Lattice homomorphisms

This file defines (bounded) lattice 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

• SupHom: Maps which preserve ⊔.
• InfHom: Maps which preserve ⊓.
• SupBotHom: Finitary supremum homomorphisms. Maps which preserve ⊔ and ⊥.
• InfTopHom: Finitary infimum homomorphisms. Maps which preserve ⊓ and ⊤.
• LatticeHom: Lattice homomorphisms. Maps which preserve ⊔ and ⊓.
• BoundedLatticeHom: Bounded lattice homomorphisms. Maps which preserve ⊤, ⊥, ⊔ and ⊓.

Typeclasses

• SupHomClass
• InfHomClass
• SupBotHomClass
• InfTopHomClass
• LatticeHomClass
• BoundedLatticeHomClass

TODO

Do we need more intersections between BotHom, TopHom and lattice homomorphisms?

structure SupHom (α : Type u_7) (β : Type u_8) [Sup α] [Sup β] : Type (max u_7 u_8)

• toFun : α → β

  The underlying function of a SupHom

• map_sup' : ∀ (a b : α), SupHom.toFun s (a ⊔ b) = SupHom.toFun s a ⊔ SupHom.toFun s b

  A SupHom preserves suprema.

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

structure InfHom (α : Type u_7) (β : Type u_8) [Inf α] [Inf β] : Type (max u_7 u_8)

• toFun : α → β

  The underlying function of an InfHom

• map_inf' : ∀ (a b : α), InfHom.toFun s (a ⊓ b) = InfHom.toFun s a ⊓ InfHom.toFun s b

  An InfHom preserves infima.

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

structure SupBotHom (α : Type u_7) (β : Type u_8) [Sup α] [Sup β] [Bot α] [Bot β] extends SupHom : Type (max u_7 u_8)

• toFun : α → β
• map_sup' : ∀ (a b : α), SupHom.toFun s.toSupHom (a ⊔ b) = SupHom.toFun s.toSupHom a ⊔ SupHom.toFun s.toSupHom b
• map_bot' : SupHom.toFun s.toSupHom ⊥ = ⊥

  A SupBotHom preserves the bottom element.

The type of finitary supremum-preserving homomorphisms from α to β.

structure InfTopHom (α : Type u_7) (β : Type u_8) [Inf α] [Inf β] [Top α] [Top β] extends InfHom : Type (max u_7 u_8)

• toFun : α → β
• map_inf' : ∀ (a b : α), InfHom.toFun s.toInfHom (a ⊓ b) = InfHom.toFun s.toInfHom a ⊓ InfHom.toFun s.toInfHom b
• map_top' : InfHom.toFun s.toInfHom ⊤ = ⊤

  An InfTopHom preserves the top element.

The type of finitary infimum-preserving homomorphisms from α to β.

structure LatticeHom (α : Type u_7) (β : Type u_8) [Lattice α] [Lattice β] extends SupHom : Type (max u_7 u_8)

• toFun : α → β
• map_sup' : ∀ (a b : α), SupHom.toFun s.toSupHom (a ⊔ b) = SupHom.toFun s.toSupHom a ⊔ SupHom.toFun s.toSupHom b
• map_inf' : ∀ (a b : α), SupHom.toFun s.toSupHom (a ⊓ b) = SupHom.toFun s.toSupHom a ⊓ SupHom.toFun s.toSupHom b

  A LatticeHom preserves infima.

The type of lattice homomorphisms from α to β.

structure BoundedLatticeHom (α : Type u_7) (β : Type u_8) [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] extends LatticeHom : Type (max u_7 u_8)

• toFun : α → β
• map_sup' : ∀ (a b : α), SupHom.toFun s.toSupHom (a ⊔ b) = SupHom.toFun s.toSupHom a ⊔ SupHom.toFun s.toSupHom b
• map_inf' : ∀ (a b : α), SupHom.toFun s.toSupHom (a ⊓ b) = SupHom.toFun s.toSupHom a ⊓ SupHom.toFun s.toSupHom b
• map_top' : SupHom.toFun s.toSupHom ⊤ = ⊤

  A BoundedLatticeHom preserves the top element.

• map_bot' : SupHom.toFun s.toSupHom ⊥ = ⊥

  A BoundedLatticeHom preserves the bottom element.

The type of bounded lattice homomorphisms from α to β.

class SupHomClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Sup α] [Sup β] extends FunLike : Type (max (max u_7 u_8) u_9)

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_sup : ∀ (f : F) (a b : α), ↑f (a ⊔ b) = ↑f a ⊔ ↑f b

  A SupHomClass morphism preserves suprema.

SupHomClass F α β states that F is a type of ⊔-preserving morphisms.

You should extend this class when you extend SupHom.

class InfHomClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Inf α] [Inf β] extends FunLike : Type (max (max u_7 u_8) u_9)

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_inf : ∀ (f : F) (a b : α), ↑f (a ⊓ b) = ↑f a ⊓ ↑f b

  An InfHomClass morphism preserves infima.

InfHomClass F α β states that F is a type of ⊓-preserving morphisms.

You should extend this class when you extend InfHom.

class SupBotHomClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Sup α] [Sup β] [Bot α] [Bot β] extends SupHomClass : Type (max (max u_7 u_8) u_9)

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

  A SupBotHomClass morphism preserves the bottom element.

SupBotHomClass F α β states that F is a type of finitary supremum-preserving morphisms.

You should extend this class when you extend SupBotHom.

class InfTopHomClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Inf α] [Inf β] [Top α] [Top β] extends InfHomClass : Type (max (max u_7 u_8) u_9)

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

  An InfTopHomClass morphism preserves the top element.

InfTopHomClass F α β states that F is a type of finitary infimum-preserving morphisms.

You should extend this class when you extend SupBotHom.

class LatticeHomClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Lattice α] [Lattice β] extends SupHomClass : Type (max (max u_7 u_8) u_9)

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_sup : ∀ (f : F) (a b : α), ↑f (a ⊔ b) = ↑f a ⊔ ↑f b
• map_inf : ∀ (f : F) (a b : α), ↑f (a ⊓ b) = ↑f a ⊓ ↑f b

  A LatticeHomClass morphism preserves infima.

LatticeHomClass F α β states that F is a type of lattice morphisms.

You should extend this class when you extend LatticeHom.

class BoundedLatticeHomClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] extends LatticeHomClass : Type (max (max u_7 u_8) u_9)

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_sup : ∀ (f : F) (a b : α), ↑f (a ⊔ b) = ↑f a ⊔ ↑f b
• map_inf : ∀ (f : F) (a b : α), ↑f (a ⊓ b) = ↑f a ⊓ ↑f b
• map_top : ∀ (f : F), ↑f ⊤ = ⊤

  A BoundedLatticeHomClass morphism preserves the top element.

• map_bot : ∀ (f : F), ↑f ⊥ = ⊥

  A BoundedLatticeHomClass morphism preserves the bottom element.

BoundedLatticeHomClass F α β states that F is a type of bounded lattice morphisms.

You should extend this class when you extend BoundedLatticeHom.

instance SupHomClass.toOrderHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] [SupHomClass F α β] : OrderHomClass F α β

instance InfHomClass.toOrderHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [SemilatticeInf β] [InfHomClass F α β] : OrderHomClass F α β

instance SupBotHomClass.toBotHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] [Bot α] [Bot β] [SupBotHomClass F α β] : BotHomClass F α β

instance InfTopHomClass.toTopHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] [Top α] [Top β] [InfTopHomClass F α β] : TopHomClass F α β

instance LatticeHomClass.toInfHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [LatticeHomClass F α β] : InfHomClass F α β

instance BoundedLatticeHomClass.toSupBotHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] [BoundedLatticeHomClass F α β] : SupBotHomClass F α β

instance BoundedLatticeHomClass.toInfTopHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] [BoundedLatticeHomClass F α β] : InfTopHomClass F α β

instance BoundedLatticeHomClass.toBoundedOrderHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] [BoundedLatticeHomClass F α β] : BoundedOrderHomClass F α β

instance OrderIsoClass.toSupHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] [OrderIsoClass F α β] : SupHomClass F α β

instance OrderIsoClass.toInfHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [SemilatticeInf β] [OrderIsoClass F α β] : InfHomClass F α β

instance OrderIsoClass.toSupBotHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [OrderBot α] [SemilatticeSup β] [OrderBot β] [OrderIsoClass F α β] : SupBotHomClass F α β

instance OrderIsoClass.toInfTopHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [OrderTop α] [SemilatticeInf β] [OrderTop β] [OrderIsoClass F α β] : InfTopHomClass F α β

instance OrderIsoClass.toLatticeHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [OrderIsoClass F α β] : LatticeHomClass F α β

instance OrderIsoClass.toBoundedLatticeHomClass {F : Type u_1} {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] [OrderIsoClass F α β] : BoundedLatticeHomClass F α β

theorem Disjoint.map {F : Type u_1} {α : Type u_3} {β : Type u_4} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] [BoundedLatticeHomClass F α β] (f : F) {a : α} {b : α} (h : Disjoint a b) : Disjoint (↑f a) (↑f b)

theorem Codisjoint.map {F : Type u_1} {α : Type u_3} {β : Type u_4} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] [BoundedLatticeHomClass F α β] (f : F) {a : α} {b : α} (h : Codisjoint a b) : Codisjoint (↑f a) (↑f b)

theorem IsCompl.map {F : Type u_1} {α : Type u_3} {β : Type u_4} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] [BoundedLatticeHomClass F α β] (f : F) {a : α} {b : α} (h : IsCompl a b) : IsCompl (↑f a) (↑f b)

theorem map_compl' {F : Type u_1} {α : Type u_3} {β : Type u_4} [BooleanAlgebra α] [BooleanAlgebra β] [BoundedLatticeHomClass F α β] (f : F) (a : α) : ↑f aᶜ = (↑f a)ᶜ

