a4b7cb7357dec1cf1371b84ae4b09de99cb24e3d096c8715 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Action.Pointwise.Finset
public import Mathlib.Algebra.Group.Subgroup.Pointwise
public import Mathlib.Data.Nat.SuccPred
Namespace:
Finset
Local ... | by
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/Group",
"family_id": "pow_right_strictmonoon",
"file_id": "mathlib/Mathlib/Geometry/Group/Growth/LinearLowerBound.lean",
"sample_id": "1a64a2331eda0da0fbaef78bea3f8bc0d6cc844d45d3bd42a1481a8c40fbd0be"
} | train |
{
"chosen_proof_sha256": "b276acfbf4317cef3c8ddc23d0eda431624fd9c228809a52c5678c7b8d9bd426",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cee550d49a61e658e839ba188228371f82... | by
refine ENNReal.eq_sub_of_add_eq hg_fin ?_
rw [← lintegral_add_right' _ hg]
exact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx) | true | {
"chosen_tokens": 28,
"rejected_tokens": 2,
"token_jaccard": 0.038462,
"token_length_ratio": 0.071429
} | invalid_tactic | invalid_tactic | e86c24faae6ec8491a439691532d09ed4719c1844a3aabe8bbdbd7e17ea70102 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.MeasureTheory.Integral.Lebesgue.Add
Namespace:
MeasureTheory
Local context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache ... | by
__oprover_invalid_tactic_79c5185e6f28 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b664c36587e0e4eda73b80984757acc7dcc47faf42493ba5e7ab78493230f5d6",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Integral",
"family_id": "lintegral_sub",
"file_id": "mathlib/Mathlib/MeasureTheory/Integral/Lebesgue/Sub.lean",
"sample_id": "79c5185e6f28d9b4a517c1d0991b691ef5318525bdf3a4fee785bfeb02cb287e"
} | train |
{
"chosen_proof_sha256": "bf3037486d0c6b30f0d1754728261534db0d266ca9de3c9d72286f831a1bc900",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "aeb2769c934ebca5ccdab77e64272ed830... | by
haveI : IsCentral K D :=
{ out := fun x ↦ show x ∈ Subalgebra.center k D → _ by
simp only [center_eq_bot, mem_bot, Set.mem_range, forall_exists_index]
rintro x rfl
exact ⟨algebraMap k K x, by simp [algebraMap_eq_smul_one, smul_assoc]⟩ }
refine ⟨FaithfulSMul.algebraMap_injective k K, fun x => ... | true | {
"chosen_tokens": 161,
"rejected_tokens": 2,
"token_jaccard": 0.016667,
"token_length_ratio": 0.012422
} | invalid_tactic | invalid_tactic | e8aef76c07f5ec5dcb28123ad7347aaf76c954448e5fe2c80991b1b9e77d6831 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Central.Defs
import Mathlib.Algebra.Module.Torsion.Field
Namespace:
Algebra.IsCentral
Local context:
/-
Copyright (c) 2024 Jujian Zhang. All rights r... | by
__oprover_invalid_tactic_53e21ad5e315 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "15f53928d893e815ce916eea7ba03424c15b673c35d96347ff26e047052004e9",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Central",
"family_id": "basefield_essentially_unique",
"file_id": "mathlib/Mathlib/Algebra/Central/Basic.lean",
"sample_id": "53e21ad5e315c317c8084e011afafd42127d23df856e46c6d95998155a1cbf26"
} | train |
{
"chosen_proof_sha256": "b62df895e5ecbe83fea44bae35972a3514a09e499e02633c6e24648777cc00bf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "69656bd82022794f70a1f8fdd7df61df1e... | by
intro contr
obtain ⟨a, b, hab, mcfab⟩ := mcf
specialize contr ![![a, b], ![b, a]]
rw [Fin.sum_univ_two', ← mcfab.left, ← two_nsmul] at contr
have sharp :
2 • ((ω.tt ![![a, b], ![b, a]]).map (fun _ => f ![a, b])).sum <
2 • ((ω.tt ![![a, b], ![b, a]]).map f).sum := by
have half_sharp :
((ω.... | false | {
"chosen_tokens": 456,
"rejected_tokens": 463,
"token_jaccard": 0.974684,
"token_length_ratio": 1.015351
} | markdown_preamble_trailing | fence | e8b20cd4e6e4b0022f47598611935f630e4af42f4878fe7afd2cc0b39f68058c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.BigOperators.Fin
public import Mathlib.Algebra.Order.BigOperators.Group.Multiset
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Linea... | ```lean
by
intro contr
obtain ⟨a, b, hab, mcfab⟩ := mcf
specialize contr ![![a, b], ![b, a]]
rw [Fin.sum_univ_two', ← mcfab.left, ← two_nsmul] at contr
have sharp :
2 • ((ω.tt ![![a, b], ![b, a]]).map (fun _ => f ![a, b])).sum <
2 • ((ω.tt ![![a, b], ![b, a]]).map f).sum := by
have half_sharp :
... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/Optimization",
"family_id": "function",
"file_id": "mathlib/Mathlib/Combinatorics/Optimization/ValuedCSP.lean",
"sample_id": "5e07b58c14b8fda2d13e55aab1c1520c2c2eed27aece7742846d0bae8fae4ed6"
} | train |
{
"chosen_proof_sha256": "c7cf3138eb7d775d3fb111ddb9cf2f6f27656c166d40a0c93cd557614d2fd3af",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0e1d06e3f61ab839dc7deda0008c234fa9... | by
cases xs
simp only [reverse, snoc, cons, toList_mk]
congr
simp [toList, append_def] | true | {
"chosen_tokens": 21,
"rejected_tokens": 2,
"token_jaccard": 0.0625,
"token_length_ratio": 0.095238
} | invalid_tactic | invalid_tactic | e906435fb5959a8730c6063ef9f71c1cd977500604fdbfc7e8fb8faa7c60b84a | 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.Vector.Basic
Namespace:
List.Vector
Local context:
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as describe... | by
__oprover_invalid_tactic_2b323eed78bd | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "ae3ba5e2b838b56637524d15611f7683482c62c5670ee010b2dbed8231cae075",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Vector",
"family_id": "reverse_snoc",
"file_id": "mathlib/Mathlib/Data/Vector/Snoc.lean",
"sample_id": "2b323eed78bdd6ac7a8d6ff6b4eecb4cf02033f76acf0f63ce6e3ba7ef4dfad0"
