Bounded order homomorphisms

This file defines (bounded) order homomorphisms.

We use the DFunLike design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.

Types of morphisms

• TopHom: Maps which preserve ⊤.
• BotHom: Maps which preserve ⊥.
• BoundedOrderHom: Bounded order homomorphisms. Monotone maps which preserve ⊤ and ⊥.

Typeclasses

• TopHomClass
• BotHomClass
• BoundedOrderHomClass

structure TopHom (α : Type u_6) (β : Type u_7) [Top α] [Top β] : Type (max u_6 u_7)

The type of ⊤-preserving functions from α to β.

• toFun : α → β

  The underlying function. The preferred spelling is DFunLike.coe.

• map_top' : self.toFun ⊤ = ⊤

  The function preserves the top element. The preferred spelling is map_top.

theorem TopHom.map_top' {α : Type u_6} {β : Type u_7} [Top α] [Top β] (self : TopHom α β) : self.toFun ⊤ = ⊤

The function preserves the top element. The preferred spelling is map_top.

structure BotHom (α : Type u_6) (β : Type u_7) [Bot α] [Bot β] : Type (max u_6 u_7)

The type of ⊥-preserving functions from α to β.

• toFun : α → β

  The underlying function. The preferred spelling is DFunLike.coe.

• map_bot' : self.toFun ⊥ = ⊥

  The function preserves the bottom element. The preferred spelling is map_bot.

theorem BotHom.map_bot' {α : Type u_6} {β : Type u_7} [Bot α] [Bot β] (self : BotHom α β) : self.toFun ⊥ = ⊥

The function preserves the bottom element. The preferred spelling is map_bot.

structure BoundedOrderHom (α : Type u_6) (β : Type u_7) [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] extends OrderHom : Type (max u_6 u_7)

The type of bounded order homomorphisms from α to β.

• toFun : α → β
• monotone' : Monotone self.toFun
• map_top' : self.toFun ⊤ = ⊤

  The function preserves the top element. The preferred spelling is map_top.

• map_bot' : self.toFun ⊥ = ⊥

  The function preserves the bottom element. The preferred spelling is map_bot.

theorem BoundedOrderHom.map_top' {α : Type u_6} {β : Type u_7} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (self : BoundedOrderHom α β) : self.toFun ⊤ = ⊤

The function preserves the top element. The preferred spelling is map_top.

theorem BoundedOrderHom.map_bot' {α : Type u_6} {β : Type u_7} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (self : BoundedOrderHom α β) : self.toFun ⊥ = ⊥

The function preserves the bottom element. The preferred spelling is map_bot.

class TopHomClass (F : Type u_6) (α : Type u_7) (β : Type u_8) [Top α] [Top β] [FunLike F α β] : Prop

TopHomClass F α β states that F is a type of ⊤-preserving morphisms.

You should extend this class when you extend TopHom.

• map_top : ∀ (f : F), f ⊤ = ⊤

  A TopHomClass morphism preserves the top element.

@[simp]

theorem TopHomClass.map_top {F : Type u_6} {α : Type u_7} {β : Type u_8} [Top α] [Top β] [FunLike F α β] [self : TopHomClass F α β] (f : F) : f ⊤ = ⊤

A TopHomClass morphism preserves the top element.

class BotHomClass (F : Type u_6) (α : Type u_7) (β : Type u_8) [Bot α] [Bot β] [FunLike F α β] : Prop

BotHomClass F α β states that F is a type of ⊥-preserving morphisms.

You should extend this class when you extend BotHom.

• map_bot : ∀ (f : F), f ⊥ = ⊥

  A BotHomClass morphism preserves the bottom element.

@[simp]

theorem BotHomClass.map_bot {F : Type u_6} {α : Type u_7} {β : Type u_8} [Bot α] [Bot β] [FunLike F α β] [self : BotHomClass F α β] (f : F) : f ⊥ = ⊥

A BotHomClass morphism preserves the bottom element.

class BoundedOrderHomClass (F : Type u_6) (α : Type u_7) (β : Type u_8) [LE α] [LE β] [BoundedOrder α] [BoundedOrder β] [FunLike F α β] extends RelHomClass : Prop

BoundedOrderHomClass F α β states that F is a type of bounded order morphisms.

You should extend this class when you extend BoundedOrderHom.

• map_rel : ∀ (f : F) {a b : α}, a ≤ b → f a ≤ f b
• map_top : ∀ (f : F), f ⊤ = ⊤

  Morphisms preserve the top element. The preferred spelling is _root_.map_top.

