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": "dad9f4770c642653b3a83e01d76fcc6253ccf22ac06ee8b959ff773f2af21b7e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b9481162d2a39b870dac6d7f3cc6ca1534... | 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] } | false | {
"chosen_tokens": 45,
"rejected_tokens": 50,
"token_jaccard": 0.885714,
"token_length_ratio": 1.111111
} | markdown_preamble_trailing | trailing | ca708eb0155b07a93ca35e3418adf0ea7e9ddef41db818b606b22f9f3fa606ca | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
public import Mathlib.CategoryTheory.Linear.Basic
public import Mathlib.Algebra.Module.LinearMap.Rat
Namespace:
Cat... | 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] }
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/Linear",
"family_id": "linear_of_iso",
"file_id": "mathlib/Mathlib/CategoryTheory/Linear/LinearFunctor.lean",
"sample_id": "5574e1cf4d2a2166ca8dd385d8105f10751922211dcf708a3a1a0c7976bbd7d3"
} | train |
{
"chosen_proof_sha256": "a57d660deed4ffd6ef566ef7d9ca6c07159f27d62499692c71e882d53f3bfa34",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
grind [= lt_iff_le_not_ge] | false | {
"chosen_tokens": 6,
"rejected_tokens": 2,
"token_jaccard": 0.142857,
"token_length_ratio": 0.333333
} | sorry_admit_sorryAx_proof_hole | admit | ca849763710d72eb71dfd88a8dd1897311af4d1f754e1be7d576616db52150f6 | 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.Tactic.Alias
public import Batteries.Tactic.Trans
public import Mathlib.Tactic.ExtendDoc
public import Mathlib.Tactic.ToDual
Namespace:
(root)
Local contex... | 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/Defs",
"family_id": "lt_iff_gt_iff_le_iff_ge",
"file_id": "mathlib/Mathlib/Order/Defs/PartialOrder.lean",
"sample_id": "211d293f722fe1130c34a732f32615615b914ff3488d05c85b24b7fea5b7d189"
} | train |
{
"chosen_proof_sha256": "c849c5df0799eac828ab19895ffb8d4649eaf0e6c519838aa60e6b56e2b819cc",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
have a := hP.card_parts_eq_average ht
lia | true | {
"chosen_tokens": 9,
"rejected_tokens": 8,
"token_jaccard": 0.230769,
"token_length_ratio": 0.888889
} | unsolved_goals | unsolved_goals | cab24ce71c642e3daea44cf8385b2e40724005a42ad988d3fb3657d624aa9d43 | 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.Nat
public import Mathlib.Data.Set.Equitable
public import Mathlib.Logic.Equiv.Fin.Basic
public import Mathlib.Order.Partition.Finpartition
... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e673fcbab29f876fe999052c806b5be7379cd98ce3e5ad9dd5d054f0dc61d4b7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Partition",
"family_id": "isequipartition",
"file_id": "mathlib/Mathlib/Order/Partition/Equipartition.lean",
"sample_id": "a6cc210366796b93c3087a5747f28bf3396e3f7d848298d6e55848e06cc71e8d"
} | train |
{
"chosen_proof_sha256": "9117fb79a21990bf1e7a81654768056689d985c1af2446daa090c5cc9faf03cd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c145cc4ee5288f2772f52444fc547753b... | by
rcases eq_or_lt_of_le hj with h | h
· rw [h]
· exact T.row_weak h cell | true | {
"chosen_tokens": 20,
"rejected_tokens": 3,
"token_jaccard": 0.117647,
"token_length_ratio": 0.15
} | unknown_identifier | unknown_identifier | cb48cd8588f65b0adf6dc28eaa4652aeec5058066f18552e861a17a481a925ca | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Combinatorics.Young.YoungDiagram
Namespace:
SemistandardYoungTableau
Local context:
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under ... | by
exact __oprover_missing_3890101cc6d5 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "f2d3a59ad39d43066788a15c26f338dae70973baa17e3a38be83a649860ca131",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/Young",
"family_id": "row_weak_of_le",
"file_id": "mathlib/Mathlib/Combinatorics/Young/SemistandardTableau.lean",
"sample_id": "3890101cc6d52342ce69e19715aee46bb30db6af8bbb140e8a5fac3718968504"
} | train |
{
"chosen_proof_sha256": "b2e750b3c5b8740fb444ea49e43055beae89260e7b9e80c29b949ec289033978",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "844d818def83ae69c024a7d8514dd9d378... | by
rw [of_apply, ← lift_symm_apply, Equiv.symm_apply_apply] | false | {
"chosen_tokens": 12,
"rejected_tokens": 19,
"token_jaccard": 0.846154,
"token_length_ratio": 1.583333
} | markdown_preamble_trailing | fence | cb65422dbddb8cffab822914ba5a7754cd65f22c5ebf7606bdf7508dc0620f62 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.SkewMonoidAlgebra.Basic
public import Mathlib.Algebra.Module.BigOperators
public import Mathlib.Algebra.Algebra.Equiv
Namespace:
SkewMonoidAlgebra
Lo... | ```lean
by
rw [of_apply, ← lift_symm_apply, Equiv.symm_apply_apply]
``` | {
"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/SkewMonoidAlgebra",
"family_id": "lift_of",
"file_id": "mathlib/Mathlib/Algebra/SkewMonoidAlgebra/Lift.lean",
"sample_id": "f0978345095eddbdb23e72c7878138dd691ea3f6ff7d3034237e21db83bc2237"
} | train |
{
"chosen_proof_sha256": "eac7599f7adbe4156b1359f0aff3dffecc04db5fb77ecf8bedd140ea7081aceb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9b4df564c00a9e22a952e465016b91d0c3... | by
induction h
simp | true | {
"chosen_tokens": 4,
"rejected_tokens": 5,
"token_jaccard": 0.125,
"token_length_ratio": 1.25
} | wrong_namespace_qualification | wrong_namespace_qualification | cba2716e5f78bacc62dd24a91b8d9f17c8bca4b7df7da04f305c8b4873d97de8 | 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.Skeleton
public import Mathlib.CategoryTheory.Subobject.MonoOver
public import Mathlib.CategoryTheory.Skeletal
public import Mathlib.Cate... | by
exact OProverMissing.missing_7ee01c83c10c | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d8c382de9011809aa3641b268fe3df78957613522a7747f9ff1f6ff44def4c83",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Subobject",
"family_id": "arrow_congr",
"file_id": "mathlib/Mathlib/CategoryTheory/Subobject/Basic.lean",
"sample_id": "7ee01c83c10c22668c1fdc800b54b271e64edd823972fd056c65d61e5211e5df"
} | train |
{
"chosen_proof_sha256": "47e39958d92c203177453f8d36c36c3e273fa3ce6ffb582eb81c95f2e2f8425a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | 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⟩
... | true | {
"chosen_tokens": 155,
"rejected_tokens": 8,
"token_jaccard": 0.044776,
"token_length_ratio": 0.051613
} | unsolved_goals | unsolved_goals | cbb74a9cdad6c5031bff3a892debaef1846c14087078e2f6cefcb8da25201d78 | 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
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "0c2c1e8d788ac4d0ef9ab5f5a9df5f6694f98c38be6f917b8ec2f35ef4a7a7d5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "75765532e9f719b9017c00ea0609cd6aeeb9a4a2bf7c2d14d50fca0b0687a396",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
obtain ⟨x, hx⟩ := h.mul_inv_cancel ha
simpa [← mul_assoc, h.mul_comm, hx] using congr_arg (x * ·) hb
mul_right_cancel_of_ne_zero ha _ _ hb := by
obtain ⟨x, hx⟩ := h.mul_inv_cancel ha
simpa [mul_assoc, hx] using congr_arg (· * x) hb
exists_pair_ne := h.exists_pair_ne | false | {
"chosen_tokens": 68,
"rejected_tokens": 2,
"token_jaccard": 0.034483,
"token_length_ratio": 0.029412
} | sorry_admit_sorryAx_proof_hole | admit | cbe64824b82ac0aee7773986f305cf77de35014f915e5ad47004cd4e851cde46 | 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.Defs
public import Mathlib.Tactic.Common
Namespace:
(root)
Local context:
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released ... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Field",
"family_id": "isfield",
"file_id": "mathlib/Mathlib/Algebra/Field/IsField.lean",
"sample_id": "e5781a280c95429bef6967a38411ca2235131ecb05d9745326c5155130e9a4ab"
} | train |
{
"chosen_proof_sha256": "9740f164ce7086b555af4b7554cbcc58f11573780fea0b6812b9fc64a3703b2b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
apply MonoidHom.eq_of_eqOn_denseM (closure_range_of _)
grind [Set.eqOn_range] | false | {
"chosen_tokens": 15,
"rejected_tokens": 2,
"token_jaccard": 0.066667,
"token_length_ratio": 0.133333
} | sorry_admit_sorryAx_proof_hole | admit | cbf36a4d3ec570060cd79be70f581da7037b98227b0973a506144a4e539ba798 | 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
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": "Algebra/PresentedMonoid",
"family_id": "ext",
"file_id": "mathlib/Mathlib/Algebra/PresentedMonoid/Basic.lean",
"sample_id": "27c7b061eb2df69ed4438c60ac705043bf7b1c99567810c2e2cbfe95dec0ba49"
} | train |
{
"chosen_proof_sha256": "3973b528cf86a6a3105014eb45539f7af155ea83fbda961dfd591ee49e04f82a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "398caa0d1490961bc830406e9179967f9e... | by
obtain ⟨p₁, p₂, n, e⟩ := P.cond
have : aeval x P.f.derivative ∣ aeval x P.g ^ n :=
⟨_, by simpa [h.1] using congr(aeval x $e.symm)⟩
exact isUnit_of_dvd_unit this (.pow _ h.2) | true | {
"chosen_tokens": 67,
"rejected_tokens": 5,
"token_jaccard": 0.071429,
"token_length_ratio": 0.074627
} | circular_dependency | circular_dependency | cbf37a365117d20c230c642370ca887a60fcf7d1f04bfe3f494844283a516d00 | 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.Bivariate
public import Mathlib.Algebra.Polynomial.Taylor
public import Mathlib.RingTheory.Etale.Basic
public import Mathlib.RingTheory.Exte... | by
exact HasMap.isUnit_derivative_f | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "cd23a21601a9aa4baddde82532dfc4648b79abbfa00258058961f43b8f1ce4be",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Etale",
"family_id": "hasmap",
"file_id": "mathlib/Mathlib/RingTheory/Etale/StandardEtale.lean",
"sample_id": "a1d25de80740477526c09d1872d0447c8000fb20b6d0ca785a16aeb63ebdfb95"
} | train |
{
"chosen_proof_sha256": "2c32995a07b48e16dcaafc728301779b447c776a69f801e281fcde5579068d01",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
refine ⟨fun h ↦ fun _ _ _ ↦ h _ _, fun h ↦ fun i₁ i₂ ↦ ?_⟩
by_cases hi : i₁ = i₂
· simp [hi]
· exact h hi | true | {
"chosen_tokens": 44,
"rejected_tokens": 8,
"token_jaccard": 0.074074,
"token_length_ratio": 0.181818
} | unsolved_goals | unsolved_goals | cc44fd93977c7d3235055dcd84e0a8fe0a6f7d1728394206c3293625a71c3ff4 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.MetricSpace.Pseudo.Defs
public import Mathlib.Topology.MetricSpace.Isometry
public import Mathlib.Topology.MetricSpace.Dilation
Namespace:
(root)
Lo... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "031b929041c799bda630701bb9bc049f8022549482549716daab2fdba835cbaa",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/MetricSpace",
"family_id": "congruent_iff_pairwise_edist_eq",
"file_id": "mathlib/Mathlib/Topology/MetricSpace/Congruence.lean",
"sample_id": "fce701ee6ff6a6606b14ec7394e9ebfdd374d824dea16656e00b6156e86509d7"
} | train |
{
"chosen_proof_sha256": "3df867611d307b2d4c52a5aeae1c5c93f72412c5fca86e32a6445afd51c4bda0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
trans fun n ↦ (n / 3 - 2) * ↑((n - 3) / 6) * exp (-4 * √(log ↑((n - 3) / 6)))
· simp_rw [sq]
refine (IsBigO.mul ?_ ?_).mul ?_
· trans fun n ↦ n / 3
· simp_rw [div_eq_inv_mul]
exact (isBigO_refl ..).const_mul_right (by simp)
refine IsLittleO.right_isBigO_sub ?_
simpa [div_eq_inv_... | false | {
"chosen_tokens": 244,
"rejected_tokens": 2,
"token_jaccard": 0.011236,
"token_length_ratio": 0.008197
} | sorry_admit_sorryAx_proof_hole | admit | cc45799f765756cd0ef42f1997f90a05b37faa6314599e8be0c2b5c2142bddcb | 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.Combinatorics.Additive.AP.Three.Behrend
public import Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite
public import Mathlib.Tactic.Rify
public import Mat... | by
admit | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/Extremal",
"family_id": "ruzsaszemeredinumbernat_asymptotic_lower_bound",
"file_id": "mathlib/Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean",
"sample_id": "fe45775e1693cac44dede504e39345f1c5dfd23b430122a30f0b82d3494c2fdd"
} | train |
{
"chosen_proof_sha256": "ad208d86006761f39c52958101642c0dd5169f84bd3c5136aba050921aea4aea",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "2ba4cc62e244d5056d28140a5aecb6bb09... | by
let ⟨g, hg⟩ := (ContinuousLinearMap.id 𝕜 S).exist_extension_of_finiteDimensional_range
exact ⟨g, DFunLike.congr_fun hg.symm⟩ | true | {
"chosen_tokens": 28,
"rejected_tokens": 2,
"token_jaccard": 0.047619,
"token_length_ratio": 0.071429
} | invalid_tactic | invalid_tactic | cc72e41710d4dbab7d78826c54d92bb37e4225ca12595beba6a667772e6d5780 | 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.Convex.Cone.Extension
public import Mathlib.Analysis.LocallyConvex.AbsConvexOpen
public import Mathlib.Analysis.LocallyConvex.WeakDual
public import M... | by
__oprover_invalid_tactic_0659aa452ee5 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e38e17756d1bf500147fbd2c190bca3685edd85c96cf9e93c947b51a3b512ac8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/LocallyConvex",
"family_id": "submodule",
"file_id": "mathlib/Mathlib/Analysis/LocallyConvex/HahnBanach.lean",
"sample_id": "0659aa452ee5b4142dc99d160a8e44415976c14c3bb5194948dedcb030d02219"
} | train |
{
"chosen_proof_sha256": "e68fac77a9fc516d7c10bdaed9dcefeaf4db7121c9ed0829fb3a283ec7ee3c03",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "76b26d55952be09aac10b04e67c37a508d... | by
rw [range, List.range_succ, ← coe_add, Multiset.add_comm, range, coe_singleton, singleton_add] | true | {
"chosen_tokens": 22,
"rejected_tokens": 3,
"token_jaccard": 0.058824,
"token_length_ratio": 0.136364
} | unknown_identifier | unknown_identifier | cc9e0ff6aea75594c951dfcf7023055b33c737de4b94c289c4f588b2e698ad3c | 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.Multiset.UnionInter
Namespace:
Multiset
Local context:
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 licen... | by
exact __oprover_missing_735bf5dce5b3 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "a2ab4480e8be77064093d36caadbc361d6e761483be451486e25a98119941bce",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Multiset",
"family_id": "range_succ",
"file_id": "mathlib/Mathlib/Data/Multiset/Range.lean",
"sample_id": "735bf5dce5b321e535055fa4c86fa49603a8028a22626fad8900aee15b3514d6"
} | train |
{
"chosen_proof_sha256": "313c9cf4c1b5eea4156e212ff571b19fbbfa824c1fd6f777992a3ed4cc19b0b1",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
by_cases hq : 1 < q
· rw [← pow_zero q, ofPowerSeries_apply_pow hq, PowerSeries.coeff_zero_eq_constantCoeff]
· simp [ofPowerSeries, dif_neg hq] | false | {
