name stringlengths 2 347 | module stringlengths 6 90 | deps listlengths 0 692 | allowCompletion bool 2
classes |
|---|---|---|---|
CategoryTheory.Pseudofunctor.ObjectProperty.noConfusionType | Mathlib.CategoryTheory.Bicategory.Functor.Cat.ObjectProperty | [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Cat.str",
"CategoryTheory.Pseudofunctor.toPrelaxFunctor",
"CategoryTheory.Bicategory.toCategoryStruct",
"CategoryTheory.Cat",
"CategoryTheory.PrelaxFunctor.toPrelaxFunctorStruct",
"CategoryTheory.Bicategory",
"CategoryTheory.Pse... | false |
SimpleGraph.Hom.map | Mathlib.Combinatorics.SimpleGraph.Maps | [
"SimpleGraph.Adj",
"Ne",
"SimpleGraph",
"SimpleGraph.Hom",
"RelHom.mk",
"SimpleGraph.Hom.map._proof_1",
"SimpleGraph.map"
] | true |
PreTilt.instCommRing._proof_27 | Mathlib.RingTheory.Perfection | [
"PreTilt.instCommRing._proof_3",
"PreTilt.instCommRing._proof_7",
"PreTilt.instCommRing._proof_15",
"Mul.mk",
"Semigroup.toMul",
"CommRing",
"Nat.Prime",
"One.mk",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"PreTilt.instCommRing._aux_1",
"CommSemiring.toSemiring",
"Add.mk",
"Perfection.instC... | false |
Std.Tactic.BVDecide.BoolExpr.toString._unsafe_rec | Std.Tactic.BVDecide.Bitblast.BoolExpr.Basic | [
"String",
"Std.Tactic.BVDecide.BoolExpr",
"ToString.toString",
"instAppendString",
"instHAppendOfAppend",
"instToStringBool",
"Std.Tactic.BVDecide.BoolExpr.toString.match_1",
"Std.Tactic.BVDecide.BoolExpr.toString._unsafe_rec",
"ToString",
"Bool",
"Std.Tactic.BVDecide.Gate",
"HAppend.hAppend",... | false |
Int.Linear.Poly.coeff_k.eq_1 | Init.Data.Int.Linear | [
"Int.Linear.Poly.rec",
"Nat.beq",
"Int",
"Int.Linear.Var",
"instOfNat",
"Bool.rec",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Eq",
"Int.Linear.Poly",
"Int.Linear.Poly.coeff_k"
] | true |
ENat.coe_ne_top._simp_2 | Mathlib.Tactic.ENatToNat | [
"False",
"eq_false",
"ENat.instNatCast",
"instTopENat",
"Nat.cast",
"Nat",
"ENat",
"ENat.coe_ne_top",
"Top.top",
"Eq"
] | false |
CategoryTheory.Discrete.productEquiv._proof_15 | Mathlib.CategoryTheory.Discrete.SumsProducts | [
"CategoryTheory.Functor",
"CategoryTheory.Discrete.productEquiv.match_5",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.eqToHom",
"CategoryTheory.Functor.mk",
"CategoryTheory.Functor.category",
"CategoryTheory.Discrete.productEquiv.match_3",
"CategoryTheory.Functor.comp",
... | false |
_private.Init.Data.String.Lemmas.Pattern.TakeDrop.Char.0.String.Slice.startsWith_char_iff_exists_append._simp_1_2 | Init.Data.String.Lemmas.Pattern.TakeDrop.Char | [
"List.head?",
"Option.some",
"Exists",
"List.cons",
"List",
"propext",
"List.head?_eq_some_iff",
"Eq",
"Option"
] | false |
CategoryTheory.MonoidalCategory.ofTensorHom._auto_9 | Mathlib.CategoryTheory.Monoidal.Category | [
"Lean.Syntax.node",
"Array.push",
"Lean.Syntax",
"Array.empty",
"Lean.Name.mkStr2",
"Lean.SourceInfo.none",
"Lean.Name.mkStr1",
"Lean.Name.mkStr4",
"Lean.mkAtom"
] | false |
CategoryTheory.Sum.associativityFunctorEquivNaturalityFunctorIso_hom_app_snd_fst | Mathlib.CategoryTheory.Sums.Products | [
"CategoryTheory.Functor",
"CategoryTheory.sum",
"CategoryTheory.sum.associativity",
"CategoryTheory.Equivalence.prod",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Equivalence.congrLeft",
"CategoryTheory.Equivalence.refl",
"CategoryTheory.Functor.category",
"CategoryTheo... | true |
Equiv.sumEmpty | Mathlib.Logic.Equiv.Sum | [
"isEmptyElim",
"Equiv.mk",
"Sum",
"id",
"Equiv",
"Sum.inl",
"IsEmpty",
"Equiv.sumEmpty._proof_2",
"Equiv.sumEmpty._proof_1",
"Sum.elim"
] | true |
AlgebraicGeometry.IsReduced.mk | Mathlib.AlgebraicGeometry.Properties | [
"AlgebraicGeometry.Scheme",
"Opposite",
"CommRingCat.carrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"TopologicalSpace.Opens.instPartialOrder",
"CommSemiring.toSemiring",
"CommRingCat",
"PartialOrder.toPreorder",
"TopologicalSpace.Opens",
"AlgebraicGeometry.IsReduced.component_reduced._auto... | true |
_private.Lean.Meta.ExprLens.0.Lean.Core.viewBindersCoord._sparseCasesOn_2 | Lean.Meta.ExprLens | [
"Nat.ne_of_beq_eq_false",
"Lean.Expr.const",
"Lean.Expr.letE",
"Lean.Expr.mvar",
"Nat.shiftRight",
"Lean.MVarId",
"Lean.Expr",
"Lean.FVarId",
"Nat.hasNotBit",
"instOfNatNat",
