name stringlengths 2 347 | module stringlengths 6 90 | deps listlengths 0 692 | allowCompletion bool 2
classes |
|---|---|---|---|
StarRingEquiv.refl_symm | Mathlib.Algebra.Star.StarRingHom | [
"Mul",
"StarRingEquiv.symm",
"StarRingEquiv.refl",
"Star",
"Eq",
"StarRingEquiv",
"Add",
"rfl"
] | true |
Topology.RelCWComplex.FiniteType.mk | Mathlib.Topology.CWComplex.Classical.Finite | [
"Topology.RelCWComplex",
"Finite",
"Topology.RelCWComplex.FiniteType.mk",
"TopologicalSpace",
"Nat",
"Topology.RelCWComplex.cell",
"Topology.RelCWComplex.FiniteType",
"Set"
] | true |
groupHomology.single_mem_cycles₁_iff | Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | [
"Submodule",
"Rep.V",
"CommRing",
"Representation",
"MonoidHom.instFunLike",
"Finsupp.module",
"DivInvOneMonoid.toInvOneClass",
"SemilinearMapClass.distribMulActionSemiHomClass",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"Finsupp.sum_single_index",
"AddCommGroup.toAddCom... | true |
Set.vadd_set_mono | Mathlib.Algebra.Group.Pointwise.Set.Scalar | [
"Set.image_mono",
"Set.vaddSet",
"VAdd",
"HasSubset.Subset",
"HVAdd.hVAdd",
"instHVAdd",
"Set.instHasSubset",
"Set"
] | true |
Subgroup.quotientEquivProdOfLE_symm_apply | Mathlib.GroupTheory.Coset.Basic | [
"Subgroup.subgroupOf",
"Equiv.instEquivLike",
"HMul.hMul",
"Monoid.toMulOneClass",
"PartialOrder.toPreorder",
"Group",
"Preorder.toLE",
"Membership.mem",
"QuotientGroup.leftRel",
"Equiv",
"MulOne.toMul",
"Subtype",
"QuotientGroup.instHasQuotientSubgroup",
"Quotient.out",
"DivInvMonoid.to... | true |
_private.Mathlib.GroupTheory.Nilpotent.0.Subgroup.ascending_central_series_le_upper.match_1_1 | Mathlib.GroupTheory.Nilpotent | [
"Unit.unit",
"instOfNatNat",
"Unit",
"Nat",
"OfNat.ofNat",
"Nat.succ",
"Nat.casesOn"
] | false |
Lean.Meta.instBEqInfoCacheKey.beq | Lean.Meta.Basic | [
"Option.instBEq",
"_private.Lean.Meta.Basic.0.Lean.Meta.instBEqInfoCacheKey.beq.match_1",
"UInt64",
"Lean.Expr.instBEq",
"Lean.Meta.InfoCacheKey",
"Lean.Expr",
"Bool.and",
"instBEqOfDecidableEq",
"BEq.beq",
"Nat",
"instDecidableEqUInt64",
"Bool",
"instDecidableEqNat",
"Bool.false",
"Opti... | true |
Nat.Partrec.Code.evaln._unary._proof_1 | Mathlib.Computability.PartrecCode | [
"invImage",
"Nat.Partrec.Code._sizeOf_inst",
"Nat.Partrec.Code",
"PSigma.casesOn",
"Prod.mk",
"sizeOfWFRel",
"Lean.opaqueId",
"Nat",
"instSizeOfNat",
"Prod.instWellFoundedRelation",
"Prod",
"WellFounded",
"PSigma"
] | false |
NonUnitalSubring.prodEquiv._proof_2 | Mathlib.RingTheory.NonUnitalSubring.Basic | [
"Set.instSProd",
"NonUnitalNonAssocRing",
"NonUnitalSubring.instSetLike",
"HMul.hMul",
"SProd.sprod",
"NonUnitalSubring.prod",
"NonUnitalSubring",
"Membership.mem",
"Set.Elem",
"Subtype",
"NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring",
"NonUnitalSubring.prodEquiv._proof_1",
"Equiv.toFun... | false |
Mathlib.Tactic.Order.AtomicFact.isSup.inj | Mathlib.Tactic.Order.CollectFacts | [
"And",
"Mathlib.Tactic.Order.AtomicFact.isSup.noConfusion",
"Mathlib.Tactic.Order.AtomicFact.isSup",
"Nat",
"And.intro",
"Mathlib.Tactic.Order.AtomicFact",
"Eq"
] | true |
HahnSeries.recOn | Mathlib.RingTheory.HahnSeries.Basic | [
"Set.IsPWO",
"PartialOrder.toPreorder",
"PartialOrder",
"HahnSeries.rec",
"HahnSeries.mk",
"Function.support",
"HahnSeries",
"Zero"
] | false |
Lean.Parser.Command.declaration._regBuiltin.Lean.Parser.Command.example.parenthesizer_289 | Lean.Parser.Command | [
"Lean.Name.mkStr5",
"IO",
"Lean.PrettyPrinter.Parenthesizer",
"Lean.PrettyPrinter.parenthesizerAttribute",
"Unit",
"Lean.Name.mkStr4",
"Lean.Parser.Command.example.parenthesizer",
"Lean.KeyedDeclsAttribute.addBuiltin"
] | false |
Finset.image_add_left_Ioc | Mathlib.Algebra.Order.Interval.Finset.Basic | [
"addLeftEmbedding",
"Eq.mpr",
"addLeftEmbedding.eq_1",
"congrArg",
"instIsLeftCancelAddOfAddLeftReflectLE",
"Finset",
"add_right_injective",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Finset.map",
"Function.Embedding.mk",
"PartialOrder",
"id",
"AddCommMonoid",
"Function.Embedding",
"... | true |
Topology.IsQuotientMap.of_comp_iff | Mathlib.Topology.Maps.Basic | [
"Eq.mpr",
"congrArg",