"chosen_tokens": 29,
"rejected_tokens": 2,
"token_jaccard": 0.045455,
"token_length_ratio": 0.068966
} | sorry_admit_sorryAx_proof_hole | sorry | ccd847e98947eacff7b6f74af58894f0896a00a7f05df35134e482f54fd07323 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.NumberTheory.ArithmeticFunction.Defs
public import Mathlib.Order.Northcott
public import Mathlib.RingTheory.PowerSeries.Basic
public import Mathlib.RingTheory.... | 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/ArithmeticFunction",
"family_id": "ofpowerseries_apply_one",
"file_id": "mathlib/Mathlib/NumberTheory/ArithmeticFunction/LFunction.lean",
"sample_id": "dd0a6c86eaf74b4453f65b352e9a02cbc6a9bc8e7367f1c0ef0f634a91f60ff2"
} | train |
{
"chosen_proof_sha256": "536352e5ddbb900c41d2280e8e59f7791e63d85f699a6f8a27e8c2d97392c7b3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
let S : Set (Ultrafilter M) := ⋂ n, { U | ∀ᶠ m in U, m ∈ FP (a.drop n) }
have h := exists_idempotent_in_compact_subsemigroup ?_ S ?_ ?_ ?_
· rcases h with ⟨U, hU, U_idem⟩
refine ⟨U, U_idem, ?_⟩
convert! Set.mem_iInter.mp hU 0
· exact Ultrafilter.continuous_mul_left
· apply IsCompact.nonempty_iInter... | true | {
"chosen_tokens": 224,
"rejected_tokens": 8,
"token_jaccard": 0.043478,
"token_length_ratio": 0.035714
} | unsolved_goals | unsolved_goals | cce7eae1f3a0b2e14810860437387f7d1427fc35ab4282f05ace605284f4690c | 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.Init
public import Mathlib.Topology.Algebra.Semigroup
public import Mathlib.Topology.Compactification.StoneCech
public import Mathlib.Algebra.BigOp... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "bd63b81ec55a775fe7c4fab35a1d24822d3e6e52f7c6720988fe9127c817cec7",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "Combinatorics",
"family_id": "exists_idempotent_ultrafilter_le_fp",
"file_id": "mathlib/Mathlib/Combinatorics/Hindman.lean",
"sample_id": "b9b65ec13f7e30e41eeaa6338022c9efbc66d8f0394c7b483c53252b38ab9a8f"
} | train |
{
"chosen_proof_sha256": "7fb3cc1775db12a3f434a99b04b7c0a0f79496406a0eb39713c9f5d39c18a786",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [sInter_eq_biInter]
exact MeasurableSet.biInter hs h | false | {
"chosen_tokens": 11,
"rejected_tokens": 3,
"token_jaccard": 0.166667,
"token_length_ratio": 0.272727
} | by_exact_placeholder | placeholder | ccf52a6698a535699465c5ffdbbd533262745c8cbda86f2fe73f1b9258540848 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Set.Countable
public import Mathlib.Order.ConditionallyCompleteLattice.Basic
public import Mathlib.Tactic.CrossRefAttribute
public import Mathlib.Tactic.F... | by
exact 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": "MeasureTheory/MeasurableSpace",
"family_id": "measurableset",
"file_id": "mathlib/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean",
"sample_id": "b9d6520e23cf7ef508708d905f3af667f91b01c1501bdfb215036dc08f82f360"
} | train |
{
"chosen_proof_sha256": "bc537854a12e226bc8123bdf85ce0d801d68aa77f9f581d84fafad195b8a7d94",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c9d53b75ff94d9ba91ecd885af73f8533c... | by
obtain ⟨n, f, hf, P, rfl⟩ := P.exists_fin_rename
rw [IsNilpotent.map_iff (rename_injective _ hf), MvPolynomial.isNilpotent_iff_of_fintype]
lift f to Fin n ↪ σ using hf
refine ⟨fun H i ↦ ?_, fun H i ↦ by simpa using H (i.embDomain f)⟩
by_cases H : i ∈ Set.range (Finsupp.embDomain f)
· aesop
· rw [coeff_... | false | {
"chosen_tokens": 97,
"rejected_tokens": 2,
"token_jaccard": 0.018519,
"token_length_ratio": 0.020619
} | sorry_admit_sorryAx_proof_hole | admit | cd392e11cc293aa5b7c5da907bb1c680b05e01190835d9268c4577a534dfcc04 | 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.MvPolynomial.Homogeneous
public import Mathlib.RingTheory.Polynomial.Nilpotent
Namespace:
MvPolynomial
Local context:
/-
Copyright (c) 2025 Andrew... | 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": "Algebra/MvPolynomial",
"family_id": "isnilpotent_iff",
"file_id": "mathlib/Mathlib/Algebra/MvPolynomial/Nilpotent.lean",
"sample_id": "7f4e05d647d8234808511f2d65c328365b205678685e9a78a56123a55f2d6582"
} | train |
{
"chosen_proof_sha256": "6c44e0bc1aac850175730ade0c94468670de6fadad8303d7beb2df51cfcf9e9e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "574a2405d1c8dc937c54d23ab5023c4ffb... | by
cases a <;> cases b
exacts [map_subset_map.2 (hf₁ _ _), disjSum_mono (hg₁ _ _) (hg₂ _ _), Subset.rfl,
map_subset_map.2 (hf₂ _ _)] | true | {
"chosen_tokens": 48,
"rejected_tokens": 2,
"token_jaccard": 0.04,
"token_length_ratio": 0.041667
} | invalid_tactic | invalid_tactic | cd403baa002656716ed358530a85bcc3e2f41fef11339397c783310126f60b91 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finset.Sum
public import Mathlib.Data.Sum.Order
public import Mathlib.Order.Interval.Finset.Defs
Namespace:
Finset
Local context:
/-
Copyright (c) 2022 ... | by
__oprover_invalid_tactic_94c18dd81e6c | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "77f09a26f72d82ea92f0f47d714b281fbaa42ef06b5115ebd558ac0504204c8d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Sum",
"family_id": "sumlexlift_mono",
"file_id": "mathlib/Mathlib/Data/Sum/Interval.lean",
"sample_id": "94c18dd81e6ce7afdea3016639b32636d0e629ff55fd5405a516ea90027133c6"
} | train |
{
"chosen_proof_sha256": "d46e9dc9cd151de5455487d42f3fd890f31162d28298263c77fbfdfebe8161e5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | 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 | true | {
"chosen_tokens": 65,
"rejected_tokens": 8,
"token_jaccard": 0.061224,
"token_length_ratio": 0.123077
} | unsolved_goals | unsolved_goals | cd771895d12affa0e554e40e004b0aaa333822760bfc2cbe14bee6ad65404848 | 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
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "4aa82f11dee5f5777fe3a740f383d8350ea75c726104aa3512cf6ee7fa4a8736",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "18e5d95a99a28ddbd089bbdf5652dbad7467c92bf9a278d6d749571d6703ac97",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9573f0056018f8a6c39ac9e88631ce207c... | by
rw [descFactorial_eq_prod_range, factorial_eq_prod_range_add_one]
simp only [cast_prod]
nth_rw 2 [← card_range n]
rw [pow_card_mul_prod]
refine prod_congr rfl ?_
intro x hx
rw [← tsub_add_eq_tsub_tsub_swap,
Nat.cast_sub <| Nat.le_trans (Nat.add_one_le_iff.mpr (List.mem_range.mp hx)) h,
CharP.ca... | true | {
"chosen_tokens": 81,
"rejected_tokens": 2,
"token_jaccard": 0.02,
"token_length_ratio": 0.024691
} | invalid_tactic | invalid_tactic | cd7851abec6e7c9cc68e1f33bdd22b706441b83f005d483353e1865197461853 | 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.Factorial.BigOperators
public import Mathlib.Data.ZMod.Basic
Namespace:
ZMod
Local context:
/-
Copyright (c) 2023 Moritz Firsching. All rights reser... | by
__oprover_invalid_tactic_8c92966f8a3d | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "cbfe661bb5791854e7b78f9b74d42cd81e7958aecf672755678555a51a7c5086",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/ZMod",
"family_id": "cast_descfactorial",
"file_id": "mathlib/Mathlib/Data/ZMod/Factorial.lean",
"sample_id": "8c92966f8a3dcfb46f7baa7761df8474960f52637455e799bcbf00ab2a68364d"
} | train |
{
"chosen_proof_sha256": "048280cfb8f07df3e6d2a0a1b3323465e6476c927989956effb4e1caeb51283e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "abf95977d4892fdcad911aa6c3299b9ebb... | by
obtain ⟨x, w₁, w₂⟩ := x
obtain ⟨⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩, -⟩ := h
simp [Prod.toFun', Prod.invFun', h₁, h₂] | true | {
"chosen_tokens": 57,
"rejected_tokens": 2,
"token_jaccard": 0.041667,
"token_length_ratio": 0.035088
} | invalid_tactic | invalid_tactic | ce18d57bf784b5d6ce27012eca5bfb287ccdb371c8b527f8c278fba6a47c99ad | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.FiberBundle.Basic
Namespace:
Bundle.Trivialization
Local context:
/-
Copyright (c) 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache... | by
__oprover_invalid_tactic_c32bfd18b1ea | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "474a9197a473e01e9da66dedf86759145ab4d652b370ff79ea221942ac3b23f8",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/FiberBundle",
"family_id": "prod",
"file_id": "mathlib/Mathlib/Topology/FiberBundle/Constructions.lean",
"sample_id": "c32bfd18b1eacc9948f2069f737c203f16059abd126861ca53c4f4fff6855765"
} | train |
{
"chosen_proof_sha256": "61c7560cab56bf0e5d3f545a1a32c8f50733fcb31780bdea8dc6b7263c7d7328",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
haveI : NumberField K := @NumberField.mk _ _ _ (IsCyclotomicExtension.finiteDimensional {n} ℚ K)
have H₁ : (aeval (hζ.powerBasis ℚ).gen) (X - 1 : ℤ[X]) = (hζ.subOnePowerBasis ℚ).gen := by simp
have H₂ : (aeval (hζ.subOnePowerBasis ℚ).gen) (X + 1 : ℤ[X]) = (hζ.powerBasis ℚ).gen := by simp
refine discr_eq_disc... | false | {
"chosen_tokens": 211,
"rejected_tokens": 2,
"token_jaccard": 0.018182,
"token_length_ratio": 0.009479
} | sorry_admit_sorryAx_proof_hole | sorry | ce7a38e3725b3b10c30c20e9024a506bbfe445ed86ef0a9b30452f3aad988fc4 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
public import Mathlib.RingTheory.DedekindDomain.Dvr
public import Mathlib.NumberTheory.NumberField.Discriminant.Defs
Na... | 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/Cyclotomic",
"family_id": "discr_zeta_eq_discr_zeta_sub_one",
"file_id": "mathlib/Mathlib/NumberTheory/Cyclotomic/Discriminant.lean",
"sample_id": "35877c6cf79fa66db90e5991ad29f2b3f093444f285a8e92079a5bb760e5281a"
} | train |
{
"chosen_proof_sha256": "e6c90bca07c73d1bcb9479a5b66cf4f9a197038e47590f39f131d2b92e478d48",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
by_contra! h
refine ((hfin.biUnion (t := Set.Ici) (by simpa using h)).subset (fun b hb ↦ ?_)).not_infinite
(ha.sdiff (Set.finite_singleton a))
obtain ⟨x, hax, hxb⟩ := ((show a ≤ b from hb.1).lt_of_ne (Ne.symm hb.2)).exists_covby_le
exact Set.mem_biUnion hax hxb | false | {
"chosen_tokens": 87,
"rejected_tokens": 3,
"token_jaccard": 0.065217,
"token_length_ratio": 0.034483
} | by_exact_placeholder | exact? | ce888ea52d396902f79c533a8004261dbad78ff57a47cc6a608bb9daff08ad0c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Set.Finite.Lattice
public import Mathlib.Order.Atoms
public import Mathlib.Order.Interval.Finset.Defs
public import Mathlib.Order.Preorder.Finite
Namespa... | 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/Atoms",
"family_id": "exists_covby_infinite_ici_of_infinite_ici",
"file_id": "mathlib/Mathlib/Order/Atoms/Finite.lean",
"sample_id": "7b81765a8031ad51037f4259b92e28a5ea92d5e155ac346ebd82828d508f44f4"
} | train |
{
"chosen_proof_sha256": "be03b33260f8a567b6bd346d2bc5d2f454ac795779b31a7801a8bdb4b9414537",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d6518471441772f8d5a85981c2aec5b7d1... | by
dsimp [Functor.HasPointwiseLeftKanExtension]
simp only [hasPointwiseLeftKanExtensionAt_iff w]
refine ⟨fun h X₄ ↦ ?_, fun h _ ↦ h _⟩
rw [← Functor.hasPointwiseLeftKanExtensionAt_iff_of_iso _ _ (B.objObjPreimageIso X₄)]
apply h | true | {
"chosen_tokens": 48,
"rejected_tokens": 3,
"token_jaccard": 0.03125,
"token_length_ratio": 0.0625
} | circular_dependency | circular_dependency | ce8d21f199f85c383f439f9c69463bff5a840c053d1da105f9efc2e311419f69 | 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.GuitartExact.Basic
public import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction
Namespace:
CategoryTheory.TwoSquare
Local context:
/-
... | by
exact hasPointwiseLeftKanExtension_iff | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "8e23c2ef8cdba39a2b013f98def7ae5ef3ccb286e07d5856e354cda107382d40",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/GuitartExact",
"family_id": "haspointwiseleftkanextension_iff",
"file_id": "mathlib/Mathlib/CategoryTheory/GuitartExact/KanExtension.lean",
"sample_id": "ca893de6923284d5337d25661aaf5c8d1221b14895d8ede2751221d59a9b92ee"
} | train |
{
"chosen_proof_sha256": "65fb0bf51735f2fd54b4e2c9d7d6b383f1f337560f0bc27999124bd7b667a991",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "18824e2814fecae627026c691166e00b0e... | by
have hA_T_nonneg : ∀ i j, 0 ≤ Aᵀ i j := fun i j => by
simpa [Matrix.transpose_apply] using hA.nonneg j i
refine ⟨hA_T_nonneg, ?_⟩
intro i j
letI : Quiver n := toQuiver A
obtain ⟨p, hp_pos⟩ := hA.connected j i
cases p with
| nil =>
simp at hp_pos
| @cons b _ q e =>
let qT := transposePath ... | false | {
"chosen_tokens": 121,
"rejected_tokens": 128,
"token_jaccard": 0.966667,
"token_length_ratio": 1.057851
} | markdown_preamble_trailing | fence | ce941dfe552bd2e8cf780933e590251e38b8c5afb4f4c974fa52823f3d70ae90 | 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.Combinatorics.Quiver.ConnectedComponent
public import Mathlib.Combinatorics.Quiver.Path.Vertices
public import Mathlib.Data.Matrix.Mul
Namespace:
Matrix
Loca... | ```lean
by
have hA_T_nonneg : ∀ i j, 0 ≤ Aᵀ i j := fun i j => by
simpa [Matrix.transpose_apply] using hA.nonneg j i
refine ⟨hA_T_nonneg, ?_⟩
intro i j
letI : Quiver n := toQuiver A
obtain ⟨p, hp_pos⟩ := hA.connected j i
cases p with
| nil =>
simp at hp_pos
| @cons b _ q e =>
let qT := transp... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Matrix",
"family_id": "isirreducible",
"file_id": "mathlib/Mathlib/LinearAlgebra/Matrix/Irreducible/Defs.lean",
"sample_id": "99fd3b1c7dbbd31133ecfe7e89f6d8fdbe8581f23efe8e63f4413b7eb18ed016"
} | train |
{
"chosen_proof_sha256": "d4687275514802a11968138ca35e457b8f90609ca53ed1eef831fff19e4d31f5",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "10f041559ce44c16bf9a3dea6dd166b47f... | by
rw [← Nat.card_prod]; exact Nat.card_congr Subgroup.groupEquivQuotientProdSubgroup | true | {
"chosen_tokens": 16,
"rejected_tokens": 3,
"token_jaccard": 0.142857,
"token_length_ratio": 0.1875
} | circular_dependency | circular_dependency | cede6e08569eefb2e5d27e82ad8d580693e24f8f7a423093b57e2ea1433a3e82 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.GroupTheory.Coset.Basic
public import Mathlib.SetTheory.Cardinal.Finite
Namespace:
Subgroup
Local context:
/-
Copyright (c) 2018 Mitchell Rowett. All rights ... | by
exact card_eq_card_quotient_mul_card_subgroup | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "18fe0ae9451f67cb096147aff3a02ef108c959d6377d57d3f2e8059e3a0c8099",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "GroupTheory/Coset",
"family_id": "card_eq_card_quotient_mul_card_subgroup",
"file_id": "mathlib/Mathlib/GroupTheory/Coset/Card.lean",
