Heyting algebra morphisms #

A Heyting homomorphism between two Heyting algebras is a bounded lattice homomorphism that preserves Heyting implication.

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 #

HeytingHom: Heyting homomorphisms.
CoheytingHom: Co-Heyting homomorphisms.
BiheytingHom: Bi-Heyting homomorphisms.

Typeclasses #

source

structure HeytingHom (α : Type u_6) (β : Type u_7) [HeytingAlgebra α] [HeytingAlgebra β] extends LatticeHom :

Type (max u_6 u_7)

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_bot' : SupHom.toFun s.toSupHom ⊥ = ⊥
The proposition that a Heyting homomorphism preserves the bottom element.
map_himp' : ∀ (a b : α), SupHom.toFun s.toSupHom (a ⇨ b) = SupHom.toFun s.toSupHom a ⇨ SupHom.toFun s.toSupHom b
The proposition that a Heyting homomorphism preserves the Heyting implication.

The type of Heyting homomorphisms from α to β. Bounded lattice homomorphisms that preserve Heyting implication.

Instances For

source

structure CoheytingHom (α : Type u_6) (β : Type u_7) [CoheytingAlgebra α] [CoheytingAlgebra β] extends LatticeHom :

Type (max u_6 u_7)

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 ⊤ = ⊤
The proposition that a co-Heyting homomorphism preserves the top element.
map_sdiff' : ∀ (a b : α), SupHom.toFun s.toSupHom (a \ b) = SupHom.toFun s.toSupHom a \ SupHom.toFun s.toSupHom b
The proposition that a co-Heyting homomorphism preserves the difference operation.

The type of co-Heyting homomorphisms from α to β. Bounded lattice homomorphisms that preserve difference.

Instances For

source

structure BiheytingHom (α : Type u_6) (β : Type u_7) [BiheytingAlgebra α] [BiheytingAlgebra β] extends LatticeHom :

Type (max u_6 u_7)

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_himp' : ∀ (a b : α), SupHom.toFun s.toSupHom (a ⇨ b) = SupHom.toFun s.toSupHom a ⇨ SupHom.toFun s.toSupHom b
The proposition that a bi-Heyting homomorphism preserves the Heyting implication.
map_sdiff' : ∀ (a b : α), SupHom.toFun s.toSupHom (a \ b) = SupHom.toFun s.toSupHom a \ SupHom.toFun s.toSupHom b
The proposition that a bi-Heyting homomorphism preserves the difference operation.

The type of bi-Heyting homomorphisms from α to β. Bounded lattice homomorphisms that preserve Heyting implication and difference.

Instances For

source

class HeytingHomClass (F : Type u_6) (α : outParam (Type u_7)) (β : outParam (Type u_8)) [HeytingAlgebra α] [HeytingAlgebra β] extends LatticeHomClass :

Type (max (max u_6 u_7) u_8)

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_bot : ∀ (f : F), ↑f ⊥ = ⊥
The proposition that a Heyting homomorphism preserves the bottom element.
map_himp : ∀ (f : F) (a b : α), ↑f (a ⇨ b) = ↑f a ⇨ ↑f b
The proposition that a Heyting homomorphism preserves the Heyting implication.

HeytingHomClass F α β states that F is a type of Heyting homomorphisms.

You should extend this class when you extend HeytingHom.

Instances

source

class CoheytingHomClass (F : Type u_6) (α : outParam (Type u_7)) (β : outParam (Type u_8)) [CoheytingAlgebra α] [CoheytingAlgebra β] extends LatticeHomClass :

Type (max (max u_6 u_7) u_8)

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 ⊤ = ⊤
The proposition that a co-Heyting homomorphism preserves the top element.
map_sdiff : ∀ (f : F) (a b : α), ↑f (a \ b) = ↑f a \ ↑f b
The proposition that a co-Heyting homomorphism preserves the difference operation.

CoheytingHomClass F α β states that F is a type of co-Heyting homomorphisms.

You should extend this class when you extend CoheytingHom.

Instances

source

class BiheytingHomClass (F : Type u_6) (α : outParam (Type u_7)) (β : outParam (Type u_8)) [BiheytingAlgebra α] [BiheytingAlgebra β] extends LatticeHomClass :

Type (max (max u_6 u_7) u_8)

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_himp : ∀ (f : F) (a b : α), ↑f (a ⇨ b) = ↑f a ⇨ ↑f b
The proposition that a bi-Heyting homomorphism preserves the Heyting implication.
map_sdiff : ∀ (f : F) (a b : α), ↑f (a \ b) = ↑f a \ ↑f b
The proposition that a bi-Heyting homomorphism preserves the difference operation.

BiheytingHomClass F α β states that F is a type of bi-Heyting homomorphisms.

You should extend this class when you extend BiheytingHom.

Instances

source

instance HeytingHomClass.toBoundedLatticeHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] :

{x : HeytingAlgebra β} → [inst : HeytingHomClass F α β] → BoundedLatticeHomClass F α β

source

instance CoheytingHomClass.toBoundedLatticeHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] :

{x : CoheytingAlgebra β} → [inst : CoheytingHomClass F α β] → BoundedLatticeHomClass F α β

source

instance BiheytingHomClass.toHeytingHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] :

{x : BiheytingAlgebra β} → [inst : BiheytingHomClass F α β] → HeytingHomClass F α β

source

instance BiheytingHomClass.toCoheytingHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] :

{x : BiheytingAlgebra β} → [inst : BiheytingHomClass F α β] → CoheytingHomClass F α β

source

instance OrderIsoClass.toHeytingHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] :

{x : HeytingAlgebra β} → [inst : OrderIsoClass F α β] → HeytingHomClass F α β

source

instance OrderIsoClass.toCoheytingHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] :

{x : CoheytingAlgebra β} → [inst : OrderIsoClass F α β] → CoheytingHomClass F α β

source

instance OrderIsoClass.toBiheytingHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] :

{x : BiheytingAlgebra β} → [inst : OrderIsoClass F α β] → BiheytingHomClass F α β

source

@[reducible]

def BoundedLatticeHomClass.toBiheytingHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [BooleanAlgebra α] [BooleanAlgebra β] [BoundedLatticeHomClass F α β] :

BiheytingHomClass F α β

This can't be an instance because of typeclass loops.