"Topology.IsQuotientMap.surjective",
"Topology.IsCoinducing.of_comp_iff",
"Iff.rfl",
"Topology.IsCoinducing",
"Function.comp",
"Topology.isQuotientMap_iff",
"id",
"TopologicalSpace",
"Function.Surjective.of_comp_iff",
"And",
"Iff",
"Topology.IsQuotientMap.isCoinducing... | true |
_private.Mathlib.Topology.Algebra.AsymptoticCone.0.Convex.smul_vadd_mem_of_mem_nhds_of_mem_asymptoticCone.match_1_1 | Mathlib.Topology.Algebra.AsymptoticCone | [
"Prod.mk",
"Prod",
"Prod.casesOn"
] | false |
CategoryTheory.instIsCardinalFilteredToTypeOrd | Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | [
"Mathlib.Tactic.Push.not_forall_eq",
"Cardinal.mk_le_of_surjective",
"Mathlib.Tactic.Push.not_exists._simp_1",
"Set.mem_range_self",
"Eq.mpr",
"Preorder.toLT",
"CategoryTheory.IsCardinalFiltered",
"Cardinal.IsRegular",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Subtype.casesOn",
"Pr... | true |
Homeomorph.coe_prodComm | Mathlib.Topology.Constructions.SumProd | [
"instTopologicalSpaceProd",
"Homeomorph.prodComm",
"TopologicalSpace",
"Homeomorph.instEquivLike",
"Homeomorph",
"Prod.swap",
"Prod",
"Eq",
"DFunLike.coe",
"rfl",
"EquivLike.toFunLike"
] | true |
QuotSMulTop.equivQuotTensor._proof_1 | Mathlib.RingTheory.QuotSMulTop | [
"CommRing",
"instSMulOfMul",
"Monoid.toMulOneClass",
"CommSemiring.toSemiring",
"AddCommGroup.toAddCommMonoid",
"IsScalarTower",
"AddCommGroup",
"MulOne.toMul",
"IsScalarTower.left",
"MulOneClass.toMulOne",
"CommRing.toCommSemiring",
"Semiring.toMonoid",
"Monoid.toSemigroup",
"Module.toDis... | false |
lightProfiniteToLightCondSetIsoTopCatToLightCondSet._proof_1 | Mathlib.Condensed.Light.Functors | [
"LightCondensed._proof_1",
"CategoryTheory.Functor",
"Continuous",
"Opposite",
"LightProfinite.toTopCat",
"CategoryTheory.ConcreteCategory.hom",
"TopCat.instCategory",
"SecondCountableTopology",
"ContinuousMap",
"CategoryTheory.Functor.category",
"ContinuousMap.instContinuousMapClass",
"Totall... | false |
_private.Lean.Elab.DocString.Builtin.Keywords.0.Lean.Doc.kindHasAtoms | Lean.Elab.DocString.Builtin.Keywords | [
"Pure.pure",
"Lean.instMonadEnvOfMonadLift",
"_private.Lean.Elab.DocString.Builtin.Keywords.0.Lean.Doc.containsAtom'.attempt.match_1",
"Lean.ConstantInfo.type",
"_private.Lean.Elab.DocString.Builtin.Keywords.0.Lean.Doc.namedParserHasAtomPrefix",
"Lean.Elab.Term.instMonadTermElabM",
"ReaderT",
"Lean.Ex... | true |
CategoryTheory.Bicategory.Adjunction.homEquiv₁._proof_5 | Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | [
"Trans.trans",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.BicategoricalCoherence.whiskerLeft",
"CategoryTheory.Bicategory.Adjunction.homEquiv₁._proof_3",
"CategoryTheory.CategoryStruct.id",
"CategoryTheory.Bicategory.whiskerRight",
"CategoryTheory.Bicategory.toCategoryStru... | false |
CategoryTheory.GrothendieckTopology.pointBotFunctor | Mathlib.CategoryTheory.Sites.Point.Presheaf | [
"CategoryTheory.Functor.flip",
"CategoryTheory.Functor",
"Lattice.toSemilatticeSup",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CompleteLattice.toLattice",
"CategoryTheory.LocallySmall",
"Quiver.Hom.op",
"CategoryTheory.GrothendieckTopology.Point",
"CategoryTheory.shrin... | true |
Set.BijOn.subset_left | Mathlib.Data.Set.Function | [
"HasSubset.Subset",
"Set.InjOn.bijOn_image",
"Set.BijOn.injOn",
"Set.BijOn",
"Set.InjOn.mono",
"Set.image",
"Set.instHasSubset",
"Set"
] | true |
hasDerivWithinAt_iff_isLittleO | Mathlib.Analysis.Calculus.Deriv.Basic | [
"NormedCommRing.toNormedRing",
"NormedCommRing.toSeminormedCommRing",
"instHSMul",
"NormedRing.toRing",
"NormedSpace.toIsBoundedSMul",
"NormedSpace",
"AddGroupWithOne.toAddGroup",
"DistribMulAction.toDistribSMul",
"AddCommGroup.toAddCommMonoid",
"nhdsWithin",
"NormedSpace.toModule",
"AddMonoid... | true |
CompHaus.limitConeIsLimit._proof_2 | Mathlib.Topology.Category.CompHaus.Basic | [
"Eq.mpr",
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor",
"CategoryTheory.Limits.Cone",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"TopCat.instCategory",
"CategoryTheory.Functor.category",
"CategoryTheory.Functor.comp",
"id",
"CategoryTheory.SmallCategory",