• map_bot : ∀ (f : F), f ⊥ = ⊥

  Morphisms preserve the bottom element. The preferred spelling is _root_.map_bot.

theorem BoundedOrderHomClass.map_top {F : Type u_6} {α : Type u_7} {β : Type u_8} [LE α] [LE β] [BoundedOrder α] [BoundedOrder β] [FunLike F α β] [self : BoundedOrderHomClass F α β] (f : F) : f ⊤ = ⊤

Morphisms preserve the top element. The preferred spelling is _root_.map_top.

theorem BoundedOrderHomClass.map_bot {F : Type u_6} {α : Type u_7} {β : Type u_8} [LE α] [LE β] [BoundedOrder α] [BoundedOrder β] [FunLike F α β] [self : BoundedOrderHomClass F α β] (f : F) : f ⊥ = ⊥

Morphisms preserve the bottom element. The preferred spelling is _root_.map_bot.

@[instance 100]

instance BoundedOrderHomClass.toTopHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [LE α] [LE β] [BoundedOrder α] [BoundedOrder β] [BoundedOrderHomClass F α β] : TopHomClass F α β

Equations

• ⋯ = ⋯

@[instance 100]

instance BoundedOrderHomClass.toBotHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [LE α] [LE β] [BoundedOrder α] [BoundedOrder β] [BoundedOrderHomClass F α β] : BotHomClass F α β

Equations

• ⋯ = ⋯

@[instance 100]

instance OrderIsoClass.toTopHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [LE α] [OrderTop α] [PartialOrder β] [OrderTop β] [OrderIsoClass F α β] : TopHomClass F α β

Equations

• ⋯ = ⋯

@[instance 100]

instance OrderIsoClass.toBotHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [LE α] [OrderBot α] [PartialOrder β] [OrderBot β] [OrderIsoClass F α β] : BotHomClass F α β

Equations

• ⋯ = ⋯

@[instance 100]

instance OrderIsoClass.toBoundedOrderHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [LE α] [BoundedOrder α] [PartialOrder β] [BoundedOrder β] [OrderIsoClass F α β] : BoundedOrderHomClass F α β

Equations

• ⋯ = ⋯

@[simp]

theorem map_eq_top_iff {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [LE α] [OrderTop α] [PartialOrder β] [OrderTop β] [OrderIsoClass F α β] (f : F) {a : α} : f a = ⊤ ↔ a = ⊤

@[simp]

theorem map_eq_bot_iff {F : Type u_1} {α : Type u_2} {β : Type u_3} [EquivLike F α β] [LE α] [OrderBot α] [PartialOrder β] [OrderBot β] [OrderIsoClass F α β] (f : F) {a : α} : f a = ⊥ ↔ a = ⊥

def TopHomClass.toTopHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [Top α] [Top β] [TopHomClass F α β] (f : F) : TopHom α β

Turn an element of a type F satisfying TopHomClass F α β into an actual TopHom. This is declared as the default coercion from F to TopHom α β.

Equations

• ↑f = { toFun := ⇑f, map_top' := ⋯ }

instance instCoeTCTopHomOfTopHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [Top α] [Top β] [TopHomClass F α β] : CoeTC F (TopHom α β)

Equations

• instCoeTCTopHomOfTopHomClass = { coe := TopHomClass.toTopHom }

def BotHomClass.toBotHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [Bot α] [Bot β] [BotHomClass F α β] (f : F) : BotHom α β

Turn an element of a type F satisfying BotHomClass F α β into an actual BotHom. This is declared as the default coercion from F to BotHom α β.

Equations

• ↑f = { toFun := ⇑f, map_bot' := ⋯ }

instance instCoeTCBotHomOfBotHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [Bot α] [Bot β] [BotHomClass F α β] : CoeTC F (BotHom α β)

Equations

• instCoeTCBotHomOfBotHomClass = { coe := BotHomClass.toBotHom }

def BoundedOrderHomClass.toBoundedOrderHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] [BoundedOrderHomClass F α β] (f : F) : BoundedOrderHom α β

Turn an element of a type F satisfying BoundedOrderHomClass F α β into an actual BoundedOrderHom. This is declared as the default coercion from F to BoundedOrderHom α β.

