name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
Lean.Grind.AC.Seq.eraseDup._sunfold | Init.Grind.AC | Lean.Grind.AC.Seq → Lean.Grind.AC.Seq |
Nat.floor_lt' | Mathlib.Algebra.Order.Floor.Semiring | ∀ {R : Type u_1} [inst : Semiring R] [inst_1 : LinearOrder R] [inst_2 : FloorSemiring R] {a : R} {n : ℕ}
[IsStrictOrderedRing R], n ≠ 0 → (⌊a⌋₊ < n ↔ a < ↑n) |
_private.Mathlib.Data.List.TFAE.0.List.exists_tfae._simp_1_2 | Mathlib.Data.List.TFAE | ∀ {α : Type u_1} {β : Type u_2} {f : α → β} {l : List α} {P : β → Prop}, (∀ i ∈ List.map f l, P i) = ∀ j ∈ l, P (f j) |
CategoryTheory.Limits.IsImage.fac_lift | Mathlib.CategoryTheory.Limits.Shapes.Images | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} {f : X ⟶ Y}
{F : CategoryTheory.Limits.MonoFactorisation f} (hF : CategoryTheory.Limits.IsImage F)
(F' : CategoryTheory.Limits.MonoFactorisation f), CategoryTheory.CategoryStruct.comp F.e (hF.lift F') = F'.e |
MulMemClass.toSemigroup.eq_1 | Mathlib.Algebra.Group.Subsemigroup.Defs | ∀ {M : Type u_4} [inst : Semigroup M] {A : Type u_5} [inst_1 : SetLike A M] [inst_2 : MulMemClass A M] (S : A),
MulMemClass.toSemigroup S = { toMul := MulMemClass.mul S, mul_assoc := ⋯ } |
_private.Mathlib.Order.Interval.Set.Disjoint.0.Set.iUnion_Ico_right._simp_1_1 | Mathlib.Order.Interval.Set.Disjoint | ∀ {α : Type u_1} [inst : Preorder α] {a b : α}, Set.Ico a b = Set.Ici a ∩ Set.Iio b |
Pi.single_apply | Mathlib.Algebra.Notation.Pi.Basic | ∀ {ι : Type u_1} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : Zero M] (i : ι) (x : M) (i' : ι),
Pi.single i x i' = if i' = i then x else 0 |
Lean.Elab.Term.PatternVarDecl.fvarId | Lean.Elab.Match | Lean.Elab.Term.PatternVarDecl → Lean.FVarId |
_private.Init.Meta.Defs.0.Lean.Syntax.instBEqPreresolved.beq._sparseCasesOn_2 | Init.Meta.Defs | {motive : Lean.Syntax.Preresolved → Sort u} →
(t : Lean.Syntax.Preresolved) →
((n : Lean.Name) → (fields : List String) → motive (Lean.Syntax.Preresolved.decl n fields)) →
(Nat.hasNotBit 2 t.ctorIdx → motive t) → motive t |
CategoryTheory.Monad.forget_creates_colimits_of_monad_preserves | Mathlib.CategoryTheory.Monad.Limits | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u}
[inst_1 : CategoryTheory.Category.{v, u} J] [CategoryTheory.Limits.PreservesColimitsOfShape J T.toFunctor]
(D : CategoryTheory.Functor J T.Algebra) [CategoryTheory.Limits.HasColimit (D.comp T.forget)],
CategoryTheory.Limits.HasColimit D |
WeierstrassCurve.Jacobian.map_polynomialZ | Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | ∀ {R : Type r} {S : Type s} [inst : CommRing R] [inst_1 : CommRing S] {W' : WeierstrassCurve.Jacobian R} (f : R →+* S),
(WeierstrassCurve.map W' f).toJacobian.polynomialZ = (MvPolynomial.map f) W'.polynomialZ |
Cardinal.toENat_lt_top._simp_1 | Mathlib.SetTheory.Cardinal.ENat | ∀ {c : Cardinal.{u}}, (Cardinal.toENat c < ⊤) = (c < Cardinal.aleph0) |
add_sub_add_right_eq_sub | Mathlib.Algebra.Group.Basic | ∀ {G : Type u_3} [inst : AddGroup G] (a b c : G), a + c - (b + c) = a - b |
Primrec.option_some_iff | Mathlib.Computability.Primrec.Basic | ∀ {α : Type u_1} {σ : Type u_3} [inst : Primcodable α] [inst_1 : Primcodable σ] {f : α → σ},
(Primrec fun a => some (f a)) ↔ Primrec f |
MultilinearMap.piFamilyₗ._proof_2 | Mathlib.LinearAlgebra.Multilinear.Pi | ∀ {ι : Type u_2} {κ : ι → Type u_3} {R : Type u_1} {N : ((i : ι) → κ i) → Type u_4} [inst : CommSemiring R]
[inst_1 : (p : (i : ι) → κ i) → AddCommMonoid (N p)] [inst_2 : (p : (i : ι) → κ i) → Module R (N p)],
SMulCommClass R R ((i : (i : ι) → κ i) → N i) |
CategoryTheory.Functor.CommShift.id_commShiftIso_inv_app | Mathlib.CategoryTheory.Shift.CommShift | ∀ (C : Type u_1) [inst : CategoryTheory.Category.{v_1, u_1} C] {A : Type u_4} [inst_1 : AddMonoid A]
[inst_2 : CategoryTheory.HasShift C A] (x : A) (X : C),
(CategoryTheory.Functor.commShiftIso (CategoryTheory.Functor.id C) x).inv.app X =
CategoryTheory.CategoryStruct.id ((CategoryTheory.shiftFunctor C x).obj X) |
Filter.atBot_eq_pure_of_isBot | Mathlib.Order.Filter.Ultrafilter.Basic | ∀ {α : Type u} [inst : PartialOrder α] {x : α}, IsBot x → Filter.atBot = pure x |
AddCommGroup.casesOn | Mathlib.Algebra.Group.Defs | {G : Type u} →
{motive : AddCommGroup G → Sort u_1} →
(t : AddCommGroup G) →
([toAddGroup : AddGroup G] →
(add_comm : ∀ (a b : G), a + b = b + a) → motive { toAddGroup := toAddGroup, add_comm := add_comm }) →
motive t |
Representation.IntertwiningMap.instAddCommGroup._proof_2 | Mathlib.RepresentationTheory.Intertwining | ∀ {A : Type u_3} {G : Type u_4} [inst : Semiring A] [inst_1 : Monoid G] {V : Type u_1} {W : Type u_2}
[inst_2 : AddCommMonoid V] [inst_3 : AddCommGroup W] [inst_4 : Module A V] [inst_5 : Module A W]
(ρ : Representation A G V) (σ : Representation A G W), ⇑0 = 0 |