} | train |
{
"chosen_proof_sha256": "792cf305c49d49d53ed8828dada761a3db9769ab9dc9ac7edc92e61179edcded",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
constructor
· rw [Ne, eq_none_iff', not_not]
exact fun h => ⟨o.get h, eq_some_iff.2 (get_mem h)⟩
· rintro ⟨x, rfl⟩
apply some_ne_none | true | {
"chosen_tokens": 38,
"rejected_tokens": 8,
"token_jaccard": 0.028571,
"token_length_ratio": 0.210526
} | unsolved_goals | unsolved_goals | e930901b112a4bd0dc8715a3cdd4be0163fea80879d4d54523dd9a732b395790 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Notation.Defs
public import Mathlib.Data.Set.Subsingleton
public import Mathlib.Logic.Equiv.Defs
Namespace:
Part
Local context:
/-
Copyright (c) 2017... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "ee3ae9ce77e19904f43c1b09679d7113197bbae3a2593580404f180b6e6e593c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data",
"family_id": "ne_none_iff",
"file_id": "mathlib/Mathlib/Data/Part.lean",
"sample_id": "1d3d2dc551d90c30ba5f909bc69c9cc130fae372a274b0debe56a42386450ace"
} | train |
{
"chosen_proof_sha256": "8beab1ef316edfb8523c7f6d8a094e22bcc81e68fa42d1a455d344f9254b12c0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simp only [MonomialOrder.lex_lt_iff, Finsupp.Lex.lt_iff_of_unique, ofLex_toLex] | false | {
"chosen_tokens": 16,
"rejected_tokens": 2,
"token_jaccard": 0.071429,
"token_length_ratio": 0.125
} | by_exact_placeholder | by? | e969e5b3c8d4e056026dca3539a6d386dbc9578f424d6fc68de98525c8b54ae1 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finsupp.Lex
public import Mathlib.Data.Finsupp.WellFounded
public import Mathlib.Data.List.TFAE
Namespace:
(root)
Local context:
/-
Copyright (c) 2024 A... | by? | {
"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/Finsupp",
"family_id": "monomialorder",
"file_id": "mathlib/Mathlib/Data/Finsupp/MonomialOrder.lean",
"sample_id": "be98270d4e641660001524cb4a6a1e9cfae5fb603cff7ae0acf7c6d8fed02aa2"
} | train |
{
"chosen_proof_sha256": "1bc941a2b3a47d131821876175410f2047d53e35df24a12c023e660701f0a1b1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a86870c89a4a8513a9ca5e86eb6ee5debb... | by
rw [← isLowerSet_compl, ← coe_compl] at h𝒜
have := h𝒜.card_inter_le_finset hℬ
rwa [card_compl, Fintype.card_finset, tsub_mul, le_tsub_iff_le_tsub, ← mul_tsub, ←
card_sdiff_of_subset inter_subset_right, sdiff_inter_self_right, sdiff_compl,
_root_.inf_comm] at this
· grw [inter_subset_right]
· grw ... | true | {
"chosen_tokens": 64,
"rejected_tokens": 3,
"token_jaccard": 0.028571,
"token_length_ratio": 0.046875
} | unknown_identifier | unknown_identifier | e99392e2f755e5bc88a364a126b3392d4a22d19725d62099c672c7a962194dee | 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.Ring.Canonical
public import Mathlib.Algebra.Order.Ring.Nat
public import Mathlib.Combinatorics.SetFamily.Compression.Down
public import Mathlib.... | by
exact __oprover_missing_f74f3c30220b | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "05eb0ee50ebd46da9d8572e1496eec440c3164e19b700dfd4a014905dc6b877f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/SetFamily",
"family_id": "isupperset",
"file_id": "mathlib/Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean",
"sample_id": "f74f3c30220b04da1fe4f1b1519530cecb6ea35343f0fc2bc9e9549863b219a6"
} | train |
{
"chosen_proof_sha256": "ff8a2edf1706af29211010fc997e519b058214e2acf8d5e15ccbda28fa02aa4d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5eeb3348906625369fd323ad89c7632971... | by
rw [Submonoid.eq_top_iff']
intro x
induction x with | _ a
induction a with
| one => exact Submonoid.one_mem _
| of x => exact subset_closure <| by simp [range, of]
| mul x y hx hy => exact Submonoid.mul_mem _ hx hy | true | {
"chosen_tokens": 55,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.054545
} | circular_dependency | circular_dependency | e9c117ba684ad29d1b46171d33c2246f35280032df588a1e6ff6caf43b0da2c6 | 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.FreeMonoid.Basic
public import Mathlib.Algebra.Group.Submonoid.Operations
public import Mathlib.GroupTheory.Congruence.Hom
Namespace:
PresentedMonoid
... | by
exact closure_range_of | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "ef9721e5fd819ac80aba9290f5eba2aed2ccdc6e8e5af40c53a0a2c41903b05b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Algebra/PresentedMonoid",
"family_id": "closure_range_of",
"file_id": "mathlib/Mathlib/Algebra/PresentedMonoid/Basic.lean",
"sample_id": "6fc83af12926d67f82e7ea31db06b0a746a421286c20a9485cd6854580deafe5"
} | train |
{
"chosen_proof_sha256": "d6d98f4113b14782373b73a8bb3f9c78ad64b883beec8e3deb78731e5a49b554",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
simp [← toList_inj] | false | {
"chosen_tokens": 6,
"rejected_tokens": 2,
"token_jaccard": 0.142857,
"token_length_ratio": 0.333333
} | sorry_admit_sorryAx_proof_hole | admit | e9fe65010e0644eeaaa8c7e18b44a4f6ae20fe20f8665c7dfc56b0edc86bd478 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Batteries.Data.String.Lemmas
public import Mathlib.Data.List.Lex
public import Mathlib.Data.Char
public import Mathlib.Algebra.Order.Group.Nat
import all Init.Data.Str... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/String",
"family_id": "oflist_eq",
"file_id": "mathlib/Mathlib/Data/String/Basic.lean",
"sample_id": "7e250c2a0a366c6dc511bcc2bdae4c79d5eb964d8cab919edc4393813525710c"
} | train |
{