"sample_id": "fd18d30e8b2b216d023535e089915b87e05c113a9e9d53b5e82b2353ec29f830"
} | train |
{
"chosen_proof_sha256": "22533728b6f9b3f9588a530b9a97fe6bf76b4cb9f09b93a8510eba1da1bd326d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "f2f0591f91cec44d93c77b4c160bc5278d... | by
ext
apply Sieve.ext
simp [← NatTrans.naturality_apply] | true | {
"chosen_tokens": 13,
"rejected_tokens": 5,
"token_jaccard": 0.142857,
"token_length_ratio": 0.384615
} | wrong_namespace_qualification | wrong_namespace_qualification | cf19d9803bca85939da30674e45760400c2439a32773c52dcff974a2cc528f32 | 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 OProverMissing.missing_1709493a1e9d | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "c9aecd76d1904089c04dd55c0b3c964bb31800176af21a9721ef85a9d8077676",
"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": "1709493a1e9d915fafb30165d87390b1f52391762f60fb9606bb3e33e9e6a8da"
} | train |
{
"chosen_proof_sha256": "b8f5229735b933ede0144ab9a48b8b3042b761eb7777cebb1d91a92fa7760495",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5b8e99c563bfea9792268681baf844de6f... | by
rw [Subgroup.rank_congr (commutator_eq_closure G)]
apply Subgroup.rank_closure_finite_le_nat_card
variable [Group.FG G] | true | {
"chosen_tokens": 22,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.136364
} | unknown_identifier | unknown_identifier | cf3e44883d43be77c53fc13b16e5bddc74e4ba9e43444737b57354098a262f18 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Subgroup.Finite
public import Mathlib.GroupTheory.Commutator.Basic
public import Mathlib.GroupTheory.Rank
public import Mathlib.GroupTheory.Index... | by
exact __oprover_missing_ad36c8eeea38 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "6f13cb3ba9bd2d6aabf75f296fbf6369e0ed1a189fe06594efdf541f02a80582",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "GroupTheory/Commutator",
"family_id": "rank_commutator_le_card",
"file_id": "mathlib/Mathlib/GroupTheory/Commutator/Finite.lean",
"sample_id": "ad36c8eeea38b3617b2fa01b60e158adc224ea844614163c66de06f8b4d2fee4"
} | train |
{
"chosen_proof_sha256": "988485a8c756571de67c1e7554cc8678471929e0b79e0159e23042969b87deb3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "696e93fb10ae259c09e9614ab7179b8e76... | by
induction p with
| nil => rfl
| cons _ _ ih => dsimp [pathToList] at *; rw [ih] | true | {
"chosen_tokens": 25,
"rejected_tokens": 2,
"token_jaccard": 0.05,
"token_length_ratio": 0.08
} | invalid_tactic | invalid_tactic | cf51ac8748508eda0ec8f466879e24a6f552127d5a7568e28f6b6b8ad2c914fb | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Combinatorics.Quiver.Cast
public import Mathlib.Combinatorics.Quiver.Symmetric
Namespace:
Quiver.SingleObj
Local context:
/-
Copyright (c) 2023 Antoine Label... | by
__oprover_invalid_tactic_5dc82f08b0f9 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "9b399d96551725639bb50a3682deb56ddde2efadef5345d3be00177463911dd5",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/Quiver",
"family_id": "listtopath_pathtolist",
"file_id": "mathlib/Mathlib/Combinatorics/Quiver/SingleObj.lean",
"sample_id": "5dc82f08b0f93626082032ca9aa64a8e948617dc0aa5acf4d820aa7f216565eb"
} | train |
{
"chosen_proof_sha256": "fd734d0795994b2df39d22bec1fc6dc26c9a299bef650b0d07eb3f85b3d6c929",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
rw [logHeight_eq_log_mulHeight]
exact log_nonneg <| x.one_le_mulHeight | false | {
"chosen_tokens": 12,
"rejected_tokens": 2,
"token_jaccard": 0.076923,
"token_length_ratio": 0.166667
} | by_exact_placeholder | by? | cf870cd02521a00895607acc736e1275673d729a4904343175a46a011008525b | 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.Projectivization.Basic
public import Mathlib.NumberTheory.Height.Basic
Namespace:
Projectivization
Local context:
/-
Copyright (c) 2026 Michael... | by? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "NumberTheory/Height",
"family_id": "logheight_nonneg",
"file_id": "mathlib/Mathlib/NumberTheory/Height/Projectivization.lean",
"sample_id": "5bf0e0681c55eec8bc5b1c759d43192dd7b331a9558e5d94174d45141766d782"
} | train |
{
"chosen_proof_sha256": "397dccc7f0b407dbb3cdb395970fa551240d91f8fe8d8b6664c36ca222b24937",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a6605051476f011bb510bc68d28ab61226... | by
simp [h]
-- used in the `nlinarith` normalization steps. The `_` argument is for uniformity. | true | {
"chosen_tokens": 25,
"rejected_tokens": 5,
"token_jaccard": 0.086957,
"token_length_ratio": 0.2
} | wrong_namespace_qualification | wrong_namespace_qualification | cfe2fd00e12037051da5cef940133085a0d98147a2d931e18f262705d3afb6f0 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Ineq
public import Mathlib.Data.Nat.Cast.Order.Ring
Namespace:
Mathlib.Tactic.Linarith
Local context:
/-
Copyright (c) 2020 Robert Y. Lewis. All rights ... | by
exact OProverMissing.missing_890928cea674 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "4455e949a0b4c849734e0fb188fdc1f6908f82e6b0858689d50ae2c0ebc0c78d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Tactic/Linarith",
"family_id": "mul_zero_eq",
"file_id": "mathlib/Mathlib/Tactic/Linarith/Lemmas.lean",
"sample_id": "890928cea67490b486f404ea924568c9df0ae3523215f6af6df2d816e57e58f7"
} | train |
{
"chosen_proof_sha256": "82023a5a0373446dea510bd79a74d7b917c1be62462ff0113fe4a4cf3e1520cd",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by simp) :
∃ b : B, b ∉ s := by
simpa [-SetLike.coe_set_eq, SetLike.ext'_iff, h_top, Set.ne_univ_iff_exists_notMem] using hs | false | {
"chosen_tokens": 33,
"rejected_tokens": 5,
"token_jaccard": 0.035714,
"token_length_ratio": 0.151515
} | sorry_admit_sorryAx_proof_hole | sorryAx | cfea20f0aa41937481e1d8180b262b43f8a562bfb860f605bf16e66117fc9a29 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Tactic.Monotonicity.Attr
public import Mathlib.Tactic.SetLike
public import Mathlib.Data.Set.Basic
Namespace:
SetLike
Local context:
/-
Copyright (c) 2021 Er... | by
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/SetLike",
"family_id": "exists_not_mem_of_ne_top",
"file_id": "mathlib/Mathlib/Data/SetLike/Basic.lean",
"sample_id": "87399eed90a7a9d71500444beb6af36405b8a0740724037a49d189fcab7e9413"
} | train |
{
"chosen_proof_sha256": "447b3ac2404a7fdae48d1d41ba080e9c5434ee1fd77cdc32e3d7e6b39eb7aa1d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
refine ⟨fun h ↦ ?_, fun h ↦ h.units_smul w⟩
convert! h.units_smul (fun i ↦ (w i)⁻¹)
simp [funext_iff] | false | {
"chosen_tokens": 37,
"rejected_tokens": 5,
"token_jaccard": 0.074074,
"token_length_ratio": 0.135135
} | sorry_admit_sorryAx_proof_hole | sorryAx | cfed9ede98240b9168b8094a80f8a6085c721c3883c577a35fa0cd3eeab34840 | 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.LinearAlgebra.LinearIndependent.Defs
public import Mathlib.Logic.Equiv.Fin.Rotate
Namespace:
(root)
Local cont... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/LinearIndependent",
"family_id": "linearindependent",
"file_id": "mathlib/Mathlib/LinearAlgebra/LinearIndependent/Basic.lean",
"sample_id": "b38b14c96b38afbba7c3259cf4c4e7b4c145e1b2ac6c153a770feeac3a7076d5"
} | train |
{
"chosen_proof_sha256": "f95ac83c92042b8eca33c773ee62f7012472f34ee0a3ebc546b077931f47c21d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
induction hs using Finset.Nonempty.cons_induction with
| singleton i => exact ⟨i, by simp⟩
| @cons i s hi hs ih =>
obtain ⟨j, hj⟩ := ih
by_cases hji : f j ≤ f i
· refine ⟨i, mem_cons_self .., ?_⟩
simp only [mem_cons, forall_eq_or_imp, imp_self, true_and]
exact fun k hk hik ↦ _root_.tran... | false | {
"chosen_tokens": 112,
"rejected_tokens": 3,
"token_jaccard": 0.033898,
"token_length_ratio": 0.026786
} | by_exact_placeholder | placeholder | d011c227b390a33e811b59dcea5a67f95c597792296f1995078b0a288432c0ab | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Set.Finite.Basic
public import Mathlib.Order.Minimal
Namespace:
Finset
Local context:
/-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released ... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/Preorder",
"family_id": "exists_maximalfor",
"file_id": "mathlib/Mathlib/Order/Preorder/Finite.lean",
"sample_id": "83bbcd0a12ed90d64932bbbf23b8435e8984a966305ad698b755df700c2893b8"
} | train |
{
"chosen_proof_sha256": "b54a12dda5879cff4ee4a1d4e164d0430717d0b1ed9dc7072b52b58f4f7965cc",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4d5c5b1473328f3a599163c0715d3d92eb... | by
refine eq_bot_iff.2 (le_trans jacobson_bot_polynomial_le_sInf_map_maximal ?_)
refine fun f hf => (Submodule.mem_bot R[X]).2 <| Polynomial.ext fun n =>
Trans.trans (?_ : coeff f n = 0) (coeff_zero n).symm
suffices f.coeff n ∈ Ideal.jacobson ⊥ by rwa [h, Submodule.mem_bot] at this
exact mem_sInf.2 fun j hj... | true | {
"chosen_tokens": 108,
"rejected_tokens": 5,
"token_jaccard": 0.054545,
"token_length_ratio": 0.046296
} | wrong_namespace_qualification | wrong_namespace_qualification | d01ab5ddd99060fda5739d8326110c4d4208c6c2bcf1064f3881d797d645d384 | 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.Jacobson.Ideal
public import Mathlib.RingTheory.Polynomial.Quotient
Namespace:
Ideal
Local context:
/-
Copyright (c) 2020 Devon Tuma. All rights r... | by
exact OProverMissing.missing_cc46ae0cd6fb | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "d71932f558fa7ea1f725c7fb319fea207849c002ae7b2bd316384f6527b37b3e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Jacobson",
"family_id": "jacobson_bot_polynomial_of_jacobson_bot",
"file_id": "mathlib/Mathlib/RingTheory/Jacobson/Polynomial.lean",
"sample_id": "cc46ae0cd6fbf8b5c482692a262adbee52fa600011e9053140abc53478aa5ddd"
} | train |
{
"chosen_proof_sha256": "76fdca8112ad5b228e441ce2f0fc6042386b4c4cdeafde082a85932a16c29411",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "77ec50a1d8bd4d1d6217637dbd7b2d3235... | by
simp_rw [← toDual_covBy_toDual_iff (α := α)] at hfin ⊢
exact exists_covby_infinite_Ici_of_infinite_Ici (α := αᵒᵈ) ha hfin | true | {
"chosen_tokens": 25,
"rejected_tokens": 2,
"token_jaccard": 0.052632,
"token_length_ratio": 0.08
} | invalid_tactic | invalid_tactic | d0900b3fbb59b835695f4a3de4eca213b7dab8d35e453a5858af0319e610148f | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Set.Finite.Lattice
public import Mathlib.Order.Atoms
public import Mathlib.Order.Interval.Finset.Defs
public import Mathlib.Order.Preorder.Finite
Namespa... | by
__oprover_invalid_tactic_8ceba723b8cf | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "502959d4e5e30853a8d25f720ed13e8bf00f0a4de75bdd5ac71520f4cdce2aa0",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Atoms",
"family_id": "exists_covby_infinite_iic_of_infinite_iic",
"file_id": "mathlib/Mathlib/Order/Atoms/Finite.lean",
"sample_id": "8ceba723b8cfca101595c85d0f256daa590d35c274c45371de9577614b0b3c30"
} | train |
{
"chosen_proof_sha256": "d7e6aa18e95f9b73927d19d13b21d8b699befd36d25410b98b9a340426e1233b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "edb8b3e01bb6880ee0bbdea1e1488404f9... | by
obtain rfl | hn := eq_or_ne n 0
· simpa [ssubset_iff_subset_not_subset, hX₁, -Finset.subset_singleton_iff]
using! hX.not_subset_singleton
refine (pow_subset_pow_right hX₁ <| n.le_add_right _).ssubset_of_ne ?_
contrapose hXclosure with hXn
rw [← closure_pow (mod_cast hX₁) hn]
wlog hn₁ : n = 1
· si... | false | {
"chosen_tokens": 307,
"rejected_tokens": 311,
"token_jaccard": 0.973451,
"token_length_ratio": 1.013029
} | missing_wrong_import | import | d0bd2e53ff72f35796e504d06461a6c94f36297019b8d5c2709badaec0ef406c | 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 ... | import Mathlib.WrongImport
by
obtain rfl | hn := eq_or_ne n 0
· simpa [ssubset_iff_subset_not_subset, hX₁, -Finset.subset_singleton_iff]
using! hX.not_subset_singleton
refine (pow_subset_pow_right hX₁ <| n.le_add_right _).ssubset_of_ne ?_
contrapose hXclosure with hXn
rw [← closure_pow (mod_cast hX₁) hn... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Geometry/Group",
"family_id": "pow_ssubset_pow_succ_of_pow_ne_closure",
"file_id": "mathlib/Mathlib/Geometry/Group/Growth/LinearLowerBound.lean",
"sample_id": "2f91e46e3433b86a1e3699a32e11802a7183cc285fe3179f472be8008081dcf8"
} | train |
{
"chosen_proof_sha256": "ad8808e6ef06989ebffa9f2a94c9e16fc819a971dc924b479659fd09f73634ab",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "d3e808dd9addf2387a7c10bc55a218ee8c... | by
simp only [Ideal.eq_top_iff_one, ← Subtype.range_val (s := s), ← Set.range_comp,
Finsupp.mem_ideal_span_range_iff_exists_finsupp] at hs' ⊢
choose f hf using hs'
have : Fintype s := hs.fintype
refine ⟨Finsupp.equivFunOnFinite.symm fun i x ↦ f x i, ?_⟩
ext i
simpa [Finsupp.sum_fintype] using hf i | false | {
"chosen_tokens": 72,
"rejected_tokens": 79,
"token_jaccard": 0.958333,
"token_length_ratio": 1.097222
} | markdown_preamble_trailing | fence | d0c009f86406e39f1e5fdab5b942aa26a244155b09535cbd10a7bd2b06b080f0 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.AlgebraicGeometry.Morphisms.Immersion
Namespace:
AlgebraicGeometry
Local context:
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apac... | ```lean
by
simp only [Ideal.eq_top_iff_one, ← Subtype.range_val (s := s), ← Set.range_comp,
Finsupp.mem_ideal_span_range_iff_exists_finsupp] at hs' ⊢
choose f hf using hs'
have : Fintype s := hs.fintype
refine ⟨Finsupp.equivFunOnFinite.symm fun i x ↦ f x i, ?_⟩
ext i
simpa [Finsupp.sum_fintype] using hf... | {
"accepted": false,
"all_failure_types": [
"markdown_fence",
"not_exact_by_body"
],
"failure_type": "markdown_fence",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "AlgebraicGeometry",
"family_id": "ideal",
"file_id": "mathlib/Mathlib/AlgebraicGeometry/PointsPi.lean",
"sample_id": "92326efcf9c32173b0a7545e06842af3e1db8144b2ac5b58b5ccfdd3ce10f509"
} | train |
{
"chosen_proof_sha256": "861cfb34c8727a0d4d437e7097497ea9d3f14804aa184568a47ee591868cf5bf",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "56d9c52271c4e055cf68149536f81a5fb9... | by
rw [Equiv.eq_symm_comp (e := Quiver.symmetrifyStar (φ.obj u))]
ext ⟨v, f | g⟩ <;>
-- Porting note (https://github.com/leanprover-community/mathlib4/issues/10745): was `simp [Quiver.symmetrifyStar]`
simp only [Quiver.symmetrifyStar, Function.comp_apply] <;>
erw [Equiv.sigmaSumDistrib_apply, Equiv.sigm... | false | {
"chosen_tokens": 96,
"rejected_tokens": 101,
"token_jaccard": 0.928571,
"token_length_ratio": 1.052083