Filter.exists_subsingleton_mem_of_forall_separating | Mathlib.Order.Filter.CountableSeparatingOn | ∀ {α : Type u_1} {l : Filter α} [CountableInterFilter l] (p : Set α → Prop) [HasCountableSeparatingOn α p Set.univ],
(∀ (U : Set α), p U → U ∈ l ∨ Uᶜ ∈ l) → ∃ s, s.Subsingleton ∧ s ∈ l |
UpperSet.mem_Ici_iff | Mathlib.Order.UpperLower.Principal | ∀ {α : Type u_1} [inst : Preorder α] {a b : α}, b ∈ UpperSet.Ici a ↔ a ≤ b |
CategoryTheory.Localization.SmallShiftedHom.mk | Mathlib.CategoryTheory.Localization.SmallShiftedHom | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
(W : CategoryTheory.MorphismProperty C) →
{M : Type w'} →
[inst_1 : AddMonoid M] →
[inst_2 : CategoryTheory.HasShift C M] →
{X Y : C} →
{m : M} →
[inst_3 : CategoryTheory.Localization.HasSmallLocalizedShiftedHom W M X Y] →
CategoryTheory.ShiftedHom X Y m → CategoryTheory.Localization.SmallShiftedHom W X Y m |
Lean.Meta.Grind.Arith.Linear.NatStruct.u | Lean.Meta.Tactic.Grind.Arith.Linear.Types | Lean.Meta.Grind.Arith.Linear.NatStruct → Lean.Level |
Qq.Impl.withUnquotedLCtx | Qq.Macro | {m : Type → Type u_1} →
{α : Type} → [MonadControlT Lean.MetaM m] → [Monad m] → [MonadLiftT Qq.Impl.QuoteM m] → m α → m α |
_private.Lean.Elab.SyntheticMVars.0.Lean.Elab.Term.reportStuckSyntheticMVar.match_3 | Lean.Elab.SyntheticMVars | (motive : Option (Lean.MVarId → Lean.Expr → Lean.Expr → Lean.MetaM Lean.MessageData) → Sort u_1) →
(mkErrorMsg? : Option (Lean.MVarId → Lean.Expr → Lean.Expr → Lean.MetaM Lean.MessageData)) →
((mkErrorMsg : Lean.MVarId → Lean.Expr → Lean.Expr → Lean.MetaM Lean.MessageData) → motive (some mkErrorMsg)) →
((x : Option (Lean.MVarId → Lean.Expr → Lean.Expr → Lean.MetaM Lean.MessageData)) → motive x) → motive mkErrorMsg? |
_private.Mathlib.RingTheory.Noetherian.Defs.0.isNoetherian_submodule.match_1_1 | Mathlib.RingTheory.Noetherian.Defs | ∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {N : Submodule R M}
(motive : IsNoetherian R ↥N → Prop) (x : IsNoetherian R ↥N),
(∀ (hn : ∀ (s : Submodule R ↥N), s.FG), motive ⋯) → motive x |
Std.IterM.Total.mk._flat_ctor | Init.Data.Iterators.Consumers.Monadic.Total | {α : Type w} → {m : Type w → Type w'} → {β : Type w} → Std.IterM m β → Std.IterM.Total m β |
Lean.Meta.DefEqCache | Lean.Meta.Basic | Type |
AlgebraicGeometry.IsImmersion.isLocallyClosed_range | Mathlib.AlgebraicGeometry.Morphisms.Immersion | ∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} [self : AlgebraicGeometry.IsImmersion f], IsLocallyClosed (Set.range ⇑f) |
OreLocalization.lift₂Expand.eq_1 | Mathlib.GroupTheory.OreLocalization.Basic | ∀ {R : Type u_1} [inst : Monoid R] {S : Submonoid R} [inst_1 : OreLocalization.OreSet S] {X : Type u_3}
[inst_2 : MulAction R X] {C : Sort u_2} (P : X → ↥S → X → ↥S → C)
(hP :
∀ (r₁ : X) (t₁ : R) (s₁ : ↥S) (ht₁ : t₁ * ↑s₁ ∈ S) (r₂ : X) (t₂ : R) (s₂ : ↥S) (ht₂ : t₂ * ↑s₂ ∈ S),
P r₁ s₁ r₂ s₂ = P (t₁ • r₁) ⟨t₁ * ↑s₁, ht₁⟩ (t₂ • r₂) ⟨t₂ * ↑s₂, ht₂⟩),
OreLocalization.lift₂Expand P hP = OreLocalization.liftExpand (fun r₁ s₁ => OreLocalization.liftExpand (P r₁ s₁) ⋯) ⋯ |
Int.sub_lt_sub_left | Init.Data.Int.Order | ∀ {a b : ℤ}, a < b → ∀ (c : ℤ), c - b < c - a |
CategoryTheory.ShortComplex.exact_map_iff_of_faithful | Mathlib.Algebra.Homology.ShortComplex.Exact | ∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_3 : CategoryTheory.Limits.HasZeroMorphisms D] (S : CategoryTheory.ShortComplex C) [S.HasHomology]
(F : CategoryTheory.Functor C D) [inst_5 : F.PreservesZeroMorphisms] [F.PreservesLeftHomologyOf S]
[F.PreservesRightHomologyOf S] [F.Faithful], (S.map F).Exact ↔ S.Exact |
DifferentiableAt.comp_mdifferentiableWithinAt | Mathlib.Geometry.Manifold.MFDeriv.NormedSpace | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
[inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {F : Type u_8} [inst_6 : NormedAddCommGroup F]
[inst_7 : NormedSpace 𝕜 F] {F' : Type u_11} [inst_8 : NormedAddCommGroup F'] [inst_9 : NormedSpace 𝕜 F'] {g : F → F'}
{f : M → F} {s : Set M} {x : M}, DifferentiableAt 𝕜 g (f x) → MDiffAt[s] f x → MDiffAt[s] (g ∘ f) x |
indicator_ae_eq_of_ae_eq_set | Mathlib.MeasureTheory.Measure.Restrict | ∀ {α : Type u_2} {β : Type u_3} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α} {f : α → β}
[inst_1 : Zero β], s =ᵐ[μ] t → s.indicator f =ᵐ[μ] t.indicator f |
commMonTypeEquivalenceCommMon._proof_4 | Mathlib.CategoryTheory.Monoidal.Internal.Types.Basic | ∀ (A : CommMonCat) (x x_1 : (CommMonTypeEquivalenceCommMon.inverse.obj A).X),
(Equiv.refl (CommMonTypeEquivalenceCommMon.inverse.obj A).X).toFun (x * x_1) =
(Equiv.refl (CommMonTypeEquivalenceCommMon.inverse.obj A).X).toFun (x * x_1) |
AffineSubspace.pointwise_vadd_direction | Mathlib.LinearAlgebra.AffineSpace.Pointwise | ∀ {k : Type u_2} {V : Type u_3} {P : Type u_4} [inst : Ring k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