Equations

• ↑f = { toFun := ⇑f, monotone' := ⋯, map_top' := ⋯, map_bot' := ⋯ }

instance instCoeTCBoundedOrderHomOfBoundedOrderHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] [BoundedOrderHomClass F α β] : CoeTC F (BoundedOrderHom α β)

Equations

• instCoeTCBoundedOrderHomOfBoundedOrderHomClass = { coe := BoundedOrderHomClass.toBoundedOrderHom }

Top homomorphisms

instance TopHom.instFunLike {α : Type u_2} {β : Type u_3} [Top α] [Top β] : FunLike (TopHom α β) α β

Equations

• TopHom.instFunLike = { coe := TopHom.toFun, coe_injective' := ⋯ }

instance TopHom.instTopHomClass {α : Type u_2} {β : Type u_3} [Top α] [Top β] : TopHomClass (TopHom α β) α β

Equations

• ⋯ = ⋯

theorem TopHom.ext_iff {α : Type u_2} {β : Type u_3} [Top α] [Top β] {f : TopHom α β} {g : TopHom α β} : f = g ↔ ∀ (a : α), f a = g a

theorem TopHom.ext {α : Type u_2} {β : Type u_3} [Top α] [Top β] {f : TopHom α β} {g : TopHom α β} (h : ∀ (a : α), f a = g a) : f = g

def TopHom.copy {α : Type u_2} {β : Type u_3} [Top α] [Top β] (f : TopHom α β) (f' : α → β) (h : f' = ⇑f) : TopHom α β

Copy of a TopHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toFun := f', map_top' := ⋯ }

@[simp]

theorem TopHom.coe_copy {α : Type u_2} {β : Type u_3} [Top α] [Top β] (f : TopHom α β) (f' : α → β) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem TopHom.copy_eq {α : Type u_2} {β : Type u_3} [Top α] [Top β] (f : TopHom α β) (f' : α → β) (h : f' = ⇑f) : f.copy f' h = f

instance TopHom.instInhabited {α : Type u_2} {β : Type u_3} [Top α] [Top β] : Inhabited (TopHom α β)

Equations

• TopHom.instInhabited = { default := { toFun := fun (x : α) => ⊤, map_top' := ⋯ } }

def TopHom.id (α : Type u_2) [Top α] : TopHom α α

id as a TopHom.

Equations

• TopHom.id α = { toFun := id, map_top' := ⋯ }

@[simp]

theorem TopHom.coe_id (α : Type u_2) [Top α] : ⇑(TopHom.id α) = id

@[simp]

theorem TopHom.id_apply {α : Type u_2} [Top α] (a : α) : (TopHom.id α) a = a

def TopHom.comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Top α] [Top β] [Top γ] (f : TopHom β γ) (g : TopHom α β) : TopHom α γ

Composition of TopHoms as a TopHom.

Equations

• f.comp g = { toFun := ⇑f ∘ ⇑g, map_top' := ⋯ }

@[simp]

theorem TopHom.coe_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Top α] [Top β] [Top γ] (f : TopHom β γ) (g : TopHom α β) : ⇑(f.comp g) = ⇑f ∘ ⇑g

@[simp]

theorem TopHom.comp_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Top α] [Top β] [Top γ] (f : TopHom β γ) (g : TopHom α β) (a : α) : (f.comp g) a = f (g a)

@[simp]

theorem TopHom.comp_assoc {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [Top α] [Top β] [Top γ] [Top δ] (f : TopHom γ δ) (g : TopHom β γ) (h : TopHom α β) : (f.comp g).comp h = f.comp (g.comp h)

@[simp]

theorem TopHom.comp_id {α : Type u_2} {β : Type u_3} [Top α] [Top β] (f : TopHom α β) : f.comp (TopHom.id α) = f

@[simp]

theorem TopHom.id_comp {α : Type u_2} {β : Type u_3} [Top α] [Top β] (f : TopHom α β) : (TopHom.id β).comp f = f

@[simp]

theorem TopHom.cancel_right {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Top α] [Top β] [Top γ] {g₁ : TopHom β γ} {g₂ : TopHom β γ} {f : TopHom α β} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂

@[simp]

theorem TopHom.cancel_left {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Top α] [Top β] [Top γ] {g : TopHom β γ} {f₁ : TopHom α β} {f₂ : TopHom α β} (hg : Function.Injective ⇑g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

instance TopHom.instLE {α : Type u_2} {β : Type u_3} [Top α] [LE β] [Top β] : LE (TopHom α β)

Equations

• TopHom.instLE = { le := fun (f g : TopHom α β) => ⇑f ≤ ⇑g }

instance TopHom.instPreorder {α : Type u_2} {β : Type u_3} [Top α] [Preorder β] [Top β] : Preorder (TopHom α β)

Equations

• TopHom.instPreorder = Preorder.lift DFunLike.coe

instance TopHom.instPartialOrder {α : Type u_2} {β : Type u_3} [Top α] [PartialOrder β] [Top β] : PartialOrder (TopHom α β)

Equations

• TopHom.instPartialOrder = PartialOrder.lift (fun (f : TopHom α β) => ⇑f) ⋯

instance TopHom.instOrderTop {α : Type u_2} {β : Type u_3} [Top α] [LE β] [OrderTop β] : OrderTop (TopHom α β)