} | markdown_preamble_trailing | preamble | d0d544deba5fd970886613891ca4e0505917097b86d24cb21751d3588db98f3d | 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.Combinatorics.Quiver.Cast
public import Mathlib.Combinatorics.Quiver.Symmetric
public import Mathlib.Data.Sigma.Basic
public import Mathlib.Data.Sum.Basic
publ... | Here is the proof:
by
rw [Equiv.eq_symm_comp (e := Quiver.symmetrifyStar (φ.obj u))]
ext ⟨v, f | g⟩ <;>
-- Porting note (https://github.com/leanprover-community/mathlib4/issues/10745): was `simp [Quiver.symmetrifyStar]`
simp only [Quiver.symmetrifyStar, Function.comp_apply] <;>
erw [Equiv.sigmaSumDistri... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Combinatorics/Quiver",
"family_id": "prefunctor",
"file_id": "mathlib/Mathlib/Combinatorics/Quiver/Covering.lean",
"sample_id": "87030b969f81b483b5bdfa5c63449397f06437d95959ae3498323790baf68ce1"
} | train |
{
"chosen_proof_sha256": "332fb89312684e7786baadfdf197587e5cdba892591555a524aedd43ed4113c6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "83d409c2d59eb7e379de832485f191b9f4... | by
induction n with
| zero =>
simp
| succ n IH =>
simp_rw [iteratedLineDerivOp_succ_left, IH, lineDerivOp_add] | true | {
"chosen_tokens": 21,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.142857
} | circular_dependency | circular_dependency | d0de58649d2aad3af2998f087a2f18f59d3026a4239f3a82d55d83ed1a32e255 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Module.Equiv.Defs
public import Mathlib.Data.Fin.Tuple.Basic
public import Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
public import Math... | by
exact iteratedLineDerivOp_add | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "be6270b8b9c29643af7aec79a9a7154679bbf9611d0230444ab04410c1ec3035",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Analysis/Distribution",
"family_id": "iteratedlinederivop_add",
"file_id": "mathlib/Mathlib/Analysis/Distribution/DerivNotation.lean",
"sample_id": "e31fb78612321e73c41c554f6e35840ea8a109f4f7b6f9fddcff3cf0f701ebac"
} | train |
{
"chosen_proof_sha256": "4fe5d002e1432ce371569b334eb570afebb4011df3c1ab6e1eb29382c5b0d1c6",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4c4c80ad0a7010c105ce75354ba15cb085... | by
haveI := Nonempty.map f ‹_›
haveI := Nonempty.map g ‹_›
exact (map_injective.and map_surjective).trans and_and_and_comm | true | {
"chosen_tokens": 29,
"rejected_tokens": 5,
"token_jaccard": 0.142857,
"token_length_ratio": 0.172414
} | wrong_namespace_qualification | wrong_namespace_qualification | d10c35ffced81b833bd78f200acf5fbaeff258aff5a748e1be8590b21440b575 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Lean.PrettyPrinter.Delaborator.Builtins
public import Mathlib.Logic.Function.Defs
public import Mathlib.Logic.Function.Iterate
public import Mathlib.Tactic.Inhabit
pub... | by
exact OProverMissing.missing_5d67e5e7b995 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "71810c68c130487596915437a09ad4fb2412e91d3ed6d930db9ef7155227d74c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Prod",
"family_id": "map_bijective",
"file_id": "mathlib/Mathlib/Data/Prod/Basic.lean",
"sample_id": "5d67e5e7b9950556ea8bd8cdfa3767dbe10a6562717cb86050afc39cdaf6a4b7"
} | train |
{
"chosen_proof_sha256": "eac7599f7adbe4156b1359f0aff3dffecc04db5fb77ecf8bedd140ea7081aceb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
induction h
simp | false | {
"chosen_tokens": 4,
"rejected_tokens": 2,
"token_jaccard": 0.2,
"token_length_ratio": 0.5
} | by_exact_placeholder | by? | d123bcc2d59669e3c0585bf169b6b41065bfe28292f95739cd0d9475e9a936bc | 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.Skeleton
public import Mathlib.CategoryTheory.Subobject.MonoOver
public import Mathlib.CategoryTheory.Skeletal
public import Mathlib.Cate... | 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/Subobject",
"family_id": "arrow_congr",
"file_id": "mathlib/Mathlib/CategoryTheory/Subobject/Basic.lean",
"sample_id": "7ee01c83c10c22668c1fdc800b54b271e64edd823972fd056c65d61e5211e5df"
} | train |
{
"chosen_proof_sha256": "3df867611d307b2d4c52a5aeae1c5c93f72412c5fca86e32a6445afd51c4bda0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "617a91f1c4527ca920c01a59a90105084b... | by
trans fun n ↦ (n / 3 - 2) * ↑((n - 3) / 6) * exp (-4 * √(log ↑((n - 3) / 6)))
· simp_rw [sq]
refine (IsBigO.mul ?_ ?_).mul ?_
· trans fun n ↦ n / 3
· simp_rw [div_eq_inv_mul]
exact (isBigO_refl ..).const_mul_right (by simp)
refine IsLittleO.right_isBigO_sub ?_
simpa [div_eq_inv_... | true | {
"chosen_tokens": 244,
"rejected_tokens": 2,
"token_jaccard": 0.011236,
"token_length_ratio": 0.008197
} | invalid_tactic | invalid_tactic | d16739fd19f6911b194a15bfe2ea6158c45bfacdc0559b514070c8eb24af3277 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Combinatorics.Additive.AP.Three.Behrend
public import Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite
public import Mathlib.Tactic.Rify
public import Mat... | by
__oprover_invalid_tactic_fe45775e1693 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "fd0405182fc02c5f97a49773c8d0449bde4e69caa5dfb8fc0778e16e360b202b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Combinatorics/Extremal",
"family_id": "ruzsaszemeredinumbernat_asymptotic_lower_bound",
"file_id": "mathlib/Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean",
"sample_id": "fe45775e1693cac44dede504e39345f1c5dfd23b430122a30f0b82d3494c2fdd"
} | train |
{
"chosen_proof_sha256": "281c14df73dbd3d58dd1e8053eb70fe2ed484fdf0d2d7eb3b862de0b5a2b7009",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "18dd8b95b77ff3462addfa3b45bcf71a82... | by
have : Nonempty J := IsFiltered.nonempty
let hc' := Under.isColimitLiftCocone D s c (p ≫ f) h hc
obtain ⟨j, q, hq⟩ := exists_hom_of_isColimit (X := Under.mk p) hc' (Under.homMk f rfl)
use j, q.right, Under.w q, congr($(hq).right) | false | {
"chosen_tokens": 72,
"rejected_tokens": 76,
"token_jaccard": 0.930233,
"token_length_ratio": 1.055556
} | missing_wrong_import | import | d1ecee38d2a85c3a90b941b171a2e9a6e6f400286791d07e3cfc50fdae8abf60 | 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.Filtered
public import Mathlib.CategoryTheory.Limits.Preserves.Filtered
public import Mathlib.CategoryTheory.Limits.Types.Filtered
public... | import Mathlib.WrongImport
by
have : Nonempty J := IsFiltered.nonempty
let hc' := Under.isColimitLiftCocone D s c (p ≫ f) h hc
obtain ⟨j, q, hq⟩ := exists_hom_of_isColimit (X := Under.mk p) hc' (Under.homMk f rfl)
use j, q.right, Under.w q, congr($(hq).right) | {
"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/Presentable",
"family_id": "isfinitelypresentable",
"file_id": "mathlib/Mathlib/CategoryTheory/Presentable/Finite.lean",
"sample_id": "e9c86bc71f86c9474c5b0e70638d93a26490f015f3c3db0caf5be6d93809c4ae"
} | train |
{
"chosen_proof_sha256": "a76192e154cb0c91a316e618a2d1566a7760c095bf189498c16a175bdc2813bb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9151830826aceec6eb28e4eaa5e54cc3c0... | by
generalize h' : [] = L'
intro h
rcases h with - | ⟨L₁, L₂⟩
simp at h' | true | {
"chosen_tokens": 25,
"rejected_tokens": 5,
"token_jaccard": 0.038462,
"token_length_ratio": 0.2
} | circular_dependency | circular_dependency | d1f30df358af3dede6967181b7731f79ed9c80fbf3352adca0bd723b44c14a8c | 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.Pi.Basic
public import Mathlib.Algebra.Group.Subgroup.Ker
public import Mathlib.Data.List.Chain
public import Mathlib.Algebra.Group.Int.Defs
publ... | by
exact Step.append_right | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "b72b9cec413a2cde1dcf1bc5ef406dcde1093df9810a38bef1e24456eca49488",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "GroupTheory/FreeGroup",
"family_id": "step",
"file_id": "mathlib/Mathlib/GroupTheory/FreeGroup/Basic.lean",
"sample_id": "e4f8810a3c97776fcd6674079eb5c88b07ed5c16b257876471689d8a0da4bfcc"
} | train |
{
"chosen_proof_sha256": "12eef625fe3216831384e04422da6efdbadfd13d594c33e0c21289712e29f94e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
classical
rw [monomial_def]
exact Pi.single_eq_same _ _ | false | {
"chosen_tokens": 12,
"rejected_tokens": 2,
"token_jaccard": 0.083333,
"token_length_ratio": 0.166667
} | by_exact_placeholder | by? | d22bb922cbc5ab42ea810114dfc460eedb61a594b273b5b6722fab3704763f32 | 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.Antidiag.Finsupp
public import Mathlib.Data.Finsupp.Weight
public import Mathlib.LinearAlgebra.Pi
public import Mathlib.Algebra.MvPolynomial.Basi... | 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/MvPowerSeries",
"family_id": "coeff_monomial_same",
"file_id": "mathlib/Mathlib/RingTheory/MvPowerSeries/Basic.lean",
"sample_id": "3a3c329ac7ef7bd97b6db71b43a7efe1eaaa001c53c4b487b813a42f94dd4a0c"
} | train |
{
"chosen_proof_sha256": "c3ee73588e36d5e89a2703d1667734bc0b58f87dcefce1c40a3828ad044e5637",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "925f0d2e6688d51b9087e7f5c0acec178d... | by
rw [ringKrullDim, ringKrullDim]
apply Order.krullDim_le_of_krullDim_preimage_le' (PrimeSpectrum.comap C) ?_ (fun p ↦ ?_)
· exact fun {a b} h ↦ Ideal.comap_mono h
· rw [show C = (algebraMap R (Polynomial R)) from rfl, Order.krullDim_eq_of_orderIso
(PrimeSpectrum.preimageOrderIsoFiber R (Polynomial R) p)... | false | {
"chosen_tokens": 111,
"rejected_tokens": 115,
"token_jaccard": 0.946429,
"token_length_ratio": 1.036036
} | missing_wrong_import | import | d265dd4b7438b1716ec0c609c1f979f3ee417741d6c699c7cf76fe83ebb9d76e | 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.FieldDivision
public import Mathlib.RingTheory.KrullDimension.PID
public import Mathlib.RingTheory.LocalRing.ResidueField.Fiber
public impor... | import Mathlib.WrongImport
by
rw [ringKrullDim, ringKrullDim]
apply Order.krullDim_le_of_krullDim_preimage_le' (PrimeSpectrum.comap C) ?_ (fun p ↦ ?_)
· exact fun {a b} h ↦ Ideal.comap_mono h
· rw [show C = (algebraMap R (Polynomial R)) from rfl, Order.krullDim_eq_of_orderIso
(PrimeSpectrum.preimageOrderI... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/KrullDimension",
"family_id": "polynomial",
"file_id": "mathlib/Mathlib/RingTheory/KrullDimension/Polynomial.lean",
"sample_id": "812ef1c18921780c46e19c5df601fc11ffc84bd9969d3bb154d739df4d388862"
} | train |
{
"chosen_proof_sha256": "7cabac5c0510d840ec215a55b2c97b8bd4ff4bb3ba7ca644c381e4c621c7c4ab",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "cd50c93d72cf5c91929bb13dde2673ec35... | by
induction rs generalizing M N with
| nil => simp
| cons x _ ih =>
simp only [List.map_cons, isWeaklyRegular_cons_iff] at reg ⊢
have e := (QuotSMulTop.algebraMapTensorEquivTensorQuotSMulTop x M S).symm ≪≫ₗ
QuotSMulTop.congr ((algebraMap R S) x) hf.equiv
have hg : IsBaseChange S <|
e.to... | true | {
"chosen_tokens": 112,
"rejected_tokens": 5,
"token_jaccard": 0.047619,
"token_length_ratio": 0.044643
} | wrong_namespace_qualification | wrong_namespace_qualification | d29e74057896feebd1a9c1fff232618ebcc965167d08999b3e817092c724277d | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
public import Mathlib.RingTheory.Flat.Localization
public import Mathlib.RingTheory.Regular.RegularSequence
Namespace:
Ri... | by
exact OProverMissing.missing_b76866134268 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ab723cea329cf8c87a6348c867aa4f26160a5c2d28f040fc58edf0f34b0d425f",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Regular",
"family_id": "isweaklyregular",
"file_id": "mathlib/Mathlib/RingTheory/Regular/Flat.lean",
"sample_id": "b76866134268ce8c1e930e844ff2aaa363a473acc169ff6b4db1a719899e0bd6"
} | train |
{
"chosen_proof_sha256": "62c5d9390a895014879439c4bd46fd00c9783796de9ef2a06a744f2c74871aea",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
apply_fun ((Sigma.desc fun a ↦ pullback.fst g₁ (Sigma.ι X a)) ≫ ·) using
(fun a b ↦ (cancel_epi _).mp)
ext a
simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_ι_app]
rw [← Category.assoc, pullback.condition]
simp only [Category.assoc, colimit.ι_desc, Cofan.mk_ι_app]
apply_fun ((Sigma.de... | true | {
"chosen_tokens": 358,
"rejected_tokens": 8,
"token_jaccard": 0.018519,
"token_length_ratio": 0.022346
} | unsolved_goals | unsolved_goals | d3374488ee30b4ea292b548c3eb7ced006637d64263dbf0c17596d355882cadd | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.EffectiveEpi.Basic
public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
public import Mathlib.Tactic.ApplyFun
Namespace:
Cat... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "0c50d6b1cc0f95d6accb96a7306cfcff78bb1f3be19aa9d1439567e546878b55",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/EffectiveEpi",
"family_id": "effectiveepifamilystructofeffectiveepidesc_aux",
"file_id": "mathlib/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean",
"sample_id": "0f39df25f86b79636ec8efd74a8a03a810910ef6427851948542bfbbe91d6d31"
} | train |
{
"chosen_proof_sha256": "d69154e0d9492b7c81193d7944884dcb10496ed41d6438469a8fe12e15c51a76",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
refine ⟨fun h => ?_, fun ⟨⟨a', m, e⟩, s⟩ => Subset.cons e m s⟩
generalize h' : Lists'.cons a l₁ = l₁' at h
obtain - | @⟨a', _, _, _, e, m, s⟩ := h
· cases a
cases h'
cases a; cases a'; cases h'; exact ⟨⟨_, m, e⟩, s⟩ | false | {
"chosen_tokens": 91,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.032967
} | by_exact_placeholder | exact? | d385ff832830447ab50576842da49a6cf4f36facfb3f9dac411653dbe7b2aa5e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Group.Nat
public import Mathlib.Algebra.Order.Monoid.NatCast
public import Mathlib.Algebra.Ring.Nat
public import Mathlib.Data.Sigma.Basic
public... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "SetTheory",
"family_id": "cons_subset",
"file_id": "mathlib/Mathlib/SetTheory/Lists.lean",
"sample_id": "6a8521660d4ef846f6eee10f1b6a7111248906a80446120366f799acf8507e31"
} | train |
{
"chosen_proof_sha256": "89a379c0af7b46c3f249bf357aea100510d92246e204cc5b623f3d9530166237",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4c75218715d8106b241c827488496f0086... | by
rw [← LieSubalgebra.toSubmodule_inj, range_toSubmodule, LieIdeal.toLieSubalgebra_toSubmodule,
ker_toSubmodule]
exact LinearMap.range_inl R L₁ L₂ | true | {
"chosen_tokens": 25,
"rejected_tokens": 2,
"token_jaccard": 0.047619,
"token_length_ratio": 0.08
} | invalid_tactic | invalid_tactic | d393e29c6c0b4b23685331b0849da0bb026d2c492afec5165fc29ee5c53ce18c | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Lie.Ideal
Namespace:
LieHom
Local context:
/-
Copyright (c) 2026 Leonid Ryvkin. All rights reserved.