[inst_3 : AddTorsor V P] (v : V) (s : AffineSubspace k P), (v +ᵥ s).direction = s.direction |
Lean.Meta.SimpEntry.toUnfold.sizeOf_spec | Lean.Meta.Tactic.Simp.SimpTheorems | ∀ (a : Lean.Name), sizeOf (Lean.Meta.SimpEntry.toUnfold a) = 1 + sizeOf a |
CategoryTheory.IsRegularMono | Mathlib.CategoryTheory.Limits.Shapes.RegularMono | {C : Type u₁} → [inst : CategoryTheory.Category.{v₁, u₁} C] → {X Y : C} → (X ⟶ Y) → Prop |
PUnit.mulSemiringAction._proof_2 | Mathlib.Algebra.Module.PUnit | ∀ {R : Type u_2} [inst : Semiring R] (r : R) (x y : PUnit.{u_1 + 1}), r • (x * y) = r • x * r • y |
Mathlib.Tactic.Order.Graph.TarjanState | Mathlib.Tactic.Order.Graph.Tarjan | Type |
_private.Mathlib.NumberTheory.LSeries.HurwitzZeta.0.HurwitzZeta.sinZeta_eq._simp_1_2 | Mathlib.NumberTheory.LSeries.HurwitzZeta | ∀ {α : Type u_2} [inst : Zero α] [inst_1 : OfNat α 2] [NeZero 2], (2 = 0) = False |
Lean.Compiler.CSimp.State.mk.injEq | Lean.Compiler.CSimpAttr | ∀ (map : Lean.SMap Lean.Name Lean.Compiler.CSimp.Entry) (thmNames : Lean.SSet Lean.Name)
(map_1 : Lean.SMap Lean.Name Lean.Compiler.CSimp.Entry) (thmNames_1 : Lean.SSet Lean.Name),
({ map := map, thmNames := thmNames } = { map := map_1, thmNames := thmNames_1 }) =
(map = map_1 ∧ thmNames = thmNames_1) |
Lean.Elab.Term.ElabElim.State.mk._flat_ctor | Lean.Elab.App | Lean.Expr →
Lean.Expr →
List Lean.Elab.Term.NamedArg →
List Lean.Elab.Term.Arg → Array Lean.MVarId → ℕ → Option Lean.Expr → Lean.Elab.Term.ElabElim.State |
Std.DHashMap.getD.congr_simp | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : BEq α} {x_1 : Hashable α} [inst : LawfulBEq α] (m m_1 : Std.DHashMap α β),
m = m_1 → ∀ (a : α) (fallback fallback_1 : β a), fallback = fallback_1 → m.getD a fallback = m_1.getD a fallback_1 |
NeZero.one_le | Mathlib.Data.Nat.Cast.NeZero | ∀ {n : ℕ} [NeZero n], 1 ≤ n |
GroupTopology.instCompleteLattice._proof_8 | Mathlib.Topology.Algebra.Group.GroupTopology | ∀ {α : Type u_1} [inst : Group α] (s : Set (GroupTopology α)), IsGLB s (sInf s) |
SubgroupClass.instZPow.eq_1 | Mathlib.Algebra.Group.Subgroup.Defs | ∀ {M : Type u_5} {S : Type u_6} [inst : DivInvMonoid M] [inst_1 : SetLike S M] [inst_2 : SubgroupClass S M] {H : S},
SubgroupClass.instZPow = { pow := fun a n => ⟨↑a ^ n, ⋯⟩ } |
_private.Lean.Compiler.LCNF.ExplicitRC.0.Lean.Compiler.LCNF.DerivedValInfo.mk.noConfusion | Lean.Compiler.LCNF.ExplicitRC | {P : Sort u} →
{parent? : Option Lean.FVarId} →
{children : Lean.FVarIdHashSet} →
{parent?' : Option Lean.FVarId} →
{children' : Lean.FVarIdHashSet} →
{ parent? := parent?, children := children } = { parent? := parent?', children := children' } →
(parent? = parent?' → children = children' → P) → P |
CategoryTheory.IsIso.eq_inv_comp | Mathlib.CategoryTheory.Iso | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y Z : C} (α : X ⟶ Y) [inst_1 : CategoryTheory.IsIso α]
{f : X ⟶ Z} {g : Y ⟶ Z},
g = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv α) f ↔ CategoryTheory.CategoryStruct.comp α g = f |
CategoryTheory.Precoverage.mk | Mathlib.CategoryTheory.Sites.Precoverage | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
((X : C) → Set (CategoryTheory.Presieve X)) → CategoryTheory.Precoverage C |
Lean.Compiler.LCNF.LetValue.updateArgs! | Lean.Compiler.LCNF.Basic | {pu : Lean.Compiler.LCNF.Purity} →
Lean.Compiler.LCNF.LetValue pu → Array (Lean.Compiler.LCNF.Arg pu) → Lean.Compiler.LCNF.LetValue pu |
PadicInt.coe_neg | Mathlib.NumberTheory.Padics.PadicIntegers | ∀ {p : ℕ} [hp : Fact (Nat.Prime p)] (z1 : ℤ_[p]), ↑(-z1) = -↑z1 |
Homeomorph.locallyConnectedSpace | Mathlib.Topology.Homeomorph.Lemmas | ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [i : LocallyConnectedSpace Y]
(h : X ≃ₜ Y), LocallyConnectedSpace X |
Lean.Elab.Term.Do.Code.jmp | Lean.Elab.Do.Legacy | Lean.Syntax → Lean.Name → Array Lean.Syntax → Lean.Elab.Term.Do.Code |
FormalMultilinearSeries.id_apply_zero | Mathlib.Analysis.Analytic.Composition | ∀ (𝕜 : Type u_1) (E : Type u_2) [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] (x : E) (v : Fin 0 → E), (FormalMultilinearSeries.id 𝕜 E x 0) v = x |
_private.Mathlib.Analysis.Meromorphic.Divisor.0.MeromorphicOn.divisor_const._simp_1_5 | Mathlib.Analysis.Meromorphic.Divisor | ∀ {p : Prop} [Decidable p], (¬¬p) = p |
LieSubalgebra.root._proof_1 | Mathlib.Algebra.Lie.Weights.Killing | ∀ {K : Type u_1} [inst : Field K], IsDomain K |
_private.Mathlib.Tactic.Ring.Common.0.Mathlib.Tactic.Ring.evalPowProd._sparseCasesOn_9 | Mathlib.Tactic.Ring.Common | {motive : ℤ → Sort u} → (t : ℤ) → ((a : ℕ) → motive (Int.ofNat a)) → (Nat.hasNotBit 1 t.ctorIdx → motive t) → motive t |