... | false |
nnnorm_ofDual | Mathlib.Analysis.Normed.Group.Constructions | [
"Equiv.instEquivLike",
"OrderDual.ofDual",
"NNNorm.nnnorm",
"OrderDual.toNNNorm",
"NNNorm",
"Equiv",
"NNReal",
"OrderDual",
"Eq",
"DFunLike.coe",
"rfl",
"EquivLike.toFunLike"
] | true |
_private.Std.Data.DHashMap.Internal.Model.0.Std.DHashMap.Internal.exists_bucket_of_uset._proof_1_5 | Std.Data.DHashMap.Internal.Model | [
"False",
"Lean.Omega.Constraint.not_sat'_of_isImpossible",
"Int.natCast_add",
"of_decide_eq_true",
"le_of_le_of_eq",
"Lean.Omega.Constraint.mk",
"Lean.Omega.Constraint.combine_sat'",
"Std.DHashMap.Internal.mkIdx",
"HSub.hSub",
"Lean.Omega.Int.add_congr",
"Lean.Omega.LinearCombo.eval",
"Option.... | false |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.isEmpty_filter_iff._simp_1_1 | Std.Data.DTreeMap.Internal.Lemmas | [
"Ord",
"Ordering",
"Ordering.eq",
"Ord.compare",
"Bool.true",
"BEq.beq",
"propext",
"Bool",
"Std.LawfulBEqOrd",
"BEq",
"Eq",
"Std.LawfulBEqOrd.compare_eq_iff_beq"
] | false |
HurwitzZeta.hurwitzOddFEPair._proof_9 | Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real",
"HMul.hMul",
"Real.lattice",
"Complex.instNormedAddCommGroup",
"Real.instZero",
"abs",
"congrArg",
"sub_zero",
"Asymptotics.IsBigO",
"Complex.instNormedField",
"HSub.hSub",
"Complex.instZero",
"Real... | false |
Fintype.coe_finsetEquivSet | Mathlib.Data.Fintype.Sets | [
"Equiv.instEquivLike",
"Finset",
"Fintype.finsetEquivSet",
"Equiv",
"Fintype",
"SetLike.coe",
"Finset.instSetLike",
"Eq",
"DFunLike.coe",
"rfl",
"EquivLike.toFunLike",
"Set"
] | true |
SBtw.recOn | Mathlib.Order.Circular | [
"SBtw.rec",
"SBtw.mk",
"SBtw"
] | false |
_private.Mathlib.Order.Interval.Finset.Nat.0.Nat.Ioc_succ_singleton._proof_1_1 | Mathlib.Order.Interval.Finset.Nat | [
"Lean.Grind.of_eq_eq_true",
"Finset.mem_singleton",
"of_eq_false",
"Lean.Grind.CommRing.le_norm_expr",
"Lean.Grind.instOrderedRingInt",
"Lean.RArray.leaf",
"False",
"Lean.Grind.and_eq_of_eq_true_right",
"Preorder.toLT",
"Lean.Grind.not_not",
"eq_false",
"Lean.Grind.iff_eq",
"Lean.Grind.CommR... | false |
_private.Init.Data.Array.Lemmas.0.Array.size_reverse.go | Init.Data.Array.Lemmas | [
"Array",
"_private.Init.Data.Array.Lemmas.0.Array.size_reverse.go._unary",
"Array.reverse.loop",
"Nat",
"PSigma.mk",
"Fin",
"PSigma",
"Eq",
"Array.size"
] | true |
_private.Mathlib.Topology.Instances.AddCircle.Defs.0.AddCircle.liftIoc_eq_liftIco._simp_1_1 | Mathlib.Topology.Instances.AddCircle.Defs | [
"Set.mem_image",
"Membership.mem",
"Exists",
"And",
"propext",
"Set.image",
"Eq",
"Set.instMembership",
"Set"
] | false |
ContinuousAffineMap.instMetricSpace._proof_1 | Mathlib.Analysis.Normed.Affine.ContinuousAffineMap | [
"PseudoMetricSpace.toUniformSpace",
"AddCommGroup.toAddGroup",
"SeminormedAddCommGroup.toPseudoMetricSpace",
"SeminormedAddCommGroup.toAddCommGroup",
"SeminormedAddCommGroup.toIsTopologicalAddGroup",
"NormedAddCommGroup.toSeminormedAddCommGroup",
"UniformSpace.toTopologicalSpace",
"NormedAddCommGroup"... | false |
CategoryTheory.Functor.ranges_directed | Mathlib.CategoryTheory.Filtered.Basic | [
"PSigma.snd",
"Set.range_comp_subset_range",
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"HEq.refl",
"Directed",
"TypeCat.instFunLikeFun",
"Function.comp",
"Exists",
"Eq.rec",
"_private.Mat... | true |
Std.DTreeMap.Internal.Impl.modify._f | Std.Data.DTreeMap.Internal.Operations | [
"Ordering.gt",
"Ord",
"congrArg",
"cast",
"Ordering",
"Ordering.eq",
"Std.DTreeMap.Internal.Impl.inner",
"Std.DTreeMap.Internal.Impl.minView!.match_1",
"Ord.compare",
"Unit",
"Std.DTreeMap.Internal.Impl.below",
"Std.DTreeMap.Internal.Impl.leaf",
"Ordering.lt",
"Nat",
"Std.DTreeMap.Intern... | false |
_private.Mathlib.MeasureTheory.Function.AEMeasurableOrder.0.MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets._proof_1_1 | Mathlib.MeasureTheory.Function.AEMeasurableOrder | [
"False",
"Lean.Grind.and_eq_of_eq_true_right",
"Set.Ioi",
"MeasureTheory.Measure",
"Preorder.toLT",
"ChainCompletePartialOrder.instOfCompleteLattice",
"MeasurableSet",
"eq_false",
"Lean.Grind.iff_eq",
"ENNReal.instAddCommMonoid",
"congrArg",