Released under Apache 2.0 license as described i... | by
__oprover_invalid_tactic_58255fb9a08e | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b7c3ccdfdf8c112fc71a01a46e82a77657cc7884027865ef065fab75887a7add",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Lie",
"family_id": "range_inl",
"file_id": "mathlib/Mathlib/Algebra/Lie/Prod.lean",
"sample_id": "58255fb9a08e3ea4853832b4b2e6754cd8d2484fe722ffff05ba10373b5631a3"
} | train |
{
"chosen_proof_sha256": "7222ee9e22761482f65e7ecb6d7dbb6c39c2c6b75d4bfee6e89b7661c7ff185a",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "609454c90f9d78f7e8599fb9bd88621144... | by
refine le_antisymm ?_ subset_span
let s' : Submodule R M :=
{ carrier := s
add_mem' := add_mem
zero_mem' := zero_mem _
smul_mem' := SMulMemClass.smul_mem }
exact span_le (p := s') |>.mpr le_rfl | false | {
"chosen_tokens": 42,
"rejected_tokens": 47,
"token_jaccard": 0.894737,
"token_length_ratio": 1.119048
} | markdown_preamble_trailing | trailing | d3b95664075e20d1bac6be66c8af4bb5cdc630adb7ee2543f4841301e23f927c | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Module.Submodule.Lattice
public import Mathlib.Algebra.Group.Pointwise.Set.Basic
Namespace:
Submodule
Local context:
/-
Copyright (c) 2017 Johannes H... | by
refine le_antisymm ?_ subset_span
let s' : Submodule R M :=
{ carrier := s
add_mem' := add_mem
zero_mem' := zero_mem _
smul_mem' := SMulMemClass.smul_mem }
exact span_le (p := s') |>.mpr le_rfl
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": "LinearAlgebra/Span",
"family_id": "coe_span_eq_self",
"file_id": "mathlib/Mathlib/LinearAlgebra/Span/Defs.lean",
"sample_id": "9a1583a6f100445b753a9ba00ceefda2c0d6b67968367089199c06ca3f1c14f0"
} | train |
{
"chosen_proof_sha256": "06d455a8b5ee3bf3c5458bfc466011b102068e1311619ad2681ed0ee2fe0c4c3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
by_cases hβ : Nonempty β
· refine small_of_surjective (f := Function.invFun g ∘ f) (fun b => ?_)
obtain ⟨a, ha⟩ := h b
exact ⟨a, by rw [Function.comp_apply, ha, Function.leftInverse_invFun hg]⟩
· simp only [not_nonempty_iff] at hβ
infer_instance
/-!
We don't define `Countable.toSmall` in this file... | true | {
"chosen_tokens": 93,
"rejected_tokens": 8,
"token_jaccard": 0.048387,
"token_length_ratio": 0.086022
} | unsolved_goals | unsolved_goals | d3dfc8db459b49e584bac58e3b8322e97023a97b80b599e146daa006fccdc23b | 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.Small.Defs
public import Mathlib.Logic.Equiv.Set
Namespace:
(root)
Local context:
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released und... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "5cc6f3952d89cc33c8cd31014488ff0fb61ac61709db41d6027ebcf3da2238e3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Logic/Small",
"family_id": "small_of_injective_of_exists",
"file_id": "mathlib/Mathlib/Logic/Small/Basic.lean",
"sample_id": "900df9f57575ae9ae5d97a81a010ab1e20e766cd1a88e2d731f34af9d7eca3f8"
} | train |
{
"chosen_proof_sha256": "7c1a95bd387fa99509421bd9a6c435e484eea5c2a9222b6585afb56f5e4cfe03",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e1b8702d9134a47fb4aaaee0e4e491b5e1... | by
apply tendsto_nhdsNE_of_tendsto_atTop _ _ h_neg
convert! h_pos | false | {
"chosen_tokens": 9,
"rejected_tokens": 14,
"token_jaccard": 0.615385,
"token_length_ratio": 1.555556
} | markdown_preamble_trailing | preamble | d40c0c30ea25630886387b3b2bb3583ea46a1ae7e6c487a896e03ed10264df82 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Topology.Algebra.Order.Field
public import Mathlib.Topology.Maps.Basic
public import Mathlib.Analysis.Asymptotics.Lemmas
Namespace:
Tactic.ComputeAsymptotics
... | Here is the proof:
by
apply tendsto_nhdsNE_of_tendsto_atTop _ _ h_neg
convert! h_pos | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Tactic/ComputeAsymptotics",
"family_id": "tendsto_nhdsne_of_tendsto_attop_nhds_of_eq",
"file_id": "mathlib/Mathlib/Tactic/ComputeAsymptotics/Lemmas.lean",
"sample_id": "fb8d70162809714734e39e1803545fb5adac8b0c13fe3dded4a257dc63c28f16"
} | train |
{
"chosen_proof_sha256": "a4055cfe4983bb68dd9d6d5e363dc43c3ee390bf3b7daa2b26130a168ae26243",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rcases eq_or_ne p 1 with rfl | hp
· simp
· rcases eq_or_ne n 0 with rfl | hn
· simp
· simp [← padicValNat_eq_emultiplicity_of_ne_one, *] | true | {
"chosen_tokens": 30,
"rejected_tokens": 8,
"token_jaccard": 0.038462,
"token_length_ratio": 0.266667
} | unsolved_goals | unsolved_goals | d41edaa9b05f5914bdf1616ef914c8e6f79d4cdbae0c12a60c0adbae466af048 | 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.MaxPowDiv
public import Mathlib.RingTheory.Multiplicity
public import Mathlib.Data.Nat.Factors
Namespace:
(root)
Local context:
/-
Copyright (c) 201... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "0726dbe9508ffb5b8389710ae5a2d03a256cb9660ad31da536e957962632166a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "NumberTheory/Padics",
"family_id": "nat",
"file_id": "mathlib/Mathlib/NumberTheory/Padics/PadicVal/Defs.lean",
"sample_id": "c6bc706d166e6b8c0ec0ad881ad47670431feaffc163f29325cfeefd84fa16cb"
} | train |
{
"chosen_proof_sha256": "5c5e46221d8b229ee79bbd41b4c1bdba3112695c6156a1e23176053e1c54427e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "036367c9e9da4688a5df880210fa2dfa4c... | by subst h; rfl | false | {
"chosen_tokens": 5,
"rejected_tokens": 9,
"token_jaccard": 0.555556,
"token_length_ratio": 1.8
} | missing_wrong_import | import | d459a671e4eb43b7e5112088509b0c6314c9b458b4b0b461df798feaef4b3627 | 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.EqToHom
public import Mathlib.CategoryTheory.Functor.Const
public import Mathlib.CategoryTheory.Opposites
public import Mathlib.Data.Prod.Basic
... | import Mathlib.WrongImport
by subst h; rfl | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Products",
"family_id": "eqtohom_snd",
"file_id": "mathlib/Mathlib/CategoryTheory/Products/Basic.lean",
"sample_id": "cb00d28e03305d658f5fd091ef6af272ded07e63d8901ab1f776e0190fa772a7"
} | train |
{
"chosen_proof_sha256": "0be2d8fff9f4e0668c3d36339336ba1af35cd1dd0eb1d09d433511d22d232ffa",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
simp [← h₀.ordConnected_iff_of_bdd (OrderBot.bddBelow I) h₂, ordConnected_iff_disjoint_Ioo_empty] | false | {
"chosen_tokens": 19,
"rejected_tokens": 5,
"token_jaccard": 0.047619,
"token_length_ratio": 0.263158
} | sorry_admit_sorryAx_proof_hole | sorryAx | d46182572fdd73679b7647720c8c82b9495822213fea1abd1fb8668c2af124c0 | 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.Int.ConditionallyCompleteOrder
public import Mathlib.Data.Int.Interval
public import Mathlib.Data.Int.SuccPred
public import Mathlib.Order.Lattice.Nat
Na... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Order/Interval",
"family_id": "set",
"file_id": "mathlib/Mathlib/Order/Interval/Set/OrdConnectedLinear.lean",
"sample_id": "43e80cf9725299057626e124ada0c7fa7af7a66a7d45d1cc2cc1eef0f94a344c"
} | train |
{
"chosen_proof_sha256": "36cd5f448af1bf1ecf44142699671ba22a07e7da26e9b39c0449f7bdf315a9b7",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
obtain ⟨P, φ, hsurj⟩ := H
obtain ⟨g, rfl⟩ := h
obtain ⟨a, ha⟩ := hsurj (algebraMap _ _ g)
have : IsLocalization.Away (f * g) (Localization.Away (φ a)) :=
ha ▸ .mul' (Localization.Away f) _ _ _
have : IsStandardEtale R (Localization.Away a) := .of_isLocalizationAway a
exact .mk _ (IsLocalization.Away.... | false | {
"chosen_tokens": 116,
"rejected_tokens": 3,
"token_jaccard": 0.04878,
"token_length_ratio": 0.025862
} | by_exact_placeholder | exact? | d47bbe6b333a43389bef7fcd7969edc55b4298ae1b53f63bc61c1e6155609e47 | 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.Etale.StandardEtale
public import Mathlib.RingTheory.LocalRing.ResidueField.Instances
public import Mathlib.RingTheory.RingHom.StandardSmooth
public... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Unramified",
"family_id": "hasstandardetalesurjectionon",
"file_id": "mathlib/Mathlib/RingTheory/Unramified/LocalStructure.lean",
"sample_id": "dfed06d389dc5d68e96360f58a96e6261e6d908c48220f135bdf99a31b4e44ba"
} | train |
{
"chosen_proof_sha256": "2038792152bb22c0a5ae377d16d2f16e94b117cceae2db86d9ede3e4d520885b",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "9c73f768c6b902af55b2843988f629261d... | by
rw [← cancel_epi ((shiftFunctorZero _ A).inv.app X), Iso.inv_hom_id_app,
pullbackShiftFunctorZero_inv_app, assoc, Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app]
rfl | false | {
"chosen_tokens": 35,
"rejected_tokens": 2,
"token_jaccard": 0.043478,
"token_length_ratio": 0.057143
} | sorry_admit_sorryAx_proof_hole | sorry | d4a3ec71c9964cbc3e870b7f0fc4112c9d1033af41e984ab29f31064950b455e | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Shift.Adjunction
public import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
Namespace:
CategoryTheory
Local context:
/-
Copyright (c) 20... | 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/Shift",
"family_id": "pullbackshiftfunctorzero_hom_app",
"file_id": "mathlib/Mathlib/CategoryTheory/Shift/Pullback.lean",
"sample_id": "59fa1cfe920c44354539a08a73d5bce71392689b4372837276159b664aba9fb8"
} | train |
{
"chosen_proof_sha256": "1246a108b01c281615de5c0ac2e3b8d9e0aa2fea03820a7bb20910f1f044ae33",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
simp_rw [he.isIdempotentElem.mem_iff, he.isIdempotentElem.range_mem_invtSubmodule_iff,
he.isIdempotentElem.ker_mem_invtSubmodule_iff, forall_and, and_iff_left_iff_imp, ← mul_def]
intro h x hx
simpa [he.isSelfAdjoint.star_eq] using! congr(star $(h _ (star_mem hx))) | false | {
"chosen_tokens": 55,
"rejected_tokens": 5,
"token_jaccard": 0.058824,
"token_length_ratio": 0.090909
} | sorry_admit_sorryAx_proof_hole | sorryAx | d4ccb20caa622b1c8645dcd0cc2fc309acbef6558cc35c991cc51017a1fd8fa7 | 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.Classes
public import Mathlib.Analysis.InnerProductSpace.Adjoint
Namespace:
VonNeumannAlgebra
Local context:
/-
Copyright (c) 2022 Kim ... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Analysis/VonNeumannAlgebra",
"family_id": "isstarprojection",
"file_id": "mathlib/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean",
"sample_id": "4180d4b77b71dc3fb2a1ccac790bfafb68b3d5a563a499f05b950658539e98e7"
} | train |
{
"chosen_proof_sha256": "ba260adf630a9d5dff8237b3bea2f33b1eff285cacd7e89777abb74bff12da31",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3ac2a51e138f18231efcaee4ea66d9527d... | by
cases x; rfl
nonrec def ExceptT.callCC {ε} [MonadCont m] {α β : Type _}
(f : Label α (ExceptT ε m) β → ExceptT ε m α) : ExceptT ε m α :=
ExceptT.mk (callCC fun x : Label _ m β => ExceptT.run <| f (ExceptT.mkLabel x)) | false | {
"chosen_tokens": 73,
"rejected_tokens": 78,
"token_jaccard": 0.897436,
"token_length_ratio": 1.068493
} | markdown_preamble_trailing | trailing | d4cf6823d0940896a28819abf16677f73619f7e6a2ca45b62597e673e005b9d4 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Control.Monad.Basic
public import Mathlib.Control.Monad.Writer
public import Mathlib.Control.Lawful
public import Batteries.Tactic.Congr
public import Batterie... | by
cases x; rfl
nonrec def ExceptT.callCC {ε} [MonadCont m] {α β : Type _}
(f : Label α (ExceptT ε m) β → ExceptT ε m α) : ExceptT ε m α :=
ExceptT.mk (callCC fun x : Label _ m β => ExceptT.run <| f (ExceptT.mkLabel x))
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": "Control/Monad",
"family_id": "exceptt",
"file_id": "mathlib/Mathlib/Control/Monad/Cont.lean",
"sample_id": "eb0438819632308cf7faaf08d1c110925f195fcd644c416f40abd06f69c1666f"
} | train |
{
"chosen_proof_sha256": "164db8212cd6dbe524941c096782dd0230af417452ad9fde38bafe58fdc70918",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4855e611094d50cafe6a8152849ab58652... | by
ext
apply IsIntegralClosure.algebraMap_injective B A L
simp | true | {
"chosen_tokens": 10,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.3
} | circular_dependency | circular_dependency | d4e4f820f136d6acab55ac283ac4deed85cf08700c8a5bb108d41d35b8a2c4e2 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.DedekindDomain.IntegralClosure
public import Mathlib.RingTheory.RingHom.Finite
public import Mathlib.RingTheory.Localization.LocalizationLocalizatio... | by
exact galRestrict'_id | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "266766fa865e14c002a877f43b0593744227e7a7688118f7e65aa14c319fad13",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/IntegralClosure",
"family_id": "galrestrict'_id",
"file_id": "mathlib/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean",
"sample_id": "470ba23482b167ed1d27d1364fdd8b69790cc7acc5d9ae4345731e59f6a235df"
} | train |
{
"chosen_proof_sha256": "0e429534a623f3db5dd3586328da30692723c2a4071b25aa658c931a2d458bab",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "3a2ce68ac4ec49658754ad114bcf7a919e... | by
rw [commutator_eq_closure G, ← image_commutatorSet_closureCommutatorRepresentatives, ←
MonoidHom.map_closure, ← commutator_eq_closure]
exact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _))
variable [Finite (commutatorSet G)] | true | {
"chosen_tokens": 42,
"rejected_tokens": 5,
"token_jaccard": 0.111111,
"token_length_ratio": 0.119048
} | wrong_namespace_qualification | wrong_namespace_qualification | d505c97f243cf4a62744d479cee6088c4bf9e53420dc377e5b80b663bbde82ba | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Subgroup.Finite
public import Mathlib.GroupTheory.Commutator.Basic
public import Mathlib.GroupTheory.Rank
public import Mathlib.GroupTheory.Index... | by
exact OProverMissing.missing_a6146b935412 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "2eddd0377156620e39225547bbddf6495e1b63b123aa59fff372655776e15e11",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "GroupTheory/Commutator",
"family_id": "card_commutator_closurecommutatorrepresentatives",
"file_id": "mathlib/Mathlib/GroupTheory/Commutator/Finite.lean",
"sample_id": "a6146b935412bf1cd41ece91f946c0daf117bb2ef301024873391e1ed5ceafb2"
} | train |
{