"chosen_proof_sha256": "c860446eb6ebd47f56bc80449291da30bdf73e713b21c6c028e07a7ce9d17e8f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "06b9a04bef694021e6ebb3add5b89c19bc... | by
rw [Iso.conj_apply, Iso.eq_inv_comp]; simp [f.hom.comm] | true | {
"chosen_tokens": 20,
"rejected_tokens": 4,
"token_jaccard": 0.2,
"token_length_ratio": 0.2
} | circular_dependency | circular_dependency | ea125adff66c8fba3e1bff37bad43de207334652656d02a93a4396d92a566b11 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Category.Grp.Basic
public import Mathlib.Algebra.Ring.PUnit
public import Mathlib.CategoryTheory.Adjunction.Limits
public import Mathlib.CategoryTheory... | by
exact Iso. | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "0526e519c502c7f3af5c8269e0ea6c9f1216ce6bcc71827b8163d76243295809",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Action",
"family_id": "iso",
"file_id": "mathlib/Mathlib/CategoryTheory/Action/Basic.lean",
"sample_id": "624cb99c0cbd1997bbdc0b1f1c47bd9e2e1caf360a81585dc2e8556116e02b79"
} | train |
{
"chosen_proof_sha256": "2ec97858c3d58fdf4fe63a79d350f35bc6aeb51e4c5540c62fc1b50ae3db70dc",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "24fdd48bb31b77f033a3807b5d2ae5cf24... | by
simp [toInt, (exists_translation val).choose_spec] | true | {
"chosen_tokens": 12,
"rejected_tokens": 2,
"token_jaccard": 0.076923,
"token_length_ratio": 0.166667
} | invalid_tactic | invalid_tactic | ea4c8e3ad74f39e0018bb681c03c79cff7077f8d8970a0a4f218fe1cbb331ae3 | 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
__oprover_invalid_tactic_0a540970c97c | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "54c1735b0c0c4233d5f05bdddf4de1dbfcb743828bc92b39fa078c67edcb2894",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/Order",
"family_id": "toint_le_toint",
"file_id": "mathlib/Mathlib/Tactic/Order/ToInt.lean",
"sample_id": "0a540970c97c25da7889f6fd2aaf6aa1ff05b98f178319841aaf0767908b62e7"
} | train |
{
"chosen_proof_sha256": "dad9f4770c642653b3a83e01d76fcc6253ccf22ac06ee8b959ff773f2af21b7e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "590ba2178e0f149ed4668d8c75b51af1b4... | by
exact
{ map_smul := fun f r => by
simp only [← NatIso.naturality_1 e (r • f), F.map_smul, Linear.smul_comp,
NatTrans.naturality, Linear.comp_smul, Iso.inv_hom_id_app_assoc] } | true | {
"chosen_tokens": 45,
"rejected_tokens": 5,
"token_jaccard": 0.090909,
"token_length_ratio": 0.111111
} | wrong_namespace_qualification | wrong_namespace_qualification | ea6afe74a3927e6d9949dfd87607a8cad111368214e3cd948799ffbc36a5a9ff | 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.Preadditive.AdditiveFunctor
public import Mathlib.CategoryTheory.Linear.Basic
public import Mathlib.Algebra.Module.LinearMap.Rat
Namespace:
Cat... | by
exact OProverMissing.missing_5574e1cf4d2a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c0db2dab9e0b841e5e0aebb9f70d1ed23e4d24f04ad165c3b475f6b8e1b0ba4c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Linear",
"family_id": "linear_of_iso",
"file_id": "mathlib/Mathlib/CategoryTheory/Linear/LinearFunctor.lean",
"sample_id": "5574e1cf4d2a2166ca8dd385d8105f10751922211dcf708a3a1a0c7976bbd7d3"
} | train |
{
"chosen_proof_sha256": "3b83d6405db8d7b67a22120881e91f0e3ce367fb9b3ca08e1bae760a47811d62",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "101bb0c8bbdc01a3ce620c24a4a892863f... | by
cases Nat.eq_zero_or_pos n with
| inl hn =>
rw [if_pos hn, hn, dp_zero]
| inr hn =>
rw [if_neg (ne_of_gt hn), ← zero_smul R (0 : M), dp_smul]
rw [zero_pow (Nat.pos_iff_ne_zero.mp hn), zero_smul] | true | {
"chosen_tokens": 57,
"rejected_tokens": 5,
"token_jaccard": 0.057143,
"token_length_ratio": 0.087719
} | wrong_namespace_qualification | wrong_namespace_qualification | eab0006b56dbe1683205adcb590b71172e17a6b15486509630dff9cd92b8dbb5 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.MvPolynomial.Eval
public import Mathlib.Algebra.RingQuot
public import Mathlib.RingTheory.DividedPowers.Basic
Namespace:
DividedPowerAlgebra
Local co... | by
exact OProverMissing.missing_ae60c1f3772d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f3de0e0817445d24df80e1e4ea3eca7b938ff0b999f9be43813ca53f1e15c930",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/DividedPowerAlgebra",
"family_id": "dp_null",
"file_id": "mathlib/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean",
"sample_id": "ae60c1f3772d8f03a4244e5c6a05fff7f8704657d4e3b6abd61ebd01d5c23d05"
} | train |
{
"chosen_proof_sha256": "5cffe96cad332b72c0ae2797275016448548d2c1b107d88e2f58540117e63993",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
obtain ⟨a', m⟩ := w.mem_essImage α
obtain ⟨m'⟩ := m
exact ⟨a', ⟨(Iso.toEquiv m').symm.trans Equiv.ulift⟩⟩ | false | {
"chosen_tokens": 38,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.052632
} | by_exact_placeholder | by? | eae35e9bc0abc7429d87da5104dbfce2911fadec5427c67c0f16daca079649aa | 0 | 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.EssentialImage
public import Mathlib.CategoryTheory.Types.Basic
public import Mathlib.Logic.UnivLE
Namespace:
(root)
Local context:
/-
Copyrig... | 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": "CategoryTheory",
"family_id": "univle",
"file_id": "mathlib/Mathlib/CategoryTheory/UnivLE.lean",
"sample_id": "501b3d3640125c7dca7fdd0dca6da79f794dca628efb68f4fea9ebb8533db64a"
} | train |
{
"chosen_proof_sha256": "a7995287f4a56e7a4f65ad1427f3f2d4f2887e143a6beb10ba5ae796bad00236",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b1bc28c81cb7045d821825b4ece0233164... | by