"Set.mem_inter_iff",
"CompletelyDistribLattice.toC... | false |
List.Cursor.tail.eq_1 | Std.Do.Triple.SpecLemmas | [
"instOfNatNat",
"List.Cursor.suffix",
"List.Cursor.tail._proof_3",
"List.Cursor.mk",
"List.tail",
"List.cons",
"instHAppendOfAppend",
"List",
"Nat",
"List.Cursor.current",
"LT.lt",
"List.Cursor",
"Eq.refl",
"List.Cursor.tail",
"instLTNat",
"List.instAppend",
"OfNat.ofNat",
"Eq",
... | true |
CategoryTheory.MonoidalClosed.whiskerLeft_curry'_ihom_ev_app | Mathlib.CategoryTheory.Monoidal.Closed.Basic | [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.MonoidalCategoryStruct.rightUnitor",
"CategoryTheory.MonoidalCategory",
"CategoryTheory.Functor.comp",
"CategoryTheory.Functor.id",
"id",
"CategoryTheory.Monoida... | true |
PosNum.commMonoid._proof_6 | Mathlib.Data.Num.Lemmas | [
"Eq.mpr",
"castPosNum",
"HMul.hMul",
"Nat.instOne",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"id",
"PosNum.to_nat_inj",
"CommMagma.toMul",
"instMulNat",
"NonUnitalCommSemiring.toNonUnitalNonAssocCommSemiring",
"PosNum.mul_to_nat",
"Nat",
"NonUnitalNonAssocCommSemiring.toCommMagm... | false |
Std.ExtHashMap.getD_diff_of_not_mem_left | Std.Data.ExtHashMap.Lemmas | [
"Std.ExtHashMap.getD",
"Std.ExtHashMap.inner",
"Membership.mem",
"Std.ExtHashMap.instSDiffOfEquivBEqOfLawfulHashable",
"LawfulHashable",
"Std.ExtHashMap",
"SDiff.sdiff",
"EquivBEq",
"Std.ExtDHashMap.Const.getD_diff_of_not_mem_left",
"Hashable",
"Std.ExtHashMap.instMembershipOfEquivBEqOfLawfulHas... | true |
Std.DTreeMap.Raw.maxKeyD_insert | Std.Data.DTreeMap.Raw.Lemmas | [
"Std.DTreeMap.Raw.maxKey?",
"Std.DTreeMap.Internal.Impl.maxKeyD_insert!",
"Std.DTreeMap.Raw.maxKeyD",
"Ord.mk",
"Ordering",
"instDecidableEqBool",
"Std.TransCmp",
"Bool.true",
"Std.DTreeMap.Raw.WF.out",
"Std.DTreeMap.Raw.inner",
"Bool",
"Std.DTreeMap.Raw.insert",
"Ordering.isLE",
"Option.e... | true |
_private.Mathlib.LinearAlgebra.Matrix.Nondegenerate.0.LinearIndependent.sum_smul_of_nondegenerate._simp_1_3 | Mathlib.LinearAlgebra.Matrix.Nondegenerate | [
"instHSMul",
"smul_assoc",
"IsScalarTower",
"SMul",
"HSMul.hSMul",
"Eq.symm",
"Eq"
] | false |
AlgebraicGeometry.Scheme.Cover.pushforwardIso | Mathlib.AlgebraicGeometry.Cover.MorphismProperty | [
"CategoryTheory.MorphismProperty",
"AlgebraicGeometry.Scheme",
"CategoryTheory.PreZeroHypercover.f",
"CategoryTheory.IsIso",
"AlgebraicGeometry.Scheme.coverOfIsIso",
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Equiv.trans",
"CategoryTheory.MorphismProperty.IsSta... | true |
CategoryTheory.Oplax.LaxTrans.Modification.instInhabited | Mathlib.CategoryTheory.Bicategory.Modification.Oplax | [
"CategoryTheory.Oplax.LaxTrans.Modification",
"CategoryTheory.Oplax.LaxTrans.instCategoryStructOplaxFunctor",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Bicategory",
"CategoryTheory.Oplax.LaxTrans.Modification.id",
"CategoryTheory.OplaxFunctor",
"Inhabited",
"Inhabited.m... | true |
Set.Pairwise.imp_on | Mathlib.Logic.Pairwise | [
"Membership.mem",
"Ne",
"Set.Pairwise",
"Set.instMembership",
"Set"
] | true |
_private.Mathlib.Data.List.Defs.0.List.map₂Left'.match_1.eq_1 | Mathlib.Data.List.Defs | [
"List.map₂Left'.match_1",
"List.cons",
"List",
"Eq.refl",
"Eq",
"List.nil"
] | true |
MDifferentiableOn.congr_mono | Mathlib.Geometry.Manifold.MFDeriv.Basic | [
"NormedSpace",
"Membership.mem",
"HasSubset.Subset",
"MDifferentiableOn",
"TopologicalSpace",
"ModelWithCorners",
"NontriviallyNormedField",
"NontriviallyNormedField.toNormedField",
"MDifferentiableWithinAt.congr_mono",
"NormedAddCommGroup.toSeminormedAddCommGroup",
"Eq",
"Set.instMembership",... | true |
_private.Mathlib.Order.Category.PartOrdEmb.0.PartOrdEmb.Hom.ext.match_1 | Mathlib.Order.Category.PartOrdEmb | [
"PartOrdEmb.str",
"PartialOrder.toPreorder",
"Preorder.toLE",
"_private.Mathlib.Order.Category.PartOrdEmb.0.PartOrdEmb.Hom.mk",
"PartOrdEmb.carrier",
"PartOrdEmb.Hom.casesOn",
"OrderEmbedding",
"PartOrdEmb",
"PartOrdEmb.Hom"
] | false |
CategoryTheory.Functor.coconeTypesEquiv_symm_apply_ι | Mathlib.CategoryTheory.Limits.Types.Colimits | [
"CategoryTheory.Functor",
"CategoryTheory.Limits.Cocone",