Equations

• TopHom.instOrderTop = OrderTop.mk ⋯

@[simp]

theorem TopHom.coe_top {α : Type u_2} {β : Type u_3} [Top α] [LE β] [OrderTop β] : ⇑⊤ = ⊤

@[simp]

theorem TopHom.top_apply {α : Type u_2} {β : Type u_3} [Top α] [LE β] [OrderTop β] (a : α) : ⊤ a = ⊤

instance TopHom.instInf {α : Type u_2} {β : Type u_3} [Top α] [SemilatticeInf β] [OrderTop β] : Inf (TopHom α β)

Equations

• TopHom.instInf = { inf := fun (f g : TopHom α β) => { toFun := ⇑f ⊓ ⇑g, map_top' := ⋯ } }

instance TopHom.instSemilatticeInf {α : Type u_2} {β : Type u_3} [Top α] [SemilatticeInf β] [OrderTop β] : SemilatticeInf (TopHom α β)

Equations

• TopHom.instSemilatticeInf = Function.Injective.semilatticeInf (fun (f : TopHom α β) => ⇑f) ⋯ ⋯

@[simp]

theorem TopHom.coe_inf {α : Type u_2} {β : Type u_3} [Top α] [SemilatticeInf β] [OrderTop β] (f : TopHom α β) (g : TopHom α β) : ⇑(f ⊓ g) = ⇑f ⊓ ⇑g

@[simp]

theorem TopHom.inf_apply {α : Type u_2} {β : Type u_3} [Top α] [SemilatticeInf β] [OrderTop β] (f : TopHom α β) (g : TopHom α β) (a : α) : (f ⊓ g) a = f a ⊓ g a

instance TopHom.instSup {α : Type u_2} {β : Type u_3} [Top α] [SemilatticeSup β] [OrderTop β] : Sup (TopHom α β)

Equations

• TopHom.instSup = { sup := fun (f g : TopHom α β) => { toFun := ⇑f ⊔ ⇑g, map_top' := ⋯ } }

instance TopHom.instSemilatticeSup {α : Type u_2} {β : Type u_3} [Top α] [SemilatticeSup β] [OrderTop β] : SemilatticeSup (TopHom α β)

Equations

• TopHom.instSemilatticeSup = Function.Injective.semilatticeSup (fun (f : TopHom α β) => ⇑f) ⋯ ⋯

@[simp]

theorem TopHom.coe_sup {α : Type u_2} {β : Type u_3} [Top α] [SemilatticeSup β] [OrderTop β] (f : TopHom α β) (g : TopHom α β) : ⇑(f ⊔ g) = ⇑f ⊔ ⇑g

@[simp]

theorem TopHom.sup_apply {α : Type u_2} {β : Type u_3} [Top α] [SemilatticeSup β] [OrderTop β] (f : TopHom α β) (g : TopHom α β) (a : α) : (f ⊔ g) a = f a ⊔ g a

instance TopHom.instLattice {α : Type u_2} {β : Type u_3} [Top α] [Lattice β] [OrderTop β] : Lattice (TopHom α β)

Equations

• TopHom.instLattice = Function.Injective.lattice (fun (f : TopHom α β) => ⇑f) ⋯ ⋯ ⋯

instance TopHom.instDistribLattice {α : Type u_2} {β : Type u_3} [Top α] [DistribLattice β] [OrderTop β] : DistribLattice (TopHom α β)

Equations

• TopHom.instDistribLattice = Function.Injective.distribLattice (fun (f : TopHom α β) => ⇑f) ⋯ ⋯ ⋯

Bot homomorphisms

instance BotHom.instFunLike {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] : FunLike (BotHom α β) α β

Equations

• BotHom.instFunLike = { coe := BotHom.toFun, coe_injective' := ⋯ }

instance BotHom.instBotHomClass {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] : BotHomClass (BotHom α β) α β

Equations

• ⋯ = ⋯

theorem BotHom.ext_iff {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] {f : BotHom α β} {g : BotHom α β} : f = g ↔ ∀ (a : α), f a = g a

theorem BotHom.ext {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] {f : BotHom α β} {g : BotHom α β} (h : ∀ (a : α), f a = g a) : f = g

def BotHom.copy {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] (f : BotHom α β) (f' : α → β) (h : f' = ⇑f) : BotHom α β

Copy of a BotHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toFun := f', map_bot' := ⋯ }

@[simp]

theorem BotHom.coe_copy {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] (f : BotHom α β) (f' : α → β) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem BotHom.copy_eq {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] (f : BotHom α β) (f' : α → β) (h : f' = ⇑f) : f.copy f' h = f

instance BotHom.instInhabited {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] : Inhabited (BotHom α β)

Equations

• BotHom.instInhabited = { default := { toFun := fun (x : α) => ⊥, map_bot' := ⋯ } }

def BotHom.id (α : Type u_2) [Bot α] : BotHom α α

id as a BotHom.