rw [← subsingleton_iff_zero_eq_one, OreLocalization.one_def,
OreLocalization.zero_def, oreDiv_eq_iff]
simp | true | {
"chosen_tokens": 17,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.176471
} | unknown_identifier | unknown_identifier | eb205cf6009981287765aa26ad6a37fad36d5e1855fa0cdb006e6a4789458974 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Submonoid.DistribMulAction
public import Mathlib.GroupTheory.OreLocalization.Basic
public import Mathlib.Algebra.GroupWithZero.Defs
Namespace:
O... | by
exact __oprover_missing_cc3728dc6e72 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "8315a55c10b2ffead848a9c09be4c3e85b84a1b56269bde6ea574d30cbedf58e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/OreLocalization",
"family_id": "subsingleton_iff",
"file_id": "mathlib/Mathlib/RingTheory/OreLocalization/Basic.lean",
"sample_id": "cc3728dc6e72e50fd255371dff3e5f651d3ce2d9c638da326516235746fae11d"
} | train |
{
"chosen_proof_sha256": "cde0cb2c54480a9a9ba9dd0ed55105a1ff6f413d177d544464139dccab437478",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a94a4e3a9c0eb0f38e2f97aea7bedf8bb5... | by
classical
simp_rw [← AlgHom.coe_toRingHom, ← coeff_map, subProdXSubC, Polynomial.map_sub,
Polynomial.map_prod, Polynomial.map_sub, map_X, map_C, toSplittingField,
AlgHom.coe_toRingHom, MvPolynomial.aeval_X, dif_pos h,
← (finEquivRoots (Monics.splits_finsetProd h)).symm.prod_comp, Equiv.apply_symm_app... | false | {
"chosen_tokens": 165,
"rejected_tokens": 169,
"token_jaccard": 0.961538,
"token_length_ratio": 1.024242
} | missing_wrong_import | import | eb3fec4fd73da96382b484712105da259587b9234422075ee99083db774f6dbb | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.CharP.Algebra
public import Mathlib.Data.Multiset.Fintype
public import Mathlib.FieldTheory.IsAlgClosed.Basic
public import Mathlib.FieldTheory.Splitti... | import Mathlib.WrongImport
by
classical
simp_rw [← AlgHom.coe_toRingHom, ← coeff_map, subProdXSubC, Polynomial.map_sub,
Polynomial.map_prod, Polynomial.map_sub, map_X, map_C, toSplittingField,
AlgHom.coe_toRingHom, MvPolynomial.aeval_X, dif_pos h,
← (finEquivRoots (Monics.splits_finsetProd h)).symm.prod... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "FieldTheory/IsAlgClosed",
"family_id": "tosplittingfield_coeff",
"file_id": "mathlib/Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean",
"sample_id": "dfc0baea2eabd164611e1d1c93cf31cf2ae01a14f1445c0b6a91023c66e00ae4"
} | train |
{
"chosen_proof_sha256": "851de4796bc656b1357eef23d637285576683be0e867f23ab28b2999ef31f864",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
letI := (Sᵀ.toNormedAddCommGroup hS.transpose)
letI := (Sᵀ.toInnerProductSpace hS.transpose.posSemidef)
ext i j
rw [LDL.lowerInv, Basis.coePiBasisFun.toMatrix_eq_transpose, coe_gramSchmidtBasis]
rfl
noncomputable instance LDL.invertibleLowerInv : Invertible (LDL.lowerInv hS) := by
rw [LDL.lowerInv_eq_gr... | false | {
"chosen_tokens": 119,
"rejected_tokens": 2,
"token_jaccard": 0.022222,
"token_length_ratio": 0.016807
} | sorry_admit_sorryAx_proof_hole | admit | eb4778c5ebae8996440892b5fa966fd54c78548577b3f955c595d04f699d89ab | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
public import Mathlib.Analysis.Matrix.PosDef
Namespace:
(root)
Local context:
/-
Copyright (c) 2022 Alexander Ben... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/Matrix",
"family_id": "ldl",
"file_id": "mathlib/Mathlib/Analysis/Matrix/LDL.lean",
"sample_id": "a95312207d0271d9516315b3dd214e2726915b26938f24e5ee114b92566a3ab3"
} | train |
{
"chosen_proof_sha256": "077e58b8ebfdb691115134cf203f73b4ed07755ad550ac73f5ae401c07fbed75",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e48c355b13d78c993332691198cc2ae1e2... | by
refine Measure.ext_of_charFunDual ?_
ext L
simp_rw [charFunDual_map, charFunDual_prod, charFunDual_eq_of_forall_strongDual_eq_zero hμ,
← Complex.exp_add]
rw [← add_div, ← add_div, ← neg_add, ← neg_add]
congr 3
norm_cast
have h1 : (L.comp (.rotation θ)).comp (.inl ℝ E E)
= Real.cos θ • L.comp ... | false | {
"chosen_tokens": 294,
"rejected_tokens": 301,
"token_jaccard": 0.975904,
"token_length_ratio": 1.02381
} | markdown_preamble_trailing | fence | eb6215e7a354d143f296d4919f5f9a4989098c8f631b655d1865a5f118552539 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Probability.Distributions.Fernique
public import Mathlib.Probability.Distributions.Gaussian.Basic
Namespace:
ProbabilityTheory.IsGaussian
Local context:
/-
C... | ```lean
by
refine Measure.ext_of_charFunDual ?_
ext L
simp_rw [charFunDual_map, charFunDual_prod, charFunDual_eq_of_forall_strongDual_eq_zero hμ,
← Complex.exp_add]
rw [← add_div, ← add_div, ← neg_add, ← neg_add]
congr 3
norm_cast
have h1 : (L.comp (.rotation θ)).comp (.inl ℝ E E)
= Real.cos θ •... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Probability/Distributions",
"family_id": "map_rotation_eq_self_of_forall_strongdual_eq_zero",
"file_id": "mathlib/Mathlib/Probability/Distributions/Gaussian/Fernique.lean",
"sample_id": "bf8d6d5204ce084429052d3f687b1b4007057ecf29fbdc62aab2b93e39615bf4"
} | train |
{
"chosen_proof_sha256": "d3371ad1ae3b9f3bec6391019a5fb1c62af2682915c2a167e06751160d75c560",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
have hb' : (0 : ℝ) < ↑(abs b) := Int.cast_pos.mpr (abs_pos.mpr hb)
have hbε : 0 < abs b • ε := by
rw [Algebra.smul_def]
exact mul_pos hb' hε
have hfloor : ∀ i, 0 ≤ floor ((A i % b : ℤ) / abs b • ε : ℝ) :=
fun _ ↦ floor_nonneg.mpr (div_nonneg (cast_nonneg (emod_nonneg _ hb)) hbε.le)