"Equiv.instEquivLike",
"CategoryTheory.ConcreteCategory.hom",
"CategoryTheory.Functor.category",
"TypeCat.instFunLikeFun",
"Equiv",
"CategoryTheory.Functor.CoconeTypes.ι",
"CategoryTheory.Limits.Cocone.ι",
"CategoryTheory.Functor.coconeType... | true |
InfiniteGalois.normalAutEquivQuotient._proof_3 | Mathlib.FieldTheory.Galois.Infinite | [
"IsGalois.to_normal",
"ClosedSubgroup.toSubgroup",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
"CommRing.toNonUnitalCommRing",
"IntermediateField",
"IntermediateField.toField",
"Algebra",
"Normal",
"Membership.mem",
"IntermediateField.fixedField",
"Algebra.toModule",
"S... | false |
_private.Mathlib.Analysis.SpecialFunctions.Artanh.0.Real.artanh_eq_half_log._proof_1_2 | Mathlib.Analysis.SpecialFunctions.Artanh | [
"instLawfulOrderLT_mathlib",
"GroupWithZero.toMonoidWithZero",
"Lean.Grind.CommRing.le_norm_expr",
"Lean.RArray.leaf",
"Lean.Grind.Field.toCommRing",
"False",
"Lean.Grind.and_eq_of_eq_true_right",
"Real.partialOrder",
"Real.instLE",
"Real",
"Lean.Grind.iff_eq",
"Lean.Grind.CommRing.Expr.var",
... | false |
ModelWithCorners.compl_boundary | Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | [
"Eq.mpr",
"NormedSpace",
"compl_compl",
"congrArg",
"Compl.compl",
"ModelWithCorners.interior",
"BooleanAlgebra.toCompl",
"id",
"ModelWithCorners.compl_interior",
"Set.instCompl",
"TopologicalSpace",
"ModelWithCorners",
"ModelWithCorners.boundary",
"Set.instBooleanAlgebra",
"Nontrivially... | true |
Std.Rii.mk.sizeOf_spec | Init.Data.Range.Polymorphic.PRange | [
"Std.Rii._sizeOf_inst",
"Std.Rii.mk",
"instOfNatNat",
"Nat",
"SizeOf.sizeOf",
"Eq.refl",
"SizeOf",
"OfNat.ofNat",
"Std.Rii",
"Eq"
] | true |
AddCommMonCat.coyonedaObjIsoForget._proof_1 | Mathlib.Algebra.Category.MonCat.ForgetCorepresentable | [
"ULift.addZeroClass",
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.coyoneda",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"AddCommMonCat.instCategory",
"Equiv.trans",
"AddMonoid.toAddZeroClass",
"CategoryTheory.Functor.category",
"ULift",
"Nat.instAddMonoid",
"AddCommMon... | false |
CharTwo.add_eq_iff_eq_add | Mathlib.Algebra.CharP.Two | [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"Iff.rfl",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
"HSub.hSub",
"AddZeroClass.toAddZero",
"id",
"Distrib.toAdd",
"instOfNatNat",
"instDistribOfSemiring",
"SubNegMonoid.toSub",
"instHAdd",
"Iff",
"instHSub",
... | true |
WeierstrassCurve.Jacobian.dblU_ne_zero_of_Y_eq | Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | [
"Finsupp.instAddZeroClass",
"WeierstrassCurve.Jacobian.nonsingular_iff_of_Y_eq_negY",
"Nat.instMulZeroClass",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"WeierstrassCurve.Jacobian.polynomialX",
"WeierstrassCurve.Jacobian.dblU",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"WeierstrassCur... | true |
MeasureTheory.condExp_stronglyMeasurable_simpleFunc_mul | Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | [
"MeasureTheory.ae",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Algebra.to_smulCommClass",
"MeasureTheory.Measure",
"MeasurableSpace.instLE",
"HMul.hMul",
"IsScalarTower.right",
"Real.denselyNormedField",
"Real.instRCLike",
"CommSemiring.toSemiring",
"P... | true |
CategoryTheory.Presieve.map.casesOn | Mathlib.CategoryTheory.Sites.Sieves | [
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Presieve",
"CategoryTheory.Functor.map",
"CategoryTheory.Presieve.map",
"CategoryTheory.Category.toCategoryStruct",
"CategoryTheory.Presieve.map.rec",
"CategoryTheory.Presieve.map.of",
"CategoryTheory... | false |
_private.Mathlib.Topology.Algebra.InfiniteSum.ENNReal.0.Summable.toNNReal._simp_1_1 | Mathlib.Topology.Algebra.InfiniteSum.ENNReal | [
"Lattice",
"le_abs_self",
"abs",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"LE.le",
"AddGroup",
"True",
"eq_true",
"Eq",
"Lattice.toSemilatticeInf"
] | false |
IsLocalization.map_smul | Mathlib.RingTheory.Localization.Defs | [
"Eq.mpr",
"RingHom.instRingHomClass",
"instHSMul",
"IsLocalization.map",
"HMul.hMul",
"IsLocalization",
"Algebra.algebraMap",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"Algebra",