Equations

• BotHom.id α = { toFun := id, map_bot' := ⋯ }

@[simp]

theorem BotHom.coe_id (α : Type u_2) [Bot α] : ⇑(BotHom.id α) = id

@[simp]

theorem BotHom.id_apply {α : Type u_2} [Bot α] (a : α) : (BotHom.id α) a = a

def BotHom.comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bot α] [Bot β] [Bot γ] (f : BotHom β γ) (g : BotHom α β) : BotHom α γ

Composition of BotHoms as a BotHom.

Equations

• f.comp g = { toFun := ⇑f ∘ ⇑g, map_bot' := ⋯ }

@[simp]

theorem BotHom.coe_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bot α] [Bot β] [Bot γ] (f : BotHom β γ) (g : BotHom α β) : ⇑(f.comp g) = ⇑f ∘ ⇑g

@[simp]

theorem BotHom.comp_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bot α] [Bot β] [Bot γ] (f : BotHom β γ) (g : BotHom α β) (a : α) : (f.comp g) a = f (g a)

@[simp]

theorem BotHom.comp_assoc {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [Bot α] [Bot β] [Bot γ] [Bot δ] (f : BotHom γ δ) (g : BotHom β γ) (h : BotHom α β) : (f.comp g).comp h = f.comp (g.comp h)

@[simp]

theorem BotHom.comp_id {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] (f : BotHom α β) : f.comp (BotHom.id α) = f

@[simp]

theorem BotHom.id_comp {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] (f : BotHom α β) : (BotHom.id β).comp f = f

@[simp]

theorem BotHom.cancel_right {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bot α] [Bot β] [Bot γ] {g₁ : BotHom β γ} {g₂ : BotHom β γ} {f : BotHom α β} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂

@[simp]

theorem BotHom.cancel_left {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bot α] [Bot β] [Bot γ] {g : BotHom β γ} {f₁ : BotHom α β} {f₂ : BotHom α β} (hg : Function.Injective ⇑g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

instance BotHom.instLE {α : Type u_2} {β : Type u_3} [Bot α] [LE β] [Bot β] : LE (BotHom α β)

Equations

• BotHom.instLE = { le := fun (f g : BotHom α β) => ⇑f ≤ ⇑g }

instance BotHom.instPreorder {α : Type u_2} {β : Type u_3} [Bot α] [Preorder β] [Bot β] : Preorder (BotHom α β)

Equations

• BotHom.instPreorder = Preorder.lift DFunLike.coe

instance BotHom.instPartialOrder {α : Type u_2} {β : Type u_3} [Bot α] [PartialOrder β] [Bot β] : PartialOrder (BotHom α β)

Equations

• BotHom.instPartialOrder = PartialOrder.lift (fun (f : BotHom α β) => ⇑f) ⋯

instance BotHom.instOrderBot {α : Type u_2} {β : Type u_3} [Bot α] [LE β] [OrderBot β] : OrderBot (BotHom α β)

Equations

• BotHom.instOrderBot = OrderBot.mk ⋯

@[simp]

theorem BotHom.coe_bot {α : Type u_2} {β : Type u_3} [Bot α] [LE β] [OrderBot β] : ⇑⊥ = ⊥

@[simp]

theorem BotHom.bot_apply {α : Type u_2} {β : Type u_3} [Bot α] [LE β] [OrderBot β] (a : α) : ⊥ a = ⊥

instance BotHom.instInf {α : Type u_2} {β : Type u_3} [Bot α] [SemilatticeInf β] [OrderBot β] : Inf (BotHom α β)

Equations

• BotHom.instInf = { inf := fun (f g : BotHom α β) => { toFun := ⇑f ⊓ ⇑g, map_bot' := ⋯ } }

instance BotHom.instSemilatticeInf {α : Type u_2} {β : Type u_3} [Bot α] [SemilatticeInf β] [OrderBot β] : SemilatticeInf (BotHom α β)

Equations

• BotHom.instSemilatticeInf = Function.Injective.semilatticeInf (fun (f : BotHom α β) => ⇑f) ⋯ ⋯

@[simp]

theorem BotHom.coe_inf {α : Type u_2} {β : Type u_3} [Bot α] [SemilatticeInf β] [OrderBot β] (f : BotHom α β) (g : BotHom α β) : ⇑(f ⊓ g) = ⇑f ⊓ ⇑g