Special case of map_compl for boolean algebras.

theorem map_sdiff' {F : Type u_1} {α : Type u_3} {β : Type u_4} [BooleanAlgebra α] [BooleanAlgebra β] [BoundedLatticeHomClass F α β] (f : F) (a : α) (b : α) : ↑f (a \ b) = ↑f a \ ↑f b

Special case of map_sdiff for boolean algebras.

theorem map_symmDiff' {F : Type u_1} {α : Type u_3} {β : Type u_4} [BooleanAlgebra α] [BooleanAlgebra β] [BoundedLatticeHomClass F α β] (f : F) (a : α) (b : α) : ↑f (a ∆ b) = ↑f a ∆ ↑f b

Special case of map_symmDiff for boolean algebras.

instance instCoeTCSupHom {F : Type u_1} {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] [SupHomClass F α β] : CoeTC F (SupHom α β)

instance instCoeTCInfHom {F : Type u_1} {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] [InfHomClass F α β] : CoeTC F (InfHom α β)

instance instCoeTCSupBotHom {F : Type u_1} {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] [Bot α] [Bot β] [SupBotHomClass F α β] : CoeTC F (SupBotHom α β)

instance instCoeTCInfTopHom {F : Type u_1} {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] [Top α] [Top β] [InfTopHomClass F α β] : CoeTC F (InfTopHom α β)

instance instCoeTCLatticeHom {F : Type u_1} {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [LatticeHomClass F α β] : CoeTC F (LatticeHom α β)

instance instCoeTCBoundedLatticeHom {F : Type u_1} {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] [BoundedLatticeHomClass F α β] : CoeTC F (BoundedLatticeHom α β)

Supremum homomorphisms

instance SupHom.instSupHomClassSupHom {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] : SupHomClass (SupHom α β) α β

instance SupHom.instFunLikeSupHom {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] : FunLike (SupHom α β) α fun x => β

Helper instance for when there's too many metavariables to apply fun_like.has_coe_to_fun directly.

@[simp]

theorem SupHom.toFun_eq_coe {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] {f : SupHom α β} : f.toFun = ↑f

theorem SupHom.ext {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] {f : SupHom α β} {g : SupHom α β} (h : ∀ (a : α), ↑f a = ↑g a) : f = g

def SupHom.copy {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : SupHom α β) (f' : α → β) (h : f' = ↑f) : SupHom α β

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

@[simp]

theorem SupHom.coe_copy {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : SupHom α β) (f' : α → β) (h : f' = ↑f) : ↑(SupHom.copy f f' h) = f'

theorem SupHom.copy_eq {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : SupHom α β) (f' : α → β) (h : f' = ↑f) : SupHom.copy f f' h = f

def SupHom.id (α : Type u_3) [Sup α] : SupHom α α

id as a SupHom.

instance SupHom.instInhabitedSupHom (α : Type u_3) [Sup α] : Inhabited (SupHom α α)

@[simp]

theorem SupHom.coe_id (α : Type u_3) [Sup α] : ↑(SupHom.id α) = id

@[simp]

theorem SupHom.id_apply {α : Type u_3} [Sup α] (a : α) : ↑(SupHom.id α) a = a

def SupHom.comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] (f : SupHom β γ) (g : SupHom α β) : SupHom α γ

Composition of SupHoms as a SupHom.

@[simp]

theorem SupHom.coe_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] (f : SupHom β γ) (g : SupHom α β) : ↑(SupHom.comp f g) = ↑f ∘ ↑g

@[simp]

theorem SupHom.comp_apply {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] (f : SupHom β γ) (g : SupHom α β) (a : α) : ↑(SupHom.comp f g) a = ↑f (↑g a)

@[simp]

theorem SupHom.comp_assoc {α : Type u_3} {β : Type u_4} {γ : Type u_5} {δ : Type u_6} [Sup α] [Sup β] [Sup γ] [Sup δ] (f : SupHom γ δ) (g : SupHom β γ) (h : SupHom α β) : SupHom.comp (SupHom.comp f g) h = SupHom.comp f (SupHom.comp g h)

@[simp]

theorem SupHom.comp_id {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : SupHom α β) : SupHom.comp f (SupHom.id α) = f

@[simp]

theorem SupHom.id_comp {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : SupHom α β) : SupHom.comp (SupHom.id β) f = f

@[simp]

theorem SupHom.cancel_right {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] {g₁ : SupHom β γ} {g₂ : SupHom β γ} {f : SupHom α β} (hf : Function.Surjective ↑f) : SupHom.comp g₁ f = SupHom.comp g₂ f ↔ g₁ = g₂

@[simp]

theorem SupHom.cancel_left {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] {g : SupHom β γ} {f₁ : SupHom α β} {f₂ : SupHom α β} (hg : Function.Injective ↑g) : SupHom.comp g f₁ = SupHom.comp g f₂ ↔ f₁ = f₂

def SupHom.const (α : Type u_3) {β : Type u_4} [Sup α] [SemilatticeSup β] (b : β) : SupHom α β

The constant function as a SupHom.

@[simp]

theorem SupHom.coe_const (α : Type u_3) {β : Type u_4} [Sup α] [SemilatticeSup β] (b : β) : ↑(SupHom.const α b) = Function.const α b

@[simp]

theorem SupHom.const_apply (α : Type u_3) {β : Type u_4} [Sup α] [SemilatticeSup β] (b : β) (a : α) : ↑(SupHom.const α b) a = b

instance SupHom.instSupSupHomToSup {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] : Sup (SupHom α β)

instance SupHom.instSemilatticeSupSupHomToSup {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] : SemilatticeSup (SupHom α β)

instance SupHom.instBotSupHomToSup {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [Bot β] : Bot (SupHom α β)

instance SupHom.instTopSupHomToSup {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [Top β] : Top (SupHom α β)

instance SupHom.instOrderBotSupHomToSupToLEToPreorderToPartialOrderInstSemilatticeSupSupHomToSup {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [OrderBot β] : OrderBot (SupHom α β)

instance SupHom.instOrderTopSupHomToSupToLEToPreorderToPartialOrderInstSemilatticeSupSupHomToSup {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [OrderTop β] : OrderTop (SupHom α β)

instance SupHom.instBoundedOrderSupHomToSupToLEToPreorderToPartialOrderInstSemilatticeSupSupHomToSup {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [BoundedOrder β] : BoundedOrder (SupHom α β)

@[simp]

theorem SupHom.coe_sup {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] (f : SupHom α β) (g : SupHom α β) : ↑(f ⊔ g) = ↑(f ⊔ g)

@[simp]

theorem SupHom.coe_bot {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [Bot β] : ↑⊥ = ⊥

@[simp]

theorem SupHom.coe_top {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [Top β] : ↑⊤ = ⊤

@[simp]

theorem SupHom.sup_apply {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] (f : SupHom α β) (g : SupHom α β) (a : α) : ↑(f ⊔ g) a = ↑f a ⊔ ↑g a

@[simp]

theorem SupHom.bot_apply {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [Bot β] (a : α) : ↑⊥ a = ⊥

@[simp]

theorem SupHom.top_apply {α : Type u_3} {β : Type u_4} [Sup α] [SemilatticeSup β] [Top β] (a : α) : ↑⊤ a = ⊤

Infimum homomorphisms

instance InfHom.instInfHomClassInfHom {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] : InfHomClass (InfHom α β) α β

instance InfHom.instFunLikeInfHom {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] : FunLike (InfHom α β) α fun x => β

Helper instance for when there's too many metavariables to apply fun_like.has_coe_to_fun directly.

