name
stringlengths
2
347
module
stringlengths
6
90
type
stringlengths
1
5.42M
Std.DTreeMap.Internal.Impl.toList_filter!
Std.Data.DTreeMap.Internal.Lemmas
∀ {α : Type u} {β : α → Type v} {instOrd : Ord α} {t : Std.DTreeMap.Internal.Impl α β} {f : (a : α) → β a → Bool}, t.WF → (Std.DTreeMap.Internal.Impl.filter! f t).toList = List.filter (fun p => f p.fst p.snd) t.toList
Equiv.sumArrowEquivProdArrow._proof_1
Mathlib.Logic.Equiv.Prod
∀ (α : Type u_3) (β : Type u_1) (γ : Type u_2) (f : α ⊕ β → γ), (fun p => Sum.elim p.1 p.2) ((fun f => (f ∘ Sum.inl, f ∘ Sum.inr)) f) = f
Module.Baer.ExtensionOfMaxAdjoin.ideal._proof_1
Mathlib.Algebra.Module.Injective
∀ {R : Type u_1} [inst : Ring R] {N : Type u_2} [inst_1 : AddCommGroup N] [inst_2 : Module R N], IsScalarTower R R N
Aesop.RuleApplication.successProbability?
Aesop.RuleTac.Basic
Aesop.RuleApplication → Option Aesop.Percent
Nat.range_succ_eq_Iic
Mathlib.Order.Interval.Finset.Nat
∀ (n : ℕ), Finset.range (n + 1) = Finset.Iic n
_private.Mathlib.Algebra.Group.Submonoid.Operations.0.AddSubmonoid.map_id.match_1_1
Mathlib.Algebra.Group.Submonoid.Operations
∀ {M : Type u_1} [inst : AddZeroClass M] (S : AddSubmonoid M) (x : M) (motive : x ∈ AddSubmonoid.map (AddMonoidHom.id M) S → Prop) (x_1 : x ∈ AddSubmonoid.map (AddMonoidHom.id M) S), (∀ (h : x ∈ ↑S), motive ⋯) → motive x_1
Lean.Elab.Structural.IndGroupInst.toMessageData
Lean.Elab.PreDefinition.Structural.IndGroupInfo
Lean.Elab.Structural.IndGroupInst → Lean.MessageData
List.all_one_of_le_one_le_of_prod_eq_one
Mathlib.Algebra.Order.BigOperators.Group.List
∀ {M : Type u_3} [inst : CommMonoid M] [inst_1 : PartialOrder M] [IsOrderedMonoid M] {l : List M}, (∀ x ∈ l, 1 ≤ x) → l.prod = 1 → ∀ {x : M}, x ∈ l → x = 1
ClopenUpperSet.noConfusion
Mathlib.Topology.Sets.Order
{P : Sort u} → {α : Type u_2} → {inst : TopologicalSpace α} → {inst_1 : LE α} → {t : ClopenUpperSet α} → {α' : Type u_2} → {inst' : TopologicalSpace α'} → {inst'_1 : LE α'} → {t' : ClopenUpperSet α'} → α = α' → inst ≍ inst' → inst_1 ≍ inst'_1 → t ≍ t' → ClopenUpperSet.noConfusionType P t t'
Urysohns.CU.approx.match_1
Mathlib.Topology.UrysohnsLemma
{X : Type u_1} → [inst : TopologicalSpace X] → {P : Set X → Set X → Prop} → (motive : ℕ → Urysohns.CU P → X → Sort u_2) → (x : ℕ) → (x_1 : Urysohns.CU P) → (x_2 : X) → ((c : Urysohns.CU P) → (x : X) → motive 0 c x) → ((n : ℕ) → (c : Urysohns.CU P) → (x : X) → motive n.succ c x) → motive x x_1 x_2
CategoryTheory.Limits.SingleObj.Types.sections.equivFixedPoints._proof_4
Mathlib.CategoryTheory.Limits.Shapes.SingleObj
∀ {M : Type u_1} [inst : Monoid M] (J : CategoryTheory.Functor (CategoryTheory.SingleObj M) (Type u_2)) (p : ↑(MulAction.fixedPoints M (J.obj (CategoryTheory.SingleObj.star M)))) {j j' : CategoryTheory.SingleObj M}, ↑p ∈ MulAction.fixedPoints M (J.obj (CategoryTheory.SingleObj.star M))
Lean.Elab.Tactic.Do.ProofMode.elabMExact
Lean.Elab.Tactic.Do.ProofMode.Exact
Lean.Elab.Tactic.Tactic
_private.Mathlib.Probability.Kernel.Composition.MeasureCompProd.0.MeasureTheory.Measure.absolutelyContinuous_of_compProd._simp_1_1
Mathlib.Probability.Kernel.Composition.MeasureCompProd
∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}, (μ Set.univ = 0) = (μ = 0)
_private.Mathlib.RingTheory.LaurentSeries.0.LaurentSeries.coe_range_dense._simp_1_2
Mathlib.RingTheory.LaurentSeries
∀ {α : Type u} (x : α), (x ∈ Set.univ) = True
Rep.diagonalOneIsoLeftRegular_inv_hom
Mathlib.RepresentationTheory.Rep
∀ (k G : Type u) [inst : CommRing k] [inst_1 : Monoid G], (Rep.diagonalOneIsoLeftRegular k G).inv.hom = ModuleCat.ofHom ↑(Finsupp.domLCongr (Equiv.funUnique (Fin 1) G).symm)
Lean.Compiler.LCNF.instantiateRangeArgs
Lean.Compiler.LCNF.Basic
{pu : Lean.Compiler.LCNF.Purity} → Lean.Expr → ℕ → ℕ → Array (Lean.Compiler.LCNF.Arg pu) → Lean.Expr
SpectrumRestricts.starAlgHom_injective
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict
∀ {R : Type u_1} {S : Type u_2} {A : Type u_3} [inst : Semifield R] [inst_1 : StarRing R] [inst_2 : MetricSpace R] [inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : Semifield S] [inst_6 : StarRing S] [inst_7 : MetricSpace S] [inst_8 : IsTopologicalSemiring S] [inst_9 : ContinuousStar S] [inst_10 : Ring A] [inst_11 : StarRing A] [inst_12 : Algebra S A] [inst_13 : Algebra R S] [inst_14 : Algebra R A] [inst_15 : IsScalarTower R S A] [inst_16 : StarModule R S] [inst_17 : ContinuousSMul R S] {a : A} {φ : C(↑(spectrum S a), S) →⋆ₐ[S] A}, Function.Injective ⇑φ → ∀ {f : C(S, R)} (h : SpectrumRestricts a ⇑f), Function.Injective ⇑(algebraMap R S) → Function.Injective ⇑(SpectrumRestricts.starAlgHom φ h)
AddCircle.homeomorphAddCircle_symm_apply_mk
Mathlib.Topology.Instances.AddCircle.Defs
∀ {𝕜 : Type u_1} [inst : Field 𝕜] (p q : 𝕜) [inst_1 : LinearOrder 𝕜] [inst_2 : IsStrictOrderedRing 𝕜] [inst_3 : TopologicalSpace 𝕜] [inst_4 : OrderTopology 𝕜] (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜), (AddCircle.homeomorphAddCircle p q hp hq).symm ↑x = ↑(x * (q⁻¹ * p))
ZeroAtInftyContinuousMap.coe_zero
Mathlib.Topology.ContinuousMap.ZeroAtInfty
∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : Zero β], ⇑0 = 0
_private.Init.Grind.Module.Envelope.0.Lean.Grind.IntModule.OfNatModule.rel
Init.Grind.Module.Envelope
{α : Type u} → [inst : Lean.Grind.NatModule α] → Equivalence (Lean.Grind.IntModule.OfNatModule.r α) → Lean.Grind.IntModule.OfNatModule.Q α → Lean.Grind.IntModule.OfNatModule.Q α → Prop
_private.Mathlib.RingTheory.Flat.EquationalCriterion.0.Module.Flat.projective_of_finitePresentation.match_1_1
Mathlib.RingTheory.Flat.EquationalCriterion
∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (motive : (∃ k h₂ h₃, LinearMap.id = h₃ ∘ₗ h₂) → Prop) (x : ∃ k h₂ h₃, LinearMap.id = h₃ ∘ₗ h₂), (∀ (w : ℕ) (f : M →ₗ[R] Fin w →₀ R) (g : (Fin w →₀ R) →ₗ[R] M) (eq : LinearMap.id = g ∘ₗ f), motive ⋯) → motive x
Module.finitePresentation_iff_finite
Mathlib.Algebra.Module.FinitePresentation
∀ (R : Type u_1) (M : Type u_2) [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [IsNoetherianRing R], Module.FinitePresentation R M ↔ Module.Finite R M
AddMonoidAlgebra.ofMagma_apply
Mathlib.Algebra.MonoidAlgebra.Defs
∀ (R : Type u_8) (M : Type u_9) [inst : Semiring R] [inst_1 : Add M] (a : Multiplicative M), (AddMonoidAlgebra.ofMagma R M) a = AddMonoidAlgebra.single (Multiplicative.toAdd a) 1
FilterBasis
Mathlib.Order.Filter.Bases.Basic
Type u_6 → Type u_6
List.le_max_of_mem
Init.Data.List.MinMax
∀ {α : Type u_1} [inst : Max α] [inst_1 : LE α] [Std.IsLinearOrder α] [Std.LawfulOrderMax α] {l : List α} {a : α} (ha : a ∈ l), a ≤ l.max ⋯
bind_pure_unit
Init.Control.Lawful.Basic
∀ {m : Type u_1 → Type u_2} [inst : Monad m] [LawfulMonad m] {x : m PUnit.{u_1 + 1}}, (do x pure PUnit.unit) = x
rank_eq_card_basis
Mathlib.LinearAlgebra.Dimension.StrongRankCondition
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [StrongRankCondition R] {ι : Type w} [inst_4 : Fintype ι] (h : Module.Basis ι R M), Module.rank R M = ↑(Fintype.card ι)
MultilinearMap.currySum._proof_7
Mathlib.LinearAlgebra.Multilinear.Curry
∀ {R : Type u_5} {ι : Type u_2} {ι' : Type u_1} {M₂ : Type u_3} [inst : CommSemiring R] [inst_1 : AddCommMonoid M₂] [inst_2 : Module R M₂] {N : ι ⊕ ι' → Type u_4} [inst_3 : (i : ι ⊕ ι') → AddCommMonoid (N i)] [inst_4 : (i : ι ⊕ ι') → Module R (N i)] (f : MultilinearMap R N M₂) [inst_5 : DecidableEq ι] (u : (i : ι) → N (Sum.inl i)) (i : ι) (x : N (Sum.inl i)) [inst_6 : DecidableEq ι'] (m : (i : ι') → N (Sum.inr i)) (i_1 : ι') (x_1 y : N (Sum.inr i_1)), (f fun t => Sum.rec (Function.update u i x) (Function.update m i_1 (x_1 + y)) t) = (f fun t => Sum.rec (Function.update u i x) (Function.update m i_1 x_1) t) + f fun t => Sum.rec (Function.update u i x) (Function.update m i_1 y) t
MeasureTheory.measurePreserving_pi_empty
Mathlib.MeasureTheory.Constructions.Pi