"Preorder.toLE",
"RingHom",
"Algebra.toSMul",
"IsLocalization.map_eq",
"id",
"Rin... | true |
Lean.AttributeImpl._sizeOf_inst | Lean.Attributes | [
"Lean.AttributeImpl",
"Lean.AttributeImpl._sizeOf_1",
"SizeOf.mk",
"SizeOf"
] | false |
_private.Mathlib.Analysis.Complex.Order.0.Mathlib.Meta.Positivity.evalComplexOfReal.match_7 | Mathlib.Analysis.Complex.Order | [
"Lean.Level.ctorIdx",
"Mathlib.Meta.Positivity.Strictness",
"Lean.Expr.const",
"Lean.Meta.MetaM",
"Nat.hasNotBit",
"Lean.Expr.sort",
"Lean.Level",
"_private.Mathlib.Analysis.Complex.Order.0.Mathlib.Meta.Positivity.evalComplexOfReal._sparseCasesOn_1",
"List.cons",
"Qq.Quoted",
"Lean.Level.zero",
... | false |
MulHom.prodMap_def | Mathlib.Algebra.Group.Prod | [
"MulHom.fst",
"MulHom",
"Prod.instMul",
"Mul",
"MulHom.comp",
"MulHom.snd",
"MulHom.prod",
"Prod",
"Eq",
"rfl",
"MulHom.prodMap"
] | true |
TopologicalSpace.Opens.inclusion'_top_functor | Mathlib.Topology.Category.TopCat.Opens | [
"Set.ext",
"CategoryTheory.Functor",
"Lattice.toSemilatticeSup",
"trivial",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CompleteLattice.toLattice",
"CategoryTheory.Functor.ext",
"TopologicalSpace.Opens.instCompleteLattice",
"TopologicalSpace.Opens.instPartialOrder",
"CategoryTheory.... | true |
Real.two_mul_sin_mul_sin | Mathlib.Analysis.Complex.Trigonometric | [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Real",
"Mathlib.Tactic.Ring.Common.mul_congr",
"HMul.hMul",
"Mathlib.Tactic.Ring.Common.add_pf_add_overlap",
"Nat.rawCast",
"Real.cos",
"congrArg",
"CommSemiring.to... | true |
_private.Mathlib.SetTheory.Ordinal.Notation.0.ONote.repr_opow_aux₂._simp_1_7 | Mathlib.SetTheory.Ordinal.Notation | [
"Preorder.toLT",
"Ordinal.partialOrder",
"PartialOrder.toPreorder",
"ONote.lt_def",
"ONote.repr",
"LT.lt",
"propext",
"ONote.instPreorder",
"Eq",
"ONote",
"Ordinal"
] | false |
Subgroup.instHasDetOneMapSpecialLinearGroupGeneralLinearGroupMapGL | Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | [
"CommRing",
"Subgroup.HasDetOne",
"Matrix.SpecialLinearGroup",
"MonoidHom.instFunLike",
"MonoidHom",
"Subgroup.map",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"Matrix",
"Algebra",
"Membership.mem",
"Units",
"Matrix.SpecialLinearGroup.instGroup",
"DivInvMonoid.toMonoi... | true |
Std.Sat.AIG.unsat_relabel | Std.Sat.AIG.Relabel | [
"Std.Sat.AIG.Decl",
"Std.Sat.AIG.Ref.mk",
"congrArg",
"Std.Sat.AIG.denote_relabel._proof_1",
"Function.comp",
"Std.Sat.AIG.unsat_relabel._proof_1",
"Std.Sat.AIG.denote_relabel",
"Std.Sat.AIG.UnsatAt",
"Std.Sat.AIG",
"Nat",
"LT.lt",
"True",
"eq_self",
"Bool",
"of_eq_true",
"congrFun'",
... | true |
retractionKerCotangentToTensorEquivSection._proof_18 | Mathlib.RingTheory.Smooth.Kaehler | [
"LinearMap.id",
"Ideal.Quotient.commSemiring",
"Eq.mpr",
"Submodule",
"RingHom.instRingHomClass",
"IsScalarTower.to_smulCommClass'",
"retractionKerCotangentToTensorEquivSection._proof_5",
"tensorKaehlerQuotKerSqEquiv._proof_3",
"tensorKaehlerQuotKerSqEquiv._proof_13",
"Ideal.Algebra.kerSquareLift"... | false |
Matrix.instLieRingToLieAlgebra._proof_7 | Mathlib.Algebra.Lie.SerreConstruction | [
"LieAlgebra.toModule",
"CommRing",
"FreeLieAlgebra.instLieRing",
"LieRing.toAddCommGroup",
"Matrix",
"Add.mk",
"LieSubmodule.toSubmodule",
"Int",
"CartanMatrix.Relations.toIdeal",
"AddSemigroup.mk",
"CartanMatrix.Generators",
"instHAdd",
"AddSemigroup.toAdd",
"Matrix.instLieRingToLieAlgebr... | false |
WittVector.frobenius_zmodp | Mathlib.RingTheory.WittVector.Frobenius | [
"Nat.Prime",
"_private.Mathlib.RingTheory.WittVector.Frobenius.0.WittVector.frobenius._simp_1",
"ZMod.commRing",
"congrArg",
"WittVector.instCommRing",
"CommSemiring.toSemiring",
"WittVector.frobenius",
"RingHom",
"instInhabitedNat",
"Fact",
"ZMod.pow_card",
"ZMod",
"WittVector.coeff_frobeni... | true |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.size_union_of_not_mem._simp_1_3 | Std.Data.DTreeMap.Internal.Lemmas | [
"Ord",
"Membership.mem",
"Bool.true",
"Std.DTreeMap.Internal.Impl.contains",
"propext",
"Bool",
"Std.DTreeMap.Internal.Impl.mem_iff_contains",
"Eq",
