The category of bounded distributive lattices

This defines BddDistLat, the category of bounded distributive lattices.

Note that this category is sometimes called DistLat when being a lattice is understood to entail having a bottom and a top element.

structure BddDistLatextends DistLat : Type (u_1 + 1)

The category of bounded distributive lattices with bounded lattice morphisms.

• carrier : Type u_1
• str : DistribLattice ↑self.toDistLat
• isBoundedOrder : BoundedOrder ↑self.toDistLat

instance BddDistLat.instCoeSortType : CoeSort BddDistLat (Type u_1)

Equations

• BddDistLat.instCoeSortType = { coe := fun (X : BddDistLat) => ↑X.toDistLat }

instance BddDistLat.instDistribLatticeCarrier (X : BddDistLat) : DistribLattice ↑X.toDistLat

Equations

• X.instDistribLatticeCarrier = X.str

@[reducible, inline]

abbrev BddDistLat.of (α : Type u_1) [DistribLattice α] [BoundedOrder α] : BddDistLat

Construct a bundled BddDistLat from a BoundedOrder DistribLattice.

Equations

• BddDistLat.of α = { carrier := α, str := inst✝¹, isBoundedOrder := inst✝ }

theorem BddDistLat.coe_of (α : Type u_1) [DistribLattice α] [BoundedOrder α] : ↑(of α).toDistLat = α

structure BddDistLat.Hom (X Y : BddDistLat) : Type u

The type of morphisms in BddDistLat R.

• hom' : BoundedLatticeHom ↑X.toDistLat ↑Y.toDistLat

  The underlying BoundedLatticeHom.

theorem BddDistLat.Hom.ext {X Y : BddDistLat} {x y : X.Hom Y} (hom' : x.hom' = y.hom') : x = y

theorem BddDistLat.Hom.ext_iff {X Y : BddDistLat} {x y : X.Hom Y} : x = y ↔ x.hom' = y.hom'

instance BddDistLat.instCategory : CategoryTheory.Category.{u, u + 1} BddDistLat

Equations

• One or more equations did not get rendered due to their size.

instance BddDistLat.instConcreteCategoryBoundedLatticeHomCarrier : CategoryTheory.ConcreteCategory BddDistLat fun (x1 x2 : BddDistLat) => BoundedLatticeHom ↑x1.toDistLat ↑x2.toDistLat

Equations

• One or more equations did not get rendered due to their size.

@[reducible, inline]

abbrev BddDistLat.Hom.hom {X Y : BddDistLat} (f : X.Hom Y) : BoundedLatticeHom ↑X.toDistLat ↑Y.toDistLat

Turn a morphism in BddDistLat back into a BoundedLatticeHom.

Equations

• f.hom = CategoryTheory.ConcreteCategory.hom f

@[reducible, inline]

abbrev BddDistLat.ofHom {X Y : Type u} [DistribLattice X] [BoundedOrder X] [DistribLattice Y] [BoundedOrder Y] (f : BoundedLatticeHom X Y) : of X ⟶ of Y

Typecheck a BoundedLatticeHom as a morphism in BddDistLat.

Equations

• BddDistLat.ofHom f = CategoryTheory.ConcreteCategory.ofHom f

def BddDistLat.Hom.Simps.hom (X Y : BddDistLat) (f : X.Hom Y) : BoundedLatticeHom ↑X.toDistLat ↑Y.toDistLat

Use the ConcreteCategory.hom projection for @[simps] lemmas.

Equations

• BddDistLat.Hom.Simps.hom X Y f = f.hom

The results below duplicate the ConcreteCategory simp lemmas, but we can keep them for dsimp.

@[simp]

theorem BddDistLat.coe_id {X : BddDistLat} : ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) = id

@[simp]

theorem BddDistLat.coe_comp {X Y Z : BddDistLat} {f : X ⟶ Y} {g : Y ⟶ Z} : ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.comp f g)) = ⇑(CategoryTheory.ConcreteCategory.hom g) ∘ ⇑(CategoryTheory.ConcreteCategory.hom f)

@[simp]

theorem BddDistLat.forget_map {X Y : BddDistLat} (f : X ⟶ Y) : (CategoryTheory.forget BddDistLat).map f = ⇑(CategoryTheory.ConcreteCategory.hom f)

theorem BddDistLat.ext {X Y : BddDistLat} {f g : X ⟶ Y} (w : ∀ (x : ↑X.toDistLat), (CategoryTheory.ConcreteCategory.hom f) x = (CategoryTheory.ConcreteCategory.hom g) x) : f = g

theorem BddDistLat.ext_iff {X Y : BddDistLat} {f g : X ⟶ Y} : f = g ↔ ∀ (x : ↑X.toDistLat), (CategoryTheory.ConcreteCategory.hom f) x = (CategoryTheory.ConcreteCategory.hom g) x

@[simp]

theorem BddDistLat.hom_id {X : BddDistLat} : Hom.hom (CategoryTheory.CategoryStruct.id X) = BoundedLatticeHom.id ↑X.toDistLat

theorem BddDistLat.id_apply (X : BddDistLat) (x : ↑X.toDistLat) : (CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) x = x