@[simp]

theorem BotHom.inf_apply {α : Type u_2} {β : Type u_3} [Bot α] [SemilatticeInf β] [OrderBot β] (f : BotHom α β) (g : BotHom α β) (a : α) : (f ⊓ g) a = f a ⊓ g a

instance BotHom.instSup {α : Type u_2} {β : Type u_3} [Bot α] [SemilatticeSup β] [OrderBot β] : Sup (BotHom α β)

Equations

• BotHom.instSup = { sup := fun (f g : BotHom α β) => { toFun := ⇑f ⊔ ⇑g, map_bot' := ⋯ } }

instance BotHom.instSemilatticeSup {α : Type u_2} {β : Type u_3} [Bot α] [SemilatticeSup β] [OrderBot β] : SemilatticeSup (BotHom α β)

Equations

• BotHom.instSemilatticeSup = Function.Injective.semilatticeSup (fun (f : BotHom α β) => ⇑f) ⋯ ⋯

@[simp]

theorem BotHom.coe_sup {α : Type u_2} {β : Type u_3} [Bot α] [SemilatticeSup β] [OrderBot β] (f : BotHom α β) (g : BotHom α β) : ⇑(f ⊔ g) = ⇑f ⊔ ⇑g

@[simp]

theorem BotHom.sup_apply {α : Type u_2} {β : Type u_3} [Bot α] [SemilatticeSup β] [OrderBot β] (f : BotHom α β) (g : BotHom α β) (a : α) : (f ⊔ g) a = f a ⊔ g a

instance BotHom.instLattice {α : Type u_2} {β : Type u_3} [Bot α] [Lattice β] [OrderBot β] : Lattice (BotHom α β)

Equations

• BotHom.instLattice = Function.Injective.lattice (fun (f : BotHom α β) => ⇑f) ⋯ ⋯ ⋯

instance BotHom.instDistribLattice {α : Type u_2} {β : Type u_3} [Bot α] [DistribLattice β] [OrderBot β] : DistribLattice (BotHom α β)

Equations

• BotHom.instDistribLattice = Function.Injective.distribLattice (fun (f : BotHom α β) => ⇑f) ⋯ ⋯ ⋯

Bounded order homomorphisms

def BoundedOrderHom.toTopHom {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (f : BoundedOrderHom α β) : TopHom α β

Reinterpret a BoundedOrderHom as a TopHom.

Equations

• f.toTopHom = { toFun := f.toFun, map_top' := ⋯ }

def BoundedOrderHom.toBotHom {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (f : BoundedOrderHom α β) : BotHom α β

Reinterpret a BoundedOrderHom as a BotHom.

Equations

• f.toBotHom = { toFun := f.toFun, map_bot' := ⋯ }

instance BoundedOrderHom.instFunLike {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] : FunLike (BoundedOrderHom α β) α β

Equations

• BoundedOrderHom.instFunLike = { coe := fun (f : BoundedOrderHom α β) => f.toFun, coe_injective' := ⋯ }

instance BoundedOrderHom.instBoundedOrderHomClass {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] : BoundedOrderHomClass (BoundedOrderHom α β) α β

Equations

• ⋯ = ⋯

theorem BoundedOrderHom.ext_iff {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] {f : BoundedOrderHom α β} {g : BoundedOrderHom α β} : f = g ↔ ∀ (a : α), f a = g a

theorem BoundedOrderHom.ext {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] {f : BoundedOrderHom α β} {g : BoundedOrderHom α β} (h : ∀ (a : α), f a = g a) : f = g

def BoundedOrderHom.copy {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (f : BoundedOrderHom α β) (f' : α → β) (h : f' = ⇑f) : BoundedOrderHom α β

Copy of a BoundedOrderHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toOrderHom := f.copy f' h, map_top' := ⋯, map_bot' := ⋯ }

@[simp]

theorem BoundedOrderHom.coe_copy {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (f : BoundedOrderHom α β) (f' : α → β) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem BoundedOrderHom.copy_eq {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (f : BoundedOrderHom α β) (f' : α → β) (h : f' = ⇑f) : f.copy f' h = f

def BoundedOrderHom.id (α : Type u_2) [Preorder α] [BoundedOrder α] : BoundedOrderHom α α

id as a BoundedOrderHom.