"Lean.Expr.sort",
"Lean.Expr.bvar",
"Lean.Level",
"Lean.Literal",
"Lean.Expr.mdata",
"Lean.Expr.fvar",
"Lean.... | false |
HomologicalComplex.mapBifunctor₂₃.D₂ | Mathlib.Algebra.Homology.BifunctorAssociator | [
"CategoryTheory.Limits.HasZeroMorphisms",
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Functor.Additive",
"CategoryTheory.Functor.category",
"HomologicalComplex.mapBifunctorAssociatorX._proof_2",
"HomologicalComplex",
"Prod.mk",
"ComplexShape.As... | true |
_private.Mathlib.MeasureTheory.Function.SimpleFuncDenseLp.0.MeasureTheory.SimpleFunc.memLp_approxOn._simp_1_2 | Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | [
"Norm.norm",
"Real",
"SeminormedAddCommGroup",
"dist_eq_norm",
"HSub.hSub",
"AddCommGroup.toAddGroup",
"SubNegMonoid.toSub",
"SeminormedAddCommGroup.toNorm",
"instHSub",
"SeminormedAddCommGroup.toPseudoMetricSpace",
"AddGroup.toSubNegMonoid",
"SeminormedAddCommGroup.toAddCommGroup",
"Dist.di... | false |
ENNReal.HolderTriple.inv_le_inv | Mathlib.Data.ENNReal.Holder | [
"LE.le",
"Inv.inv",
"ENNReal.HolderTriple.le",
"ENNReal.inv_le_inv._simp_1",
"True",
"ENNReal.instLE",
"ENNReal",
"eq_true",
"of_eq_true",
"ENNReal.instInv",
"ENNReal.HolderTriple",
"Eq.trans"
] | true |
CovBy.coe_fin | Mathlib.Data.Nat.SuccPred | [
"CovBy",
"Fin.val",
"Nat",
"instLTFin",
"Iff.mp",
"instLTNat",
"Fin",
"Fin.covBy_iff"
] | true |
List.isSome_max?_of_mem | Init.Data.List.MinMax | [
"False",
"False.elim",
"Membership.mem",
"Max",
"Eq.mp",
"List.not_mem_nil._simp_1",
"List.max?",
"List.cons",
"Bool.true",
"List",
"List.instMembership",
"List.casesOn",
"eq_self",
"Bool",
"of_eq_true",
"Eq.ndrec",
"Eq.refl",
"Option.isSome",
"Eq.symm",
"Eq",
"List.nil"
] | true |
_private.Lean.Server.CodeActions.UnknownIdentifier.0.Lean.Server.FileWorker.handleUnknownIdentifierCodeAction.match_1 | Lean.Server.CodeActions.UnknownIdentifier | [
"Unit.unit",
"Option.casesOn",
"Option.some",
"String.Pos.Raw",
"Option.none",
"Unit",
"Option"
] | false |
instSetLikeSubRootedTreeα._proof_3 | Mathlib.Order.SuccPred.Tree | [
"Set.Ici",
"RootedTree.α",
"SubRootedTree",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"Eq.mp",
"RootedTree.semilatticeInf",
"SubRootedTree.root",
"RootedTree",
"Eq",
"_private.Mathlib.Order.SuccPred.Tree.0.instSetLikeSubRootedTreeα._simp_1",
"Eq.trans",
"_private.Mathlib.Or... | false |
IsCentralVAdd.mk._flat_ctor | Mathlib.Algebra.Group.Action.Defs | [
"IsCentralVAdd.mk",
"AddOpposite",
"VAdd",
"HVAdd.hVAdd",
"instHVAdd",
"IsCentralVAdd",
"Eq",
"AddOpposite.op"
] | false |
Lean.Elab.TermInfo.recOn | Lean.Elab.InfoTree.Types | [
"Lean.Elab.ElabInfo",
"Lean.Elab.TermInfo.rec",
"Lean.Expr",
"Lean.Elab.TermInfo",
"Bool",
"Lean.Elab.TermInfo.mk",
"Lean.LocalContext",
"Option"
] | false |
_private.Mathlib.RingTheory.Localization.NumDen.0.IsFractionRing.isUnit_den_zero._simp_1_2 | Mathlib.RingTheory.Localization.NumDen | [
"IsLocalization.IsInteger",
"CommSemiring.toSemiring",
"Algebra",
"CommSemiring",
"True",
"IsLocalization.isInteger_zero",
"eq_true",
"Zero.toOfNat0",
"OfNat.ofNat",
"Eq",
"MulZeroClass.toZero",
"instMulZeroClassOfSemiring"
] | false |
IsAlgebraic.algebraMap | Mathlib.RingTheory.Algebraic.Basic | [
"Eq.mpr",
"RingHom.instRingHomClass",
"CommRing",
"Algebra.algebraMap",
"IsAlgebraic",
"congrArg",
"CommSemiring.toSemiring",
"AlgHom",
"IsScalarTower",
"AlgHom.funLike",
"Polynomial.algebraOfAlgebra",
"Algebra",
"RingHom",
"Algebra.toSMul",
"map_zero",
"Algebra.id",
"id",
"RingHom... | true |
CategoryTheory.Precoverage.ZeroHypercover.Small.casesOn | Mathlib.CategoryTheory.Sites.Hypercover.Zero | [
"CategoryTheory.Precoverage",
"CategoryTheory.Presieve",
"CategoryTheory.Precoverage.ZeroHypercover.Small.rec",
"CategoryTheory.PreZeroHypercover.presieve₀",
"Membership.mem",
"Exists",
"CategoryTheory.PreZeroHypercover.restrictIndex",
"CategoryTheory.Precoverage.ZeroHypercover",
"CategoryTheory.Pre... | false |
CategoryTheory.Limits.epi_image_of_epi | Mathlib.CategoryTheory.Limits.Shapes.Images | [
"CategoryTheory.Limits.factorThruImage",
"CategoryTheory.Epi",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.epi_of_epi",
"CategoryTheory.Limits.image.ι",
"Eq.mp",
"CategoryTheory.Limits.image.fac",
"CategoryTheory.Limits.image",