_private.Mathlib.Analysis.Convex.Side.0.Affine.Simplex.wSameSide_affineSpan_faceOpposite_point_left_iff._simp_1_3 | Mathlib.Analysis.Convex.Side | ∀ {a b : Prop}, (a ∨ b) = (b ∨ a) |
Topology.IsInducing.hasSum_iff | Mathlib.Topology.Algebra.InfiniteSum.Basic | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : AddCommMonoid α] [inst_1 : TopologicalSpace α]
{L : SummationFilter β} [inst_2 : AddCommMonoid γ] [inst_3 : TopologicalSpace γ] {G : Type u_4}
[inst_4 : FunLike G α γ] [AddMonoidHomClass G α γ] {g : G},
Topology.IsInducing ⇑g → ∀ (f : β → α) (a : α), HasSum (⇑g ∘ f) (g a) L ↔ HasSum f a L |
OrderDual.instCancelCommMonoid.eq_1 | Mathlib.Algebra.Order.Group.Synonym | ∀ {α : Type u_1} [h : CancelCommMonoid α], OrderDual.instCancelCommMonoid = h |
_private.Init.Data.Vector.InsertIdx.0.Vector.getElem_insertIdx_of_gt._proof_1 | Init.Data.Vector.InsertIdx | ∀ {n i k : ℕ}, k ≤ n → k > i → ¬i ≤ n → False |
_private.Mathlib.Algebra.Category.CommBialgCat.0.CommBialgCat.mk.inj | Mathlib.Algebra.Category.CommBialgCat | ∀ {R : Type u} {inst : CommRing R} {carrier : Type v} {commRing : CommRing carrier} {bialgebra : Bialgebra R carrier}
{carrier_1 : Type v} {commRing_1 : CommRing carrier_1} {bialgebra_1 : Bialgebra R carrier_1},
{ carrier := carrier, commRing := commRing, bialgebra := bialgebra } =
{ carrier := carrier_1, commRing := commRing_1, bialgebra := bialgebra_1 } →
carrier = carrier_1 ∧ commRing ≍ commRing_1 ∧ bialgebra ≍ bialgebra_1 |
CategoryTheory.Join.mkNatIso._proof_1 | Mathlib.CategoryTheory.Join.Basic | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_5, u_1} C] {D : Type u_2}
[inst_1 : CategoryTheory.Category.{u_6, u_2} D] {E : Type u_4} [inst_2 : CategoryTheory.Category.{u_3, u_4} E]
{F G : CategoryTheory.Functor (CategoryTheory.Join C D) E}
(eₗ : (CategoryTheory.Join.inclLeft C D).comp F ≅ (CategoryTheory.Join.inclLeft C D).comp G)
(eᵣ : (CategoryTheory.Join.inclRight C D).comp F ≅ (CategoryTheory.Join.inclRight C D).comp G),
CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F)
((CategoryTheory.Prod.snd C D).isoWhiskerLeft eᵣ).hom =
CategoryTheory.CategoryStruct.comp ((CategoryTheory.Prod.fst C D).isoWhiskerLeft eₗ).hom
(CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) G) →
CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) G)
((CategoryTheory.Prod.snd C D).whiskerLeft eᵣ.inv) =
CategoryTheory.CategoryStruct.comp ((CategoryTheory.Prod.fst C D).whiskerLeft eₗ.inv)
(CategoryTheory.Functor.whiskerRight (CategoryTheory.Join.edgeTransform C D) F) |
CategoryTheory.Equalizer.Sieve.firstMap | Mathlib.CategoryTheory.Sites.EqualizerSheafCondition | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
(P : CategoryTheory.Functor Cᵒᵖ (Type (max v u))) →
{X : C} →
(S : CategoryTheory.Sieve X) →
CategoryTheory.Equalizer.FirstObj P S.arrows ⟶ CategoryTheory.Equalizer.Sieve.SecondObj P S |
Filter.atBot_countable_basis | Mathlib.Order.Filter.AtTopBot.CountablyGenerated | ∀ {α : Type u_1} [inst : Preorder α] [IsCodirectedOrder α] [Nonempty α] [Countable α],
Filter.atBot.HasCountableBasis (fun x => True) Set.Iic |
Lean.ParserCompiler.Context.rec | Lean.ParserCompiler | {α : Type} →
{motive : Lean.ParserCompiler.Context α → Sort u} →
((varName : Lean.Name) →
(categoryAttr : Lean.KeyedDeclsAttribute α) →
(combinatorAttr : Lean.ParserCompiler.CombinatorAttribute) →
motive { varName := varName, categoryAttr := categoryAttr, combinatorAttr := combinatorAttr }) →
(t : Lean.ParserCompiler.Context α) → motive t |
Filter.subseq_tendsto_of_neBot | Mathlib.Order.Filter.AtTopBot.CountablyGenerated | ∀ {α : Type u_1} {f : Filter α} [f.IsCountablyGenerated] {u : ℕ → α},
(f ⊓ Filter.map u Filter.atTop).NeBot → ∃ θ, StrictMono θ ∧ Filter.Tendsto (u ∘ θ) Filter.atTop f |
Std.DTreeMap.Internal.Impl.getKeyGT.eq_2 | Std.Data.DTreeMap.Internal.Model | ∀ {α : Type u} {β : α → Type v} [inst : Ord α] [inst_1 : Std.TransOrd α] (k : α) (size : ℕ) (ky : α) (y : β ky)
(l r : Std.DTreeMap.Internal.Impl α β) (ho : (Std.DTreeMap.Internal.Impl.inner size ky y l r).Ordered)
(he : ∃ a ∈ Std.DTreeMap.Internal.Impl.inner size ky y l r, compare a k = Ordering.gt),
Std.DTreeMap.Internal.Impl.getKeyGT k (Std.DTreeMap.Internal.Impl.inner size ky y l r) ho he =
if hkky : compare k ky = Ordering.lt then Std.DTreeMap.Internal.Impl.getKeyGTD k l ky
else Std.DTreeMap.Internal.Impl.getKeyGT k r ⋯ ⋯ |
CategoryTheory.Limits.isBilimitOfTotal._proof_2 | Mathlib.CategoryTheory.Preadditive.Biproducts | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Preadditive C] {J : Type u_3}
[inst_2 : Fintype J] {f : J → C} (b : CategoryTheory.Limits.Bicone f),
∑ j, CategoryTheory.CategoryStruct.comp (b.π j) (b.ι j) = CategoryTheory.CategoryStruct.id b.pt →
∀ (s : CategoryTheory.Limits.Cone (CategoryTheory.Discrete.functor f)) (m : s.pt ⟶ b.toCone.pt),