@[simp]

theorem InfHom.toFun_eq_coe {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] {f : InfHom α β} : f.toFun = ↑f

theorem InfHom.ext {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] {f : InfHom α β} {g : InfHom α β} (h : ∀ (a : α), ↑f a = ↑g a) : f = g

def InfHom.copy {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : InfHom α β) (f' : α → β) (h : f' = ↑f) : InfHom α β

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

@[simp]

theorem InfHom.coe_copy {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : InfHom α β) (f' : α → β) (h : f' = ↑f) : ↑(InfHom.copy f f' h) = f'

theorem InfHom.copy_eq {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : InfHom α β) (f' : α → β) (h : f' = ↑f) : InfHom.copy f f' h = f

def InfHom.id (α : Type u_3) [Inf α] : InfHom α α

id as an InfHom.

instance InfHom.instInhabitedInfHom (α : Type u_3) [Inf α] : Inhabited (InfHom α α)

@[simp]

theorem InfHom.coe_id (α : Type u_3) [Inf α] : ↑(InfHom.id α) = id

@[simp]

theorem InfHom.id_apply {α : Type u_3} [Inf α] (a : α) : ↑(InfHom.id α) a = a

def InfHom.comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] (f : InfHom β γ) (g : InfHom α β) : InfHom α γ

Composition of InfHoms as an InfHom.

@[simp]

theorem InfHom.coe_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] (f : InfHom β γ) (g : InfHom α β) : ↑(InfHom.comp f g) = ↑f ∘ ↑g

@[simp]

theorem InfHom.comp_apply {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] (f : InfHom β γ) (g : InfHom α β) (a : α) : ↑(InfHom.comp f g) a = ↑f (↑g a)

@[simp]

theorem InfHom.comp_assoc {α : Type u_3} {β : Type u_4} {γ : Type u_5} {δ : Type u_6} [Inf α] [Inf β] [Inf γ] [Inf δ] (f : InfHom γ δ) (g : InfHom β γ) (h : InfHom α β) : InfHom.comp (InfHom.comp f g) h = InfHom.comp f (InfHom.comp g h)

@[simp]

theorem InfHom.comp_id {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : InfHom α β) : InfHom.comp f (InfHom.id α) = f

@[simp]

theorem InfHom.id_comp {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : InfHom α β) : InfHom.comp (InfHom.id β) f = f

@[simp]

theorem InfHom.cancel_right {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] {g₁ : InfHom β γ} {g₂ : InfHom β γ} {f : InfHom α β} (hf : Function.Surjective ↑f) : InfHom.comp g₁ f = InfHom.comp g₂ f ↔ g₁ = g₂

@[simp]

theorem InfHom.cancel_left {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] {g : InfHom β γ} {f₁ : InfHom α β} {f₂ : InfHom α β} (hg : Function.Injective ↑g) : InfHom.comp g f₁ = InfHom.comp g f₂ ↔ f₁ = f₂

def InfHom.const (α : Type u_3) {β : Type u_4} [Inf α] [SemilatticeInf β] (b : β) : InfHom α β

The constant function as an InfHom.

@[simp]

theorem InfHom.coe_const (α : Type u_3) {β : Type u_4} [Inf α] [SemilatticeInf β] (b : β) : ↑(InfHom.const α b) = Function.const α b

@[simp]

theorem InfHom.const_apply (α : Type u_3) {β : Type u_4} [Inf α] [SemilatticeInf β] (b : β) (a : α) : ↑(InfHom.const α b) a = b

instance InfHom.instInfInfHomToInf {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] : Inf (InfHom α β)

instance InfHom.instSemilatticeInfInfHomToInf {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] : SemilatticeInf (InfHom α β)

instance InfHom.instBotInfHomToInf {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [Bot β] : Bot (InfHom α β)

instance InfHom.instTopInfHomToInf {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [Top β] : Top (InfHom α β)

instance InfHom.instOrderBotInfHomToInfToLEToPreorderToPartialOrderInstSemilatticeInfInfHomToInf {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [OrderBot β] : OrderBot (InfHom α β)

instance InfHom.instOrderTopInfHomToInfToLEToPreorderToPartialOrderInstSemilatticeInfInfHomToInf {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [OrderTop β] : OrderTop (InfHom α β)

instance InfHom.instBoundedOrderInfHomToInfToLEToPreorderToPartialOrderInstSemilatticeInfInfHomToInf {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [BoundedOrder β] : BoundedOrder (InfHom α β)

@[simp]

theorem InfHom.coe_inf {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] (f : InfHom α β) (g : InfHom α β) : ↑(f ⊓ g) = ↑(f ⊓ g)

@[simp]

theorem InfHom.coe_bot {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [Bot β] : ↑⊥ = ⊥

@[simp]

theorem InfHom.coe_top {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [Top β] : ↑⊤ = ⊤

@[simp]

theorem InfHom.inf_apply {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] (f : InfHom α β) (g : InfHom α β) (a : α) : ↑(f ⊓ g) a = ↑f a ⊓ ↑g a

@[simp]

theorem InfHom.bot_apply {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [Bot β] (a : α) : ↑⊥ a = ⊥

@[simp]

theorem InfHom.top_apply {α : Type u_3} {β : Type u_4} [Inf α] [SemilatticeInf β] [Top β] (a : α) : ↑⊤ a = ⊤

Finitary supremum homomorphisms

def SupBotHom.toBotHom {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β) : BotHom α β

Reinterpret a SupBotHom as a BotHom.

instance SupBotHom.instSupBotHomClassSupBotHom {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] : SupBotHomClass (SupBotHom α β) α β

instance SupBotHom.instFunLikeSupBotHom {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] : FunLike (SupBotHom α β) α fun x => β

@[simp]

theorem SupBotHom.coe_toSupHom {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] {f : SupBotHom α β} : ↑f.toSupHom = ↑f

@[simp]

theorem SupBotHom.coe_toBotHom {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] {f : SupBotHom α β} : ↑(SupBotHom.toBotHom f) = ↑f

theorem SupBotHom.toFun_eq_coe {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] {f : SupBotHom α β} : f.toFun = ↑f

theorem SupBotHom.ext {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] {f : SupBotHom α β} {g : SupBotHom α β} (h : ∀ (a : α), ↑f a = ↑g a) : f = g

def SupBotHom.copy {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β) (f' : α → β) (h : f' = ↑f) : SupBotHom α β

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

@[simp]

theorem SupBotHom.coe_copy {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β) (f' : α → β) (h : f' = ↑f) : ↑(SupBotHom.copy f f' h) = f'

theorem SupBotHom.copy_eq {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β) (f' : α → β) (h : f' = ↑f) : SupBotHom.copy f f' h = f

@[simp]

theorem SupBotHom.id_toSupHom (α : Type u_3) [Sup α] [Bot α] : (SupBotHom.id α).toSupHom = SupHom.id α

def SupBotHom.id (α : Type u_3) [Sup α] [Bot α] : SupBotHom α α

id as a SupBotHom.

instance SupBotHom.instInhabitedSupBotHom (α : Type u_3) [Sup α] [Bot α] : Inhabited (SupBotHom α α)

@[simp]

theorem SupBotHom.coe_id (α : Type u_3) [Sup α] [Bot α] : ↑(SupBotHom.id α) = id

@[simp]

theorem SupBotHom.id_apply {α : Type u_3} [Sup α] [Bot α] (a : α) : ↑(SupBotHom.id α) a = a

def SupBotHom.comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] (f : SupBotHom β γ) (g : SupBotHom α β) : SupBotHom α γ

Composition of SupBotHoms as a SupBotHom.

@[simp]

theorem SupBotHom.coe_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] (f : SupBotHom β γ) (g : SupBotHom α β) : ↑(SupBotHom.comp f g) = ↑f ∘ ↑g

@[simp]

theorem SupBotHom.comp_apply {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] (f : SupBotHom β γ) (g : SupBotHom α β) (a : α) : ↑(SupBotHom.comp f g) a = ↑f (↑g a)

@[simp]

theorem SupBotHom.comp_assoc {α : Type u_3} {β : Type u_4} {γ : Type u_5} {δ : Type u_6} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] [Sup δ] [Bot δ] (f : SupBotHom γ δ) (g : SupBotHom β γ) (h : SupBotHom α β) : SupBotHom.comp (SupBotHom.comp f g) h = SupBotHom.comp f (SupBotHom.comp g h)

@[simp]

theorem SupBotHom.comp_id {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β) : SupBotHom.comp f (SupBotHom.id α) = f

@[simp]

theorem SupBotHom.id_comp {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] (f : SupBotHom α β) : SupBotHom.comp (SupBotHom.id β) f = f

@[simp]

theorem SupBotHom.cancel_right {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] {g₁ : SupBotHom β γ} {g₂ : SupBotHom β γ} {f : SupBotHom α β} (hf : Function.Surjective ↑f) : SupBotHom.comp g₁ f = SupBotHom.comp g₂ f ↔ g₁ = g₂

@[simp]

theorem SupBotHom.cancel_left {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] {g : SupBotHom β γ} {f₁ : SupBotHom α β} {f₂ : SupBotHom α β} (hg : Function.Injective ↑g) : SupBotHom.comp g f₁ = SupBotHom.comp g f₂ ↔ f₁ = f₂

instance SupBotHom.instSupSupBotHomToSupToBotToLEToPreorderToPartialOrder {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [SemilatticeSup β] [OrderBot β] : Sup (SupBotHom α β)

instance SupBotHom.instSemilatticeSupSupBotHomToSupToBotToLEToPreorderToPartialOrder {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [SemilatticeSup β] [OrderBot β] : SemilatticeSup (SupBotHom α β)

instance SupBotHom.instOrderBotSupBotHomToSupToBotToLEToPreorderToPartialOrderToLEToPreorderToPartialOrderInstSemilatticeSupSupBotHomToSupToBotToLEToPreorderToPartialOrder {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [SemilatticeSup β] [OrderBot β] : OrderBot (SupBotHom α β)