refine ⟨fun i ... | false | {
"chosen_tokens": 376,
"rejected_tokens": 5,
"token_jaccard": 0.02521,
"token_length_ratio": 0.013298
} | sorry_admit_sorryAx_proof_hole | sorryAx | eb82db3972614fd1a12feaf2694e4d6711a28ff6374c54141a44eff3f057b946 | 0 | 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.Basic
public import Mathlib.Algebra.Order.Archimedean.Real.Basic
public import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
Namesp... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/ClassNumber",
"family_id": "exists_partition_int",
"file_id": "mathlib/Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean",
"sample_id": "f5c3570decc7db710ef7c794c68d9c0b62e8e6967dc4c3bfe5c338cfbcf456b8"
} | train |
{
"chosen_proof_sha256": "e483053df1cef9a9899f381185253ce0593370fe3c8c56da6f87d63a4270c28a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c12cbb39a416f92d3d0c274fc8bf118a1a... | by
dsimp [localization]
simp only [Localization.liftNatTrans_app, Functor.id_obj, Functor.whiskerRight_app,
NatTrans.naturality, Functor.comp_map, Functor.id_map, Iso.hom_inv_id_app_assoc]
include W | true | {
"chosen_tokens": 38,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.052632
} | invalid_tactic | invalid_tactic | eb86e79b2950c3380aa89f2ee59348bb338b32d1c84232b24246fbe1b64042f6 | 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.CategoryTheory.Localization.Predicate
public import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
... | by
__oprover_invalid_tactic_81608c9debe1 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b2623d2650365934678f7940e494b7457db9b2f160a04704689df2836048eb34",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Center",
"family_id": "localization_app",
"file_id": "mathlib/Mathlib/CategoryTheory/Center/Localization.lean",
"sample_id": "81608c9debe142f31aa7efa37f974504e0e8f8ceff0f0397335ad80e4df5060e"
} | train |
{
"chosen_proof_sha256": "3dbc0fabaa57897ff1e49ce5f7faadddfa1be34ceeabe2d7c9fe2b87f820ba31",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
refine .mk _ Etale.respectsIso fun R S T _ _ _ _ _ h h' ↦ ?_
rw [etale_algebraMap] at h' ⊢
rw [faithfullyFlat_algebraMap_iff] at h
exact .of_etale_tensorProduct_of_faithfullyFlat S | false | {
"chosen_tokens": 39,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.051282
} | by_exact_placeholder | by? | eb93870a44729af10ae612e457be11de757a19bf8b0c561a89d091ea074a365d | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.RingHom.Etale
public import Mathlib.RingTheory.Finiteness.Descent
public import Mathlib.RingTheory.Extension.Cotangent.BaseChange
Namespace:
RingHo... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Etale",
"family_id": "etale",
"file_id": "mathlib/Mathlib/RingTheory/Etale/Descent.lean",
"sample_id": "3087fe77077a4804fd4adc39f3b2f67cf5b7a70da11ffe0c839d1c6a058f86e4"
} | train |
{
"chosen_proof_sha256": "105decf31a73d8f03c28fa8bbcf536ea59b795c7d4d6c1c8dd7fade7fd71c4b3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
refine ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => ?_⟩
have a0 := K0.trans_le ha
have b0 := K0.trans_le hb
rw [inv_sub_inv' ((abv_pos abv).1 a0) ((abv_pos abv).1 b0), abv_mul abv, abv_mul abv, abv_inv abv,
abv_inv abv, abv_sub abv]
refine lt_of_mul_lt_mul_left (lt_of_mul_lt_mul_right ?... | false | {
"chosen_tokens": 131,
"rejected_tokens": 3,
"token_jaccard": 0.037037,
"token_length_ratio": 0.022901
} | by_exact_placeholder | placeholder | ebe7d5c250a824506d86b6ec2b7ce46c08b6730d51e2ea6fb6a628ec3b75bd64 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Action.Pi
public import Mathlib.Algebra.Order.AbsoluteValue.Basic
public import Mathlib.Algebra.Order.Field.Basic
public import Mathlib.Algebra.O... | 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/Order",
"family_id": "rat_inv_continuous_lemma",
"file_id": "mathlib/Mathlib/Algebra/Order/CauSeq/Basic.lean",
"sample_id": "2ecf221e60a1de5319a947fdeadd18115cd6ed88d23420eaeb85a40fe0314ea2"
} | train |
{
"chosen_proof_sha256": "0bd39f4463d7de80fb25819949e6b30d4389ae1eaeb6979f0ff98507bcbb9bbf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "858a6efb9e7b2421f0ba9ed1937a4f310a... | by
cat_disch
variable {D : Type u'} [𝒟 : Category.{u'} D] (I J : D ⥤ C) (ρ : I ⟶ J) {W : D} | true | {
"chosen_tokens": 39,
"rejected_tokens": 3,
"token_jaccard": 0.04,
"token_length_ratio": 0.076923
} | circular_dependency | circular_dependency | ec14c38aace76d9a3771dc710abd0d3e27f84d637c896af208c67309c3629161 | 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.Elementwise
public import Mathlib.CategoryTheory.EpiMono
public import Mathlib.Tactic.PPWithUniv
public import Mathlib.Tactic.ToAdditive
Namesp... | by
exact eqToHom_map_comp_apply | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "65f2a877d7f430bd6963e7a5c69524377ed8f3293c7aca38aae8583b39265536",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Types",
"family_id": "eqtohom_map_comp_apply",
"file_id": "mathlib/Mathlib/CategoryTheory/Types/Basic.lean",
"sample_id": "f97af12d0eda400dcad66bb72ae73c904310f1b01ffca3c206a7820ced091c83"
} | train |
{
"chosen_proof_sha256": "bdd21497f8131a5f5a6c2c2e48cfeb6b69583b6be9858ce150b32d0a5afe6448",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d0483fc373e622a8e612626bc8876496e0... | by
rw [← Representation.IntertwiningMap.toLinearMap_apply, d_eq]; rfl | true | {
"chosen_tokens": 14,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.214286