(∀ (j : CategoryTheory.Discrete J), CategoryTheory.CategoryStruct.comp m (b.toCone.π.app j) = s.π.app j) →
m = ∑ j, CategoryTheory.CategoryStruct.comp (s.π.app { as := j }) (b.ι j) |
RingCon.instNonAssocSemiringQuotient._proof_4 | Mathlib.RingTheory.Congruence.Defs | ∀ {R : Type u_1} [inst : NonAssocSemiring R] (c : RingCon R) (x x_1 : R),
Quotient.mk'' (x * x_1) = Quotient.mk'' (x * x_1) |
Matrix.smul_eq_diagonal_mul | Mathlib.Data.Matrix.Mul | ∀ {m : Type u_2} {n : Type u_3} {α : Type v} [inst : NonUnitalNonAssocSemiring α] [inst_1 : Fintype m]
[inst_2 : DecidableEq m] (M : Matrix m n α) (a : α), a • M = (Matrix.diagonal fun x => a) * M |
ContinuousLinearMap.uncurryBilinear._proof_9 | Mathlib.Analysis.Analytic.Linear | ∀ {G : Type u_1} [inst : NormedAddCommGroup G], ContinuousAdd G |
MonoidAlgebra.support_single_mul_subset | Mathlib.Algebra.MonoidAlgebra.Support | ∀ {k : Type u₁} {G : Type u₂} [inst : Semiring k] [inst_1 : Mul G] [inst_2 : DecidableEq G] (f : MonoidAlgebra k G)
(r : k) (a : G), (MonoidAlgebra.single a r * f).support ⊆ Finset.image (fun x => a * x) f.support |
_private.Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms.0.SimplexCategoryGenRel.IsAdmissible.head._proof_1 | Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms | ∀ {a : ℕ} {L : List ℕ}, 0 < (a :: L).length |
Mathlib.Tactic.Coherence.LiftObj_tensor | Mathlib.Tactic.CategoryTheory.Coherence | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
[inst_1 : CategoryTheory.MonoidalCategory C] →
(X Y : C) →
[Mathlib.Tactic.Coherence.LiftObj X] →
[Mathlib.Tactic.Coherence.LiftObj Y] →
Mathlib.Tactic.Coherence.LiftObj (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y) |
Complex.nnnorm_intCast | Mathlib.Analysis.Complex.Basic | ∀ (n : ℤ), ‖↑n‖₊ = ‖n‖₊ |
_private.Mathlib.Combinatorics.SimpleGraph.Walks.Decomp.0.SimpleGraph.Walk.dropUntil_eq_drop._proof_1_4 | Mathlib.Combinatorics.SimpleGraph.Walks.Decomp | ∀ {V : Type u_1} {G : SimpleGraph V} {w u_1 : V}, w ∈ SimpleGraph.Walk.nil.support → G.Walk u_1 u_1 = G.Walk w u_1 |
MeasureTheory.hahn_decomposition | Mathlib.MeasureTheory.Measure.Decomposition.Hahn | ∀ {α : Type u_1} {mα : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ]
[MeasureTheory.IsFiniteMeasure ν],
∃ s,
MeasurableSet s ∧
(∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t |
HasDerivAt.cos | Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv | ∀ {f : ℝ → ℝ} {f' x : ℝ}, HasDerivAt f f' x → HasDerivAt (fun x => Real.cos (f x)) (-Real.sin (f x) * f') x |
Lean.Declaration.getNames | Lean.Declaration | Lean.Declaration → List Lean.Name |
CategoryTheory.Limits.inv_prodComparison_map_snd | Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{w, u₂} D]
(F : CategoryTheory.Functor C D) {A B : C} [inst_2 : CategoryTheory.Limits.HasBinaryProduct A B]
[inst_3 : CategoryTheory.Limits.HasBinaryProduct (F.obj A) (F.obj B)]
[inst_4 : CategoryTheory.IsIso (CategoryTheory.Limits.prodComparison F A B)],
CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (CategoryTheory.Limits.prodComparison F A B))
(F.map CategoryTheory.Limits.prod.snd) =
CategoryTheory.Limits.prod.snd |
padicValInt.eq_zero_of_not_dvd | Mathlib.NumberTheory.Padics.PadicVal.Basic | ∀ {p : ℕ} {z : ℤ}, ¬↑p ∣ z → padicValInt p z = 0 |
ENat.toNat_zero | Mathlib.Data.ENat.Basic | ENat.toNat 0 = 0 |
SeparationQuotient.continuousOn_lift._simp_1 | Mathlib.Topology.Inseparable | ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y}
{hf : ∀ (x y : X), Inseparable x y → f x = f y} {s : Set (SeparationQuotient X)},
ContinuousOn (SeparationQuotient.lift f hf) s = ContinuousOn f (SeparationQuotient.mk ⁻¹' s) |
compactumToCompHaus.isoOfTopologicalSpace._proof_6 | Mathlib.Topology.Category.Compactum | ∀ {D : CompHaus}, T2Space (Compactum.ofTopologicalSpace ↑D.toTop).A |
wrapped._@.Mathlib.MeasureTheory.Integral.CurveIntegral.Basic.4195796962._hygCtx._hyg.2 | Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | Subtype (Eq @definition✝) |
Std.TreeMap.Raw.getKey_union_of_not_mem_left | Std.Data.TreeMap.Raw.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t₁ t₂ : Std.TreeMap.Raw α β cmp} [inst : Std.TransCmp cmp]
(h₁ : t₁.WF) (h₂ : t₂.WF) {k : α} (not_mem : k ∉ t₁) {h' : k ∈ t₁ ∪ t₂}, (t₁ ∪ t₂).getKey k h' = t₂.getKey k ⋯ |
Nat.toList_rcc_eq_cons_iff | Init.Data.Range.Polymorphic.NatLemmas | ∀ {xs : List ℕ} {m n a : ℕ}, (m...=n).toList = a :: xs ↔ m = a ∧ m ≤ n ∧ ((m + 1)...=n).toList = xs |
CategoryTheory.MonoidalCategory.MonoidalRightAction.mk.noConfusion | Mathlib.CategoryTheory.Monoidal.Action.Basic | {C : Type u_1} →
{D : Type u_2} →
{inst : CategoryTheory.Category.{v_1, u_1} C} →
{inst_1 : CategoryTheory.Category.{v_2, u_2} D} →
{inst_2 : CategoryTheory.MonoidalCategory C} →
{P : Sort u} →
{toMonoidalRightActionStruct : CategoryTheory.MonoidalCategory.MonoidalRightActionStruct C D} →