@[simp]

theorem SupBotHom.coe_sup {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [SemilatticeSup β] [OrderBot β] (f : SupBotHom α β) (g : SupBotHom α β) : ↑(f ⊔ g) = ↑(f ⊔ g)

@[simp]

theorem SupBotHom.coe_bot {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [SemilatticeSup β] [OrderBot β] : ↑⊥ = ⊥

@[simp]

theorem SupBotHom.sup_apply {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [SemilatticeSup β] [OrderBot β] (f : SupBotHom α β) (g : SupBotHom α β) (a : α) : ↑(f ⊔ g) a = ↑f a ⊔ ↑g a

@[simp]

theorem SupBotHom.bot_apply {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [SemilatticeSup β] [OrderBot β] (a : α) : ↑⊥ a = ⊥

Finitary infimum homomorphisms

def InfTopHom.toTopHom {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β) : TopHom α β

Reinterpret an InfTopHom as a TopHom.

instance InfTopHom.instInfTopHomClassInfTopHom {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] : InfTopHomClass (InfTopHom α β) α β

instance InfTopHom.instFunLikeInfTopHom {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] : FunLike (InfTopHom α β) α fun x => β

Helper instance for when there's too many metavariables to apply fun_like.has_coe_to_fun directly.

@[simp]

theorem InfTopHom.coe_toInfHom {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] {f : InfTopHom α β} : ↑f.toInfHom = ↑f

@[simp]

theorem InfTopHom.coe_toTopHom {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] {f : InfTopHom α β} : ↑(InfTopHom.toTopHom f) = ↑f

theorem InfTopHom.toFun_eq_coe {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] {f : InfTopHom α β} : f.toFun = ↑f

theorem InfTopHom.ext {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] {f : InfTopHom α β} {g : InfTopHom α β} (h : ∀ (a : α), ↑f a = ↑g a) : f = g

def InfTopHom.copy {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β) (f' : α → β) (h : f' = ↑f) : InfTopHom α β

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

@[simp]

theorem InfTopHom.coe_copy {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β) (f' : α → β) (h : f' = ↑f) : ↑(InfTopHom.copy f f' h) = f'

theorem InfTopHom.copy_eq {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β) (f' : α → β) (h : f' = ↑f) : InfTopHom.copy f f' h = f

@[simp]

theorem InfTopHom.id_toInfHom (α : Type u_3) [Inf α] [Top α] : (InfTopHom.id α).toInfHom = InfHom.id α

def InfTopHom.id (α : Type u_3) [Inf α] [Top α] : InfTopHom α α

id as an InfTopHom.

instance InfTopHom.instInhabitedInfTopHom (α : Type u_3) [Inf α] [Top α] : Inhabited (InfTopHom α α)

@[simp]

theorem InfTopHom.coe_id (α : Type u_3) [Inf α] [Top α] : ↑(InfTopHom.id α) = id

@[simp]

theorem InfTopHom.id_apply {α : Type u_3} [Inf α] [Top α] (a : α) : ↑(InfTopHom.id α) a = a

def InfTopHom.comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] (f : InfTopHom β γ) (g : InfTopHom α β) : InfTopHom α γ

Composition of InfTopHoms as an InfTopHom.

@[simp]

theorem InfTopHom.coe_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] (f : InfTopHom β γ) (g : InfTopHom α β) : ↑(InfTopHom.comp f g) = ↑f ∘ ↑g

@[simp]

theorem InfTopHom.comp_apply {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] (f : InfTopHom β γ) (g : InfTopHom α β) (a : α) : ↑(InfTopHom.comp f g) a = ↑f (↑g a)

@[simp]

theorem InfTopHom.comp_assoc {α : Type u_3} {β : Type u_4} {γ : Type u_5} {δ : Type u_6} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] [Inf δ] [Top δ] (f : InfTopHom γ δ) (g : InfTopHom β γ) (h : InfTopHom α β) : InfTopHom.comp (InfTopHom.comp f g) h = InfTopHom.comp f (InfTopHom.comp g h)

@[simp]

theorem InfTopHom.comp_id {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β) : InfTopHom.comp f (InfTopHom.id α) = f

@[simp]

theorem InfTopHom.id_comp {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β) : InfTopHom.comp (InfTopHom.id β) f = f

@[simp]

theorem InfTopHom.cancel_right {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] {g₁ : InfTopHom β γ} {g₂ : InfTopHom β γ} {f : InfTopHom α β} (hf : Function.Surjective ↑f) : InfTopHom.comp g₁ f = InfTopHom.comp g₂ f ↔ g₁ = g₂

@[simp]

theorem InfTopHom.cancel_left {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] {g : InfTopHom β γ} {f₁ : InfTopHom α β} {f₂ : InfTopHom α β} (hg : Function.Injective ↑g) : InfTopHom.comp g f₁ = InfTopHom.comp g f₂ ↔ f₁ = f₂

instance InfTopHom.instInfInfTopHomToInfToTopToLEToPreorderToPartialOrder {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [SemilatticeInf β] [OrderTop β] : Inf (InfTopHom α β)

instance InfTopHom.instSemilatticeInfInfTopHomToInfToTopToLEToPreorderToPartialOrder {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [SemilatticeInf β] [OrderTop β] : SemilatticeInf (InfTopHom α β)

instance InfTopHom.instOrderTopInfTopHomToInfToTopToLEToPreorderToPartialOrderToLEToPreorderToPartialOrderInstSemilatticeInfInfTopHomToInfToTopToLEToPreorderToPartialOrder {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [SemilatticeInf β] [OrderTop β] : OrderTop (InfTopHom α β)

@[simp]

theorem InfTopHom.coe_inf {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [SemilatticeInf β] [OrderTop β] (f : InfTopHom α β) (g : InfTopHom α β) : ↑(f ⊓ g) = ↑(f ⊓ g)

@[simp]

theorem InfTopHom.coe_top {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [SemilatticeInf β] [OrderTop β] : ↑⊤ = ⊤

@[simp]

theorem InfTopHom.inf_apply {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [SemilatticeInf β] [OrderTop β] (f : InfTopHom α β) (g : InfTopHom α β) (a : α) : ↑(f ⊓ g) a = ↑f a ⊓ ↑g a

@[simp]

theorem InfTopHom.top_apply {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [SemilatticeInf β] [OrderTop β] (a : α) : ↑⊤ a = ⊤

Lattice homomorphisms

def LatticeHom.toInfHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) : InfHom α β

Reinterpret a LatticeHom as an InfHom.

instance LatticeHom.instLatticeHomClassLatticeHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] : LatticeHomClass (LatticeHom α β) α β

instance LatticeHom.instFunLikeLatticeHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] : FunLike (LatticeHom α β) α fun x => β

Helper instance for when there's too many metavariables to apply fun_like.has_coe_to_fun directly.

@[simp]

theorem LatticeHom.coe_toSupHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] {f : LatticeHom α β} : ↑f.toSupHom = ↑f

@[simp]

theorem LatticeHom.coe_toInfHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] {f : LatticeHom α β} : ↑(LatticeHom.toInfHom f) = ↑f

theorem LatticeHom.toFun_eq_coe {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] {f : LatticeHom α β} : f.toFun = ↑f

theorem LatticeHom.ext {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] {f : LatticeHom α β} {g : LatticeHom α β} (h : ∀ (a : α), ↑f a = ↑g a) : f = g

def LatticeHom.copy {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) (f' : α → β) (h : f' = ↑f) : LatticeHom α β

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

@[simp]

theorem LatticeHom.coe_copy {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) (f' : α → β) (h : f' = ↑f) : ↑(LatticeHom.copy f f' h) = f'

theorem LatticeHom.copy_eq {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) (f' : α → β) (h : f' = ↑f) : LatticeHom.copy f f' h = f

def LatticeHom.id (α : Type u_3) [Lattice α] : LatticeHom α α

id as a LatticeHom.

instance LatticeHom.instInhabitedLatticeHom (α : Type u_3) [Lattice α] : Inhabited (LatticeHom α α)

@[simp]

theorem LatticeHom.coe_id (α : Type u_3) [Lattice α] : ↑(LatticeHom.id α) = id

@[simp]

theorem LatticeHom.id_apply {α : Type u_3} [Lattice α] (a : α) : ↑(LatticeHom.id α) a = a

def LatticeHom.comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β) : LatticeHom α γ

Composition of LatticeHoms as a LatticeHom.