"CategoryTheory.CategoryStru... | true |
Nat.infinite_odd_deficient | Mathlib.NumberTheory.FactorisationProperties | [
"Iff.mpr",
"Eq.mpr",
"Nat.Prime",
"Preorder.toLT",
"congrArg",
"_private.Mathlib.NumberTheory.FactorisationProperties.0.Nat.infinite_odd_deficient._proof_1_1",
"PartialOrder.toPreorder",
"Odd",
"setOf",
"Nat.instLocallyFiniteOrder",
"instInhabitedNat",
"Membership.mem",
"Exists",
"Semilatt... | true |
Ordinal.nfp_zero | Mathlib.SetTheory.Ordinal.FixedPoint | [
"id",
"Pi.instZero",
"Ordinal.nfp",
"funext",
"Ordinal.nfp_zero_left",
"Ordinal.zero",
"Zero.toOfNat0",
"OfNat.ofNat",
"Eq",
"Ordinal"
] | true |
String.eq_empty_iff_forall_eq | Init.Data.String.Lemmas.IsEmpty | [
"String.toSlice",
"Eq.mpr",
"congrArg",
"String.Pos.ofToSlice_inj._simp_1",
"String",
"String.Slice.isEmpty_iff_forall_eq",
"String.Pos.toSlice",
"Eq.mp",
"id",
"String.isEmpty_toSlice_iff",
"String.Slice.isEmpty",
"Bool.true",
"String.Pos.toSlice_inj._simp_1",
"Iff",
"String.Pos.ofToSli... | true |
Std.Internal.List.getKey?_eq_getEntry? | Std.Data.Internal.List.Associative | [
"Eq.mpr",
"Std.Internal.List.getEntry?_cons_of_true",
"congrArg",
"Option.some",
"id",
"Std.Internal.List.assoc_induction",
"Std.Internal.List.getKey?_cons_of_false",
"Sigma.fst",
"Std.Internal.List.getKey?_cons_of_true",
"List.cons",
"Bool.true",
"Option.none",
"List",
"Option.map",
"BE... | true |
Int.radical_natCast | Mathlib.RingTheory.Radical.NatInt | [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instNormalizedGCDMonoidNat",
"congrArg",
"CommSemiring.toSemiring",
"Int.euclideanDomain",
"Nat.radical_eq_prod_primeFactors",
"CommSemiring.toCommMonoidWithZero",
"PrincipalIdealRing.to_uniqueFactorizationMonoid",
"AddMonoidWithOne.toNatCast",
"Nat.inst... | true |
DilationEquiv.smulTorsor._proof_3 | Mathlib.Analysis.Normed.Affine.AddTorsor | [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"ENNReal.ofNNReal",
"instHSMul",
"SeminormedAddCommGroup",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrArg",
"CommSemiring.toSemiring",
"DistribMulAction.toDistribSMul",
"AddCommGroup.toAddCommMonoid",
"SeminormedAddGroup.toNNNorm",
... | false |
Sym.fill | Mathlib.Data.Sym.Basic | [
"Equiv.instEquivLike",
"Sym.replicate",
"HSub.hSub",
"Equiv",
"instSubNat",
"instOfNatNat",
"Sym.append",
"Fin.val",
"instHAdd",
"instHSub",
"HAdd.hAdd",
"Nat",
"Sym",
"Sym.cast",
"Sym.fill._proof_1",
"instAddNat",
"OfNat.ofNat",
"Fin",
"DFunLike.coe",
"EquivLike.toFunLike"
] | true |
Matrix.toLinearMap₂'Aux._proof_2 | Mathlib.LinearAlgebra.Matrix.SesquilinearForm | [
"Pi.Function.module",
"instHSMul",
"Semiring.toModule",
"Pi.addCommMonoid",
"HMul.hMul",
"Finset.univ",
"congrArg",
"DistribMulAction.toDistribSMul",
"Matrix",
"Finset",
"SMul",
"AddMonoid.toAddZeroClass",
"RingHom",
"Membership.mem",
"SemigroupWithZero.toSemigroup",
"AddZeroClass.toAd... | false |
FreeLieAlgebra.Rel.lie_self | Mathlib.Algebra.Lie.Free | [
"CommRing",
"HMul.hMul",
"FreeNonUnitalNonAssocAlgebra",
"CommSemiring.toSemiring",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"MonoidAlgebra.instMul",
"CommRing.toCommSemiring",
"Zero.toOfNat0",
"MonoidAlgebra.nonUnitalNonAssocSemiring",
"FreeMagma.instMul",
"FreeLieAlgebra.Rel",
"OfNat.ofNa... | true |
DirectLimit.instSemifieldOfRingHomClass._proof_3 | Mathlib.Algebra.Colimit.DirectLimit | [
"DivisionSemiring.div_eq_mul_inv",
"Semigroup.toMul",
"instHDiv",
"HMul.hMul",
"DivisionSemiring.toInv",
"Preorder.toLE",
"inferInstance",
"Semifield",
"HDiv.hDiv",
"LE.le",
"DirectLimit.instDivisionSemiring",
"DirectLimit",
"DivisionSemiring.toDiv",
"DivisionSemiring",
"Inv.inv",
"Sem... | false |
CochainComplex.mappingCone.map.congr_simp | Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated | [
"HomologicalComplex.instCategory",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"AddGroupWithOne.toAddMonoidWithOne",
"AddRightCancelSemigroup.toAddSemigroup",
"AddCancelMonoid.toAddRightCancelMonoid",
"CochainComplex.mappingCone",
"CochainComplex.instHasHomotopyCofiberOfHasBinaryBiproductX... | true |
iteratedFDerivWithin_eq_iteratedFDeriv | Mathlib.Analysis.Calculus.ContDiff.Defs | [
"ENat.natCast_ne_coe_top._simp_1",
"ContDiffAt",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"False",