∀ {ι : Type u} {α : ι → Type v} [inst : Fintype ι] [inst_1 : IsEmpty ι] {m : (i : ι) → MeasurableSpace (α i)} (μ : (i : ι) → MeasureTheory.Measure (α i)), MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.ofUniqueOfUnique ((i : ι) → α i) Unit)) (MeasureTheory.Measure.pi μ) (MeasureTheory.Measure.dirac ())
definition._proof_2._@.Mathlib.Analysis.InnerProductSpace.PiL2.1554134833._hygCtx._hyg.2
Mathlib.Analysis.InnerProductSpace.PiL2
∀ {ι : Type u_1} {𝕜 : Type u_2} [inst : RCLike 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E] [inst_3 : Fintype ι] [FiniteDimensional 𝕜 E] {n : ℕ}, Module.finrank 𝕜 E = n → ∀ [inst_5 : DecidableEq ι] {V : ι → Submodule 𝕜 E}, DirectSum.IsInternal V → (OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ) → Fintype.card ((i : ι) × Fin (Module.finrank 𝕜 ↥(V i))) = n
BoundedContinuousFunction.rec
Mathlib.Topology.ContinuousMap.Bounded.Basic
{α : Type u} → {β : Type v} → [inst : TopologicalSpace α] → [inst_1 : PseudoMetricSpace β] → {motive : BoundedContinuousFunction α β → Sort u_1} → ((toContinuousMap : C(α, β)) → (map_bounded' : ∃ C, ∀ (x y : α), dist (toContinuousMap.toFun x) (toContinuousMap.toFun y) ≤ C) → motive { toContinuousMap := toContinuousMap, map_bounded' := map_bounded' }) → (t : BoundedContinuousFunction α β) → motive t
CategoryTheory.ShortComplex.opcyclesFunctor
Mathlib.Algebra.Homology.ShortComplex.RightHomology
(C : Type u_1) → [inst : CategoryTheory.Category.{v_1, u_1} C] → [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] → [CategoryTheory.Limits.HasKernels C] → [CategoryTheory.Limits.HasCokernels C] → CategoryTheory.Functor (CategoryTheory.ShortComplex C) C
monovaryOn_neg
Mathlib.Algebra.Order.Monovary
∀ {ι : Type u_1} {α : Type u_2} {β : Type u_3} [inst : AddCommGroup α] [inst_1 : PartialOrder α] [IsOrderedAddMonoid α] [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] {s : Set ι} {f : ι → α} {g : ι → β}, MonovaryOn (-f) (-g) s ↔ MonovaryOn f g s
Lean.Elab.Term.elabLetDelayedDecl
Lean.Elab.Binders
Lean.Elab.Term.TermElab
Array.appendCore.loop
Init.Prelude
{α : Type u} → Array α → ℕ → ℕ → Array α → Array α
Subgroup.FG.eq_1
Mathlib.GroupTheory.Finiteness
∀ {G : Type u_3} [inst : Group G] (P : Subgroup G), P.FG = ∃ S, Subgroup.closure ↑S = P
instDecidableEqColex
Mathlib.Order.Synonym
(α : Type u_2) → [h : DecidableEq α] → DecidableEq (Colex α)
AlgEquiv.aut._proof_8
Mathlib.Algebra.Algebra.Equiv
∀ {R : Type u_1} {A₁ : Type u_2} [inst : CommSemiring R] [inst_1 : Semiring A₁] [inst_2 : Algebra R A₁] (x : A₁ ≃ₐ[R] A₁), 1 * x = x
Lean.Server.Test.Runner.rpcRequest
Lean.Server.Test.Runner
{α : Type u_1} → Lean.Name → [Lean.ToJson α] → α → (β : Type) → [Lean.FromJson β] → Lean.Server.Test.Runner.RunnerM β
MulEquiv.toMonCatIso
Mathlib.Algebra.Category.MonCat.Basic
{X Y : Type u} → [inst : Monoid X] → [inst_1 : Monoid Y] → X ≃* Y → (MonCat.of X ≅ MonCat.of Y)
InfClosed.infClosure_eq
Mathlib.Order.SupClosed
∀ {α : Type u_3} [inst : SemilatticeInf α] {s : Set α}, InfClosed s → infClosure s = s
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjTransformObjSquare_iso_hom_comp
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
∀ {A : Type u₁} {B : Type u₂} {C : Type u₃} [inst : CategoryTheory.Category.{v₁, u₁} A] [inst_1 : CategoryTheory.Category.{v₂, u₂} B] [inst_2 : CategoryTheory.Category.{v₃, u₃} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {A₁ : Type u₄} {B₁ : Type u₅} {C₁ : Type u₆} [inst_3 : CategoryTheory.Category.{v₄, u₄} A₁] [inst_4 : CategoryTheory.Category.{v₅, u₅} B₁] [inst_5 : CategoryTheory.Category.{v₆, u₆} C₁] {F₁ : CategoryTheory.Functor A₁ B₁} {G₁ : CategoryTheory.Functor C₁ B₁} {X : Type u₇} {Y : Type u₈} {Z : Type u₉} [inst_6 : CategoryTheory.Category.{v₇, u₇} X] [inst_7 : CategoryTheory.Category.{v₈, u₈} Y] [inst_8 : CategoryTheory.Category.