"chosen_proof_sha256": "bbbd68780f20b0315d7a36f5a0356a461835a08a2fef3351d5866ce71a89c492",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "37064592f892514616569c3cb0eac4f45c... | by
simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_pow] at hxy ⊢
rw [hxy] | false | {
"chosen_tokens": 23,
"rejected_tokens": 28,
"token_jaccard": 0.772727,
"token_length_ratio": 1.217391
} | markdown_preamble_trailing | preamble | d510268167f816313ec88d4c6465146cc7a8c348db9faa500803269afd3eced3 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Module.Submodule.Map
public import Mathlib.Algebra.Polynomial.Eval.Defs
public import Mathlib.RingTheory.Ideal.Quotient.Defs
public import Mathlib.Alge... | Here is the proof:
by
simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_pow] at hxy ⊢
rw [hxy] | {
"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/SModEq",
"family_id": "pow",
"file_id": "mathlib/Mathlib/LinearAlgebra/SModEq/Basic.lean",
"sample_id": "550eea517b6f57ead3df265df6cf73d2563e67f3ae71b6c49159805a9ce8003d"
} | train |
{
"chosen_proof_sha256": "0726e99880ae39e0c58cb38c38f992cfa6af39d47cf8ffe5ea36831ca442aef8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "c379f0ea095080b6167d0f938b9ad3f769... | by
obtain ⟨Us, hUs, heq⟩ := TopologicalSpace.Opens.isBasis_iff_cover.mp
(PrespectralSpace.isBasis_opens X) ⟨f ⁻¹' U, hU.preimage hfc⟩
obtain ⟨t, ht⟩ := by
refine hc.elim_finite_subcover (fun s : Us ↦ f '' s.1) (fun s ↦ h _ s.1.2) (fun x hx ↦ ?_)
obtain ⟨x, rfl⟩ := hs hx
obtain ⟨i, hi, hx⟩ := mem_sSu... | true | {
"chosen_tokens": 252,
"rejected_tokens": 5,
"token_jaccard": 0.037037,
"token_length_ratio": 0.019841
} | wrong_namespace_qualification | wrong_namespace_qualification | d5277a335d09abbbbedda64e08211f91c1ec3d35a94c8249310fec3959835981 | 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
exact OProverMissing.missing_06e54fa94e04 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "e0d8c79aeba6d4b88d11e7e5a6ae32701f2c0a5e59919f7a512a5316382a34ec",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Topology/Spectral",
"family_id": "isopenmap",
"file_id": "mathlib/Mathlib/Topology/Spectral/Prespectral.lean",
"sample_id": "06e54fa94e0402146a1056a8e38c4177d4d7f6c2eb72f119267197e2349465f8"
} | train |
{
"chosen_proof_sha256": "5a49665e0c5d2151828a184678146c90b5369415677dd342dff1e4392eda48b8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
simp [*] | false | {
"chosen_tokens": 5,
"rejected_tokens": 3,
"token_jaccard": 0.142857,
"token_length_ratio": 0.6
} | by_exact_placeholder | exact? | d5579f7c9e9f306460c26c6fb40c36a2d9e61093d6ec0e271b3fd19b232b2843 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Ineq
public import Mathlib.Data.Nat.Cast.Order.Ring
Namespace:
Mathlib.Tactic.Linarith
Local context:
/-
Copyright (c) 2020 Robert Y. Lewis. All rights ... | by
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": "Tactic/Linarith",
"family_id": "eq_of_eq_of_eq",
"file_id": "mathlib/Mathlib/Tactic/Linarith/Lemmas.lean",
"sample_id": "83ec59d8b8358fb8a455e8f01681be42384d6a03b957fc77c05e2d58d2ad540d"
} | train |
{
"chosen_proof_sha256": "2851538f3a7c40f88dfdee204d38577a3a636017ca196f0ed156206cf7eba4b2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8bbc6f2b397a8cc463b8503d4ac88ef365... | by
induction hx using span_induction with
| mem x h => obtain ⟨i, rfl⟩ := h; cases j <;> simp [h]
| zero => simp
| add u v _ _ hu hv => simp [hu, hv]
| smul t u _ hu => simp [hu] | true | {
"chosen_tokens": 59,
"rejected_tokens": 3,
"token_jaccard": 0.027027,
"token_length_ratio": 0.050847
} | unknown_identifier | unknown_identifier | d5a6e83da85dcaee0dd3b77ab61f0b5839c52c0af2c7c3c2b717b30993c5ff14 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Lie.Matrix
public import Mathlib.Algebra.Lie.OfAssociative
public import Mathlib.Algebra.Lie.Weights.Basic
public import Mathlib.LinearAlgebra.Eigenspa... | by
exact __oprover_missing_06f9768559f2 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "af2478621552e6fd0d65aa2949052d08477f0c4af0da82df09dd1b51e8daa920",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/RootSystem",
"family_id": "apply_sum_inl_eq_zero_of_mem_span_h",
"file_id": "mathlib/Mathlib/LinearAlgebra/RootSystem/GeckConstruction/Basic.lean",
"sample_id": "06f9768559f2fe8088fc3798603424c8e8c34edf39453ccd8c0d887cb067503b"
} | train |
{
"chosen_proof_sha256": "e79e7e0dc18cc849a7cc458c4f413b9fd00b3758b5b3041d108cd5fc7c5dd925",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
refine ⟨fun _ ↦ ?_, fun _ ↦ inferInstance⟩
let e : G.inv ⋙ G ⋙ F ≅ F := (associator _ _ _).symm ≪≫
isoWhiskerRight (G.asEquivalence.counitIso) _ ≪≫ F.leftUnitor
exact of_iso e | false | {
"chosen_tokens": 54,
"rejected_tokens": 3,
"token_jaccard": 0.09375,
"token_length_ratio": 0.055556
} | by_exact_placeholder | exact? | d6189f58a46af849cf19871b1782fa0855d7b13edb85caa98302eab34fdebc41 | 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.Functor.KanExtension.DenseAt
public import Mathlib.CategoryTheory.Limits.Presheaf
public import Mathlib.CategoryTheory.Generator.StrongGenerator... | by
exact? | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Functor",
"family_id": "isdense",
"file_id": "mathlib/Mathlib/CategoryTheory/Functor/KanExtension/Dense.lean",
"sample_id": "05a81ca678f684a0e635006515f905d4791271bb1386b472a3d4d7bf0d040e70"
} | train |
{
"chosen_proof_sha256": "c165c49ef010a17e120cfdcd1a6cc6afaacc95eed29ab44ee041da167fa437e8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "bc6b5ae320fe350ef6424ba0b4da9753b0... | by
-- this follows from a covering argument using the sets satisfying `ρ a ≤ ν a`.
apply ENNReal.le_of_forall_pos_le_add fun ε εpos _ => ?_
obtain ⟨U, sU, U_open, νU⟩ : ∃ (U : Set α), s ⊆ U ∧ IsOpen U ∧ ν U ≤ ν s + ε :=
exists_isOpen_le_add s ν (ENNReal.coe_pos.2 εpos).ne'
let f : α → Set (Set α) := fun _ =... | true | {
"chosen_tokens": 324,
"rejected_tokens": 3,
"token_jaccard": 0.020619,
"token_length_ratio": 0.009259
} | circular_dependency | circular_dependency | d66dcb7709be2eb89578ad64882996984efa2422571e7721296777ed54536487 | 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.Covering.VitaliFamily
public import Mathlib.MeasureTheory.Function.AEMeasurableOrder
public import Mathlib.MeasureTheory.Integral.Average
public ... | by
exact measure_le_of_frequently_le | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "d1a9cbad5c1ec41b19af4e0eabb3ae5601ad6a18569cbef2d6111783fc2137eb",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Covering",
"family_id": "measure_le_of_frequently_le",
"file_id": "mathlib/Mathlib/MeasureTheory/Covering/Differentiation.lean",
"sample_id": "c0fbb14cf7d5c8d77a74ea9851f8a6d43a29d0444f221fd7a52f0350a1b23692"
} | train |
{
"chosen_proof_sha256": "6bc6459c3657e743f6b65ab2f495aaf259c49629a6dfb3dcfa555bfa29fc68bb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b6ed2b9a81e9433eac524aab243c09c5cb... | by
have hf : Monotone fun n => ⨅ i ≥ n, f i := fun n m h => biInf_mono fun i => h.trans
rw [← Monotone.iSup_nat_add hf k]
· simp_rw [iInf_ge_eq_iInf_nat_add, ← Nat.add_assoc]
-- Not `@[simp]` since the subterm `?f (i + ?k)` produces an ugly higher-order unification problem.
-- (Although the `simpNF` linter does ... | true | {
"chosen_tokens": 143,
"rejected_tokens": 5,
"token_jaccard": 0.02439,
"token_length_ratio": 0.034965
} | wrong_namespace_qualification | wrong_namespace_qualification | d6db77a1d0ef97448f0a8576c54de9b71ec2c97c961bc819b20b5f95561df51b | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Bool.Set
public import Mathlib.Data.Nat.Set
public import Mathlib.Order.CompleteLattice.Basic
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Joh... | by
exact OProverMissing.missing_56582689395b | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "05d49dd2c544ee0b85a30592f3bd7ba0ed052e1d08f0005e5e6516189ab25d35",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/CompleteLattice",
"family_id": "isup_iinf_ge_nat_add",
"file_id": "mathlib/Mathlib/Order/CompleteLattice/Lemmas.lean",
"sample_id": "56582689395b3fd4d53bdd9aa4d7c0b70282e024f7b7737ed578d8946fb979bf"
} | train |
{
"chosen_proof_sha256": "5403bbdd6b8db64d0404aea07e7fcb0b3cc956807fed1c2cfdeeee9bac43ff23",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
constructor
simp [h.neg_to_eq, div_nonneg] | false | {
"chosen_tokens": 10,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.3
} | by_exact_placeholder | placeholder | d6dc05f4c00067c0306a3dcfa5493a5f4324c0752440de05c941dd06fe682b26 | 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.Floor.Semiring
public import Mathlib.Data.NNRat.Order
public import Mathlib.Data.Rat.Floor
Namespace:
Mathlib.Meta.NormNum
Local context:
/-
Co... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Data/NNRat",
"family_id": "israt",
"file_id": "mathlib/Mathlib/Data/NNRat/Floor.lean",
"sample_id": "a3a4139f0052b0a32136e52937242d5f55eef48c36e8634d6c5292950c12886e"
} | train |
{
"chosen_proof_sha256": "3816114db1c077f621bfeb2d3b1b98b7b671ab19074e0d158d715632b3696809",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
intro x y hxy
let xy : G.obj (pullback π π) := (PreservesPullback.iso G π π).inv <|
(TopCat.pullbackIsoProdSubtype (G.map π) (G.map π)).inv ⟨(x, y), hxy⟩
have ha' := congr_fun ha xy
dsimp at ha'
have h₁ : ∀ y, G.map (pullback.fst _ _) ((PreservesPullback.iso G π π).inv y) =
pullback.fst (G.map π)... | false | {
"chosen_tokens": 211,
"rejected_tokens": 5,
"token_jaccard": 0.034483,
"token_length_ratio": 0.023697
} | sorry_admit_sorryAx_proof_hole | sorryAx | d6e121eb35031ac83c85f775b6057b146539bacf320891cbf3da106951c09f84 | 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.Preserves.Opposites
public import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
public import Mathlib.Condensed.Basic
public impo... | 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": "Condensed",
"family_id": "factorsthrough_of_pullbackcondition",
"file_id": "mathlib/Mathlib/Condensed/TopComparison.lean",
"sample_id": "9584eeedec12f1270f773e97e07803ac6bace4ed2fabdce1f1ae60147a31ccc0"
} | train |
{
"chosen_proof_sha256": "7098d4ede92516c6303541ad23377423cb7f62e04005055a6e4b359a31869d3c",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
intro ⟨h1, h2⟩; use y.drop x.length, h2
nth_rw 1 [← List.take_append_drop x.length y]
simpa [-List.take_append_drop, List.prefix_iff_eq_take, hl] using h1
mpr := by simp +contextual [pullSub] | true | {
"chosen_tokens": 53,
"rejected_tokens": 8,
"token_jaccard": 0.054054,
"token_length_ratio": 0.150943
} | unsolved_goals | unsolved_goals | d6e38178d798945ba66f5aab8461267cf1e02c8eb17d233755c0b057250e3009 | 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
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "ea0de1a29252f2d6a594ccb340c5d91e082e190c949187667db7765671193290",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "SetTheory/Descriptive",
"family_id": "mem_pullsub_long",
"file_id": "mathlib/Mathlib/SetTheory/Descriptive/Tree.lean",
"sample_id": "2b9acd22df8b3e0b46d29d9433ec7adfa97c3ca8da662c2868b5f6dac8fb61d2"
} | train |
{
"chosen_proof_sha256": "05c1e20ccd52d20617d06c19bb3d65322091abd072a38ef854172815fc9654f9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "b6f984148eb97ed2790857133a651dc0f8... | by
refine ⟨?_, fun h ↦ (isCyclic_of_card_le_two h).isMulCommutative⟩
classical
rintro ⟨⟨h⟩⟩
rw [← not_lt, ← Set.ncard_univ, Set.two_lt_ncard_iff]
rintro ⟨a, b, c, _, _, _, hab, hac, hbc⟩
apply hbc
simp_rw [Perm.ext_iff] at h
simpa [swap_apply_of_ne_of_ne hab hac] using h (swap a b) (swap b c) a | true | {
"chosen_tokens": 86,
"rejected_tokens": 3,
"token_jaccard": 0.02381,
"token_length_ratio": 0.034884
} | unknown_identifier | unknown_identifier | d6f5a7bebf6480677ec74e26742a1f4ca268ad683f839a13ff91de346a6f2b85 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Fintype.Perm
public import Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic
public import Mathlib.SetTheory.Cardinal.Finite
Namespace:
Equiv.Perm
Local c... | by
exact __oprover_missing_15e83a19ac7a | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "792ce9c247b7905e4fbbc8ebeea7a2f08bb351b4fb5cf55e27a87c8b4a9ab80b",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Finite",
"family_id": "ismulcommutative_iff_card_le_two",
"file_id": "mathlib/Mathlib/Data/Finite/Perm.lean",
"sample_id": "15e83a19ac7a4ffb734f70f90cc5cd72e116c7c5b68a279718e9f6a477143d88"
} | train |
{
"chosen_proof_sha256": "084e758470934f76ebf17be847788f3818be1118ff79efcd27a52969227aebb9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
have h₁ : x⁻¹ ≠ 1 := by rwa [inv_eq_one_div, Ne, div_eq_iff_mul_eq hx0, one_mul]
have h₂ : x⁻¹ - 1 ≠ 0 := mt sub_eq_zero.1 h₁
have h₃ : x - 1 ≠ 0 := mt sub_eq_zero.1 hx1
have h₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x :=
Nat.recOn n (by simp) fun n h => by
rw [pow_succ', mul_inv_rev, ← mul_assoc, h, mul_as... | false | {
"chosen_tokens": 194,
"rejected_tokens": 5,
"token_jaccard": 0.029851,
"token_length_ratio": 0.025773
} | sorry_admit_sorryAx_proof_hole | sorryAx | d6fc59f1efd36b12d3fe2695991cc2ed6bac84539e65aabead8f59c272da08b2 | 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.Ring.GeomSum
Namespace:
(root)
Local context:
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
R... | by
exact sorryAx _ true | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Algebra/Field",
"family_id": "geom_sum_inv",
"file_id": "mathlib/Mathlib/Algebra/Field/GeomSum.lean",
"sample_id": "2357e78341a07f48e92532f87e0bfabcf4fba0a7567069cbcd02ca8502dd6972"
} | train |
{
"chosen_proof_sha256": "c1390084cb2580392cbfe88281f49b6df9fd83f38d46b0fc57029acaa73ff3f3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [← span_span_of_tower R, finrank_span_eq_finrank] | true | {
"chosen_tokens": 9,
"rejected_tokens": 8,
"token_jaccard": 0.066667,
"token_length_ratio": 0.888889
} | unsolved_goals | unsolved_goals | d72f0646f153fca669402859e8812380009159611cdc44d39118c88132681fc2 | 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.Epi
public import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
public import Mathlib.LinearAlgebra.Finsupp.LinearCombination