} | circular_dependency | circular_dependency | ec5067fac237b1d528f0ee34433fbcb8da489bc7edf5fa3afd39b34b0ea8a162 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Category.ModuleCat.Adjunctions
public import Mathlib.AlgebraicTopology.ExtraDegeneracy
public import Mathlib.CategoryTheory.Abelian.Ext
public import M... | by
exact d_apply | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "8d3599e7d736eda53fd168596994dc260fb83738990a5a85a1abb47430d8a6d3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RepresentationTheory/Homological",
"family_id": "d_apply",
"file_id": "mathlib/Mathlib/RepresentationTheory/Homological/Resolution.lean",
"sample_id": "2f1e37c134e39703c11252b834c1e076724924538d8af83dbd0bc1c8dab12e37"
} | train |
{
"chosen_proof_sha256": "6695cef3757a420cfb80a318d6ef88b082eb4eef77453a026e398af8fe7ac543",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [iSup, Bool.range_eq, sSup_pair, sup_comm] | false | {
"chosen_tokens": 13,
"rejected_tokens": 3,
"token_jaccard": 0.076923,
"token_length_ratio": 0.230769
} | by_exact_placeholder | exact? | ec6a1df1373504910d7477ce1d9df77133967f4aea7dd6e0da648d2233a31c49 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Bool.Set
public import Mathlib.Data.Nat.Set
public import Mathlib.Order.CompleteLattice.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Joh... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/CompleteLattice",
"family_id": "isup_bool_eq",
"file_id": "mathlib/Mathlib/Order/CompleteLattice/Lemmas.lean",
"sample_id": "37f235bb48e86794e4b09884c23d8d2d7e5709381d2c53948b9b73b1480e3fa6"
} | train |
{
"chosen_proof_sha256": "8e7840e8c83d8f557e8647b534950399050e6f0e066df7ccc73d874d601c7f9a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rcases exists_isRoot_of_odd_natDegree (f := X ^ n - C x) (by simp [hn]) with ⟨r, hr⟩
exact ⟨r, by linear_combination - (by simpa using hr : r ^ n - x = 0)⟩ | true | {
"chosen_tokens": 48,
"rejected_tokens": 8,
"token_jaccard": 0.088235,
"token_length_ratio": 0.166667
} | unsolved_goals | unsolved_goals | ec79520bbc966b95c638fbf29e1b7366c17c717c3e587043647c86a6077aa7f4 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Polynomial.Degree.Domain
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.Algebra.Ring.Semireal.Defs
public import Mathlib.Tact... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "6b172166f97b5999080c8c0ba1efd10a29e400e20c72f0d46c89a86c80a7d551",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "FieldTheory/IsRealClosed",
"family_id": "exists_eq_pow_of_odd",
"file_id": "mathlib/Mathlib/FieldTheory/IsRealClosed/Basic.lean",
"sample_id": "4791ed0fbefe25e9f05c1dec233b6ddc5e09c117965a9cb9936eb95639f9e525"
} | train |
{
"chosen_proof_sha256": "693a609014e39cacfdbaa86bb0c471973c1d57e2dd7fdea8e3f828ddfc7bc6a5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
refine RelEmbedding.wellFounded_iff_isEmpty.mpr ⟨fun a => ?_⟩
suffices sSup (Set.range a) ∈ Set.range a by
obtain ⟨n, hn⟩ := Set.mem_range.mp this
have h' : sSup (Set.range a) < a (n + 1) := by
change _ > _
simp [← hn, a.map_rel_iff]
apply lt_irrefl (a (n + 1))
apply l... | false | {
"chosen_tokens": 141,
"rejected_tokens": 2,
"token_jaccard": 0.016393,
"token_length_ratio": 0.014184
} | sorry_admit_sorryAx_proof_hole | admit | ec80c49cd0f348cdad817d2c0b27f9daff6180144b0ea5be4525821da4b4230c | 0 | 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.Atoms
public import Mathlib.Order.OrderIsoNat
public import Mathlib.Order.RelIso.Set
public import Mathlib.Order.SupClosed
public import Mathlib.Order.Su... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/CompactlyGenerated",
"family_id": "issupclosedcompact",
"file_id": "mathlib/Mathlib/Order/CompactlyGenerated/Basic.lean",
"sample_id": "535e482a49088acaa3ff943705a11086429404a3e1464c08016c546a7309ccb5"
} | train |
{
"chosen_proof_sha256": "ee1a0ff1d116d0272d387712ee44ccfa46821cca224349b1934d9628b210bb5f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rcases isEmpty_or_nonempty ι
· simp
rcases isEmpty_or_nonempty ι'
· simp
refine ciSup_le fun j ↦ ?_
grw [hv.apply_sum_le]
simp only [map_mul]
have (f : ι → ℝ) : ⨆ i : ↥Finset.univ, f i.val = ⨆ i, f i :=
Function.Surjective.iSup_comp (fun i ↦ ⟨⟨i, Finset.mem_univ i⟩, rfl⟩) f
rw [this fun i ↦ v (... | true | {
"chosen_tokens": 136,
"rejected_tokens": 8,
"token_jaccard": 0.065574,
"token_length_ratio": 0.058824
} | unsolved_goals | unsolved_goals | ec82a054689e58ee915aee959c5c38192815f472375f1e685c01e34725989459 | 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.Homogenize
public import Mathlib.NumberTheory.Height.Basic
import Mathlib.Algebra.Order.Ring.IsNonarchimedean
import Mathlib.Data.Fintype.Or... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e7b325d3971bc618b22eaceb8f99c56e33102df74382887a5499a5012a35b981",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Height",
"family_id": "isnonarchimedean",
"file_id": "mathlib/Mathlib/NumberTheory/Height/MvPolynomial.lean",
"sample_id": "f97002eb6f5349cba0b0cba353ad0fc3306516a27d9c160813ec6a889f48cb83"
} | train |
{
"chosen_proof_sha256": "df3d8d4ac17ee91cfc6c4a0a0a21cbff1216a809c22aaf032985f2bc0080f0a1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "bd5e201734617f1764adce1a82681aaccc... | by
simpa using coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal' (I := ⊤) (f := f) (by simp) hg
variable [I.IsTwoSided] | true | {
"chosen_tokens": 25,
"rejected_tokens": 5,
"token_jaccard": 0.1,
"token_length_ratio": 0.2