{v₉, u₉} Z] (ψ : CategoryTheory.Limits.CatCospanTransform F G F₁ G₁) (U : CategoryTheory.Functor X Y) (V : CategoryTheory.Functor Y Z), CategoryTheory.CategoryStruct.comp (CategoryTheory.CatCommSq.iso ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj (U.comp V)) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Z).obj ψ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj (U.comp V))).hom (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Z).obj ψ).whiskerLeft (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjComp F₁ G₁ U V).hom) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjComp F G U V).hom ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ)) (CategoryTheory.CategoryStruct.comp (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj V).associator ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ)).hom (CategoryTheory.CategoryStruct.comp (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj V).whiskerLeft (CategoryTheory.CatCommSq.iso ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ψ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ψ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj U)).hom) (CategoryTheory.CategoryStruct.comp (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj V).associator ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ψ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj U)).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.CatCommSq.iso ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj V) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Z).obj ψ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ψ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj V)).hom ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj U)) (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Z).obj ψ).associator ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj V) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F₁ G₁).obj U)).hom))))
AlgebraicGeometry.IsLocallyArtinian.discreteTopology_of_isAffine
Mathlib.AlgebraicGeometry.Artinian
∀ {X : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsLocallyArtinian X], DiscreteTopology ↥X
Lean.Meta.Tactic.Cbv.getMatchTheorems
Lean.Meta.Tactic.Cbv.TheoremsLookup
Lean.Name → Lean.MetaM Lean.Meta.Sym.Simp.Theorems
CategoryTheory.Bicategory.Adj.iso₂Mk._proof_1
Mathlib.CategoryTheory.Bicategory.Adjunction.Adj
∀ {B : Type u_3} [inst : CategoryTheory.Bicategory B] {a b : CategoryTheory.Bicategory.Adj B} {α β : a ⟶ b} (el : α.l ≅ β.l) (er : β.r ≅ α.r), (CategoryTheory.Bicategory.conjugateEquiv β.adj α.adj) el.hom = er.hom → (CategoryTheory.Bicategory.conjugateEquiv α.adj β.adj) el.inv = er.inv
String.Slice.Pos.byte.eq_1
Init.Data.String.Basic
∀ {s : String.Slice} (pos : s.Pos) (h : pos ≠ s.endPos), pos.byte h = s.getUTF8Byte pos.offset ⋯
Mathlib.Linter.linter.style.setOption
Mathlib.Tactic.Linter.Style
Lean.Option Bool
CategoryTheory.Monoidal.InducingFunctorData.rec
Mathlib.CategoryTheory.Monoidal.Transport
{C : Type u₁} → [inst : CategoryTheory.Category.{v₁, u₁} C] → [inst_1 : CategoryTheory.MonoidalCategory C] → {D : Type u₂} → [inst_2 : CategoryTheory.Category.{v₂, u₂} D] → [inst_3 : CategoryTheory.MonoidalCategoryStruct D] → {F : CategoryTheory.Functor D C} → {motive : CategoryTheory.Monoidal.InducingFunctorData F → Sort u} → ((μIso : (X Y : D) → CategoryTheory.MonoidalCategoryStruct.tensorObj (F.obj X) (F.obj Y) ≅ F.obj (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y)) → (whiskerLeft_eq : ∀ (X : D) {Y₁ Y₂ : D} (f : Y₁ ⟶ Y₂), F.map (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X f) = CategoryTheory.CategoryStruct.comp (μIso X Y₁).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft (F.obj X) (F.map f)) (μIso X Y₂).hom)) → (whiskerRight_eq : ∀ {X₁ X₂ : D} (f : X₁ ⟶ X₂) (Y : D), F.map (CategoryTheory.MonoidalCategoryStruct.whiskerRight f Y) = CategoryTheory.CategoryStruct.comp (μIso X₁ Y).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight (F.map f) (F.obj Y)) (μIso X₂ Y).hom)) → (tensorHom_eq : ∀ {X₁ Y₁ X₂ Y₂ : D} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂), F.map (CategoryTheory.MonoidalCategoryStruct.tensorHom f g) = CategoryTheory.CategoryStruct.comp (μIso X₁ X₂).