@[simp]

theorem LatticeHom.coe_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β) : ↑(LatticeHom.comp f g) = ↑f ∘ ↑g

@[simp]

theorem LatticeHom.comp_apply {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β) (a : α) : ↑(LatticeHom.comp f g) a = ↑f (↑g a)

@[simp]

theorem LatticeHom.coe_comp_sup_hom' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β) : { toFun := ↑f ∘ ↑g, map_sup' := (_ : ∀ (a b : α), ↑(LatticeHom.comp f g) (a ⊔ b) = ↑(LatticeHom.comp f g) a ⊔ ↑(LatticeHom.comp f g) b) } = SupHom.comp { toFun := ↑f, map_sup' := (_ : ∀ (a b : β), ↑f (a ⊔ b) = ↑f a ⊔ ↑f b) } { toFun := ↑g, map_sup' := (_ : ∀ (a b : α), ↑g (a ⊔ b) = ↑g a ⊔ ↑g b) }

theorem LatticeHom.coe_comp_sup_hom {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β) : { toFun := ↑(LatticeHom.comp f g), map_sup' := (_ : ∀ (a b : α), ↑(LatticeHom.comp f g) (a ⊔ b) = ↑(LatticeHom.comp f g) a ⊔ ↑(LatticeHom.comp f g) b) } = SupHom.comp { toFun := ↑f, map_sup' := (_ : ∀ (a b : β), ↑f (a ⊔ b) = ↑f a ⊔ ↑f b) } { toFun := ↑g, map_sup' := (_ : ∀ (a b : α), ↑g (a ⊔ b) = ↑g a ⊔ ↑g b) }

@[simp]

theorem LatticeHom.coe_comp_inf_hom' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β) : { toFun := ↑f ∘ ↑g, map_inf' := (_ : ∀ (a b : α), ↑(LatticeHom.comp f g) (a ⊓ b) = ↑(LatticeHom.comp f g) a ⊓ ↑(LatticeHom.comp f g) b) } = InfHom.comp { toFun := ↑f, map_inf' := (_ : ∀ (a b : β), ↑f (a ⊓ b) = ↑f a ⊓ ↑f b) } { toFun := ↑g, map_inf' := (_ : ∀ (a b : α), ↑g (a ⊓ b) = ↑g a ⊓ ↑g b) }

theorem LatticeHom.coe_comp_inf_hom {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β) : { toFun := ↑(LatticeHom.comp f g), map_inf' := (_ : ∀ (a b : α), ↑(LatticeHom.comp f g) (a ⊓ b) = ↑(LatticeHom.comp f g) a ⊓ ↑(LatticeHom.comp f g) b) } = InfHom.comp { toFun := ↑f, map_inf' := (_ : ∀ (a b : β), ↑f (a ⊓ b) = ↑f a ⊓ ↑f b) } { toFun := ↑g, map_inf' := (_ : ∀ (a b : α), ↑g (a ⊓ b) = ↑g a ⊓ ↑g b) }

@[simp]

theorem LatticeHom.comp_assoc {α : Type u_3} {β : Type u_4} {γ : Type u_5} {δ : Type u_6} [Lattice α] [Lattice β] [Lattice γ] [Lattice δ] (f : LatticeHom γ δ) (g : LatticeHom β γ) (h : LatticeHom α β) : LatticeHom.comp (LatticeHom.comp f g) h = LatticeHom.comp f (LatticeHom.comp g h)

@[simp]

theorem LatticeHom.comp_id {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) : LatticeHom.comp f (LatticeHom.id α) = f

@[simp]

theorem LatticeHom.id_comp {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) : LatticeHom.comp (LatticeHom.id β) f = f

@[simp]

theorem LatticeHom.cancel_right {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] {g₁ : LatticeHom β γ} {g₂ : LatticeHom β γ} {f : LatticeHom α β} (hf : Function.Surjective ↑f) : LatticeHom.comp g₁ f = LatticeHom.comp g₂ f ↔ g₁ = g₂

@[simp]

theorem LatticeHom.cancel_left {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] {g : LatticeHom β γ} {f₁ : LatticeHom α β} {f₂ : LatticeHom α β} (hg : Function.Injective ↑g) : LatticeHom.comp g f₁ = LatticeHom.comp g f₂ ↔ f₁ = f₂

instance OrderHomClass.toLatticeHomClass {F : Type u_1} (α : Type u_3) (β : Type u_4) [LinearOrder α] [Lattice β] [OrderHomClass F α β] : LatticeHomClass F α β

An order homomorphism from a linear order is a lattice homomorphism.

def OrderHomClass.toLatticeHom {F : Type u_1} (α : Type u_3) (β : Type u_4) [LinearOrder α] [Lattice β] [OrderHomClass F α β] (f : F) : LatticeHom α β

Reinterpret an order homomorphism to a linear order as a LatticeHom.

@[simp]

theorem OrderHomClass.coe_to_lattice_hom {F : Type u_1} (α : Type u_3) (β : Type u_4) [LinearOrder α] [Lattice β] [OrderHomClass F α β] (f : F) : ↑(OrderHomClass.toLatticeHom α β f) = ↑f

@[simp]

theorem OrderHomClass.to_lattice_hom_apply {F : Type u_1} (α : Type u_3) (β : Type u_4) [LinearOrder α] [Lattice β] [OrderHomClass F α β] (f : F) (a : α) : ↑(OrderHomClass.toLatticeHom α β f) a = ↑f a

Bounded lattice homomorphisms

def BoundedLatticeHom.toSupBotHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β) : SupBotHom α β

Reinterpret a BoundedLatticeHom as a SupBotHom.

def BoundedLatticeHom.toInfTopHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β) : InfTopHom α β

Reinterpret a BoundedLatticeHom as an InfTopHom.

def BoundedLatticeHom.toBoundedOrderHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β) : BoundedOrderHom α β

Reinterpret a BoundedLatticeHom as a BoundedOrderHom.

instance BoundedLatticeHom.instBoundedLatticeHomClass {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] : BoundedLatticeHomClass (BoundedLatticeHom α β) α β

@[simp]

theorem BoundedLatticeHom.coe_toLatticeHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] {f : BoundedLatticeHom α β} : ↑f.toLatticeHom = ↑f

@[simp]

theorem BoundedLatticeHom.toFun_eq_coe {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] {f : BoundedLatticeHom α β} : f.toFun = ↑f

theorem BoundedLatticeHom.ext {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] {f : BoundedLatticeHom α β} {g : BoundedLatticeHom α β} (h : ∀ (a : α), ↑f a = ↑g a) : f = g

def BoundedLatticeHom.copy {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β) (f' : α → β) (h : f' = ↑f) : BoundedLatticeHom α β

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

@[simp]

theorem BoundedLatticeHom.coe_copy {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β) (f' : α → β) (h : f' = ↑f) : ↑(BoundedLatticeHom.copy f f' h) = f'

theorem BoundedLatticeHom.copy_eq {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β) (f' : α → β) (h : f' = ↑f) : BoundedLatticeHom.copy f f' h = f

def BoundedLatticeHom.id (α : Type u_3) [Lattice α] [BoundedOrder α] : BoundedLatticeHom α α

id as a BoundedLatticeHom.

instance BoundedLatticeHom.instInhabitedBoundedLatticeHom (α : Type u_3) [Lattice α] [BoundedOrder α] : Inhabited (BoundedLatticeHom α α)

@[simp]

theorem BoundedLatticeHom.coe_id (α : Type u_3) [Lattice α] [BoundedOrder α] : ↑(BoundedLatticeHom.id α) = id

@[simp]

theorem BoundedLatticeHom.id_apply {α : Type u_3} [Lattice α] [BoundedOrder α] (a : α) : ↑(BoundedLatticeHom.id α) a = a

def BoundedLatticeHom.comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) : BoundedLatticeHom α γ

Composition of BoundedLatticeHoms as a BoundedLatticeHom.

@[simp]

theorem BoundedLatticeHom.coe_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) : ↑(BoundedLatticeHom.comp f g) = ↑f ∘ ↑g

@[simp]

theorem BoundedLatticeHom.comp_apply {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) (a : α) : ↑(BoundedLatticeHom.comp f g) a = ↑f (↑g a)