"le_rfl",
"NormedSpace.toIsBoundedSMul",
"instAddMonoidWithOneENat",
"NormedSpace",
"ENat.instNatCast",
"UniqueDiffOn",
"instTopENat",
"WithTop.instPreorder",
"congrArg",
"DistribMu... | true |
Algebra.IsStandardSmooth.mk | Mathlib.RingTheory.Smooth.StandardSmooth | [
"CommRing",
"Algebra.IsStandardSmooth",
"CommSemiring.toSemiring",
"Finite",
"Algebra",
"Exists",
"And",
"CommRing.toCommSemiring",
"Algebra.SubmersivePresentation",
"Algebra.IsStandardSmooth.mk",
"Nonempty"
] | true |
CategoryTheory.shrinkYonedaEquiv_shrinkYoneda_map | Mathlib.CategoryTheory.ShrinkYoneda | [
"CategoryTheory.instSmallOppositeObjFunctorTypeYoneda",
"CategoryTheory.Functor",
"Opposite",
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Equiv.symm_apply_apply",
"CategoryTheory.LocallySmall",
"CategoryTheory.shrinkYoneda",
"CategoryTheory.Functor... | true |
skyscraperPresheafCoconeOfSpecializes | Mathlib.Topology.Sheaves.Skyscraper | [
"CategoryTheory.Functor.op",
"CategoryTheory.Functor",
"CategoryTheory.Limits.Cocone",
"Opposite",
"Specializes",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"TopologicalSpace.Opens.instPartialOrder",
"CategoryTheory.eqToHom",
"PartialOrder.toPreorder",
"CategoryTheory.Functor.catego... | true |
CategoryTheory.instInhabitedFreeMonoidalCategory.default | Mathlib.CategoryTheory.Monoidal.Free.Basic | [
"CategoryTheory.FreeMonoidalCategory",
"CategoryTheory.FreeMonoidalCategory.unit"
] | true |
TestFunction.instZero._proof_1 | Mathlib.Analysis.Distribution.TestFunction | [
"Real",
"NormedSpace",
"Real.denselyNormedField",
"ContDiff",
"SubtractionMonoid.toSubNegZeroMonoid",
"SubNegZeroMonoid.toNegZeroClass",
"SubtractionCommMonoid.toSubtractionMonoid",
"WithTop.some",
"ENat",
"Zero.toOfNat0",
"NormedAddCommGroup.toAddCommGroup",
"Real.normedField",
"DenselyNorm... | false |
CompactlySupportedContinuousMap.toReal_apply | Mathlib.Topology.ContinuousMap.CompactlySupported | [
"NNReal.instTopologicalSpace",
"Real",
"Real.instZero",
"CompactlySupportedContinuousMap",
"ContinuousMap",
"CompactlySupportedContinuousMap.toReal",
"PseudoMetricSpace.toUniformSpace",
"CompactlySupportedContinuousMap.compLeft_apply",
"NNReal",
"ContinuousMap.coeNNRealReal",
"NNReal.instZero",
... | true |
Aesop.mvarIdToSubgoal | Aesop.RuleTac.Basic | [
"Pure.pure",
"Aesop.BaseM",
"StateRefT'.instAlternativeOfMonad",
"Aesop.diffGoals",
"Lean.MVarId",
"Lean.Meta.MetaM",
"IO.RealWorld",
"Aesop.Subgoal",
"Applicative.toPure",
"Aesop.Subgoal.mk",
"Lean.Meta.instMonadMetaM",
"StateRefT'.instMonad",
"Aesop.GoalDiff",
"Aesop.BaseM.State",
"Alt... | true |
Std.HashMap.Raw.Equiv.of_forall_getKey?_unit_eq | Std.Data.HashMap.RawLemmas | [
"Std.HashMap.Raw.getKey?",
"Std.HashMap.Raw.WF.out",
"Std.HashMap.Raw.WF",
"LawfulHashable",
"Std.HashMap.Raw",
"Unit",
"EquivBEq",
"Std.HashMap.Raw.inner",
"Hashable",
"Std.HashMap.Raw.Equiv",
"BEq",
"Eq",
"Std.HashMap.Raw.Equiv.mk",
"Std.DHashMap.Raw.Equiv.of_forall_getKey?_unit_eq",
"... | true |
OpenNormalAddSubgroup.instSetLike | Mathlib.Topology.Algebra.OpenSubgroup | [
"SetLike",
"OpenNormalAddSubgroup.instSetLike._proof_1",
"OpenNormalAddSubgroup.toOpenAddSubgroup",
"SetLike.mk",
"TopologicalSpace",
"OpenAddSubgroup",
"AddGroup",
"SetLike.coe",
"OpenAddSubgroup.instSetLike",
"OpenNormalAddSubgroup"
] | true |
ContinuousAlternatingMap.toContinuousMultilinearMap_zero | Mathlib.Topology.Algebra.Module.Alternating.Basic | [
"ContinuousMultilinearMap",
"AddCommMonoid",
"TopologicalSpace",
"ContinuousAlternatingMap.toContinuousMultilinearMap",
"Semiring",
"Zero.toOfNat0",
"Module",
"OfNat.ofNat",
"ContinuousAlternatingMap.instZero",
"Eq",
"ContinuousMultilinearMap.instZero",
"ContinuousAlternatingMap",
"rfl"
] | true |
ArchimedeanClass.FiniteResidueField.instArchimedean | Mathlib.Algebra.Order.Ring.StandardPart | [
"ArchimedeanOrder.of",
"Eq.mpr",
"AddValuation.toValuation",
"instHSMul",
"IsDomain.to_noZeroDivisors",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Equiv.instEquivLike",
"CommRing.toNonUnitalCommRing",
"ArchimedeanClass.instFieldFiniteResidueField",
"outParam",
"IsOrdere... | true |
Interval.instPreorder._proof_4 | Mathlib.Order.Interval.Basic | [
"NonemptyInterval.instPreorder",