@[simp]

theorem BddDistLat.hom_comp {X Y Z : BddDistLat} (f : X ⟶ Y) (g : Y ⟶ Z) : Hom.hom (CategoryTheory.CategoryStruct.comp f g) = (Hom.hom g).comp (Hom.hom f)

theorem BddDistLat.comp_apply {X Y Z : BddDistLat} (f : X ⟶ Y) (g : Y ⟶ Z) (x : ↑X.toDistLat) : (CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.comp f g)) x = (CategoryTheory.ConcreteCategory.hom g) ((CategoryTheory.ConcreteCategory.hom f) x)

theorem BddDistLat.hom_ext {X Y : BddDistLat} {f g : X ⟶ Y} (hf : Hom.hom f = Hom.hom g) : f = g

theorem BddDistLat.hom_ext_iff {X Y : BddDistLat} {f g : X ⟶ Y} : f = g ↔ Hom.hom f = Hom.hom g

@[simp]

theorem BddDistLat.hom_ofHom {X Y : Type u} [DistribLattice X] [BoundedOrder X] [DistribLattice Y] [BoundedOrder Y] (f : BoundedLatticeHom X Y) : Hom.hom (ofHom f) = f

@[simp]

theorem BddDistLat.ofHom_hom {X Y : BddDistLat} (f : X ⟶ Y) : ofHom (Hom.hom f) = f

@[simp]

theorem BddDistLat.ofHom_id {X : Type u} [DistribLattice X] [BoundedOrder X] : ofHom (BoundedLatticeHom.id X) = CategoryTheory.CategoryStruct.id (of X)

@[simp]

theorem BddDistLat.ofHom_comp {X Y Z : Type u} [DistribLattice X] [BoundedOrder X] [DistribLattice Y] [BoundedOrder Y] [DistribLattice Z] [BoundedOrder Z] (f : BoundedLatticeHom X Y) (g : BoundedLatticeHom Y Z) : ofHom (g.comp f) = CategoryTheory.CategoryStruct.comp (ofHom f) (ofHom g)

theorem BddDistLat.ofHom_apply {X Y : Type u} [DistribLattice X] [BoundedOrder X] [DistribLattice Y] [BoundedOrder Y] (f : BoundedLatticeHom X Y) (x : X) : (CategoryTheory.ConcreteCategory.hom (ofHom f)) x = f x

theorem BddDistLat.inv_hom_apply {X Y : BddDistLat} (e : X ≅ Y) (x : ↑X.toDistLat) : (CategoryTheory.ConcreteCategory.hom e.inv) ((CategoryTheory.ConcreteCategory.hom e.hom) x) = x

theorem BddDistLat.hom_inv_apply {X Y : BddDistLat} (e : X ≅ Y) (s : ↑Y.toDistLat) : (CategoryTheory.ConcreteCategory.hom e.hom) ((CategoryTheory.ConcreteCategory.hom e.inv) s) = s

instance BddDistLat.instInhabited : Inhabited BddDistLat

Equations

• BddDistLat.instInhabited = { default := BddDistLat.of PUnit.{?u.2 + 1} }

def BddDistLat.toBddLat (X : BddDistLat) : BddLat

Turn a BddDistLat into a BddLat by forgetting it is distributive.

Equations

• X.toBddLat = BddLat.of ↑X.toDistLat

@[simp]

theorem BddDistLat.coe_toBddLat (X : BddDistLat) : ↑X.toBddLat.toLat = ↑X.toDistLat

instance BddDistLat.hasForgetToDistLat : CategoryTheory.HasForget₂ BddDistLat DistLat

Equations

• One or more equations did not get rendered due to their size.

instance BddDistLat.hasForgetToBddLat : CategoryTheory.HasForget₂ BddDistLat BddLat

Equations

• One or more equations did not get rendered due to their size.

theorem BddDistLat.forget_bddLat_lat_eq_forget_distLat_lat : (CategoryTheory.forget₂ BddDistLat BddLat).comp (CategoryTheory.forget₂ BddLat Lat) = (CategoryTheory.forget₂ BddDistLat DistLat).comp (CategoryTheory.forget₂ DistLat Lat)

def BddDistLat.Iso.mk {α β : BddDistLat} (e : ↑α.toDistLat ≃o ↑β.toDistLat) : α ≅ β

Constructs an equivalence between bounded distributive lattices from an order isomorphism between them.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem BddDistLat.Iso.mk_hom {α β : BddDistLat} (e : ↑α.toDistLat ≃o ↑β.toDistLat) : (mk e).hom = ofHom (have __src := { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ }; { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_bot' := ⋯ })

@[simp]

theorem BddDistLat.Iso.mk_inv {α β : BddDistLat} (e : ↑α.toDistLat ≃o ↑β.toDistLat) : (mk e).inv = ofHom (have __src := { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ }; { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_bot' := ⋯ })

def BddDistLat.dual : CategoryTheory.Functor BddDistLat BddDistLat

OrderDual as a functor.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem BddDistLat.dual_map {X✝ Y✝ : BddDistLat} (f : X✝ ⟶ Y✝) : dual.map f = ofHom (BoundedLatticeHom.dual (Hom.hom f))

def BddDistLat.dualEquiv : BddDistLat ≌ BddDistLat

The equivalence between BddDistLat and itself induced by OrderDual both ways.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem BddDistLat.dualEquiv_functor : dualEquiv.functor = dual

@[simp]

theorem BddDistLat.dualEquiv_inverse : dualEquiv.inverse = dual

theorem bddDistLat_dual_comp_forget_to_distLat : BddDistLat.dual.comp (CategoryTheory.forget₂ BddDistLat DistLat) = (CategoryTheory.forget₂ BddDistLat DistLat).comp DistLat.dual