Equations

• BoundedOrderHom.id α = { toOrderHom := OrderHom.id, map_top' := ⋯, map_bot' := ⋯ }

instance BoundedOrderHom.instInhabited (α : Type u_2) [Preorder α] [BoundedOrder α] : Inhabited (BoundedOrderHom α α)

Equations

• BoundedOrderHom.instInhabited α = { default := BoundedOrderHom.id α }

@[simp]

theorem BoundedOrderHom.coe_id (α : Type u_2) [Preorder α] [BoundedOrder α] : ⇑(BoundedOrderHom.id α) = id

@[simp]

theorem BoundedOrderHom.id_apply {α : Type u_2} [Preorder α] [BoundedOrder α] (a : α) : (BoundedOrderHom.id α) a = a

def BoundedOrderHom.comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β) : BoundedOrderHom α γ

Composition of BoundedOrderHoms as a BoundedOrderHom.

Equations

• f.comp g = { toOrderHom := f.comp g.toOrderHom, map_top' := ⋯, map_bot' := ⋯ }

@[simp]

theorem BoundedOrderHom.coe_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β) : ⇑(f.comp g) = ⇑f ∘ ⇑g

@[simp]

theorem BoundedOrderHom.comp_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β) (a : α) : (f.comp g) a = f (g a)

@[simp]

theorem BoundedOrderHom.coe_comp_orderHom {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β) : ↑(f.comp g) = (↑f).comp ↑g

@[simp]

theorem BoundedOrderHom.coe_comp_topHom {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β) : ↑(f.comp g) = (↑f).comp ↑g

@[simp]

theorem BoundedOrderHom.coe_comp_botHom {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedOrderHom β γ) (g : BoundedOrderHom α β) : ↑(f.comp g) = (↑f).comp ↑g

@[simp]

theorem BoundedOrderHom.comp_assoc {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [Preorder α] [Preorder β] [Preorder γ] [Preorder δ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] [BoundedOrder δ] (f : BoundedOrderHom γ δ) (g : BoundedOrderHom β γ) (h : BoundedOrderHom α β) : (f.comp g).comp h = f.comp (g.comp h)

@[simp]

theorem BoundedOrderHom.comp_id {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (f : BoundedOrderHom α β) : f.comp (BoundedOrderHom.id α) = f

@[simp]

theorem BoundedOrderHom.id_comp {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] (f : BoundedOrderHom α β) : (BoundedOrderHom.id β).comp f = f

@[simp]

theorem BoundedOrderHom.cancel_right {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] {g₁ : BoundedOrderHom β γ} {g₂ : BoundedOrderHom β γ} {f : BoundedOrderHom α β} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂

@[simp]

theorem BoundedOrderHom.cancel_left {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] {g : BoundedOrderHom β γ} {f₁ : BoundedOrderHom α β} {f₂ : BoundedOrderHom α β} (hg : Function.Injective ⇑g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

Dual homs

@[simp]

theorem TopHom.dual_apply_apply {α : Type u_2} {β : Type u_3} [LE α] [OrderTop α] [LE β] [OrderTop β] (f : TopHom α β) (a : α) : (TopHom.dual f) a = f a

@[simp]

theorem TopHom.dual_symm_apply_apply {α : Type u_2} {β : Type u_3} [LE α] [OrderTop α] [LE β] [OrderTop β] (f : BotHom αᵒᵈ βᵒᵈ) (a : αᵒᵈ) : (TopHom.dual.symm f) a = f a

def TopHom.dual {α : Type u_2} {β : Type u_3} [LE α] [OrderTop α] [LE β] [OrderTop β] : TopHom α β ≃ BotHom αᵒᵈ βᵒᵈ

Reinterpret a top homomorphism as a bot homomorphism between the dual lattices.

Equations

• TopHom.dual = { toFun := fun (f : TopHom α β) => { toFun := ⇑f, map_bot' := ⋯ }, invFun := fun (f : BotHom αᵒᵈ βᵒᵈ) => { toFun := ⇑f, map_top' := ⋯ }, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem TopHom.dual_id {α : Type u_2} [LE α] [OrderTop α] : TopHom.dual (TopHom.id α) = BotHom.id αᵒᵈ

@[simp]

theorem TopHom.dual_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [LE α] [OrderTop α] [LE β] [OrderTop β] [LE γ] [OrderTop γ] (g : TopHom β γ) (f : TopHom α β) : TopHom.dual (g.comp f) = (TopHom.dual g).comp (TopHom.dual f)

@[simp]

theorem TopHom.symm_dual_id {α : Type u_2} [LE α] [OrderTop α] : TopHom.dual.symm (BotHom.id αᵒᵈ) = TopHom.id α

@[simp]

theorem TopHom.symm_dual_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [LE α] [OrderTop α] [LE β] [OrderTop β] [LE γ] [OrderTop γ] (g : BotHom βᵒᵈ γᵒᵈ) (f : BotHom αᵒᵈ βᵒᵈ) : TopHom.dual.symm (g.comp f) = (TopHom.dual.symm g).comp (TopHom.dual.symm f)