"WithBot",
"Preorder.toLE",
"WithBot.instPreorder._proof_2",
"Interval",
"LE.le",
"instLEInterval",
"Preorder",
"NonemptyInterval"
] | false |
SimplexCategory.δ₀Iter_apply | Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter | [
"CategoryTheory.ConcreteCategory.hom",
"SimplexCategory.instConcreteCategoryOrderHomFinHAddNatLenOfNat",
"PartialOrder.toPreorder",
"SimplexCategory.δ₀Iter_apply._auto_1",
"Fin.mk",
"instOfNatNat",
"Fin.val",
"autoParam",
"instHAdd",
"OrderHom.instFunLike",
"Fin.instPartialOrder",
"HAdd.hAdd",... | true |
_private.Lean.Elab.MutualInductive.0.Lean.Elab.Command.FinalizeContext.lctx | Lean.Elab.MutualInductive | [
"_private.Lean.Elab.MutualInductive.0.Lean.Elab.Command.FinalizeContext",
"Lean.LocalContext"
] | true |
FirstOrder.Language.Theory.Model.realize_of_mem | Mathlib.ModelTheory.Semantics | [
"FirstOrder.Language.Sentence.Realize",
"Membership.mem",
"FirstOrder.Language.Theory",
"FirstOrder.Language.Theory.Model",
"FirstOrder.Language.Structure",
"FirstOrder.Language",
"FirstOrder.Language.Sentence",
"Set.instMembership"
] | true |
and_imp._simp_1 | Init.SimpLemmas | [
"and_imp",
"And",
"propext",
"Eq"
] | false |
_private.Lean.Meta.DecLevel.0.Lean.Meta.decAux?.match_8 | Lean.Meta.DecLevel | [
"Unit.unit",
"Lean.Level.ctorIdx",
"Lean.Level.mvar",
"_private.Lean.Meta.DecLevel.0.Lean.Meta.decAux?._sparseCasesOn_8",
"Lean.Level.param",
"Nat.hasNotBit",
"Lean.Level",
"Unit",
"Lean.Level.zero",
"Lean.Name",
"Lean.Level.succ",
"Lean.LMVarId"
] | false |
_private.Lean.DocString.Extension.0.Lean.VersoModuleDocs.DocFrame.mk.sizeOf_spec | Lean.DocString.Extension | [
"Lean.Doc.Inline._sizeOf_inst",
"Lean.ElabInline._sizeOf_inst",
"String",
"Lean.Doc.Block",
"instSizeOfDefault",
"instOfNatNat",
"_private.Lean.DocString.Extension.0.Lean.VersoModuleDocs.DocFrame.mk",
"_private.Lean.DocString.Extension.0.Lean.VersoModuleDocs.DocFrame._sizeOf_inst",
"Array._sizeOf_in... | true |
Std.TreeMap.Raw.modify | Std.Data.TreeMap.Raw.Basic | [
"Std.TreeMap.Raw.mk",
"Ordering",
"Std.TreeMap.Raw.inner",
"Std.TreeMap.Raw",
"Std.DTreeMap.Raw.Const.modify"
] | true |
AlgebraicGeometry.Scheme.AffineZariskiSite.mem_grothendieckTopology | Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.Scheme.AffineZariskiSite",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"TopologicalSpace.Opens.instPartialOrder",
"AlgebraicGeome... | true |
List.isPrefixOfAux_toArray_succ' | Init.Data.List.ToArray | [
"Eq.mpr",
"Nat.recAux",
"Array.isPrefixOfAux.congr_simp",
"congrArg",
"List.drop_tail",
"id",
"instOfNatNat",
"LE.le",
"instLENat",
"List.tail",
"List.toArray",
"Array",
"List.isPrefixOfAux_toArray_succ._proof_2",
"List",
"instHAdd",
"List.isPrefixOfAux_toArray_succ",
"List.drop_one"... | true |
AdjoinRoot.coe_ofAlgHom | Mathlib.RingTheory.AdjoinRoot | [
"CommRing",
"AdjoinRoot",
"CommSemiring.toSemiring",
"AlgHom",
"AlgHom.funLike",
"Algebra",
"RingHom",
"Polynomial",
"CommSemiring",
"AdjoinRoot.instAlgebra",
"CommRing.toCommSemiring",
"RingHom.instFunLike",
"AdjoinRoot.instCommRing",
"AdjoinRoot.of",
"Semiring.toNonAssocSemiring",
"E... | true |
ProofWidgets.HtmlDisplayProps.mk.sizeOf_spec | ProofWidgets.Component.HtmlDisplay | [
"ProofWidgets.Html",
"instOfNatNat",
"ProofWidgets.HtmlDisplayProps.mk",
"ProofWidgets.HtmlDisplayProps",
"instHAdd",
"HAdd.hAdd",
"Nat",
"SizeOf.sizeOf",
"ProofWidgets.Html._sizeOf_inst",
"instAddNat",
"Eq.refl",
"OfNat.ofNat",
"Eq",
"ProofWidgets.HtmlDisplayProps._sizeOf_inst"
] | true |
MeasureTheory.Filtration.IsRightContinuous.RC | Mathlib.Probability.Process.Filtration | [
"MeasureTheory.Filtration.rightCont",
"MeasureTheory.Filtration.instLE",
"MeasureTheory.Filtration.IsRightContinuous",
"PartialOrder.toPreorder",
"PartialOrder",
"MeasureTheory.Filtration",
"LE.le",
"MeasurableSpace"
] | true |
LinearEquiv.domMulActCongrRight._proof_5 | Mathlib.Algebra.Module.Equiv.Basic | [
"Function.LeftInverse",
"RingHom",
"RingHomCompTriple",
"LinearMap.instAdd",
"Equiv.toFun",
"LinearMap",
"AddCommMonoid",
"RingHomInvPair",
"LinearEquiv.arrowCongrAddEquiv",
"RingHomInvPair.ids",
"LinearEquiv",
"Semiring",
"LinearEquiv.refl",
"AddEquiv.toEquiv",
"Module",
"RingHom.id",... | false |