} | wrong_namespace_qualification | wrong_namespace_qualification | eca6970ad9aadcc1afffef302d502587e09e85eda9b0edd13173bcb155a7bfad | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Ideal.Operations
public import Mathlib.RingTheory.Ideal.BigOperators
public import Mathlib.RingTheory.PowerSeries.Basic
Namespace:
PowerSeries
Loc... | by
exact OProverMissing.missing_502c601dc704 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "e42d7e59da61f7268c6e9e0b8dca29c058ce195a98cfcea47ca1ebb7c6b9cc8d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/PowerSeries",
"family_id": "coeff_mul_mem_ideal_of_coeff_right_mem_ideal",
"file_id": "mathlib/Mathlib/RingTheory/PowerSeries/CoeffMulMem.lean",
"sample_id": "502c601dc704b016dbc827a9cd8efa7a22333a98a499271c49028df90994ce61"
} | train |
{
"chosen_proof_sha256": "1791f6bfaef40a4ae8b1dcdf06c489870e6b8a0fe97f09d60381a63c54a541cd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "46398d714eff44fac1bfd9b634185d6379... | by
rw [irreducibleComponents_eq_singleton, Set.mem_singleton_iff] at hZ
simp [irreducibleComponentOpen, irreducibleComponents_eq_singleton, hZ] | true | {
"chosen_tokens": 19,
"rejected_tokens": 5,
"token_jaccard": 0.125,
"token_length_ratio": 0.263158
} | wrong_namespace_qualification | wrong_namespace_qualification | ecb6357fd6320f4962ce6c2e526d6444c358f27803cc7ca519595441f5090e28 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
public import Mathlib.AlgebraicGeometry.Noetherian
Namespace:
AlgebraicGeometry.Scheme
Local context:
/-
Copyrigh... | by
exact OProverMissing.missing_9b3f45e52741 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "4cf93d311610720b133c971c5ed08d8ca386dc7ca2e746028c092b8b7b921294",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "536871cc5d7d8fd19b1680026dd5fcc3bae6cf4953adabbd17e9ab3ecd407260",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "225045c52101b712c12567b33713784630... | by
refine ⟨fun h ↦ ⟨?_, ?_, ?_⟩, fun h ↦ ?_⟩
any_goals rintro a b rfl rfl; exact map_eq_empty.1 h
· rintro a b rfl rfl; exact disjSum_eq_empty.1 h
cases a <;> cases b
· exact map_eq_empty.2 (h.1 _ _ rfl rfl)
· simp [h.2.1 _ _ rfl rfl]
· rfl
· exact map_eq_empty.2 (h.2.2 _ _ rfl rfl) | false | {
"chosen_tokens": 99,
"rejected_tokens": 104,
"token_jaccard": 0.861111,
"token_length_ratio": 1.050505
} | markdown_preamble_trailing | preamble | ecc3e599ab5a5e434ba137e69f212d3077cca7f202dcbc8785e0019f6c8c1bbb | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finset.Sum
public import Mathlib.Data.Sum.Order
public import Mathlib.Order.Interval.Finset.Defs
Namespace:
Finset
Local context:
/-
Copyright (c) 2022 ... | Here is the proof:
by
refine ⟨fun h ↦ ⟨?_, ?_, ?_⟩, fun h ↦ ?_⟩
any_goals rintro a b rfl rfl; exact map_eq_empty.1 h
· rintro a b rfl rfl; exact disjSum_eq_empty.1 h
cases a <;> cases b
· exact map_eq_empty.2 (h.1 _ _ rfl rfl)
· simp [h.2.1 _ _ rfl rfl]
· rfl
· exact map_eq_empty.2 (h.2.2 _ _ rfl rfl) | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/Sum",
"family_id": "sumlexlift_eq_empty",
"file_id": "mathlib/Mathlib/Data/Sum/Interval.lean",
"sample_id": "7fa1587a449af9bd025ff7fcf8e9e6160d9d026e5e59b7be7c90ab324dc63b3b"
} | train |
{
"chosen_proof_sha256": "c0bd9ef2de4bda624d2c87bea71eb5ca5fdf0a8bf96e99ba0215cfd11acebae5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9eff0f1367e2de98b6a2bc05edad8e2501... | by
constructor
· intro h
constructor
rintro _ _ f (hf : epimorphisms C f)
rw [epi_iff_surjective]
rw [← h] at hf
exact hf
· intro
apply le_antisymm (surjective_le_epimorphisms C)
intro _ _ f hf
have : Epi f := hf
change Function.Surjective ((forget C).map f)
rw [← epi_iff_s... | false | {
"chosen_tokens": 68,
"rejected_tokens": 73,
"token_jaccard": 0.894737,
"token_length_ratio": 1.073529
} | markdown_preamble_trailing | trailing | ed2d54c72fc21ebe02584e0e8b9df9d408b23d7bcc91801cf22381e32e81ad2d | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
public import Mathlib.CategoryTheory.Limits.Preserves.Basic
public import Mathlib.CategoryTheory.Limits.Shapes.Imag... | by
constructor
· intro h
constructor
rintro _ _ f (hf : epimorphisms C f)
rw [epi_iff_surjective]
rw [← h] at hf
exact hf
· intro
apply le_antisymm (surjective_le_epimorphisms C)
intro _ _ f hf
have : Epi f := hf
change Function.Surjective ((forget C).map f)
rw [← epi_iff_s... | {
"accepted": false,
"all_failure_types": [
"prose_or_trailing_explanation"
],
"failure_type": "prose_or_trailing_explanation",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/ConcreteCategory",
"family_id": "surjective_eq_epimorphisms_iff",
"file_id": "mathlib/Mathlib/CategoryTheory/ConcreteCategory/EpiMono.lean",
"sample_id": "57e63da4c8d7ec6a478c9fdfc7ae432be0adeebf4eec412381dc418d3098ca3d"
} | train |
{
"chosen_proof_sha256": "3fa3b1641e3a223c1b1c2acb29844761f01c0429d5fdec1fe9c7e636fcc33dde",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
suffices h₁ : n ≤ sInf {m | p (m - n)} by
convert! sInf_add h₁
simp_rw [Nat.add_sub_cancel_right]
obtain ⟨m, hm⟩ := nonempty_of_pos_sInf h
refine
le_csInf ⟨m + n, ?_⟩ fun b hb ↦
le_of_not_gt fun hbn ↦
ne_of_mem_of_not_mem ?_ (notMem_of_lt_sInf h) (Nat.sub_eq_zero_of_le hbn.le)
· dsi... | false | {
"chosen_tokens": 83,
"rejected_tokens": 2,
"token_jaccard": 0.019231,
"token_length_ratio": 0.024096
} | sorry_admit_sorryAx_proof_hole | sorry | ed827a658dd554d5a0cec33e2121789c7a69cd2b0b6fb3a8725cbd5322796a02 | 0 | 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.... | by
sorry | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/Lattice",