@[simp]

theorem BoundedLatticeHom.coe_comp_lattice_hom' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) : { toSupHom := SupHom.comp { toFun := ↑f, map_sup' := (_ : ∀ (a b : β), ↑f (a ⊔ b) = ↑f a ⊔ ↑f b) } { toFun := ↑g, map_sup' := (_ : ∀ (a b : α), ↑g (a ⊔ b) = ↑g a ⊔ ↑g b) }, map_inf' := (_ : ∀ (a b : α), ↑(BoundedLatticeHom.comp f g) (a ⊓ b) = ↑(BoundedLatticeHom.comp f g) a ⊓ ↑(BoundedLatticeHom.comp f g) b) } = LatticeHom.comp { toSupHom := { toFun := ↑f, map_sup' := (_ : ∀ (a b : β), ↑f (a ⊔ b) = ↑f a ⊔ ↑f b) }, map_inf' := (_ : ∀ (a b : β), ↑f (a ⊓ b) = ↑f a ⊓ ↑f b) } { toSupHom := { toFun := ↑g, map_sup' := (_ : ∀ (a b : α), ↑g (a ⊔ b) = ↑g a ⊔ ↑g b) }, map_inf' := (_ : ∀ (a b : α), ↑g (a ⊓ b) = ↑g a ⊓ ↑g b) }

theorem BoundedLatticeHom.coe_comp_lattice_hom {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) : { toSupHom := { toFun := ↑(BoundedLatticeHom.comp f g), map_sup' := (_ : ∀ (a b : α), ↑(BoundedLatticeHom.comp f g) (a ⊔ b) = ↑(BoundedLatticeHom.comp f g) a ⊔ ↑(BoundedLatticeHom.comp f g) b) }, map_inf' := (_ : ∀ (a b : α), ↑(BoundedLatticeHom.comp f g) (a ⊓ b) = ↑(BoundedLatticeHom.comp f g) a ⊓ ↑(BoundedLatticeHom.comp f g) b) } = LatticeHom.comp { toSupHom := { toFun := ↑f, map_sup' := (_ : ∀ (a b : β), ↑f (a ⊔ b) = ↑f a ⊔ ↑f b) }, map_inf' := (_ : ∀ (a b : β), ↑f (a ⊓ b) = ↑f a ⊓ ↑f b) } { toSupHom := { toFun := ↑g, map_sup' := (_ : ∀ (a b : α), ↑g (a ⊔ b) = ↑g a ⊔ ↑g b) }, map_inf' := (_ : ∀ (a b : α), ↑g (a ⊓ b) = ↑g a ⊓ ↑g b) }

@[simp]

theorem BoundedLatticeHom.coe_comp_sup_hom' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) : { toFun := ↑f ∘ ↑g, map_sup' := (_ : ∀ (a b : α), ↑(BoundedLatticeHom.comp f g) (a ⊔ b) = ↑(BoundedLatticeHom.comp f g) a ⊔ ↑(BoundedLatticeHom.comp f g) b) } = SupHom.comp { toFun := ↑f, map_sup' := (_ : ∀ (a b : β), ↑f (a ⊔ b) = ↑f a ⊔ ↑f b) } { toFun := ↑g, map_sup' := (_ : ∀ (a b : α), ↑g (a ⊔ b) = ↑g a ⊔ ↑g b) }

theorem BoundedLatticeHom.coe_comp_sup_hom {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) : { toFun := ↑(BoundedLatticeHom.comp f g), map_sup' := (_ : ∀ (a b : α), ↑(BoundedLatticeHom.comp f g) (a ⊔ b) = ↑(BoundedLatticeHom.comp f g) a ⊔ ↑(BoundedLatticeHom.comp f g) b) } = SupHom.comp { toFun := ↑f, map_sup' := (_ : ∀ (a b : β), ↑f (a ⊔ b) = ↑f a ⊔ ↑f b) } { toFun := ↑g, map_sup' := (_ : ∀ (a b : α), ↑g (a ⊔ b) = ↑g a ⊔ ↑g b) }

@[simp]

theorem BoundedLatticeHom.coe_comp_inf_hom' {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) : { toFun := ↑f ∘ ↑g, map_inf' := (_ : ∀ (a b : α), ↑(BoundedLatticeHom.comp f g) (a ⊓ b) = ↑(BoundedLatticeHom.comp f g) a ⊓ ↑(BoundedLatticeHom.comp f g) b) } = InfHom.comp { toFun := ↑f, map_inf' := (_ : ∀ (a b : β), ↑f (a ⊓ b) = ↑f a ⊓ ↑f b) } { toFun := ↑g, map_inf' := (_ : ∀ (a b : α), ↑g (a ⊓ b) = ↑g a ⊓ ↑g b) }

theorem BoundedLatticeHom.coe_comp_inf_hom {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] (f : BoundedLatticeHom β γ) (g : BoundedLatticeHom α β) : { toFun := ↑(BoundedLatticeHom.comp f g), map_inf' := (_ : ∀ (a b : α), ↑(BoundedLatticeHom.comp f g) (a ⊓ b) = ↑(BoundedLatticeHom.comp f g) a ⊓ ↑(BoundedLatticeHom.comp f g) b) } = InfHom.comp { toFun := ↑f, map_inf' := (_ : ∀ (a b : β), ↑f (a ⊓ b) = ↑f a ⊓ ↑f b) } { toFun := ↑g, map_inf' := (_ : ∀ (a b : α), ↑g (a ⊓ b) = ↑g a ⊓ ↑g b) }

@[simp]

theorem BoundedLatticeHom.comp_assoc {α : Type u_3} {β : Type u_4} {γ : Type u_5} {δ : Type u_6} [Lattice α] [Lattice β] [Lattice γ] [Lattice δ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] [BoundedOrder δ] (f : BoundedLatticeHom γ δ) (g : BoundedLatticeHom β γ) (h : BoundedLatticeHom α β) : BoundedLatticeHom.comp (BoundedLatticeHom.comp f g) h = BoundedLatticeHom.comp f (BoundedLatticeHom.comp g h)

@[simp]

theorem BoundedLatticeHom.comp_id {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β) : BoundedLatticeHom.comp f (BoundedLatticeHom.id α) = f

@[simp]

theorem BoundedLatticeHom.id_comp {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder α] [BoundedOrder β] (f : BoundedLatticeHom α β) : BoundedLatticeHom.comp (BoundedLatticeHom.id β) f = f

@[simp]

theorem BoundedLatticeHom.cancel_right {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] {g₁ : BoundedLatticeHom β γ} {g₂ : BoundedLatticeHom β γ} {f : BoundedLatticeHom α β} (hf : Function.Surjective ↑f) : BoundedLatticeHom.comp g₁ f = BoundedLatticeHom.comp g₂ f ↔ g₁ = g₂

@[simp]

theorem BoundedLatticeHom.cancel_left {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] [BoundedOrder α] [BoundedOrder β] [BoundedOrder γ] {g : BoundedLatticeHom β γ} {f₁ : BoundedLatticeHom α β} {f₂ : BoundedLatticeHom α β} (hg : Function.Injective ↑g) : BoundedLatticeHom.comp g f₁ = BoundedLatticeHom.comp g f₂ ↔ f₁ = f₂

Dual homs

@[simp]

theorem SupHom.dual_symm_apply_toFun {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : InfHom αᵒᵈ βᵒᵈ) (a : αᵒᵈ) : ↑(↑SupHom.dual.symm f) a = ↑f a

@[simp]

theorem SupHom.dual_apply_toFun {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] (f : SupHom α β) (a : α) : ↑(↑SupHom.dual f) a = ↑f a

def SupHom.dual {α : Type u_3} {β : Type u_4} [Sup α] [Sup β] : SupHom α β ≃ InfHom αᵒᵈ βᵒᵈ

Reinterpret a supremum homomorphism as an infimum homomorphism between the dual lattices.

@[simp]

theorem SupHom.dual_id {α : Type u_3} [Sup α] : ↑SupHom.dual (SupHom.id α) = InfHom.id αᵒᵈ

@[simp]

theorem SupHom.dual_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] (g : SupHom β γ) (f : SupHom α β) : ↑SupHom.dual (SupHom.comp g f) = InfHom.comp (↑SupHom.dual g) (↑SupHom.dual f)

@[simp]

theorem SupHom.symm_dual_id {α : Type u_3} [Sup α] : ↑SupHom.dual.symm (InfHom.id αᵒᵈ) = SupHom.id α

@[simp]

theorem SupHom.symm_dual_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Sup β] [Sup γ] (g : InfHom βᵒᵈ γᵒᵈ) (f : InfHom αᵒᵈ βᵒᵈ) : ↑SupHom.dual.symm (InfHom.comp g f) = SupHom.comp (↑SupHom.dual.symm g) (↑SupHom.dual.symm f)

@[simp]

theorem InfHom.dual_apply_toFun {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : InfHom α β) (a : α) : ↑(↑InfHom.dual f) a = ↑f a

@[simp]

theorem InfHom.dual_symm_apply_toFun {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] (f : SupHom αᵒᵈ βᵒᵈ) (a : αᵒᵈ) : ↑(↑InfHom.dual.symm f) a = ↑f a

def InfHom.dual {α : Type u_3} {β : Type u_4} [Inf α] [Inf β] : InfHom α β ≃ SupHom αᵒᵈ βᵒᵈ

Reinterpret an infimum homomorphism as a supremum homomorphism between the dual lattices.

@[simp]

theorem InfHom.dual_id {α : Type u_3} [Inf α] : ↑InfHom.dual (InfHom.id α) = SupHom.id αᵒᵈ

@[simp]

theorem InfHom.dual_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] (g : InfHom β γ) (f : InfHom α β) : ↑InfHom.dual (InfHom.comp g f) = SupHom.comp (↑InfHom.dual g) (↑InfHom.dual f)

@[simp]

theorem InfHom.symm_dual_id {α : Type u_3} [Inf α] : ↑InfHom.dual.symm (SupHom.id αᵒᵈ) = InfHom.id α

@[simp]

theorem InfHom.symm_dual_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Inf β] [Inf γ] (g : SupHom βᵒᵈ γᵒᵈ) (f : SupHom αᵒᵈ βᵒᵈ) : ↑InfHom.dual.symm (SupHom.comp g f) = InfHom.comp (↑InfHom.dual.symm g) (↑InfHom.dual.symm f)

def SupBotHom.dual {α : Type u_3} {β : Type u_4} [Sup α] [Bot α] [Sup β] [Bot β] : SupBotHom α β ≃ InfTopHom αᵒᵈ βᵒᵈ

Reinterpret a finitary supremum homomorphism as a finitary infimum homomorphism between the dual lattices.