AddMagma.FreeAddSemigroup.map.eq_1 | Mathlib.Algebra.Free | [
"Equiv.instEquivLike",
"AddMagma.FreeAddSemigroup.lift",
"AddMagma.FreeAddSemigroup",
"AddHom",
"Equiv",
"AddSemigroup.toAdd",
"AddMagma.FreeAddSemigroup.map",
"AddMagma.FreeAddSemigroup.instAddSemigroup",
"AddMagma.FreeAddSemigroup.of",
"Eq.refl",
"AddHom.comp",
"Eq",
"DFunLike.coe",
"Add... | true |
AntitoneOn.map_bddBelow | Mathlib.Order.Bounds.Image | [
"OrderDual.toDual",
"Equiv.instEquivLike",
"lowerBounds",
"Preorder.toLE",
"Function.comp",
"Equiv",
"HasSubset.Subset",
"AntitoneOn.dual_right",
"Set.instInter",
"BddAbove",
"Inter.inter",
"Set.Nonempty",
"MonotoneOn.map_bddBelow",
"Set.image",
"AntitoneOn",
"OrderDual",
"OrderDual.... | true |
Lean.Grind.Linarith.Poly.NonnegCoeffs.add | Init.Grind.Module.NatModuleNorm | [
"GE.ge",
"Int",
"Lean.Grind.Linarith.Poly.NonnegCoeffs.add",
"Lean.Grind.Linarith.Var",
"instOfNat",
"Lean.Grind.Linarith.Poly.NonnegCoeffs",
"OfNat.ofNat",
"Lean.Grind.Linarith.Poly.add",
"Int.instLEInt",
"Lean.Grind.Linarith.Poly"
] | true |
Function.Surjective.mulActionWithZero | Mathlib.Algebra.GroupWithZero.Action.Defs | [
"Function.Surjective.smulWithZero",
"ZeroHom.funLike",
"instHSMul",
"SMulWithZero.toSMulZeroClass",
"SMul",
"Function.Surjective.mulActionWithZero._proof_2",
"Function.Surjective.mulActionWithZero._proof_1",
"Function.Surjective.mulAction",
"MulActionWithZero",
"ZeroHom",
"MonoidWithZero",
"Mo... | true |
Finsupp.split_apply | Mathlib.Data.Finsupp.Basic | [
"Finsupp.instFunLike",
"Eq.mpr",
"congrArg",
"id",
"Finsupp.comapDomain_apply",
"Finsupp.split._proof_1",
"Eq.refl",
"Sigma.mk",
"Finsupp.split.eq_1",
"Finsupp.comapDomain",
"Eq",
"DFunLike.coe",
"Finsupp",
"Sigma",
"Finsupp.split",
"Zero"
] | true |
Ideal.associatesEquivIsPrincipal._proof_3 | Mathlib.RingTheory.Ideal.IsPrincipal | [
"CommRing",
"Semiring.toModule",
"CommSemiring.toSemiring",
"Subtype",
"Ideal",
"NonUnitalNonAssocSemiring.toAddCommMonoid",
"CommRing.toCommSemiring",
"NonAssocSemiring.toNonUnitalNonAssocSemiring",
"Subtype.val",
"Semiring.toNonAssocSemiring",
"Submodule.IsPrincipal",
"Subtype.property"
] | false |
LipschitzWith.mapsTo_closedBall | Mathlib.Topology.MetricSpace.Lipschitz | [
"Real",
"LipschitzWith",
"HMul.hMul",
"LipschitzWith.dist_le_mul_of_le",
"Membership.mem",
"NNReal",
"PseudoMetricSpace",
"Set.MapsTo",
"Real.instMul",
"Metric.closedBall",
"Set.instMembership",
"NNReal.toReal",
"instHMul",
"PseudoMetricSpace.toPseudoEMetricSpace",
"Set"
] | true |
Nat.Coprime.sum_divisors_mul | Mathlib.NumberTheory.ArithmeticFunction.Misc | [
"Nat.Coprime",
"Nat.instMulZeroClass",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"MulZeroClass.toMul",
"congrArg",
"instMulNat",
"instOfNatNat",
"Nat.divisors",
"Nat.instMonoidWithZero",
"MonoidWithZero.toMulZeroOneClass",
"ArithmeticFunction.isMultiplicative_sigma",
"Nat",
"congr"... | true |
Real.isTheta_exp_comp_exp_comp | Mathlib.Analysis.SpecialFunctions.Exp | [
"Real.isBigO_exp_comp_exp_comp._simp_1",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"neg_sub",
"Real.lattice",
"abs",
"congrArg",
"Real.instSub",
"PartialOrder.toPreorder",
"Asymptotics.IsBigO",
"HSub.hSub",
"Preorder.toLE",
"AddCommGroup.toAddGroup",
"SemilatticeInf.toPartialOrder",
... | true |
Std.ExtDTreeMap.union_insert_right_eq_insert_union | Std.Data.ExtDTreeMap.Lemmas | [
"Std.DTreeMap",
"Std.ExtDTreeMap.instUnionOfTransCmp",
"Std.DTreeMap.insert",
"Std.ExtDTreeMap.insert",
"Std.ExtDTreeMap.inductionOn₂",
"Ordering",
"Std.DTreeMap.union",
"Sigma.fst",
"Std.TransCmp",
"Std.ExtDTreeMap.sound",
"Std.ExtDTreeMap",
"Std.DTreeMap.union_insert_right_equiv_insert_union... | true |
_private.Mathlib.Algebra.Homology.Factorizations.CM5a.0.CochainComplex.Plus.modelCategoryQuillen.cm5a_cof.step₁.instMonoHomologyMapIntι._proof_2 | Mathlib.Algebra.Homology.Factorizations.CM5a | [
"False",
"congrArg",
"Classical.byContradiction",
"AddGroupWithOne.toAddMonoidWithOne",
"HSub.hSub",
"False.casesOn",
"Int",
"SubNegMonoid.toSub",
"AddMonoidWithOne.toOne",
"instHSub",
"AddGroup.toSubNegMonoid",
"instOfNat",
"Int.instRing",
"True",
"eq_self",
"Int.instAddGroup",
"Lea... | false |
