Bounded order homomorphisms

This file defines (bounded) order homomorphisms.

We use the FunLike 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)

• toFun : α → β

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

• map_top' : TopHom.toFun s ⊤ = ⊤

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

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

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

• toFun : α → β

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

• map_bot' : BotHom.toFun s ⊥ = ⊥

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

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

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)

• toFun : α → β
• monotone' : Monotone s.toFun
• map_top' : OrderHom.toFun s.toOrderHom ⊤ = ⊤

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

• map_bot' : OrderHom.toFun s.toOrderHom ⊥ = ⊥

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

The type of bounded order homomorphisms from α to β.

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

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_top : ∀ (f : F), ↑f ⊤ = ⊤

  A TopHomClass morphism preserves the top element.

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

You should extend this class when you extend TopHom.

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

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_bot : ∀ (f : F), ↑f ⊥ = ⊥

  A BotHomClass morphism preserves the bottom element.

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

You should extend this class when you extend BotHom.

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

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• 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.

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

You should extend this class when you extend BoundedOrderHom.

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

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

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

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

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

@[simp]

theorem map_eq_top_iff {F : Type u_1} {α : Type u_2} {β : Type u_3} [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} [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} [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.

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

def BotHomClass.toBotHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [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.

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

def BoundedOrderHomClass.toBoundedOrderHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [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.

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

Top homomorphisms

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

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.

@[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) : ↑(TopHom.copy f 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) : TopHom.copy f f' h = f

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

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

id as a TopHom.

@[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.

@[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) : ↑(TopHom.comp f 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) : ↑(TopHom.comp f 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) : TopHom.comp (TopHom.comp f g) h = TopHom.comp f (TopHom.comp g h)

@[simp]

theorem TopHom.comp_id {α : Type u_2} {β : Type u_3} [Top α] [Top β] (f : TopHom α β) : TopHom.comp f (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.comp (TopHom.id β) 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) : TopHom.comp g₁ f = TopHom.comp g₂ 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) : TopHom.comp g f₁ = TopHom.comp g f₂ ↔ f₁ = f₂

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

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

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

@[simp]

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

@[simp]

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

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

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

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

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

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

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

Bot homomorphisms

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

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.

@[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) : ↑(BotHom.copy f 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) : BotHom.copy f f' h = f

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

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

id as a BotHom.

@[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.

@[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) : ↑(BotHom.comp f 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) : ↑(BotHom.comp f 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) : BotHom.comp (BotHom.comp f g) h = BotHom.comp f (BotHom.comp g h)

@[simp]

theorem BotHom.comp_id {α : Type u_2} {β : Type u_3} [Bot α] [Bot β] (f : BotHom α β) : BotHom.comp f (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.comp (BotHom.id β) 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) : BotHom.comp g₁ f = BotHom.comp g₂ 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) : BotHom.comp g f₁ = BotHom.comp g f₂ ↔ f₁ = f₂

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

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

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

@[simp]

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

@[simp]

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

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

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

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

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

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

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

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.

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.

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

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.

@[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) : ↑(BoundedOrderHom.copy f 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) : BoundedOrderHom.copy f f' h = f

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

id as a BoundedOrderHom.

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

@[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) : ↑(BoundedOrderHom.comp f 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) : ↑(BoundedOrderHom.comp f 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) : ↑(BoundedOrderHom.comp f g) = OrderHom.comp ↑f ↑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) : ↑(BoundedOrderHom.comp f g) = TopHom.comp ↑f ↑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) : ↑(BoundedOrderHom.comp f g) = BotHom.comp ↑f ↑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) : BoundedOrderHom.comp (BoundedOrderHom.comp f g) h = BoundedOrderHom.comp f (BoundedOrderHom.comp g 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) : BoundedOrderHom.comp f (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.comp (BoundedOrderHom.id β) 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) : BoundedOrderHom.comp g₁ f = BoundedOrderHom.comp g₂ 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) : BoundedOrderHom.comp g f₁ = BoundedOrderHom.comp g f₂ ↔ f₁ = f₂

Dual homs

@[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

@[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

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.

@[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 (TopHom.comp g f) = BotHom.comp (↑TopHom.dual g) (↑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 (BotHom.comp g f) = TopHom.comp (↑TopHom.dual.symm g) (↑TopHom.dual.symm f)

@[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

@[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

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.

@[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 (BotHom.comp g f) = TopHom.comp (↑BotHom.dual g) (↑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 (TopHom.comp g f) = BotHom.comp (↑BotHom.dual.symm g) (↑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.

@[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 (BoundedOrderHom.comp g f) = BoundedOrderHom.comp (↑BoundedOrderHom.dual g) (↑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 (BoundedOrderHom.comp g f) = BoundedOrderHom.comp (↑BoundedOrderHom.dual.symm g) (↑BoundedOrderHom.dual.symm f)