"Std.DTreeMap.Internal.Impl",
"Std.DTreeMap.Internal.Impl.instMembershipOfOrd"
] | false |
PLift.instNonempty_mathlib | Mathlib.Data.ULift | [
"PLift",
"Equiv.plift",
"Nonempty",
"Equiv.nonempty"
] | true |
_private.Mathlib.Data.ZMod.Basic.0.ZModModule.two_le_char._proof_1_1 | Mathlib.Data.ZMod.Basic | [
"Int.Linear.not_eq_norm_expr",
"Lean.RArray.leaf",
"False",
"Nat.instMulZeroClass",
"of_decide_eq_true",
"Int.Linear.le_norm_expr",
"Int.Linear.le_unsat",
"Classical.byContradiction",
"HSub.hSub",
"Nat.not_le_eq",
"Int.Linear.le_of_le_diseq",
"Nat.ToInt.natCast_ofNat",
"id",
"Lean.RArray",... | false |
Std.Http.Method.propfind.elim | Std.Internal.Http.Data.Method | [
"Std.Http.Method",
"PULift.up",
"Std.Http.Method.ctorIdx",
"Nat",
"Std.Http.Method.propfind",
"Eq.symm",
"Std.Http.Method.ctorElim",
"Eq"
] | false |
Nat.factorization_choose_eq_zero_of_lt | Mathlib.Data.Nat.Choose.Factorization | [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.instMulZeroClass",
"Nat.instOrderedSub",
"Nat.choose",
"instHDiv",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"HSub.hSub",
"Finsupp.tsub",
"Preorder.toLE",
"AddZeroClass.toAddZero"... | true |
Nonneg.instIsOrderedModule | Mathlib.Algebra.Order.Nonneg.Module | [
"IsOrderedModule.toPosSMulMono",
"SMulWithZero.toSMulZeroClass",
"IsOrderedModule.toSMulPosMono",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"PosSMulMono.mk",
"Preorder.toLE",
"IsOrderedModule",
"AddZeroClass.toAddZero",
"PartialOrder",
"PosSMulMono.smul_le_smul_of_nonneg_left",
"S... | true |
Real.tanh_arsinh | Mathlib.Analysis.SpecialFunctions.Arsinh | [
"Eq.mpr",
"Real",
"instHDiv",
"Real.tanh",
"congrArg",
"Real.tanh_eq_sinh_div_cosh",
"Real.instDivInvMonoid",
"Real.sinh",
"Real.sinh_arsinh",
"id",
"HDiv.hDiv",
"instOfNatNat",
"Real.arsinh",
"Monoid.toPow",
"Real.instAdd",
"Real.instOne",
"Real.instMonoid",
"instHAdd",
"HPow.hP... | true |
Lean.Literal.type | Lean.Expr | [
"_private.Lean.Expr.0.Lean.instReprLiteral.repr.match_1",
"String",
"Lean.Expr",
"Lean.Level",
"Lean.Literal",
"Lean.mkConst",
"Nat",
"Lean.Name.mkStr1",
"List.nil"
] | true |
Mathlib.Tactic.Algebra.algebra | Mathlib.Tactic.Algebra.Basic | [
"Lean.ParserDescr.nonReservedSymbol",
"instOfNatNat",
"Lean.ParserDescr",
"Lean.ParserDescr.node",
"Nat",
"OfNat.ofNat",
"Bool.false",
"Lean.Name.mkStr4"
] | true |
_private.Mathlib.Combinatorics.SimpleGraph.Walk.Basic.0.SimpleGraph.Walk.mem_darts_iff_infix_support._proof_1_26 | Mathlib.Combinatorics.SimpleGraph.Walk.Basic | [
"Nat.succ_lt_succ_iff",
"Lean.Grind.nestedProof",
"HSub.hSub",
"_private.Mathlib.Combinatorics.SimpleGraph.Walk.Basic.0.SimpleGraph.Walk.mem_darts_iff_infix_support._proof_1_25",
"instSubNat",
"instOfNatNat",
"List.cons",
"instHAdd",
"instHSub",
"HAdd.hAdd",
"Nat",
"LT.lt",
"Iff.mp",
"inst... | false |
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionAssocNatIso_hom_app_app_app | Mathlib.CategoryTheory.Monoidal.Action.Basic | [
"CategoryTheory.Functor.flip",
"CategoryTheory.Functor",
"CategoryTheory.MonoidalCategory.MonoidalRightAction.actionAssocNatIso",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.MonoidalCategory",
"CategoryTheory.Functor.category",
"CategoryTheory.MonoidalCategory.MonoidalRight... | true |
Set.Icc_eq_empty_of_lt | Mathlib.Order.Interval.Set.Basic | [
"Preorder.toLT",
"Set.Icc_eq_empty",
"Set.Icc",
"LT.lt",
"Set.instEmptyCollection",
"LT.lt.not_ge",
"EmptyCollection.emptyCollection",
"Eq",
"Preorder",
"Set"
] | true |
Lean.Meta.Grind.closeGoal | Lean.Meta.Tactic.Grind.Types | [
"Pure.pure",
"Lean.Meta.Grind.GrindM",
"Lean.instMonadMCtxOfMonadLift",
"Lean.Meta.Grind.Goal",
"StateRefT'.instAlternativeOfMonad",
"Lean.Core.instMonadLiftIOCoreM",
"Lean.Meta.Grind.State",
"IO.instMonadLiftSTRealWorldBaseIO",
"ReaderT",
"Lean.Meta.State",
"instMonadLiftT",
"MonadState.get",... | true |
HasFDerivAt.of_isLittleOTVS | Mathlib.Analysis.Calculus.FDeriv.Defs | [
"Iff.mpr",
"HasFDerivAt",
"NormedCommRing.toSeminormedCommRing",
"DistribMulAction.toDistribSMul",
"AddCommGroup.toAddCommMonoid",
"ContinuousLinearMap.funLike",
"AddMonoid.toAddZeroClass",
"Asymptotics.IsLittleOTVS",