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom (F.map f) (F.map g)) (μIso Y₁ Y₂).hom)) → (εIso : CategoryTheory.MonoidalCategoryStruct.tensorUnit C ≅ F.obj (CategoryTheory.MonoidalCategoryStruct.tensorUnit D)) → (associator_eq : ∀ (X Y Z : D), F.map (CategoryTheory.MonoidalCategoryStruct.associator X Y Z).hom = (((μIso (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y) Z).symm ≪≫ CategoryTheory.MonoidalCategory.tensorIso (μIso X Y).symm (CategoryTheory.Iso.refl (F.obj Z))) ≪≫ CategoryTheory.MonoidalCategoryStruct.associator (F.obj X) (F.obj Y) (F.obj Z) ≪≫ CategoryTheory.MonoidalCategory.tensorIso (CategoryTheory.Iso.refl (F.obj X)) (μIso Y Z) ≪≫ μIso X (CategoryTheory.MonoidalCategoryStruct.tensorObj Y Z)).hom) → (leftUnitor_eq : ∀ (X : D), F.map (CategoryTheory.MonoidalCategoryStruct.leftUnitor X).hom = (((μIso (CategoryTheory.MonoidalCategoryStruct.tensorUnit D) X).symm ≪≫ CategoryTheory.MonoidalCategory.tensorIso εIso.symm (CategoryTheory.Iso.refl (F.obj X))) ≪≫ CategoryTheory.MonoidalCategoryStruct.leftUnitor (F.obj X)).hom) → (rightUnitor_eq : ∀ (X : D), F.map (CategoryTheory.MonoidalCategoryStruct.rightUnitor X).hom = (((μIso X (CategoryTheory.MonoidalCategoryStruct.tensorUnit D)).symm ≪≫ CategoryTheory.MonoidalCategory.tensorIso (CategoryTheory.Iso.refl (F.obj X)) εIso.symm) ≪≫ CategoryTheory.MonoidalCategoryStruct.rightUnitor (F.obj X)).hom) → motive { μIso := μIso, whiskerLeft_eq := whiskerLeft_eq, whiskerRight_eq := whiskerRight_eq, tensorHom_eq := tensorHom_eq, εIso := εIso, associator_eq := associator_eq, leftUnitor_eq := leftUnitor_eq, rightUnitor_eq := rightUnitor_eq }) → (t : CategoryTheory.Monoidal.InducingFunctorData F) → motive t
CategoryTheory.ULift.upFunctor
Mathlib.CategoryTheory.Category.ULift
{C : Type u₁} → [inst : CategoryTheory.Category.{v₁, u₁} C] → CategoryTheory.Functor C (ULift.{u₂, u₁} C)
List.chain'_map
Mathlib.Data.List.Chain
∀ {α : Type u} {β : Type v} {R : α → α → Prop} (f : β → α) {l : List β}, List.IsChain R (List.map f l) ↔ List.IsChain (fun a b => R (f a) (f b)) l
Mathlib.Tactic.filterUpwards
Mathlib.Order.Filter.Defs
Lean.ParserDescr
instTotallyDisconnectedSpaceMultiplicative
Mathlib.Topology.Connected.TotallyDisconnected
∀ {α : Type u} [inst : TopologicalSpace α] [TotallyDisconnectedSpace α], TotallyDisconnectedSpace (Multiplicative α)
BooleanAlgebra.toBooleanRing._proof_5
Mathlib.Algebra.Ring.BooleanRing
∀ {α : Type u_1} [inst : BooleanAlgebra α] (n : ℕ), (↑n).castDef = ↑n
Module.Finite.addMonoidHom
Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
∀ (M : Type v) (N : Type w) [inst : AddCommGroup M] [Module.Finite ℤ M] [Module.Free ℤ M] [inst_3 : AddCommGroup N] [Module.Finite ℤ N], Module.Finite ℤ (M →+ N)
Lean.Meta.Grind.Arith.Cutsat.DiseqCnstrProof.reorder.injEq
Lean.Meta.Tactic.Grind.Arith.Cutsat.Types
∀ (c c_1 : Lean.Meta.Grind.Arith.Cutsat.DiseqCnstr), (Lean.Meta.Grind.Arith.Cutsat.DiseqCnstrProof.reorder c = Lean.Meta.Grind.Arith.Cutsat.DiseqCnstrProof.reorder c_1) = (c = c_1)
Lean.Core.Context.cancelTk?
Lean.CoreM
Lean.Core.Context → Option IO.CancelToken
_private.Batteries.Data.List.Lemmas.0.List.dropPrefix?.match_1.eq_1
Batteries.Data.List.Lemmas
∀ {α : Type u_1} (motive : List α → List α → Sort u_2) (list : List α) (h_1 : (list : List α) → motive list []) (h_2 : (head : α) → (tail : List α) → motive [] (head :: tail)) (h_3 : (a : α) → (as : List α) → (b : α) → (bs : List α) → motive (a :: as) (b :: bs)), (match list, [] with | list, [] => h_1 list | [], head :: tail => h_2 head tail | a :: as, b :: bs => h_3 a as b bs) = h_1 list
SimpleGraph.Subgraph.Connected.mono'
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph
∀ {V : Type u} {G : SimpleGraph V} {H H' : G.Subgraph}, (∀ (v w : V), H.Adj v w → H'.Adj v w) → H.verts = H'.verts → H.Connected → H'.Connected
dvd_mul_gcd_of_dvd_mul
Mathlib.Algebra.GCDMonoid.Basic
∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : GCDMonoid α] {m n k : α}, k ∣ m * n → k ∣ m * gcd k n
CategoryTheory.GlueData.mk
Mathlib.CategoryTheory.GlueData