{actionHom_def :
autoParam
(∀ {c c' : C} {d d' : D} (f : d ⟶ d') (g : c ⟶ c'),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom f g =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomLeft f c)
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight d' g))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHom_def._autoParam} →
{actionHomRight_id :
autoParam
(∀ (c : C) (d : D),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight d
(CategoryTheory.CategoryStruct.id c) =
CategoryTheory.CategoryStruct.id
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionObj d c))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHomRight_id._autoParam} →
{id_actionHomLeft :
autoParam
(∀ (c : C) (d : D),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomLeft
(CategoryTheory.CategoryStruct.id d) c =
CategoryTheory.CategoryStruct.id
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionObj d c))
CategoryTheory.MonoidalCategory.MonoidalRightAction.id_actionHomLeft._autoParam} →
{actionHom_comp :
autoParam
(∀ {c c' c'' : C} {d d' d'' : D} (f₁ : d ⟶ d') (f₂ : d' ⟶ d'') (g₁ : c ⟶ c') (g₂ : c' ⟶ c''),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom
(CategoryTheory.CategoryStruct.comp f₁ f₂) (CategoryTheory.CategoryStruct.comp g₁ g₂) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom f₁ g₁)
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom f₂ g₂))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHom_comp._autoParam} →
{actionAssocIso_hom_naturality :
autoParam
(∀ {d₁ d₂ : D} {c₁ c₂ c₃ c₄ : C} (f : d₁ ⟶ d₂) (g : c₁ ⟶ c₂) (h : c₃ ⟶ c₄),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom f
(CategoryTheory.MonoidalCategoryStruct.tensorHom g h))
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso d₂ c₂
c₄).hom =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso d₁ c₁
c₃).hom
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom f g) h))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionAssocIso_hom_naturality._autoParam} →
{actionUnitIso_hom_naturality :
autoParam
(∀ {d d' : D} (f : d ⟶ d'),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionUnitIso d).hom f =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomLeft f
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C))
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionUnitIso d').hom)
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionUnitIso_hom_naturality._autoParam} →
{actionHomRight_whiskerRight :
autoParam
(∀ {c' c'' : C} (f : c' ⟶ c'') (c : C) (d : D),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight d
(CategoryTheory.MonoidalCategoryStruct.whiskerRight f c) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso d c'
c).hom
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomLeft
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight d f)
c)
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso d c''
c).inv))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHomRight_whiskerRight._autoParam} →
{whiskerRight_actionHomLeft :
autoParam
(∀ (c : C) {c' c'' : C} (f : c' ⟶ c'') (d : D),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight d
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft c f) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso d c
c').hom
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionObj d c) f)
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso d c
c'').inv))
CategoryTheory.MonoidalCategory.MonoidalRightAction.whiskerRight_actionHomLeft._autoParam} →
{actionHom_associator :
autoParam
(∀ (c₁ c₂ c₃ : C) (d : D),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight d
(CategoryTheory.MonoidalCategoryStruct.associator c₁ c₂ c₃).hom)
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso d
c₁ (CategoryTheory.MonoidalCategoryStruct.tensorObj c₂ c₃)).hom
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionObj d
c₁)
c₂ c₃).hom) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso d
(CategoryTheory.MonoidalCategoryStruct.tensorObj c₁ c₂) c₃).hom
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomLeft
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso d
c₁ c₂).hom
c₃))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHom_associator._autoParam} →
{actionHom_leftUnitor :
autoParam
(∀ (c : C) (d : D),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight d
(CategoryTheory.MonoidalCategoryStruct.leftUnitor c).hom =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso d
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C) c).hom
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomLeft
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionUnitIso
d).hom
c))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHom_leftUnitor._autoParam} →