@[simp]

theorem SupBotHom.dual_id {α : Type u_3} [Sup α] [Bot α] : ↑SupBotHom.dual (SupBotHom.id α) = InfTopHom.id αᵒᵈ

@[simp]

theorem SupBotHom.dual_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] (g : SupBotHom β γ) (f : SupBotHom α β) : ↑SupBotHom.dual (SupBotHom.comp g f) = InfTopHom.comp (↑SupBotHom.dual g) (↑SupBotHom.dual f)

@[simp]

theorem SupBotHom.symm_dual_id {α : Type u_3} [Sup α] [Bot α] : ↑SupBotHom.dual.symm (InfTopHom.id αᵒᵈ) = SupBotHom.id α

@[simp]

theorem SupBotHom.symm_dual_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Sup α] [Bot α] [Sup β] [Bot β] [Sup γ] [Bot γ] (g : InfTopHom βᵒᵈ γᵒᵈ) (f : InfTopHom αᵒᵈ βᵒᵈ) : ↑SupBotHom.dual.symm (InfTopHom.comp g f) = SupBotHom.comp (↑SupBotHom.dual.symm g) (↑SupBotHom.dual.symm f)

@[simp]

theorem InfTopHom.dual_symm_apply_toInfHom {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : SupBotHom αᵒᵈ βᵒᵈ) : (↑InfTopHom.dual.symm f).toInfHom = ↑InfHom.dual.symm f.toSupHom

@[simp]

theorem InfTopHom.dual_apply_toSupHom {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] (f : InfTopHom α β) : (↑InfTopHom.dual f).toSupHom = ↑InfHom.dual f.toInfHom

def InfTopHom.dual {α : Type u_3} {β : Type u_4} [Inf α] [Top α] [Inf β] [Top β] : InfTopHom α β ≃ SupBotHom αᵒᵈ βᵒᵈ

Reinterpret a finitary infimum homomorphism as a finitary supremum homomorphism between the dual lattices.

@[simp]

theorem InfTopHom.dual_id {α : Type u_3} [Inf α] [Top α] : ↑InfTopHom.dual (InfTopHom.id α) = SupBotHom.id αᵒᵈ

@[simp]

theorem InfTopHom.dual_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] (g : InfTopHom β γ) (f : InfTopHom α β) : ↑InfTopHom.dual (InfTopHom.comp g f) = SupBotHom.comp (↑InfTopHom.dual g) (↑InfTopHom.dual f)

@[simp]

theorem InfTopHom.symm_dual_id {α : Type u_3} [Inf α] [Top α] : ↑InfTopHom.dual.symm (SupBotHom.id αᵒᵈ) = InfTopHom.id α

@[simp]

theorem InfTopHom.symm_dual_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Inf α] [Top α] [Inf β] [Top β] [Inf γ] [Top γ] (g : SupBotHom βᵒᵈ γᵒᵈ) (f : SupBotHom αᵒᵈ βᵒᵈ) : ↑InfTopHom.dual.symm (SupBotHom.comp g f) = InfTopHom.comp (↑InfTopHom.dual.symm g) (↑InfTopHom.dual.symm f)

@[simp]

theorem LatticeHom.dual_symm_apply_toSupHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom αᵒᵈ βᵒᵈ) : (↑LatticeHom.dual.symm f).toSupHom = ↑SupHom.dual.symm (LatticeHom.toInfHom f)

@[simp]

theorem LatticeHom.dual_apply_toSupHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) : (↑LatticeHom.dual f).toSupHom = ↑InfHom.dual (LatticeHom.toInfHom f)

def LatticeHom.dual {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] : LatticeHom α β ≃ LatticeHom αᵒᵈ βᵒᵈ

Reinterpret a lattice homomorphism as a lattice homomorphism between the dual lattices.

@[simp]

theorem LatticeHom.dual_id {α : Type u_3} [Lattice α] : ↑LatticeHom.dual (LatticeHom.id α) = LatticeHom.id αᵒᵈ

@[simp]

theorem LatticeHom.dual_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (g : LatticeHom β γ) (f : LatticeHom α β) : ↑LatticeHom.dual (LatticeHom.comp g f) = LatticeHom.comp (↑LatticeHom.dual g) (↑LatticeHom.dual f)

@[simp]

theorem LatticeHom.symm_dual_id {α : Type u_3} [Lattice α] : ↑LatticeHom.dual.symm (LatticeHom.id αᵒᵈ) = LatticeHom.id α

@[simp]

theorem LatticeHom.symm_dual_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (g : LatticeHom βᵒᵈ γᵒᵈ) (f : LatticeHom αᵒᵈ βᵒᵈ) : ↑LatticeHom.dual.symm (LatticeHom.comp g f) = LatticeHom.comp (↑LatticeHom.dual.symm g) (↑LatticeHom.dual.symm f)

@[simp]

theorem BoundedLatticeHom.dual_symm_apply_toLatticeHom {α : Type u_3} {β : Type u_4} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] (f : BoundedLatticeHom αᵒᵈ βᵒᵈ) : (↑BoundedLatticeHom.dual.symm f).toLatticeHom = ↑LatticeHom.dual.symm f.toLatticeHom

@[simp]

theorem BoundedLatticeHom.dual_apply_toLatticeHom {α : Type u_3} {β : Type u_4} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] (f : BoundedLatticeHom α β) : (↑BoundedLatticeHom.dual f).toLatticeHom = ↑LatticeHom.dual f.toLatticeHom

def BoundedLatticeHom.dual {α : Type u_3} {β : Type u_4} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] : BoundedLatticeHom α β ≃ BoundedLatticeHom αᵒᵈ βᵒᵈ

Reinterpret a bounded lattice homomorphism as a bounded lattice homomorphism between the dual bounded lattices.

@[simp]

theorem BoundedLatticeHom.dual_id {α : Type u_3} [Lattice α] [BoundedOrder α] : ↑BoundedLatticeHom.dual (BoundedLatticeHom.id α) = BoundedLatticeHom.id αᵒᵈ

@[simp]

theorem BoundedLatticeHom.dual_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] [Lattice γ] [BoundedOrder γ] (g : BoundedLatticeHom β γ) (f : BoundedLatticeHom α β) : ↑BoundedLatticeHom.dual (BoundedLatticeHom.comp g f) = BoundedLatticeHom.comp (↑BoundedLatticeHom.dual g) (↑BoundedLatticeHom.dual f)

@[simp]

theorem BoundedLatticeHom.symm_dual_id {α : Type u_3} [Lattice α] [BoundedOrder α] : ↑BoundedLatticeHom.dual.symm (BoundedLatticeHom.id αᵒᵈ) = BoundedLatticeHom.id α

@[simp]

theorem BoundedLatticeHom.symm_dual_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [BoundedOrder α] [Lattice β] [BoundedOrder β] [Lattice γ] [BoundedOrder γ] (g : BoundedLatticeHom βᵒᵈ γᵒᵈ) (f : BoundedLatticeHom αᵒᵈ βᵒᵈ) : ↑BoundedLatticeHom.dual.symm (BoundedLatticeHom.comp g f) = BoundedLatticeHom.comp (↑BoundedLatticeHom.dual.symm g) (↑BoundedLatticeHom.dual.symm f)

WithTop, WithBot

@[simp]

theorem SupHom.withTop_toFun {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] (f : SupHom α β) : ∀ (a : WithTop α), ↑(SupHom.withTop f) a = WithTop.map (↑f) a

def SupHom.withTop {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] (f : SupHom α β) : SupHom (WithTop α) (WithTop β)

Adjoins a ⊤ to the domain and codomain of a SupHom.

@[simp]

theorem SupHom.withTop_id {α : Type u_3} [SemilatticeSup α] : SupHom.withTop (SupHom.id α) = SupHom.id (WithTop α)

@[simp]

theorem SupHom.withTop_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [SemilatticeSup α] [SemilatticeSup β] [SemilatticeSup γ] (f : SupHom β γ) (g : SupHom α β) : SupHom.withTop (SupHom.comp f g) = SupHom.comp (SupHom.withTop f) (SupHom.withTop g)

@[simp]

theorem SupHom.withBot_toSupHom_toFun {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] (f : SupHom α β) : ∀ (a : Option α), ↑(SupHom.withBot f).toSupHom a = Option.map (↑f) a

def SupHom.withBot {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] (f : SupHom α β) : SupBotHom (WithBot α) (WithBot β)

Adjoins a ⊥ to the domain and codomain of a SupHom.