"HSub.hSub",
"PseudoMetricSpace.toUniformSpace",
"AddCommGroup.toAddGroup",
... | true |
Mathlib.Tactic.FieldSimp.Sign.div | Mathlib.Tactic.FieldSimp.Lemmas | [
"Qq.QuotedDefEq",
"Pure.pure",
"Mathlib.Tactic.FieldSimp.Sign.mulRight._proof_1",
"Lean.Meta.State",
"Lean.Expr.const",
"Mathlib.Tactic.FieldSimp.Sign.mul._proof_1",
"Lean.Meta.MetaM",
"IO.RealWorld",
"Lean.Expr.sort",
"Lean.Level",
"Applicative.toPure",
"Mathlib.Tactic.FieldSimp.Sign.minus",
... | true |
CommRingCat.limitCommRing._aux_1 | Mathlib.Algebra.Category.Ring.Limits | [
"CategoryTheory.Functor",
"CommRing",
"Equiv.instEquivLike",
"CategoryTheory.Limits.Types.Small.limitCone",
"CommRingCat.carrier",
"CommSemiring.toSemiring",
"CommRingCat",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
"CommRingCat.instConcreteCategoryRingHomCarrier",
"Catego... | false |
CategoryTheory.Oplax.StrongTrans.Modification.casesOn | Mathlib.CategoryTheory.Bicategory.Modification.Oplax | [
"CategoryTheory.Oplax.StrongTrans.naturality",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.OplaxFunctor.toPrelaxFunctor",
"CategoryTheory.Bicategory.whiskerRight",
"CategoryTheory.Oplax.StrongTrans.app",
"CategoryTheory.Bicategory.toCategoryStruct",
"CategoryTheory.PrelaxFu... | false |
Bipointed.X | Mathlib.CategoryTheory.Category.Bipointed | [
"Bipointed"
] | true |
Std.DTreeMap.Raw.Equiv.getKeyGTD_eq | Std.Data.DTreeMap.Raw.Lemmas | [
"Std.DTreeMap.Raw.getKeyGTD",
"Ord.mk",
"Ordering",
"Std.DTreeMap.Raw.Equiv",
"Std.TransCmp",
"Std.DTreeMap.Raw.WF.out",
"Std.DTreeMap.Raw.inner",
"Std.DTreeMap.Raw.Equiv.inner",
"Std.DTreeMap.Internal.Impl.Equiv.getKeyGTD_eq",
"Eq",
"Std.DTreeMap.Raw.WF",
"Std.DTreeMap.Raw"
] | true |
CategoryTheory.Abelian.SpectralObject.isoMapFourδ₁Toδ₀'._auto_1 | Mathlib.Algebra.Homology.SpectralObject.EpiMono | [
"Lean.Syntax.node",
"Array.push",
"Lean.Syntax",
"Array.empty",
"Lean.SourceInfo.none",
"Lean.Name.mkStr1",
"Lean.Name.mkStr4",
"Lean.mkAtom"
] | false |
Subgroup.strictPeriods_eq_zmultiples_one_of_T_mem | Mathlib.NumberTheory.ModularForms.Cusps | [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Units.val",
"Eq.mpr",
"NegZeroClass.toNeg",
"instNeZeroNatHAdd_1",
"zsmul_eq_mul",
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"Subgroup.instSubgroupClass",
"MonoidHom.instFunLike",
"Fintype.elems",
"Equiv.instEquivLike",
"HMul.hMul",
... | true |
AlgebraicGeometry.Scheme.coprodPresheafObjIso._proof_2 | Mathlib.AlgebraicGeometry.Limits | [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"Lattice.toSemilatticeSup",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"TopologicalSpace.O... | false |
SimpleGraph.chromaticNumber_bddBelow | Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | [
"zero_le",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"lowerBounds",
"setOf",
"Membership.mem",
"instOfNatNat",
"instLENat",
"SimpleGraph",
"Nat",
"BddBelow",
"Exists.intro",
"Nat.instLinearOrderedCommMonoidWithZero",
"SimpleGraph.Colorable",
"OfNat.ofNat... | true |
Lean.Meta.Grind.Arith.Linear.EqCnstrProof.coreOfNat.inj | Lean.Meta.Tactic.Grind.Arith.Linear.Types | [
"Lean.Meta.Grind.Arith.Linear.EqCnstrProof.coreOfNat",
"Lean.Meta.Grind.Arith.Linear.LinExpr",
"Lean.Expr",
"And",
"Nat",
"And.intro",
"Lean.Meta.Grind.Arith.Linear.EqCnstrProof",
"Lean.Meta.Grind.Arith.Linear.EqCnstrProof.coreOfNat.noConfusion",
"Eq"
] | true |
instCommutativeInt64HAnd | Init.Data.SInt.Bitwise | [
"Std.Commutative",
"Int64.and_comm",
"Int64",
"instAndOpInt64",
"HAnd.hAnd",
"Std.Commutative.mk",
"instHAndOfAndOp"
] | true |
CategoryTheory.MorphismProperty.CostructuredArrow.isoMk._auto_1 | Mathlib.CategoryTheory.MorphismProperty.Comma | [
"Lean.Syntax.node",
"Array.push",
"Lean.Syntax",
"Array.empty",
"Lean.Name.mkStr2",
"Lean.SourceInfo.none",
"Lean.Name.mkStr1",
"Lean.Name.mkStr4",
"Lean.mkAtom"
] | false |
Equiv.equivCongr_refl_left | Mathlib.Logic.Equiv.Defs | [
"Equiv.instEquivLike",
"Equiv.equivCongr",
"Equiv.trans",
"Equiv",
"Equiv.refl",
"Eq",
"DFunLike.coe",
"rfl",
"EquivLike.toFunLike"
] | true |
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