{C : Type u₁} → [inst : CategoryTheory.Category.{v, u₁} C] → (J : Type v) → (U : J → C) → (V : J × J → C) → (f : (i j : J) → V (i, j) ⟶ U i) → autoParam (∀ (i j : J), CategoryTheory.Mono (f i j)) CategoryTheory.GlueData.f_mono._autoParam → (f_hasPullback : autoParam (∀ (i j k : J), CategoryTheory.Limits.HasPullback (f i j) (f i k)) CategoryTheory.GlueData.f_hasPullback._autoParam) → autoParam (∀ (i : J), CategoryTheory.IsIso (f i i)) CategoryTheory.GlueData.f_id._autoParam → (t : (i j : J) → V (i, j) ⟶ V (j, i)) → (∀ (i : J), t i i = CategoryTheory.CategoryStruct.id (V (i, i))) → (t' : (i j k : J) → CategoryTheory.Limits.pullback (f i j) (f i k) ⟶ CategoryTheory.Limits.pullback (f j k) (f j i)) → (∀ (i j k : J), CategoryTheory.CategoryStruct.comp (t' i j k) (CategoryTheory.Limits.pullback.snd (f j k) (f j i)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.fst (f i j) (f i k)) (t i j)) → (∀ (i j k : J), CategoryTheory.CategoryStruct.comp (t' i j k) (CategoryTheory.CategoryStruct.comp (t' j k i) (t' k i j)) = CategoryTheory.CategoryStruct.id (CategoryTheory.Limits.pullback (f i j) (f i k))) → CategoryTheory.GlueData C
Std.Rio.mem_iff_mem_Rco
Init.Data.Range.Polymorphic.Lemmas
∀ {α : Type u_1} {r : Std.Rio α} [inst : LE α] [inst_1 : LT α] [inst_2 : Std.PRange.Least? α] [inst_3 : Std.PRange.UpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLE α] [inst_6 : Std.PRange.LawfulUpwardEnumerableLeast? α] [Std.Rxo.IsAlwaysFinite α] {a : α}, a ∈ r ↔ a ∈ Std.PRange.UpwardEnumerable.least...r.upper
Filter.div_le_div_right
Mathlib.Order.Filter.Pointwise
∀ {α : Type u_2} [inst : Div α] {f₁ f₂ g : Filter α}, f₁ ≤ f₂ → f₁ / g ≤ f₂ / g
_private.Mathlib.Analysis.PSeries.0.Real.summable_nat_rpow_inv._simp_1_1
Mathlib.Analysis.PSeries
∀ (x : ℝ) (n : ℕ), x ^ n = x ^ ↑n
MeasureTheory.memLp_zero_iff_aestronglyMeasurable._simp_1
Mathlib.MeasureTheory.Function.LpSeminorm.Basic
∀ {α : Type u_1} {ε : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : ENorm ε] [inst_1 : TopologicalSpace ε] {f : α → ε}, MeasureTheory.MemLp f 0 μ = MeasureTheory.AEStronglyMeasurable f μ
MulOpposite.instMulOneClass._proof_1
Mathlib.Algebra.Group.Opposite
∀ {α : Type u_1} [inst : MulOneClass α] (x : αᵐᵒᵖ), 1 * x = x
Submodule.range_inclusion
Mathlib.Algebra.Module.Submodule.Range
∀ {R : Type u_1} {M : Type u_5} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p q : Submodule R M) (h : p ≤ q), (Submodule.inclusion h).range = Submodule.comap q.subtype p
MulEquiv.AddMonoid.End._proof_1
Mathlib.Algebra.Group.Equiv.TypeTags
∀ {M : Type u_1} [inst : AddMonoid M] (x x_1 : AddMonoid.End M), AddMonoidHom.toMultiplicative.toFun (x * x_1) = AddMonoidHom.toMultiplicative.toFun (x * x_1)
CommRingCat.tensorProd_map_right
Mathlib.Algebra.Category.Ring.Under.Basic
∀ (R S : CommRingCat) [inst : Algebra ↑R ↑S] {X Y : CategoryTheory.Under R} (f : X ⟶ Y), ((R.tensorProd S).map f).right = CommRingCat.ofHom ↑(Algebra.TensorProduct.map (AlgHom.id ↑S ↑S) (CommRingCat.toAlgHom f))
Std.Tactic.BVDecide.Normalize.Bool.ite_else_ite'
Std.Tactic.BVDecide.Normalize.Bool
∀ {α : Sort u_1} (c0 c1 : Bool) {a b : α}, (bif c0 then a else bif c1 then a else b) = bif !c0 && !c1 then b else a
InitialSeg.transPrincipal_apply
Mathlib.Order.InitialSeg
∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} [inst : IsWellOrder β s] [inst_1 : IsTrans γ t] (f : InitialSeg r s) (g : PrincipalSeg s t) (a : α), (f.transPrincipal g).toRelEmbedding a = g.toRelEmbedding (f a)
ModuleCat.HasLimits.limitCone._proof_1
Mathlib.Algebra.Category.ModuleCat.Limits
∀ {R : Type u_4} [inst : Ring R] {J : Type u_2} [inst_1 : CategoryTheory.Category.{u_1, u_2} J] (F : CategoryTheory.Functor J (ModuleCat R)) [inst_2 : Small.{u_3, max u_2 u_3} ↑(F.comp (CategoryTheory.forget (ModuleCat R))).sections] (x x_1 : J) (f : x ⟶ x_1), CategoryTheory.CategoryStruct.comp (((CategoryTheory.Functor.const J).obj (ModuleCat.of R (CategoryTheory.Limits.Types.Small.limitCone (F.comp (CategoryTheory.forget (ModuleCat R)))).pt)).map f) (ModuleCat.ofHom (ModuleCat.limitπLinearMap F x_1)) = CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom (ModuleCat.limitπLinearMap F x)) (F.map f)