"family_id": "sinf_add",
"file_id": "mathlib/Mathlib/Order/Lattice/Nat.lean",
"sample_id": "717673c42ec9cf536eab52497d443d7e188d9ccddc0b2dc34242dbb48af892e4"
} | train |
{
"chosen_proof_sha256": "1656a7f84780a26168e97da85997b830e60c4a9c21bd50667f35238ce2172d26",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ef209941b3f109c06162c2f7b35734faae... | by
let U := f ⁻¹' {f y}
suffices x ∉ Uᶜ from Classical.not_not.1 this
intro hxV
specialize hs U Uᶜ (hf {f y}) (hf {f y}ᶜ) _ ⟨y, ⟨hy, rfl⟩⟩ ⟨x, ⟨hx, hxV⟩⟩
· simp only [union_compl_self, subset_univ]
· simp only [inter_empty, Set.not_nonempty_empty, inter_compl_self] at hs | true | {
"chosen_tokens": 87,
"rejected_tokens": 2,
"token_jaccard": 0.020833,
"token_length_ratio": 0.022989
} | invalid_tactic | invalid_tactic | ed97b22deebdb842b3c5b937fc0062d8ae2ab14856db9da5d82ac0b1d98e6a56 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Notation.Indicator
public import Mathlib.Topology.Connected.LocallyConnected
public import Mathlib.Topology.Sets.Closeds
Namespace:
IsLocallyConstant
... | by
__oprover_invalid_tactic_0e4b1666a2df | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "66e0c5e66c7c99602389f5f8be187515a655cf0d45a469631b313073d226049a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/LocallyConstant",
"family_id": "apply_eq_of_ispreconnected",
"file_id": "mathlib/Mathlib/Topology/LocallyConstant/Basic.lean",
"sample_id": "0e4b1666a2df18b8f82460264078354bcb621eb24caecc149f02733eb9e19be1"
} | train |
{
"chosen_proof_sha256": "d0424b22b522a25f022cebb1346824da37ff580ade98ae91173e442bd2f6ba06",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "dcd5259bf50f78916804d2a00f0af0f070... | by
apply IsPullback.of_right (t := (inf_isPullback ((«exists» f).obj A') B').flip)
(p := by simp [frobeniusMorphism])
simpa [frobeniusMorphism, IsPullback.lift_fst, ← imageFactorisation_F_m,
(isPullback f B').paste_horiz_iff] using
(inf_isPullback A' ((Subobject.pullback f).obj B')).flip | false | {
"chosen_tokens": 72,
"rejected_tokens": 76,
"token_jaccard": 0.916667,
"token_length_ratio": 1.055556
} | missing_wrong_import | import | ed9b6d443a5e9104fd27b9c6d1caa8c73bcbd5de0637891306aded607c82a6a3 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.ExtremalEpi
public import Mathlib.CategoryTheory.MorphismProperty.Limits
public import Mathlib.CategoryTheory.Sites.Coherent.Basic
Namespace:
C... | import Mathlib.WrongImport
by
apply IsPullback.of_right (t := (inf_isPullback ((«exists» f).obj A') B').flip)
(p := by simp [frobeniusMorphism])
simpa [frobeniusMorphism, IsPullback.lift_fst, ← imageFactorisation_F_m,
(isPullback f B').paste_horiz_iff] using
(inf_isPullback A' ((Subobject.pullback f).ob... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/RegularCategory",
"family_id": "frobeniusmorphism_ispullback",
"file_id": "mathlib/Mathlib/CategoryTheory/RegularCategory/Basic.lean",
"sample_id": "093333b00fcbe2feec5384aaaba84c83717794caa26f2c6955f67a6005b04fa4"
} | train |
{
"chosen_proof_sha256": "e1fced99958acc84dd8fa2ad1ee6184c3708ee54a7e8d9105ea3938d1bcfadce",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cb0ad84f124428492c2b06be9a3022ca3c... | by
classical
refine ⟨fun h₂ ↦ by simp [isBigO_of_le' (c := 0), h₂], ?_⟩
contrapose
intro h₁
obtain ⟨e, he⟩ := exists_single_le_pos (lt_of_le_of_ne h (h₁ ·.symm))
rw [isBigO_iff'']
push Not
intro a ha
simp only [Pi.one_apply, norm_eq_abs, frequently_atTop, abs_one]
intro b
obtain ⟨c, hc⟩ := eventua... | true | {
"chosen_tokens": 264,
"rejected_tokens": 3,
"token_jaccard": 0.011765,
"token_length_ratio": 0.011364
} | circular_dependency | circular_dependency | edefe588435b2baa98bf0e451b0e25f6c9ce8e9d94ee06703b3d0950abbc8f27 | 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.Complex.ValueDistribution.LogCounting.Basic
Namespace:
Function.locallyFinsuppWithin
Local context:
/-
Copyright (c) 2026 Stefan Kebekus. All rights... | by
exact zero_iff_logCounting_bounded | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "54a81d5d82b0000aef0e5cc67afa757acf92564980c3205b3689f7f278c69bb3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Complex",
"family_id": "zero_iff_logcounting_bounded",
"file_id": "mathlib/Mathlib/Analysis/Complex/ValueDistribution/LogCounting/Asymptotic.lean",
"sample_id": "8ccb2339aba41c643ce9ab6237c697b1faecc53decbb2a2025ef71895c9c0e9f"
} | train |
{
"chosen_proof_sha256": "911bfcaad6d4d642cf91de9a824237ea9c93ccf06c46a0091532a586f961081e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
simp_rw [SetLike.ext_iff, mem_stabilizer_set]
refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_right ha⟩
have : 1 * MulOpposite.op a ∈ s := (h 1).2 s.one_mem
simpa only [op_smul_eq_mul, SetLike.mem_coe, one_mul] using! this | false | {
"chosen_tokens": 63,
"rejected_tokens": 2,
"token_jaccard": 0.047619,
"token_length_ratio": 0.031746
} | by_exact_placeholder | by? | edfbdcd260d1183aa211d99b518b5a2b79a9b540becd5bdef9fdf38273cb9d34 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Action.Pointwise.Finset
public import Mathlib.GroupTheory.QuotientGroup.Defs
public import Mathlib.Order.ConditionallyCompleteLattice.Basic
Name... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Pointwise",
"family_id": "stabilizer_subgroup_op",
"file_id": "mathlib/Mathlib/Algebra/Pointwise/Stabilizer.lean",
"sample_id": "61d172a6c1c40adedfc16e781f60f95fac4d1f6ec88c11c9a5b57c8a1f496302"
} | train |
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