@[simp]

theorem SupHom.withBot_id {α : Type u_3} [SemilatticeSup α] : SupHom.withBot (SupHom.id α) = SupBotHom.id (WithBot α)

@[simp]

theorem SupHom.withBot_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [SemilatticeSup α] [SemilatticeSup β] [SemilatticeSup γ] (f : SupHom β γ) (g : SupHom α β) : SupHom.withBot (SupHom.comp f g) = SupBotHom.comp (SupHom.withBot f) (SupHom.withBot g)

@[simp]

theorem SupHom.withTop'_toFun {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] [OrderTop β] (f : SupHom α β) (a : WithTop α) : ↑(SupHom.withTop' f) a = Option.elim a ⊤ ↑f

def SupHom.withTop' {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] [OrderTop β] (f : SupHom α β) : SupHom (WithTop α) β

Adjoins a ⊤ to the codomain of a SupHom.

@[simp]

theorem SupHom.withBot'_toSupHom_toFun {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] [OrderBot β] (f : SupHom α β) (a : WithBot α) : ↑(SupHom.withBot' f).toSupHom a = Option.elim a ⊥ ↑f

def SupHom.withBot' {α : Type u_3} {β : Type u_4} [SemilatticeSup α] [SemilatticeSup β] [OrderBot β] (f : SupHom α β) : SupBotHom (WithBot α) β

Adjoins a ⊥ to the domain of a SupHom.

@[simp]

theorem InfHom.withTop_toInfHom_toFun {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [SemilatticeInf β] (f : InfHom α β) : ∀ (a : Option α), ↑(InfHom.withTop f).toInfHom a = Option.map (↑f) a

def InfHom.withTop {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [SemilatticeInf β] (f : InfHom α β) : InfTopHom (WithTop α) (WithTop β)

Adjoins a ⊤ to the domain and codomain of an InfHom.

@[simp]

theorem InfHom.withTop_id {α : Type u_3} [SemilatticeInf α] : InfHom.withTop (InfHom.id α) = InfTopHom.id (WithTop α)

@[simp]

theorem InfHom.withTop_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [SemilatticeInf α] [SemilatticeInf β] [SemilatticeInf γ] (f : InfHom β γ) (g : InfHom α β) : InfHom.withTop (InfHom.comp f g) = InfTopHom.comp (InfHom.withTop f) (InfHom.withTop g)

@[simp]

theorem InfHom.withBot_toFun {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [SemilatticeInf β] (f : InfHom α β) : ∀ (a : Option α), ↑(InfHom.withBot f) a = Option.map (↑f) a

def InfHom.withBot {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [SemilatticeInf β] (f : InfHom α β) : InfHom (WithBot α) (WithBot β)

Adjoins a ⊥ to the domain and codomain of an InfHom.

@[simp]

theorem InfHom.withBot_id {α : Type u_3} [SemilatticeInf α] : InfHom.withBot (InfHom.id α) = InfHom.id (WithBot α)

@[simp]

theorem InfHom.withBot_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [SemilatticeInf α] [SemilatticeInf β] [SemilatticeInf γ] (f : InfHom β γ) (g : InfHom α β) : InfHom.withBot (InfHom.comp f g) = InfHom.comp (InfHom.withBot f) (InfHom.withBot g)

@[simp]

theorem InfHom.withTop'_toInfHom_toFun {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [SemilatticeInf β] [OrderTop β] (f : InfHom α β) (a : WithTop α) : ↑(InfHom.withTop' f).toInfHom a = Option.elim a ⊤ ↑f

def InfHom.withTop' {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [SemilatticeInf β] [OrderTop β] (f : InfHom α β) : InfTopHom (WithTop α) β

Adjoins a ⊤ to the codomain of an InfHom.

@[simp]

theorem InfHom.withBot'_toFun {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [SemilatticeInf β] [OrderBot β] (f : InfHom α β) (a : WithBot α) : ↑(InfHom.withBot' f) a = Option.elim a ⊥ ↑f

def InfHom.withBot' {α : Type u_3} {β : Type u_4} [SemilatticeInf α] [SemilatticeInf β] [OrderBot β] (f : InfHom α β) : InfHom (WithBot α) β

Adjoins a ⊥ to the codomain of an InfHom.

@[simp]

theorem LatticeHom.withTop_toSupHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) : (LatticeHom.withTop f).toSupHom = SupHom.withTop f.toSupHom

def LatticeHom.withTop {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) : LatticeHom (WithTop α) (WithTop β)

Adjoins a ⊤ to the domain and codomain of a LatticeHom.

@[simp]

theorem LatticeHom.coe_withTop {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) : ↑(LatticeHom.withTop f) = WithTop.map ↑f

theorem LatticeHom.withTop_apply {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) (a : WithTop α) : ↑(LatticeHom.withTop f) a = WithTop.map (↑f) a

@[simp]

theorem LatticeHom.withTop_id {α : Type u_3} [Lattice α] : LatticeHom.withTop (LatticeHom.id α) = LatticeHom.id (WithTop α)

@[simp]

theorem LatticeHom.withTop_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β) : LatticeHom.withTop (LatticeHom.comp f g) = LatticeHom.comp (LatticeHom.withTop f) (LatticeHom.withTop g)

@[simp]

theorem LatticeHom.withBot_toSupHom_toFun {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) (a : WithBot α) : ↑(LatticeHom.withBot f).toSupHom a = ↑(SupHom.withBot f.toSupHom) a

def LatticeHom.withBot {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) : LatticeHom (WithBot α) (WithBot β)

Adjoins a ⊥ to the domain and codomain of a LatticeHom.

@[simp]

theorem LatticeHom.coe_withBot {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) : ↑(LatticeHom.withBot f) = Option.map ↑f

theorem LatticeHom.withBot_apply {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) (a : WithBot α) : ↑(LatticeHom.withBot f) a = WithBot.map (↑f) a

@[simp]

theorem LatticeHom.withBot_id {α : Type u_3} [Lattice α] : LatticeHom.withBot (LatticeHom.id α) = LatticeHom.id (WithBot α)

@[simp]

theorem LatticeHom.withBot_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β) : LatticeHom.withBot (LatticeHom.comp f g) = LatticeHom.comp (LatticeHom.withBot f) (LatticeHom.withBot g)

@[simp]

theorem LatticeHom.withTopWithBot_toLatticeHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) : (LatticeHom.withTopWithBot f).toLatticeHom = LatticeHom.withTop (LatticeHom.withBot f)

def LatticeHom.withTopWithBot {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) : BoundedLatticeHom (WithTop (WithBot α)) (WithTop (WithBot β))

Adjoins a ⊤ and ⊥ to the domain and codomain of a LatticeHom.

@[simp]

theorem LatticeHom.coe_withTopWithBot {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) : ↑(LatticeHom.withTopWithBot f) = Option.map (Option.map ↑f)

theorem LatticeHom.withTopWithBot_apply {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] (f : LatticeHom α β) (a : WithTop (WithBot α)) : ↑(LatticeHom.withTopWithBot f) a = WithTop.map (Option.map ↑f) a

@[simp]

theorem LatticeHom.withTopWithBot_id {α : Type u_3} [Lattice α] : LatticeHom.withTopWithBot (LatticeHom.id α) = BoundedLatticeHom.id (WithTop (WithBot α))

@[simp]

theorem LatticeHom.withTopWithBot_comp {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Lattice α] [Lattice β] [Lattice γ] (f : LatticeHom β γ) (g : LatticeHom α β) : LatticeHom.withTopWithBot (LatticeHom.comp f g) = BoundedLatticeHom.comp (LatticeHom.withTopWithBot f) (LatticeHom.withTopWithBot g)

@[simp]

theorem LatticeHom.withTop'_toSupHom_toFun {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [OrderTop β] (f : LatticeHom α β) : ∀ (a : WithTop α), ↑(LatticeHom.withTop' f).toSupHom a = SupHom.toFun (SupHom.withTop' f.toSupHom) a

def LatticeHom.withTop' {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [OrderTop β] (f : LatticeHom α β) : LatticeHom (WithTop α) β

Adjoins a ⊥ to the codomain of a LatticeHom.

@[simp]

theorem LatticeHom.withBot'_toSupHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [OrderBot β] (f : LatticeHom α β) : (LatticeHom.withBot' f).toSupHom = (SupHom.withBot' f.toSupHom).toSupHom

def LatticeHom.withBot' {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [OrderBot β] (f : LatticeHom α β) : LatticeHom (WithBot α) β

Adjoins a ⊥ to the domain and codomain of a LatticeHom.

@[simp]

theorem LatticeHom.withTopWithBot'_toLatticeHom {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder β] (f : LatticeHom α β) : (LatticeHom.withTopWithBot' f).toLatticeHom = LatticeHom.withTop' (LatticeHom.withBot' f)

def LatticeHom.withTopWithBot' {α : Type u_3} {β : Type u_4} [Lattice α] [Lattice β] [BoundedOrder β] (f : LatticeHom α β) : BoundedLatticeHom (WithTop (WithBot α)) β

Adjoins a ⊤ and ⊥ to the codomain of a LatticeHom.