@[simp]

theorem BotHom.dual_symm_apply_apply {α : Type u_2} {β : Type u_3} [LE α] [OrderBot α] [LE β] [OrderBot β] (f : TopHom αᵒᵈ βᵒᵈ) (a : αᵒᵈ) : (BotHom.dual.symm f) a = f a

@[simp]

theorem BotHom.dual_apply_apply {α : Type u_2} {β : Type u_3} [LE α] [OrderBot α] [LE β] [OrderBot β] (f : BotHom α β) (a : α) : (BotHom.dual f) a = f a

def BotHom.dual {α : Type u_2} {β : Type u_3} [LE α] [OrderBot α] [LE β] [OrderBot β] : BotHom α β ≃ TopHom αᵒᵈ βᵒᵈ

Reinterpret a bot homomorphism as a top homomorphism between the dual lattices.

Equations

• BotHom.dual = { toFun := fun (f : BotHom α β) => { toFun := ⇑f, map_top' := ⋯ }, invFun := fun (f : TopHom αᵒᵈ βᵒᵈ) => { toFun := ⇑f, map_bot' := ⋯ }, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem BotHom.dual_id {α : Type u_2} [LE α] [OrderBot α] : BotHom.dual (BotHom.id α) = TopHom.id αᵒᵈ

@[simp]

theorem BotHom.dual_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [LE α] [OrderBot α] [LE β] [OrderBot β] [LE γ] [OrderBot γ] (g : BotHom β γ) (f : BotHom α β) : BotHom.dual (g.comp f) = (BotHom.dual g).comp (BotHom.dual f)

@[simp]

theorem BotHom.symm_dual_id {α : Type u_2} [LE α] [OrderBot α] : BotHom.dual.symm (TopHom.id αᵒᵈ) = BotHom.id α

@[simp]

theorem BotHom.symm_dual_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [LE α] [OrderBot α] [LE β] [OrderBot β] [LE γ] [OrderBot γ] (g : TopHom βᵒᵈ γᵒᵈ) (f : TopHom αᵒᵈ βᵒᵈ) : BotHom.dual.symm (g.comp f) = (BotHom.dual.symm g).comp (BotHom.dual.symm f)

@[simp]

theorem BoundedOrderHom.dual_apply_toOrderHom {α : Type u_2} {β : Type u_3} [Preorder α] [BoundedOrder α] [Preorder β] [BoundedOrder β] (f : BoundedOrderHom α β) : (BoundedOrderHom.dual f).toOrderHom = OrderHom.dual f.toOrderHom

@[simp]

theorem BoundedOrderHom.dual_symm_apply_toOrderHom {α : Type u_2} {β : Type u_3} [Preorder α] [BoundedOrder α] [Preorder β] [BoundedOrder β] (f : BoundedOrderHom αᵒᵈ βᵒᵈ) : (BoundedOrderHom.dual.symm f).toOrderHom = OrderHom.dual.symm f.toOrderHom

def BoundedOrderHom.dual {α : Type u_2} {β : Type u_3} [Preorder α] [BoundedOrder α] [Preorder β] [BoundedOrder β] : BoundedOrderHom α β ≃ BoundedOrderHom αᵒᵈ βᵒᵈ

Reinterpret a bounded order homomorphism as a bounded order homomorphism between the dual orders.

Equations

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

@[simp]

theorem BoundedOrderHom.dual_id {α : Type u_2} [Preorder α] [BoundedOrder α] : BoundedOrderHom.dual (BoundedOrderHom.id α) = BoundedOrderHom.id αᵒᵈ

@[simp]

theorem BoundedOrderHom.dual_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [BoundedOrder α] [Preorder β] [BoundedOrder β] [Preorder γ] [BoundedOrder γ] (g : BoundedOrderHom β γ) (f : BoundedOrderHom α β) : BoundedOrderHom.dual (g.comp f) = (BoundedOrderHom.dual g).comp (BoundedOrderHom.dual f)

@[simp]

theorem BoundedOrderHom.symm_dual_id {α : Type u_2} [Preorder α] [BoundedOrder α] : BoundedOrderHom.dual.symm (BoundedOrderHom.id αᵒᵈ) = BoundedOrderHom.id α

@[simp]

theorem BoundedOrderHom.symm_dual_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [BoundedOrder α] [Preorder β] [BoundedOrder β] [Preorder γ] [BoundedOrder γ] (g : BoundedOrderHom βᵒᵈ γᵒᵈ) (f : BoundedOrderHom αᵒᵈ βᵒᵈ) : BoundedOrderHom.dual.symm (g.comp f) = (BoundedOrderHom.dual.symm g).comp (BoundedOrderHom.dual.symm f)