{actionHom_rightUnitor :
autoParam
(∀ (c : C) (d : D),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight d
(CategoryTheory.MonoidalCategoryStruct.rightUnitor c).hom =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
d c (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)).hom
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionUnitIso
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionObj d
c)).hom)
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHom_rightUnitor._autoParam} →
{toMonoidalRightActionStruct' :
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct C D} →
{actionHom_def' :
autoParam
(∀ {c c' : C} {d d' : D} (f : d ⟶ d') (g : c ⟶ c'),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom f g =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomLeft
f c)
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight
d' g))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHom_def._autoParam} →
{actionHomRight_id' :
autoParam
(∀ (c : C) (d : D),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight
d (CategoryTheory.CategoryStruct.id c) =
CategoryTheory.CategoryStruct.id
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionObj
d c))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHomRight_id._autoParam} →
{id_actionHomLeft' :
autoParam
(∀ (c : C) (d : D),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomLeft
(CategoryTheory.CategoryStruct.id d) c =
CategoryTheory.CategoryStruct.id
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionObj
d c))
CategoryTheory.MonoidalCategory.MonoidalRightAction.id_actionHomLeft._autoParam} →
{actionHom_comp' :
autoParam
(∀ {c c' c'' : C} {d d' d'' : D} (f₁ : d ⟶ d') (f₂ : d' ⟶ d'')
(g₁ : c ⟶ c') (g₂ : c' ⟶ c''),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom
(CategoryTheory.CategoryStruct.comp f₁ f₂)
(CategoryTheory.CategoryStruct.comp g₁ g₂) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom
f₁ g₁)
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom
f₂ g₂))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHom_comp._autoParam} →
{actionAssocIso_hom_naturality' :
autoParam
(∀ {d₁ d₂ : D} {c₁ c₂ c₃ c₄ : C} (f : d₁ ⟶ d₂) (g : c₁ ⟶ c₂)
(h : c₃ ⟶ c₄),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom
f (CategoryTheory.MonoidalCategoryStruct.tensorHom g h))
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
d₂ c₂ c₄).hom =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
d₁ c₁ c₃).hom
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHom
f g)
h))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionAssocIso_hom_naturality._autoParam} →
{actionUnitIso_hom_naturality' :
autoParam
(∀ {d d' : D} (f : d ⟶ d'),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionUnitIso
d).hom
f =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomLeft
f (CategoryTheory.MonoidalCategoryStruct.tensorUnit C))
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionUnitIso
d').hom)
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionUnitIso_hom_naturality._autoParam} →
{actionHomRight_whiskerRight' :
autoParam
(∀ {c' c'' : C} (f : c' ⟶ c'') (c : C) (d : D),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight
d
(CategoryTheory.MonoidalCategoryStruct.whiskerRight f c) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
d c' c).hom
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomLeft
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight
d f)
c)
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
d c'' c).inv))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHomRight_whiskerRight._autoParam} →
{whiskerRight_actionHomLeft' :
autoParam
(∀ (c : C) {c' c'' : C} (f : c' ⟶ c'') (d : D),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight
d
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft c
f) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
d c c').hom
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionObj
d c)
f)
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
d c c'').inv))
CategoryTheory.MonoidalCategory.MonoidalRightAction.whiskerRight_actionHomLeft._autoParam} →
{actionHom_associator' :
autoParam
(∀ (c₁ c₂ c₃ : C) (d : D),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight
d
(CategoryTheory.MonoidalCategoryStruct.associator c₁
c₂ c₃).hom)
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
d c₁
(CategoryTheory.MonoidalCategoryStruct.tensorObj
c₂ c₃)).hom
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionObj
d c₁)
c₂ c₃).hom) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
d
(CategoryTheory.MonoidalCategoryStruct.tensorObj
c₁ c₂)
c₃).hom
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomLeft
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
d c₁ c₂).hom