public impo... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "d7b290445319af70fe27fcdc7fa8d5a53873f01da9d97260db5f6cec71fa2cd2",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Span",
"family_id": "finrank_span_eq_finrank_span",
"file_id": "mathlib/Mathlib/LinearAlgebra/Span/TensorProduct.lean",
"sample_id": "f378af96f4b655711415908ce319014735dc5bb779d91dee4115105bae787859"
} | train |
{
"chosen_proof_sha256": "536352e5ddbb900c41d2280e8e59f7791e63d85f699a6f8a27e8c2d97392c7b3",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "e6432d29ef4ba24cfd7ea5c2ca105f906c... | by
let S : Set (Ultrafilter M) := ⋂ n, { U | ∀ᶠ m in U, m ∈ FP (a.drop n) }
have h := exists_idempotent_in_compact_subsemigroup ?_ S ?_ ?_ ?_
· rcases h with ⟨U, hU, U_idem⟩
refine ⟨U, U_idem, ?_⟩
convert! Set.mem_iInter.mp hU 0
· exact Ultrafilter.continuous_mul_left
· apply IsCompact.nonempty_iInter... | false | {
"chosen_tokens": 224,
"rejected_tokens": 5,
"token_jaccard": 0.032967,
"token_length_ratio": 0.022321
} | sorry_admit_sorryAx_proof_hole | sorryAx | d76b9b34356b855c1a406a2693056e9986d986aa828b02dd3b6549022fea3c34 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Stream.Init
public import Mathlib.Topology.Algebra.Semigroup
public import Mathlib.Topology.Compactification.StoneCech
public import Mathlib.Algebra.BigOp... | 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": "Combinatorics",
"family_id": "exists_idempotent_ultrafilter_le_fp",
"file_id": "mathlib/Mathlib/Combinatorics/Hindman.lean",
"sample_id": "b9b65ec13f7e30e41eeaa6338022c9efbc66d8f0394c7b483c53252b38ab9a8f"
} | train |
{
"chosen_proof_sha256": "158b158a558370a66bb2762fb6673634c41f0d235bb36659662b90d5aa0cb875",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "8612ceda65694614a4d71f561ffbd968d0... | by
contrapose! +distrib
rw [Set.infinite_image2 hfs hft]
grind only [Set.Infinite.nonempty] | true | {
"chosen_tokens": 22,
"rejected_tokens": 5,
"token_jaccard": 0.1,
"token_length_ratio": 0.227273
} | wrong_namespace_qualification | wrong_namespace_qualification | d76bc76c1ad27c830388eaa7e43f163dd05fff942a8ab139ea6ceacb0356630d | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Set.Finite.Basic
public import Mathlib.Data.Fintype.Prod
public import Mathlib.Data.Fintype.Pi
public import Mathlib.Algebra.Order.Group.Multiset
public i... | by
exact OProverMissing.missing_335fd55413b4 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "cfdd3ec414294a277a56cc8a1108925e669261b4ef1f757484c85f9793598d15",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Finite",
"family_id": "finite_image",
"file_id": "mathlib/Mathlib/Data/Finite/Prod.lean",
"sample_id": "335fd55413b4f96955767217c73d17f78f377c305db1438e043702f98d245c46"
} | train |
{
"chosen_proof_sha256": "ac66c6e721cc952e221200b69cfe94cb7c90e16c08e83be7b74b05ee3c9aa39f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "00dd289d0a927f45e5d74bbf587cb38856... | by aesop
variable [Fintype ι] | true | {
"chosen_tokens": 7,
"rejected_tokens": 3,
"token_jaccard": 0.111111,
"token_length_ratio": 0.428571
} | unknown_identifier | unknown_identifier | d79c2ed443f68b76e27ed1e6bc7fe16a2d20dee638685f1b7d7d9a92e1ba3ead | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Data.Finset.Sigma
public import Mathlib.Data.Fintype.OfMap
Namespace:
(root)
Local context:
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Releas... | by
exact __oprover_missing_b7150f0dc88e | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "ba2429c9d797e8900081ef5cd5fec30396a39367bf1bada1f4c8e3620fe8156c",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Data/Fintype",
"family_id": "set",
"file_id": "mathlib/Mathlib/Data/Fintype/Sigma.lean",
"sample_id": "b7150f0dc88e25f3382430b108a0281da820a234916871dd325c0904c8b12aaf"
} | train |
{
"chosen_proof_sha256": "021873745f79b607afe2480d1a3ec9b92a92bc4d1a53d9616a37575a2e4fbf1f",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "fd18fa607c6f84a743396868ee103846e4... | by rw [← top_sdiff', ← top_sdiff', map_sdiff, map_top] | true | {
"chosen_tokens": 13,
"rejected_tokens": 3,
"token_jaccard": 0.090909,
"token_length_ratio": 0.230769
} | circular_dependency | circular_dependency | d7ac4155f5abb850288d89d59118ae06d499891ce24c34e66843a8d1fe85af1c | 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.Hom.BoundedLattice
Namespace:
(root)
Local context:
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as descr... | by
exact map_hnot | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "a27d149a9beb24a22ef2df2790ecc2d675ac796a059b5d3f801a349847061103",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Order/Heyting",
"family_id": "map_hnot",
"file_id": "mathlib/Mathlib/Order/Heyting/Hom.lean",
"sample_id": "88184057873a0a559cc6d5510520e38470290a3d57980287af7d4cf570e8c138"
} | train |
{
"chosen_proof_sha256": "de67f9b6451b5db3ba017889f07c8f6879cb98f3fc2736a86445d62c14d1c664",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ccd1b6ebbfc7925ea5a53c372ff24897d1... | by
rw [Squarefree, not_forall]
exact ⟨0, by simp⟩ | true | {
"chosen_tokens": 14,
"rejected_tokens": 3,
"token_jaccard": 0.153846,
"token_length_ratio": 0.214286
} | circular_dependency | circular_dependency | d7c860fc6318f5c13edf1a3b94d79a08a4f56f7b42ebc19e035e475bfda2fe8e | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.RingTheory.Coprime.Lemmas
public import Mathlib.RingTheory.Nilpotent.Basic
public import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid
public import M... | by
exact not_squarefree_zero | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "11f7d20af14dcfb0a445b11b6308d279e981fd0bf57915cbfdabf4bdec103a6e",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Algebra/Squarefree",
"family_id": "not_squarefree_zero",
"file_id": "mathlib/Mathlib/Algebra/Squarefree/Basic.lean",
"sample_id": "ba8a1ee7158309c944c89fb3fff36b668ca7628b981cba7184536316677b8d1f"
} | train |
{
"chosen_proof_sha256": "61f4608135cc80db3b144a8f0b385332ee17b58f1c351db640ceccbb2a9d7cc8",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "149a9c772317371fc2b6f906fcbdef5bbd... | by
ext x; grind [eq_one_or_toFreeMonoid x] | false | {
"chosen_tokens": 9,
"rejected_tokens": 13,
"token_jaccard": 0.666667,
"token_length_ratio": 1.444444
} | missing_wrong_import | import | d7d9b5951413722646760d813fd90bca977a651fd25b09e061471c9614065fb6 | 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.Free
public import Mathlib.Algebra.Group.WithOne.Basic
public import Mathlib.Algebra.Group.Units.Basic
p... | import Mathlib.WrongImport
by
ext x; grind [eq_one_or_toFreeMonoid 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": "Algebra/FreeMonoid",
"family_id": "range_tofreemonoid",
"file_id": "mathlib/Mathlib/Algebra/FreeMonoid/FreeSemigroup.lean",
"sample_id": "6a4d757b91c5500aa6ea19c37733ffcdf411c7c5b0b8869100541b0d79adff6c"
} | train |
{
"chosen_proof_sha256": "c3ee73588e36d5e89a2703d1667734bc0b58f87dcefce1c40a3828ad044e5637",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4b54c5b2c0a2b38c02774c2f2d109212cb... | by
rw [ringKrullDim, ringKrullDim]
apply Order.krullDim_le_of_krullDim_preimage_le' (PrimeSpectrum.comap C) ?_ (fun p ↦ ?_)
· exact fun {a b} h ↦ Ideal.comap_mono h
· rw [show C = (algebraMap R (Polynomial R)) from rfl, Order.krullDim_eq_of_orderIso
(PrimeSpectrum.preimageOrderIsoFiber R (Polynomial R) p)... | true | {
"chosen_tokens": 111,
"rejected_tokens": 5,
"token_jaccard": 0.074074,
"token_length_ratio": 0.045045
} | circular_dependency | circular_dependency | d7def9b2d1fdb8891a271096c6e19d91fc62aef814052678c16faef4dc1ad72b | 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.FieldDivision
public import Mathlib.RingTheory.KrullDimension.PID
public import Mathlib.RingTheory.LocalRing.ResidueField.Fiber
public impor... | by
exact Polynomial.ringKrullDim_le | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "05f8c76094614921265b6d75219be5df87c573ab92ef1fa2980fca8354e5f3ee",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/KrullDimension",
"family_id": "polynomial",
"file_id": "mathlib/Mathlib/RingTheory/KrullDimension/Polynomial.lean",
"sample_id": "812ef1c18921780c46e19c5df601fc11ffc84bd9969d3bb154d739df4d388862"
} | train |
{
"chosen_proof_sha256": "0e69e1ffdf18797894f9d6a7bd81718318be487366506848fac73b20c0a23f03",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "78f61c99b43e2dea3859f08d8e1ca140e6... | by
rw [internallyProjective_iff_tensor_condition]
refine ⟨fun h A B e he S g ↦ ?_, fun h A B e he S g ↦ ?_⟩
· specialize h e S ((β_ _ _).hom ≫ g)
obtain ⟨S', π, hπ, g', hh⟩ := h
refine ⟨S', π, hπ, (β_ _ _).inv ≫ g', ?_⟩
simp [← hh]
· specialize h e S ((β_ _ _).inv ≫ g)
obtain ⟨S', π, hπ, g', hh⟩... | true | {
"chosen_tokens": 151,
"rejected_tokens": 3,
"token_jaccard": 0.052632,
"token_length_ratio": 0.019868
} | circular_dependency | circular_dependency | d7e671c167c5a85dacc2edad6bc9a8c9461032af9d93adbb384fa08bbe5f61cb | 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.Projective.Internal
public import Mathlib.Condensed.Light.Epi
public import Mathlib.Condensed.Light.Functors
public import Mathlib.C... | by
exact internallyProjective_iff_tensor_condition | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "c096491035f5ce3dddee76453d5b0f89fb32e29fd0957800c7d092760c8155ad",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Condensed/Light",
"family_id": "internallyprojective_iff_tensor_condition",
"file_id": "mathlib/Mathlib/Condensed/Light/InternallyProjective.lean",
"sample_id": "e1cca0717031d58aaaf848a8d08234c2dff2f8f85f108568fd9abbcb33e5e463"
} | train |
{
"chosen_proof_sha256": "3f65d3f4792abc1e42ea24793b4621de9bedda45327f9e9eb8d4aee686a05437",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "148d4bb481d5fb6dd96abb5cffc2629c9e... | by
rw [← LinearOrderedCommGroupWithZero.wellFoundedOn_setOf_ge_gt_iff_nonempty_discrete_of_ne_zero
one_ne_zero, ← Set.wellFoundedOn_range]
classical
refine ⟨fun h ↦ (h.mapsTo Subtype.val ?_).mono' (by simp), fun h ↦ (h.mapsTo ?_ ?_).mono' ?_⟩
· rintro ⟨_, x, rfl⟩
simp only [← Subtype.coe_le_coe, OneMemC... | false | {
"chosen_tokens": 173,
"rejected_tokens": 3,
"token_jaccard": 0.032258,
"token_length_ratio": 0.017341
} | by_exact_placeholder | placeholder | d7e693be6455f4408655cb63864e00ad1a5880554a0d577ef0843ac416529a48 | 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.ArchimedeanDensely
public import Mathlib.RingTheory.Valuation.ValuationRing
Namespace:
Valuation.Integers
Local context:
/-
Copyright (c) 2024 Ya... | by
exact placeholder | {
"accepted": false,
"all_failure_types": [
"proof_hole_or_search_placeholder"
],
"failure_type": "proof_hole_or_search_placeholder",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "RingTheory/Valuation",
"family_id": "wellfounded_gt_on_v_iff_discrete_mrange",
"file_id": "mathlib/Mathlib/RingTheory/Valuation/Archimedean.lean",
"sample_id": "45bf8f219b5d1572a7b7bfc82220b21da6d9c0dadf354c8db833eeedf98c284d"
} | train |
{
"chosen_proof_sha256": "974123677968aef5cec49d697aa3b1eb6faa233c3680735631e7ed936621e266",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "5b17a6a744755a2e6fb4017210906255fc... | by
rw [integral_fintype .of_finite]
congr with x
rw [measureReal_def]
congr 2
exact PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _) | true | {
"chosen_tokens": 28,
"rejected_tokens": 3,
"token_jaccard": 0.090909,
"token_length_ratio": 0.107143
} | unknown_identifier | unknown_identifier | d84b95db5a15767d1f9b7be3e5461acb24cd0a857587283ce9cafda591ad3157 | 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.ProbabilityMassFunction.Basic
public import Mathlib.Probability.ProbabilityMassFunction.Constructions
public import Mathlib.MeasureTheory.Integral.... | by
exact __oprover_missing_aab605044ac1 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "16b6d9253327d213169b422bdddba1982f944f595a03a61524e43c858fa8a25d",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "Probability/ProbabilityMassFunction",
"family_id": "integral_eq_sum",
"file_id": "mathlib/Mathlib/Probability/ProbabilityMassFunction/Integrals.lean",
"sample_id": "aab605044ac1a0cc179875c4807508cccde8f4829eb26975d9537a751ad57521"
} | train |
{
"chosen_proof_sha256": "8cf4a63a26dc0e5b281fd2da3d3e4050e7d2b711edcf9188ce1a11ef8a2b0753",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "21e6b6874ff31a22dfe9207b446fa97b3b... | by
unfold mkFinCons
exact coe_mk (v := Fin.cons y (N.subtype ∘ b)) _ _
/-- Let `b` be a basis for a submodule `N ≤ O`. If `y ∈ O` is linear independent of `N`
and `y` and `N` together span the whole of `O`, then there is a basis for `O`
whose basis vectors are given by `Fin.cons y b`. -/
noncomputable def mkFinCon... | false | {
"chosen_tokens": 262,
"rejected_tokens": 266,
"token_jaccard": 0.967033,
"token_length_ratio": 1.015267
} | missing_wrong_import | import | d87158316733db7657f547ef8bf00e6a0812c6a62fcd3580f0eda65e9517a730 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.Basis.Basic
public import Mathlib.LinearAlgebra.Pi
Namespace:
Module.Basis
Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights rese... | import Mathlib.WrongImport
by
unfold mkFinCons
exact coe_mk (v := Fin.cons y (N.subtype ∘ b)) _ _
/-- Let `b` be a basis for a submodule `N ≤ O`. If `y ∈ O` is linear independent of `N`
and `y` and `N` together span the whole of `O`, then there is a basis for `O`
whose basis vectors are given by `Fin.cons y b`. -/... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body",
"top_level_command"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "LinearAlgebra/Basis",
"family_id": "coe_mkfincons",
"file_id": "mathlib/Mathlib/LinearAlgebra/Basis/Fin.lean",
"sample_id": "4814cbf7f3508d036f5e5a6a7155650097bbec6785ba9dace923119e89bbbe59"
} | train |
{
"chosen_proof_sha256": "82dcb0744887ee0b2b5e66ebed4e1d1db29ac81ed20eb128a4882cf8f95fc26d",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [aeval_def, aeval_def, ← coe_eval₂Hom, ← coe_eval₂Hom, map_eval₂Hom, ←
IsScalarTower.algebraMap_eq, Function.comp_def] | false | {
"chosen_tokens": 30,
"rejected_tokens": 3,
"token_jaccard": 0.055556,