_private.Std.Tactic.BVDecide.LRAT.Internal.CompactLRATChecker.0.Std.Tactic.BVDecide.LRAT.Internal.compactLratChecker.go.match_3.eq_1 | Std.Tactic.BVDecide.LRAT.Internal.CompactLRATChecker | [
"Std.Tactic.BVDecide.LRAT.Internal.DefaultClauseAction",
"Unit.unit",
"Std.Tactic.BVDecide.LRAT.Action.addEmpty",
"Option.some",
"Std.Tactic.BVDecide.LRAT.Action.del",
"Std.Tactic.BVDecide.LRAT.Action.addRat",
"Array",
"Option.none",
"Unit",
"Nat",
"Eq.refl",
"Std.Tactic.BVDecide.LRAT.Internal... | true |
Vector.insertIdx_eraseIdx._proof_3 | Init.Data.Vector.InsertIdx | [
"instOfNatNat",
"Nat.add_lt_add_iff_right._simp_1",
"instHAdd",
"HAdd.hAdd",
"Nat",
"LT.lt",
"True",
"eq_true",
"of_eq_true",
"instAddNat",
"instLTNat",
"OfNat.ofNat",
"Eq.trans"
] | false |
_private.Batteries.Data.List.Lemmas.0.List.getElem_idxOf_eq_idxOfNth_add._proof_1_5 | Batteries.Data.List.Lemmas | [
"List.idxsOf",
"Lean.Grind.nestedProof",
"_private.Batteries.Data.List.Lemmas.0.List.getElem_idxOf_eq_idxOfNth_add._proof_1_3",
"HSub.hSub",
"Lean.Grind.Nat.lt_eq",
"instSubNat",
"instOfNatNat",
"LE.le",
"instLENat",
"GetElem.getElem",
"List",
"instHAdd",
"instHSub",
"HAdd.hAdd",
"Nat",
... | false |
_private.Mathlib.AlgebraicGeometry.StructureSheaf.0.AlgebraicGeometry.StructureSheaf.toStalkₗ'.eq_1 | Mathlib.AlgebraicGeometry.StructureSheaf | [
"CategoryTheory.Functor",
"CommRing",
"Lattice.toSemilatticeSup",
"trivial",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"_private.Mathlib.AlgebraicGeometry.StructureSheaf.0.AlgebraicGeometry.StructureSheaf.modulePresheafStalkIso._proof_5",
"Quiver.Hom",
"CompleteLattice.toLattice",
"Mod... | true |
List.length_erase | Init.Data.List.Erase | [
"Eq.mpr",
"False",
"Decidable.casesOn",
"eq_false",
"List.any_eq_true._simp_1",
"congrArg",
"False.elim",
"HSub.hSub",
"Decidable",
"LawfulBEq",
"Membership.mem",
"Exists",
"List.erase_eq_eraseP",
"Eq.mp",
"not_true_eq_false",
"id",
"instDecidableEqBool",
"instSubNat",
"instOfNat... | true |
Nat.Partition.recOn | Mathlib.Combinatorics.Enumerative.Partition.Basic | [
"Multiset.sum",
"Nat.Partition.rec",
"Nat.Partition.mk",
"Membership.mem",
"Multiset",
"instOfNatNat",
"Multiset.instMembership",
"Nat.Partition",
"Nat",
"LT.lt",
"Nat.instAddCommMonoid",
"instLTNat",
"OfNat.ofNat",
"Eq"
] | false |
Mathlib.Meta.FunProp.instInhabitedFunPropDecl | Mathlib.Tactic.FunProp.Decl | [
"Mathlib.Meta.FunProp.FunPropDecl",
"Inhabited",
"Mathlib.Meta.FunProp.instInhabitedFunPropDecl.default",
"Inhabited.mk"
] | true |
_private.Mathlib.Tactic.ComputeAsymptotics.Multiseries.Basis.0.Tactic.ComputeAsymptotics.WellFormedBasis.eventually_pos._simp_1_2 | Mathlib.Tactic.ComputeAsymptotics.Multiseries.Basis | [
"List.Pairwise",
"Membership.mem",
"List.cons",
"List.pairwise_cons",
"List",
"And",
"List.instMembership",
"propext",
"Eq"
] | false |
_private.Mathlib.Data.List.Triplewise.0.List.triplewise_iff_getElem._proof_1_63 | Mathlib.Data.List.Triplewise | [
"Lean.Grind.CommRing.le_norm_expr",
"Lean.Grind.instOrderedRingInt",
"Lean.RArray.leaf",
"HMul.hMul",
"Int.Linear.norm_le",
"Lean.Grind.CommRing.Expr.var",
"Int.Linear.Expr.eq_of_norm_eq",
"congrArg",
"Lean.Grind.Semiring.mul_one",
"Lean.Grind.Order.le_of_eq_2_k",
"Std.IsLinearPreorder.toIsPreor... | false |
AddAction.orbitRel | Mathlib.GroupTheory.GroupAction.Defs | [
"AddAction.orbit",
"AddMonoid.toAddSemigroup",
"Membership.mem",
"Setoid",
"AddAction",
"AddAction.toAddSemigroupAction",
"AddGroup",
"AddSemigroupAction.toVAdd",
"AddGroup.toSubNegMonoid",
"Setoid.mk",
"SubNegMonoid.toAddMonoid",
"Set.instMembership",
"AddAction.orbitRel._proof_3",
"Set"
... | true |
CategoryTheory.MorphismProperty.IsStableUnderBraiding.braiding_hom_mem | Mathlib.CategoryTheory.Monoidal.Widesubcategory | [
"CategoryTheory.MorphismProperty",
"CategoryTheory.MonoidalCategory",
"CategoryTheory.MorphismProperty.IsStableUnderBraiding",
"CategoryTheory.BraidedCategory",
"CategoryTheory.MonoidalCategoryStruct.tensorObj",
"CategoryTheory.MonoidalCategory.toMonoidalCategoryStruct",
"CategoryTheory.BraidedCategory.... | true |
Lean.Name.getRoot._sunfold | Init.Meta.Defs | [
"String",
"Lean.Name.num",
"Lean.Name.str",
"Lean.Name.anonymous",
"Lean.Name.getRoot",
"Unit",
"Nat",
"Lean.Name",
"Eq",
"_private.Init.Meta.Defs.0.Lean.Name.getRoot.match_1"
] | false |