MulEquiv.piCongrRight_symm
Mathlib.Algebra.Group.Equiv.Basic
∀ {η : Type u_16} {Ms : η → Type u_17} {Ns : η → Type u_18} [inst : (j : η) → Mul (Ms j)] [inst_1 : (j : η) → Mul (Ns j)] (es : (j : η) → Ms j ≃* Ns j), (MulEquiv.piCongrRight es).symm = MulEquiv.piCongrRight fun i => (es i).symm
LieSubalgebra.exists_nested_lieIdeal_coe_eq_iff
Mathlib.Algebra.Lie.Ideal
∀ (R : Type u) {L : Type v} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L] (K : LieSubalgebra R L) {K' : LieSubalgebra R L} (h : K ≤ K'), (∃ I, LieIdeal.toLieSubalgebra R (↥K') I = LieSubalgebra.ofLe h) ↔ ∀ (x y : L), x ∈ K' → y ∈ K → ⁅x, y⁆ ∈ K
_private.Init.Data.List.MinMaxIdx.0.List.minIdxOn_nil_eq_iff_true.match_1_1
Init.Data.List.MinMaxIdx
∀ {α : Type u_1} (motive : [] ≠ [] → Prop) (h : [] ≠ []), motive h
Submodule.ClosedComplemented.complement
Mathlib.Topology.Algebra.Module.LinearMap
{R : Type u_1} → [inst : Ring R] → {M : Type u_2} → [inst_1 : TopologicalSpace M] → [inst_2 : AddCommGroup M] → [inst_3 : Module R M] → {p : Submodule R M} → [T1Space ↥p] → p.ClosedComplemented → Submodule R M
EquivFunctor.mapEquiv_apply
Mathlib.Control.EquivFunctor
∀ (f : Type u₀ → Type u₁) [inst : EquivFunctor f] {α β : Type u₀} (e : α ≃ β) (x : f α), (EquivFunctor.mapEquiv f e) x = EquivFunctor.map e x
PosNum.testBit.eq_5
Mathlib.Data.Num.Lemmas
∀ (a : PosNum), a.bit1.testBit 0 = true
_private.Lean.Elab.Import.0.Lean.Elab.parseImports.match_1
Lean.Elab.Import
(motive : Lean.TSyntax `Lean.Parser.Module.header × Lean.Parser.ModuleParserState × Lean.MessageLog → Sort u_1) → (__discr : Lean.TSyntax `Lean.Parser.Module.header × Lean.Parser.ModuleParserState × Lean.MessageLog) → ((header : Lean.TSyntax `Lean.Parser.Module.header) → (parserState : Lean.Parser.ModuleParserState) → (messages : Lean.MessageLog) → motive (header, parserState, messages)) → motive __discr
CategoryTheory.Epi.left_cancellation
Mathlib.CategoryTheory.Category.Basic
∀ {C : Type u} {inst : CategoryTheory.Category.{v, u} C} {X Y : C} {f : X ⟶ Y} [self : CategoryTheory.Epi f] {Z : C} (g h : Y ⟶ Z), CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp f h → g = h
MeasureTheory.convolution_zero
Mathlib.Analysis.Convolution
∀ {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [inst : NormedAddCommGroup E] [inst_1 : NormedAddCommGroup E'] [inst_2 : NormedAddCommGroup F] {f : G → E} [inst_3 : NontriviallyNormedField 𝕜] [inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜 E'] [inst_6 : NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [inst_7 : MeasurableSpace G] {μ : MeasureTheory.Measure G} [inst_8 : NormedSpace ℝ F] [inst_9 : AddGroup G], MeasureTheory.convolution f 0 L μ = 0
CategoryTheory.Subobject.ofLEMk.eq_1
Mathlib.CategoryTheory.Subobject.Basic
∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {B A : C} (X : CategoryTheory.Subobject B) (f : A ⟶ B) [inst_1 : CategoryTheory.Mono f] (h : X ≤ CategoryTheory.Subobject.mk f), X.ofLEMk f h = CategoryTheory.CategoryStruct.comp (X.ofLE (CategoryTheory.Subobject.mk f) h) (CategoryTheory.Subobject.underlyingIso f).hom
_private.Mathlib.GroupTheory.Index.0.Subgroup.finiteIndex_iInf'.match_1_1
Mathlib.GroupTheory.Index
∀ {ι : Type u_1} {s : Finset ι} (motive : Subtype (Membership.mem s) → Prop) (x : Subtype (Membership.mem s)), (∀ (i : ι) (hi : i ∈ s), motive ⟨i, hi⟩) → motive x
Std.DTreeMap.keys
Std.Data.DTreeMap.Basic
{α : Type u} → {β : α → Type v} → {cmp : α → α → Ordering} → Std.DTreeMap α β cmp → List α
_private.Mathlib.Algebra.Group.Units.Opposite.0.IsAddUnit.op.match_1_1
Mathlib.Algebra.Group.Units.Opposite
∀ {M : Type u_1} [inst : AddMonoid M] {m : M} (motive : IsAddUnit m → Prop) (h : IsAddUnit m), (∀ (u : AddUnits M) (hu : ↑u = m), motive ⋯) → motive h
_private.Lean.DocString.Parser.0.Lean.Doc.Parser.codeBlock.withIndentColumn._sparseCasesOn_4