"token_length_ratio": 0.1
} | by_exact_placeholder | exact? | d8736f7c9f6736ca56971777ae615b67b5447ec1c57f7895c97411c72a95b597 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Algebra.Subalgebra.Tower
public import Mathlib.Algebra.MvPolynomial.Eval
Namespace:
MvPolynomial
Local context:
/-
Copyright (c) 2022 Yuyang Zhao. Al... | 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/MvPolynomial",
"family_id": "aeval_algebramap_apply",
"file_id": "mathlib/Mathlib/RingTheory/MvPolynomial/Tower.lean",
"sample_id": "f073ff6b550aeff5c73e243e84ebf2d91d1d169b0b3b2c4fe873846d588b6590"
} | train |
{
"chosen_proof_sha256": "05b7a960b09bb29c3bdb02159f0748a158edd33fe39806e604a0595973e7a4e9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "87930c51804421981909023af82af6c0ef... | by
rw [isSemilinearSet_iff] at hs
rcases hs with ⟨S, hS, rfl⟩
choose φ hφ using fun s : S => (hS s.1 s.2).definable
refine ⟨Formula.iSup φ, ?_⟩
ext x
have := fun s hs x => Set.ext_iff.1 (hφ ⟨s, hs⟩).symm x
simp only [mem_setOf_eq] at this
simp [this] | true | {
"chosen_tokens": 85,
"rejected_tokens": 3,
"token_jaccard": 0.021277,
"token_length_ratio": 0.035294
} | unknown_identifier | unknown_identifier | d8a2f212b600d258da347dfa0539f57b51f68742221c8cf664b02d02eebc8e9f | 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.ModelTheory.Arithmetic.Presburger.Basic
public import Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic
public import Mathlib.ModelTheory.Definability... | by
exact __oprover_missing_861b7d8a1fe1 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "2a94c432c40aaa0aba8cb92a60d95b10abb52590df992e99cc2868658811769a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "ModelTheory/Arithmetic",
"family_id": "issemilinearset",
"file_id": "mathlib/Mathlib/ModelTheory/Arithmetic/Presburger/Definability.lean",
"sample_id": "861b7d8a1fe1c1ac90ef7cb56d31d4772a1a7fac4746b22305f0316ae345cc5c"
} | train |
{
"chosen_proof_sha256": "63ac599b07e7436f21a63f1b73d4a7f54c73a40c1b4a50f48b5455d9ae2693f2",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "6a0f03b458ed50734a3025d61af5335127... | by
ext j
dsimp
rw [mapBifunctorLeftUnitor_inv_apply, mapBifunctorLeftUnitor_inv_apply, assoc, assoc,
ι_mapBifunctorMapMap]
dsimp
rw [Functor.map_id, NatTrans.id_app, id_comp, ← NatTrans.naturality_assoc,
← NatTrans.naturality_assoc]
rfl | false | {
"chosen_tokens": 41,
"rejected_tokens": 2,
"token_jaccard": 0.045455,
"token_length_ratio": 0.04878
} | by_exact_placeholder | by? | d8a350d3c1398f256d304df6a3c403c53734ff00717d729efa9f689d743e7d01 | 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.GradedObject.Associator
public import Mathlib.CategoryTheory.GradedObject.Single
Namespace:
CategoryTheory.GradedObject
Local context:
/-
Copy... | 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/GradedObject",
"family_id": "mapbifunctorleftunitor_inv_naturality",
"file_id": "mathlib/Mathlib/CategoryTheory/GradedObject/Unitor.lean",
"sample_id": "9614d072a9105eea2ab6f57e5011e3855902130db93743053ebaa70a564b484b"
} | train |
{
"chosen_proof_sha256": "b9120dc0c9841f5cc0966a2e218d4adcc087ef91568585fb5a852de6e1d93e10",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d306a3f4f2440500e38cd155adf08ac6a... | by
apply Quotient.lift_unique
apply Paths.lift_unique
fapply @Quiver.Symmetrify.lift_unique _ _ _ _ _ _ _ _ _
· rw [← Functor.toPrefunctor_comp]
exact hΦ
· rintro X Y f
simp only [← Functor.toPrefunctor_comp, Prefunctor.comp_map, Paths.of_map]
change Φ.map (Groupoid.inv ((Quotient.functor redStep)... | true | {
"chosen_tokens": 139,
"rejected_tokens": 2,
"token_jaccard": 0.018868,
"token_length_ratio": 0.014388
} | invalid_tactic | invalid_tactic | d8d9cc740d7ada3478fc0d6809d3ff3117b1a7c000f3e4ef02aef562b4cfa75d | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Groupoid
public import Mathlib.CategoryTheory.PathCategory.Basic
Namespace:
Quiver.FreeGroupoid
Local context:
/-
Copyright (c) 2022 Rémi Bott... | by
__oprover_invalid_tactic_3e937a169766 | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "4dc30413705b6c0ac90a7eb0a2fa52fe2b255819df0dad627d99a6b42f858190",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Groupoid",
"family_id": "lift_unique",
"file_id": "mathlib/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean",
"sample_id": "3e937a169766b398011457da21839b47e8a3e949c8dec6560a9e40da808f1d10"
} | train |
{
"chosen_proof_sha256": "6ec31665b33411506cda64640e21f6774b047c685e35cff9b518e65219ef78f0",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "a2717c01644181afb92dd63d96aa9ca9ce... | by
refine c.isColimit.hom_ext (fun j ↦ ?_)
dsimp
induction j using SuccOrder.limitRecOn with
| isMin j hj =>
obtain rfl := hj.eq_bot
simpa [← cancel_epi c.isoBot.inv] using h₀
| succ j hj hj' =>
apply (c.attachCells j hj).hom_ext
· simpa using hj'
· intro i
simpa only [Category.assoc... | true | {
"chosen_tokens": 129,
"rejected_tokens": 3,
"token_jaccard": 0.050847,
"token_length_ratio": 0.023256
} | circular_dependency | circular_dependency | d947f5e436279310eacea7956ed43d6233726b3a2a0bd95276b6c19f914ad5b1 | 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.RelativeCellComplex.AttachCells
public import Mathlib.CategoryTheory.MorphismProperty.TransfiniteComposition
Namespace:
HomotopicalAlgebra.R... | by
exact hom_ext | {
"accepted": false,
"failure_type": "lean_elaboration_error",
"output_sha256": "f1cd0e069abadb8a5f95105164c9d75a38a4e9e481983e6bbe2893f443685f83",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "AlgebraicTopology/RelativeCellComplex",
"family_id": "hom_ext",
"file_id": "mathlib/Mathlib/AlgebraicTopology/RelativeCellComplex/Basic.lean",
"sample_id": "96fed0cd3e0fdbf1ac8825b9ea3ac8717dd91342a9fea198dd325c9dbb8823cc"
} | train |
{
"chosen_proof_sha256": "da195f4abae7ab0a2c1c536bf1f2974db5c532e362792146fd16a70b703b9aa9",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "7b6515db8de5c99b6ea544300d4f908ec7... | by
obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf
obtain ⟨cg, rfl⟩ := Code.exists_code.1 hg
have : Nat.Partrec fun n => Nat.rfindOpt fun k => cf.evaln k n <|> cg.evaln k n :=
Partrec.nat_iff.1
(Partrec.rfindOpt <|
Primrec.option_orElse.to_comp.comp
(Code.primrec_evaln.to_comp.comp <| (snd.p... | false | {
"chosen_tokens": 350,
"rejected_tokens": 355,
"token_jaccard": 0.950617,
"token_length_ratio": 1.014286
} | markdown_preamble_trailing | preamble | d9858a395ec41d8db259717a9d6ffb63ec259591c7f15029b954ac3127578e25 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Computability.PartrecCode
Namespace:
Nat.Partrec
Local context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license ... | Here is the proof:
by
obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf
obtain ⟨cg, rfl⟩ := Code.exists_code.1 hg
have : Nat.Partrec fun n => Nat.rfindOpt fun k => cf.evaln k n <|> cg.evaln k n :=
Partrec.nat_iff.1
(Partrec.rfindOpt <|
Primrec.option_orElse.to_comp.comp
(Code.primrec_evaln.to_... | {
"accepted": false,
"all_failure_types": [
"not_exact_by_body"
],
"failure_type": "not_exact_by_body",
"rejection_layer": "strict_parser"
} | true | 1 | {
"dependency_family_id": "Computability",
"family_id": "merge",
"file_id": "mathlib/Mathlib/Computability/RE.lean",
"sample_id": "f4c589a7f4818f438b23af657c2db224655a94001f25229b505dee753deb8fe1"
} | train |
{
"chosen_proof_sha256": "507db4142d79d356282c11878756b5338f2f435787c1651165c4326a0009c857",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ec54f6d939d178c00cf294ecb4fb203e2e... | by
cases x; rw [Zsqrtd.norm, normSq]; simp | true | {
"chosen_tokens": 14,
"rejected_tokens": 3,
"token_jaccard": 0.066667,
"token_length_ratio": 0.214286
} | unknown_identifier | unknown_identifier | da399269a39da501ee63f70f7235812cefcc4cb5c6f055a8e3890b7d093cc3dc | 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.Archimedean.Real.Basic
public import Mathlib.Data.Complex.Basic
public import Mathlib.Data.Nat.Prime.Basic
public import Mathlib.NumberTheory.Zsq... | by
exact __oprover_missing_2eaa4c3a7a89 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "8edab4bc7b7599fbfb0f4dbf116b372c1d60f04dd638f87e3843688cd37edf09",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | 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": "73fbc60842c9edf00faf984e5736df9245f2ebd275179f441c2133f339aab829",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
by_cases H : ∃ s : Finset S, Nonempty (Basis s R S)
· rw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some]
· simp [trace_eq_zero_of_not_exists_basis R H, finrank_eq_zero_of_not_exists_basis_finset H] | true | {
"chosen_tokens": 44,
"rejected_tokens": 8,
"token_jaccard": 0.064516,
"token_length_ratio": 0.181818
} | unsolved_goals | unsolved_goals | da39f368ee49de12e46d00531694af219415e6aa74eb3875c9915cb7f3c5b8cf | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
public import Mathlib.LinearAlgebra.Matrix.BilinearForm
public import Mathlib.LinearAlgebra.Trace
Namespace:
Algebra
L... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "b10e727d1b894d0dc5840f098008a9a689fb8692fea504e91c338123141c6a0a",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "RingTheory/Trace",
"family_id": "trace_algebramap",
"file_id": "mathlib/Mathlib/RingTheory/Trace/Defs.lean",
"sample_id": "46dbe9e477ddbb7606c3f29bdeda37a0f7a2816233719f84ab33136a865943b2"
} | train |
{
"chosen_proof_sha256": "7727f8d1a6315f3170b243f902f0e2ff87bac72c93a1e273349cdbafdd693264",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "ca2fb92b9a27f6c7dccff0a61d2e985ccf... | by
dsimp [homEquiv]
ext j
simp [← uncurry_natural_left] | true | {
"chosen_tokens": 12,
"rejected_tokens": 3,
"token_jaccard": 0.083333,
"token_length_ratio": 0.25
} | unknown_identifier | unknown_identifier | da52b4cdaad8444c0bdc25def01d49962168f4ad613971ad6a49b091f0376dc9 | 1 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.CategoryTheory.Monoidal.Closed.Enrichment
public import Mathlib.CategoryTheory.Enriched.FunctorCategory
Namespace:
CategoryTheory.MonoidalClosed.FunctorCatego... | by
exact __oprover_missing_1205eb9e1bd5 | {
"accepted": false,
"failure_type": "unknown_identifier",
"output_sha256": "b72b318e26b4e21a4fa5bf047eb1636e011f725c15ff6bdb5bfd29d95c349169",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": false
} | true | 1 | {
"dependency_family_id": "CategoryTheory/Monoidal",
"family_id": "homequiv_naturality_two_symm",
"file_id": "mathlib/Mathlib/CategoryTheory/Monoidal/Closed/FunctorCategory/Basic.lean",
"sample_id": "1205eb9e1bd541c4f97a6b6d682d3d9f05b7c621a6ffc31f4b5fc185468fec86"
} | train |
{
"chosen_proof_sha256": "9342ddf1a3670b64a91f411236ca705a4f712a410fef1d38cc662b4389a061cb",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "704efeeee592c3ee6480336bbb69c4f75d... | by
rw [← image_commutatorSet_closureCommutatorRepresentatives G]
exact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _)) | false | {
"chosen_tokens": 24,
"rejected_tokens": 3,
"token_jaccard": 0.105263,
"token_length_ratio": 0.125
} | by_exact_placeholder | exact? | daa470165dbee0295d2c5556c9eb3b972c9bb4964873bde1292449fc02bf7054 | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Group.Subgroup.Finite
public import Mathlib.GroupTheory.Commutator.Basic
public import Mathlib.GroupTheory.Rank
public import Mathlib.GroupTheory.Index... | 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": "GroupTheory/Commutator",
"family_id": "card_commutatorset_closurecommutatorrepresentatives",
"file_id": "mathlib/Mathlib/GroupTheory/Commutator/Finite.lean",
"sample_id": "83c5bc19a8475efdecefba11a168fc6bb576240b8897c80b4ded959f3ff7248c"
} | train |
{
"chosen_proof_sha256": "50b6a5e16d3844921370e0938982952c322ae0fcb0f107c81a94eedb5977aaa4",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "0d002db9cc8388d0bb048c967b6cc0ba1b... | by
rw [← lintegral_sum_measure, h.sum_restrict_of_ac hν] | true | {
"chosen_tokens": 12,
"rejected_tokens": 8,
"token_jaccard": 0.058824,
"token_length_ratio": 0.666667
} | unsolved_goals | unsolved_goals | dac8859800f1833a7520afb0d9b70478e26970cb634411885de31c2d1dfd65f5 | 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.Group.Action
public import Mathlib.MeasureTheory.Group.Pointwise
public import Mathlib.MeasureTheory.Integral.Lebesgue.Map
public import Mathlib.... | by
have __oprover_safe_h : True := by trivial | {
"accepted": false,
"failure_type": "unsolved_goals",
"output_sha256": "e4d8209d607dde102bdf2fc0b5f89a5283191c9a08be3c51adfeeb599f60b0f3",
"rejection_layer": "lean",
"returncode": 1,
"timed_out": false,
"warning": true
} | true | 1 | {
"dependency_family_id": "MeasureTheory/Group",
"family_id": "lintegral_eq_tsum_of_ac",
"file_id": "mathlib/Mathlib/MeasureTheory/Group/FundamentalDomain.lean",
"sample_id": "70a4170ae953a085b2ee61b70a1cefea9bde7f6d81dae6658409edfa420b8c2e"
} | train |
{
"chosen_proof_sha256": "3fa4e6a02899a81962f514978e80ca0d79529b0b1a773bcbf2dd1c22e7f2b19e",
"mathlib_revision": "360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56",
"negative_generator_hash": "8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9",
"rejected_proof_sha256": "4d0c2d672852344cf952109d7ab97db759... | by
rw [Order.covBy_iff_add_one_eq, Order.covBy_iff_add_one_eq]
exact Int.natCast_inj | false | {
"chosen_tokens": 15,
"rejected_tokens": 20,
"token_jaccard": 0.6875,
"token_length_ratio": 1.333333
} | markdown_preamble_trailing | preamble | daf82d8b75c71f3e108cb7c038daf2ca7b6db7b7c8b85eb7d14c7b84b7f5bcba | 0 | Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.
Imports:
public import Mathlib.Algebra.Order.Ring.Int
public import Mathlib.Data.Nat.SuccPred
Namespace:
Int
Local context:
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Release... | Here is the proof:
by
rw [Order.covBy_iff_add_one_eq, Order.covBy_iff_add_one_eq]
exact Int.natCast_inj | {
"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/Int",
"family_id": "natcast_covby",
"file_id": "mathlib/Mathlib/Data/Int/SuccPred.lean",
"sample_id": "6402417a187a534989eab3a2c52dac736091b114295a76d321088d940f0ef62a"
} | train |
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