TypeVec.typevecCasesNil₂._proof_2 | Mathlib.Data.TypeVec | [
"congrArg",
"id",
"instOfNatNat",
"Fin2.elim0",
"Nat",
"TypeVec.nilFun",
"OfNat.ofNat",
"TypeVec.Arrow",
"Eq",
"Fin2"
] | false |
Std.TreeMap.mem_union_of_left | Std.Data.TreeMap.Lemmas | [
"Std.DTreeMap.mem_union_of_left",
"Std.TreeMap.inner",
"Membership.mem",
"Ordering",
"Std.TransCmp",
"Std.TreeMap.instMembership",
"Union.union",
"Std.TreeMap.instUnion",
"Std.TreeMap"
] | true |
SimpleGraph.Coloring.sumEquiv._proof_1 | Mathlib.Combinatorics.SimpleGraph.Sum | [
"congrArg",
"Sum",
"Prod.mk",
"Prod.fst",
"SimpleGraph.Coloring",
"SimpleGraph",
"SimpleGraph.Coloring.sumRight",
"True",
"eq_self",
"of_eq_true",
"SimpleGraph.Coloring.sum_sumLeft_sumRight",
"SimpleGraph.sum",
"congrFun'",
"SimpleGraph.Coloring.sumLeft",
"Prod",
"Eq",
"Prod.snd",
... | false |
FixedDetMatrices.instSMulSpecialLinearGroupFixedDetMatrix | Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | [
"CommRing",
"Matrix.SpecialLinearGroup",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"CommSemiring.toSemiring",
"Matrix",
"Matrix.instHMulOfFintypeOfMulOfAddCommMonoid",
"SMul",
"AddGroupWithOne.toAddMonoidWithOne",
"NonUnitalNonAssocRing.toNonUn... | true |
Lean.Meta.RecursorInfo | Lean.Meta.RecursorInfo | [
"Lean.Meta.RecursorInfo.mk"
] | true |
RootedTree.mk.noConfusion | Mathlib.Order.SuccPred.Tree | [
"RootedTree.noConfusion",
"PartialOrder.toPreorder",
"Preorder.toLE",
"OrderBot",
"SemilatticeInf.toPartialOrder",
"id",
"RootedTree.mk",
"RootedTree",
"IsPredArchimedean",
"SemilatticeInf",
"HEq",
"PredOrder",
"Eq"
] | false |
Hyperreal.instField._aux_70 | Mathlib.Analysis.Real.Hyperreal | [
"Real",
"Rat",
"DivisionRing.qsmul",
"Field.toDivisionRing",
"Filter.hyperfilter",
"Filter.Germ",
"Nat",
"Hyperreal",
"Real.instField",
"Filter.Germ.instDivisionRing",
"Ultrafilter.toFilter",
"instInfiniteNat"
] | false |
Std.Roi.size.eq_1 | Init.Data.Range.Polymorphic.Lemmas | [
"Std.PRange.UpwardEnumerable",
"Std.Roi.lower",
"instOfNatNat",
"Std.PRange.UpwardEnumerable.succ?",
"Unit",
"Nat",
"Std.Rxi.HasSize",
"Eq.refl",
"Std.Rxi.HasSize.size",
"Std.Roi.size",
"Std.Roc.size.match_1",
"OfNat.ofNat",
"Eq",
"Std.Roi",
"Option"
] | true |
_private.Mathlib.Computability.RegularExpressions.0.RegularExpression.matches'.match_1.eq_4 | Mathlib.Computability.RegularExpressions | [
"RegularExpression.matches'.match_1",
"RegularExpression",
"RegularExpression.char",
"RegularExpression.comp",
"RegularExpression.plus",
"RegularExpression.epsilon",
"Unit",
"RegularExpression.star",
"RegularExpression.zero",
"Eq.refl",
"Eq"
] | true |
Nat.forall_lt_succ_left'._proof_2 | Init.Data.Nat.Lemmas | [
"_private.Init.Data.Nat.Lemmas.0.Nat.forall_lt_succ_left'._proof_1",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Nat",
"LT.lt",
"Decidable.byContradiction",
"Nat.decLt",
"instAddNat",
"instLTNat",
"OfNat.ofNat",
"Not"
] | false |
AddRightCancelMonoid.add_eq_zero | Mathlib.Algebra.Group.Units.Basic | [
"AddMonoid.toAddZeroClass",
"AddRightCancelMonoid.eq_zero_of_add_left",
"AddZeroClass.toAddZero",
"AddRightCancelMonoid.eq_zero_of_add_right",
"And.casesOn",
"AddRightCancelMonoid",
"AddZero.toZero",
"instHAdd",
"And",
"Iff",
"HAdd.hAdd",
"AddRightCancelMonoid.toAddMonoid",
"And.intro",
"I... | true |
CategoryTheory.SingleFunctors.hom_ext_iff | Mathlib.CategoryTheory.Shift.SingleFunctors | [
"CategoryTheory.Functor",
"CategoryTheory.SingleFunctors.instCategory",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"HEq.refl",
"CategoryTheory.Functor.category",
"Eq.casesOn",
"CategoryTheory.SingleFunctors.functor",
"CategoryTheory.SingleFunctors.Hom.hom",
"Iff",
"CategoryTheory.Si... | true |
CategoryTheory.Adjunction.ofNatIsoLeft | Mathlib.CategoryTheory.Adjunction.Basic | [
"CategoryTheory.Functor",
"CategoryTheory.Functor.category",
"CategoryTheory.Functor.comp",
"CategoryTheory.Adjunction.ofNatIsoLeft._proof_2",
"CategoryTheory.Adjunction.unit",
"CategoryTheory.Functor.id",
"CategoryTheory.Functor.whiskerLeft",
"CategoryTheory.Iso",
"CategoryTheory.Adjunction.counit"... | true |
Lean.Elab.CommandContextInfo.ctorIdx | Lean.Elab.InfoTree.Types | [
"Lean.Elab.CommandContextInfo",
"Nat"
] | false |
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