Lean.DocString.Parser
{motive : Lean.Name → Sort u} → (t : Lean.Name) → ((pre : Lean.Name) → (str : String) → motive (pre.str str)) → (Nat.hasNotBit 2 t.ctorIdx → motive t) → motive t
ULift.divInvMonoid.eq_1
Mathlib.Algebra.Group.ULift
∀ {α : Type u} [inst : DivInvMonoid α], ULift.divInvMonoid = Function.Injective.divInvMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
IsInvariantSubfield.mk._flat_ctor
Mathlib.FieldTheory.Fixed
∀ {M : Type u} [inst : Monoid M] {F : Type v} [inst_1 : Field F] [inst_2 : MulSemiringAction M F] {S : Subfield F}, (∀ (m : M) {x : F}, x ∈ S → m • x ∈ S) → IsInvariantSubfield M S
Aesop.LocalRuleSet.mk.sizeOf_spec
Aesop.RuleSet
∀ (toBaseRuleSet : Aesop.BaseRuleSet) (simpTheoremsArray : Array (Lean.Name × Lean.Meta.SimpTheorems)) (simpTheoremsArrayNonempty : 0 < simpTheoremsArray.size) (simprocsArray : Array (Lean.Name × Lean.Meta.Simprocs)) (simprocsArrayNonempty : 0 < simprocsArray.size) (localNormSimpRules : Array Aesop.LocalNormSimpRule), sizeOf { toBaseRuleSet := toBaseRuleSet, simpTheoremsArray := simpTheoremsArray, simpTheoremsArrayNonempty := simpTheoremsArrayNonempty, simprocsArray := simprocsArray, simprocsArrayNonempty := simprocsArrayNonempty, localNormSimpRules := localNormSimpRules } = 1 + sizeOf toBaseRuleSet + sizeOf simpTheoremsArray + sizeOf simpTheoremsArrayNonempty + sizeOf simprocsArray + sizeOf simprocsArrayNonempty + sizeOf localNormSimpRules
Matroid.cRk_map_image._auto_1
Mathlib.Combinatorics.Matroid.Rank.Cardinal
Lean.Syntax
VectorAll._unsafe_rec
Mathlib.Data.Vector3
{α : Type u_1} → (k : ℕ) → (Vector3 α k → Prop) → Prop
Std.DTreeMap.Raw.minKeyD_erase_eq_of_not_compare_minKeyD_eq
Std.Data.DTreeMap.Raw.Lemmas
∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap.Raw α β cmp} [Std.TransCmp cmp], t.WF → ∀ {k fallback : α}, (t.erase k).isEmpty = false → ¬cmp k (t.minKeyD fallback) = Ordering.eq → (t.erase k).minKeyD fallback = t.minKeyD fallback
CategoryTheory.MonoOver.mapIso._proof_5
Mathlib.CategoryTheory.Subobject.MonoOver
∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {A B : C} (e : A ≅ B) (X : CategoryTheory.MonoOver A), CategoryTheory.CategoryStruct.comp ((CategoryTheory.MonoOver.map e.hom).map (((CategoryTheory.MonoOver.mapComp e.hom e.inv).symm ≪≫ CategoryTheory.eqToIso ⋯ ≪≫ CategoryTheory.MonoOver.mapId A).symm.hom.app X)) (((CategoryTheory.MonoOver.mapComp e.inv e.hom).symm ≪≫ CategoryTheory.eqToIso ⋯ ≪≫ CategoryTheory.MonoOver.mapId B).hom.app ((CategoryTheory.MonoOver.map e.hom).obj X)) = CategoryTheory.CategoryStruct.id ((CategoryTheory.MonoOver.map e.hom).obj X)
pow_nonneg._simp_1
Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
∀ {M₀ : Type u_2} [inst : MonoidWithZero M₀] [inst_1 : Preorder M₀] {a : M₀} [ZeroLEOneClass M₀] [PosMulMono M₀], 0 ≤ a → ∀ (n : ℕ), (0 ≤ a ^ n) = True
Lean.Data.AC.evalList._sparseCasesOn_1.else_eq
Init.Data.AC
∀ {α : Type u} {motive : List α → Sort u_1} (t : List α) (nil : motive []) («else» : Nat.hasNotBit 1 t.ctorIdx → motive t) (h : Nat.hasNotBit 1 t.ctorIdx), Lean.Data.AC.evalList._sparseCasesOn_1 t nil «else» = «else» h
Int.le_of_not_le
Init.Data.Int.Order
∀ {a b : ℤ}, ¬a ≤ b → b ≤ a
Topology.IsLowerSet.WithLowerSetHomeomorph
Mathlib.Topology.Order.UpperLowerSetTopology
{α : Type u_1} → [inst : Preorder α] → [inst_1 : TopologicalSpace α] → [Topology.IsLowerSet α] → Topology.WithLowerSet α ≃ₜ α
Nat.eq_or_lt_of_le._unsafe_rec
Init.Prelude
∀ {n m : ℕ}, n ≤ m → n = m ∨ n < m
ShareCommon.StateFactoryImpl.setFind?
Init.ShareCommon
(self : ShareCommon.StateFactoryImpl) → self.Set → ShareCommon.Object → Option ShareCommon.Object
Equiv.Perm.isCycleOn_swap
Mathlib.GroupTheory.Perm.Cycle.Basic
∀ {α : Type u_2} {a b : α} [inst : DecidableEq α], a ≠ b → (Equiv.swap a b).IsCycleOn {a, b}
Mathlib.Tactic.Ring.evalZSMul._proof_2
Mathlib.Tactic.Ring.Common
∀ {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring «$α»)), «$sα» =Q Int.instCommSemiring