c₃))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHom_associator._autoParam} →
{actionHom_leftUnitor' :
autoParam
(∀ (c : C) (d : D),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight
d
(CategoryTheory.MonoidalCategoryStruct.leftUnitor
c).hom =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
d
(CategoryTheory.MonoidalCategoryStruct.tensorUnit
C)
c).hom
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomLeft
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionUnitIso
d).hom
c))
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHom_leftUnitor._autoParam} →
{actionHom_rightUnitor' :
autoParam
(∀ (c : C) (d : D),
CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionHomRight
d
(CategoryTheory.MonoidalCategoryStruct.rightUnitor
c).hom =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionAssocIso
d c
(CategoryTheory.MonoidalCategoryStruct.tensorUnit
C)).hom
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionUnitIso
(CategoryTheory.MonoidalCategory.MonoidalRightActionStruct.actionObj
d c)).hom)
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionHom_rightUnitor._autoParam} →
{
toMonoidalRightActionStruct :=
toMonoidalRightActionStruct,
actionHom_def := actionHom_def,
actionHomRight_id := actionHomRight_id,
id_actionHomLeft := id_actionHomLeft,
actionHom_comp := actionHom_comp,
actionAssocIso_hom_naturality :=
actionAssocIso_hom_naturality,
actionUnitIso_hom_naturality :=
actionUnitIso_hom_naturality,
actionHomRight_whiskerRight :=
actionHomRight_whiskerRight,
whiskerRight_actionHomLeft :=
whiskerRight_actionHomLeft,
actionHom_associator := actionHom_associator,
actionHom_leftUnitor := actionHom_leftUnitor,
actionHom_rightUnitor := actionHom_rightUnitor } =
{
toMonoidalRightActionStruct :=
toMonoidalRightActionStruct',
actionHom_def := actionHom_def',
actionHomRight_id := actionHomRight_id',
id_actionHomLeft := id_actionHomLeft',
actionHom_comp := actionHom_comp',
actionAssocIso_hom_naturality :=
actionAssocIso_hom_naturality',
actionUnitIso_hom_naturality :=
actionUnitIso_hom_naturality',
actionHomRight_whiskerRight :=
actionHomRight_whiskerRight',
whiskerRight_actionHomLeft :=
whiskerRight_actionHomLeft',
actionHom_associator := actionHom_associator',
actionHom_leftUnitor := actionHom_leftUnitor',
actionHom_rightUnitor := actionHom_rightUnitor' } →
(toMonoidalRightActionStruct ≍
toMonoidalRightActionStruct' →
P) →
P |
_private.Mathlib.NumberTheory.FLT.Four.0.Fermat42.not_minimal._proof_1_7 | Mathlib.NumberTheory.FLT.Four | ∀ {a : ℤ},
a % 2 = 1 → ∀ (m r s : ℤ), a = r ^ 2 - s ^ 2 → m = r ^ 2 + s ^ 2 → ∀ (k : ℤ), s = k ^ 2 ∨ s = -k ^ 2 → s ^ 2 = k ^ 4 |
Lean.Lsp.instToJsonLeanModule.toJson | Lean.Data.Lsp.Extra | Lean.Lsp.LeanModule → Lean.Json |
LocallyConnectedSpace.open_connected_basis | Mathlib.Topology.Connected.LocallyConnected | ∀ {α : Type u_3} {inst : TopologicalSpace α} [self : LocallyConnectedSpace α] (x : α),
(nhds x).HasBasis (fun s => IsOpen s ∧ x ∈ s ∧ IsConnected s) id |
LinearEquiv.toSpanNonzeroSingleton._proof_3 | Mathlib.LinearAlgebra.Span.Basic | ∀ (R : Type u_1) [inst : Ring R], RingHomInvPair (RingHom.id R) (RingHom.id R) |
ENNReal.toNNReal_toReal_eq | Mathlib.Data.ENNReal.Basic | ∀ (z : ENNReal), z.toReal.toNNReal = z.toNNReal |
_private.Mathlib.Order.Interval.Finset.Basic.0.Finset.Ioc_erase_right._simp_1_1 | Mathlib.Order.Interval.Finset.Basic | ∀ {α : Type u_1} {s₁ s₂ : Finset α}, (s₁ = s₂) = (↑s₁ = ↑s₂) |
TypeVec.toSubtype'._proof_3 | Mathlib.Data.TypeVec | ∀ (n : ℕ) (α : TypeVec.{u_1} (n + 1)) (p : (α.prod α).Arrow (TypeVec.repeat (n + 1) Prop))
(x : (fun i => { x // TypeVec.ofRepeat (p i (TypeVec.prod.mk i x.1 x.2)) }) Fin2.fz),
TypeVec.ofRepeat (p Fin2.fz (TypeVec.prod.mk Fin2.fz (↑x).1 (↑x).2)) = p Fin2.fz ↑x |
AlgebraicGeometry.Scheme.GlueData.ι_isoLocallyRingedSpace_inv | Mathlib.AlgebraicGeometry.Gluing | ∀ (D : AlgebraicGeometry.Scheme.GlueData) (i : D.J),
CategoryTheory.CategoryStruct.comp (D.toLocallyRingedSpaceGlueData.ι i) D.isoLocallyRingedSpace.inv =
AlgebraicGeometry.Scheme.Hom.toLRSHom (D.ι i) |
_private.Mathlib.Analysis.Matrix.Order.0.Matrix.PosSemidef.matrixPreInnerProductSpace._proof_1 | Mathlib.Analysis.Matrix.Order | ∀ {𝕜 : Type u_2} {n : Type u_1} [inst : RCLike 𝕜] [inst_1 : Fintype n] {M : Matrix n n 𝕜},
M.PosSemidef →
∀ (x x_1 : Matrix n n 𝕜), (starRingEnd 𝕜) (x * M * x_1.conjTranspose).trace = (x_1 * M * x.conjTranspose).trace |
Polynomial.toContinuousMapOnAlgHom._proof_6 | Mathlib.Topology.ContinuousMap.Polynomial | ∀ {R : Type u_1} [inst : CommSemiring R] [inst_1 : TopologicalSpace R] [inst_2 : IsTopologicalSemiring R] (X : Set R)
(x : R), ((algebraMap R (Polynomial R)) x).toContinuousMapOn X = (algebraMap R C(↑X, R)) x |
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