Instances For

source

@[simp]

theorem map_compl {F : Type u_1} {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingHomClass F α β] (f : F) (a : α) :

↑f aᶜ = (↑f a)ᶜ

source

@[simp]

theorem map_bihimp {F : Type u_1} {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingHomClass F α β] (f : F) (a : α) (b : α) :

↑f (a ⇔ b) = ↑f a ⇔ ↑f b

source

@[simp]

theorem map_hnot {F : Type u_1} {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingHomClass F α β] (f : F) (a : α) :

↑f (￢a) = ￢↑f a

source

@[simp]

theorem map_symmDiff {F : Type u_1} {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingHomClass F α β] (f : F) (a : α) (b : α) :

↑f (a ∆ b) = ↑f a ∆ ↑f b

source

instance instCoeTCHeytingHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingHomClass F α β] :

CoeTC F (HeytingHom α β)

source

instance instCoeTCCoheytingHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingHomClass F α β] :

CoeTC F (CoheytingHom α β)

source

instance instCoeTCBiheytingHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] [BiheytingHomClass F α β] :

CoeTC F (BiheytingHom α β)

source

instance HeytingHom.instHeytingHomClass {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] :

HeytingHomClass (HeytingHom α β) α β

source

theorem HeytingHom.toFun_eq_coe {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] {f : HeytingHom α β} :

f.toFun = ↑f

source

@[simp]

theorem HeytingHom.toFun_eq_coe_aux {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] {f : HeytingHom α β} :

↑f.toLatticeHom = ↑f

source

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

f = g

source

def HeytingHom.copy {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] (f : HeytingHom α β) (f' : α → β) (h : f' = ↑f) :

HeytingHom α β

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

Instances For

source

@[simp]

theorem HeytingHom.coe_copy {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] (f : HeytingHom α β) (f' : α → β) (h : f' = ↑f) :

↑(HeytingHom.copy f f' h) = f'

source

theorem HeytingHom.copy_eq {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] (f : HeytingHom α β) (f' : α → β) (h : f' = ↑f) :

HeytingHom.copy f f' h = f

source

def HeytingHom.id (α : Type u_2) [HeytingAlgebra α] :

HeytingHom α α

id as a HeytingHom.

Instances For

source

@[simp]

theorem HeytingHom.coe_id (α : Type u_2) [HeytingAlgebra α] :

↑(HeytingHom.id α) = id

source

@[simp]

theorem HeytingHom.id_apply {α : Type u_2} [HeytingAlgebra α] (a : α) :

↑(HeytingHom.id α) a = a

source

instance HeytingHom.instInhabitedHeytingHom {α : Type u_2} [HeytingAlgebra α] :

Inhabited (HeytingHom α α)

source

instance HeytingHom.instPartialOrderHeytingHom {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] :

PartialOrder (HeytingHom α β)

source

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

HeytingHom α γ

Composition of HeytingHoms as a HeytingHom.

Instances For

source

@[simp]

theorem HeytingHom.coe_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingAlgebra γ] (f : HeytingHom β γ) (g : HeytingHom α β) :

↑(HeytingHom.comp f g) = ↑f ∘ ↑g

source

@[simp]

theorem HeytingHom.comp_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingAlgebra γ] (f : HeytingHom β γ) (g : HeytingHom α β) (a : α) :

↑(HeytingHom.comp f g) a = ↑f (↑g a)

source

@[simp]

theorem HeytingHom.comp_assoc {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingAlgebra γ] [HeytingAlgebra δ] (f : HeytingHom γ δ) (g : HeytingHom β γ) (h : HeytingHom α β) :

HeytingHom.comp (HeytingHom.comp f g) h = HeytingHom.comp f (HeytingHom.comp g h)

source

@[simp]

theorem HeytingHom.comp_id {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] (f : HeytingHom α β) :

HeytingHom.comp f (HeytingHom.id α) = f

source

@[simp]

theorem HeytingHom.id_comp {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] (f : HeytingHom α β) :

HeytingHom.comp (HeytingHom.id β) f = f

source

@[simp]

theorem HeytingHom.cancel_right {α : Type u_2} {β : Type u_3} {γ : Type u_4} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingAlgebra γ] {f : HeytingHom α β} {g₁ : HeytingHom β γ} {g₂ : HeytingHom β γ} (hf : Function.Surjective ↑f) :

HeytingHom.comp g₁ f = HeytingHom.comp g₂ f ↔ g₁ = g₂

source

@[simp]

theorem HeytingHom.cancel_left {α : Type u_2} {β : Type u_3} {γ : Type u_4} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingAlgebra γ] {f₁ : HeytingHom α β} {f₂ : HeytingHom α β} {g : HeytingHom β γ} (hg : Function.Injective ↑g) :

HeytingHom.comp g f₁ = HeytingHom.comp g f₂ ↔ f₁ = f₂

source

instance CoheytingHom.instCoheytingHomClassCoheytingHom {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] :

CoheytingHomClass (CoheytingHom α β) α β

source

theorem CoheytingHom.toFun_eq_coe {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] {f : CoheytingHom α β} :

f.toFun = ↑f

source

@[simp]

theorem CoheytingHom.toFun_eq_coe_aux {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] {f : CoheytingHom α β} :

↑f.toLatticeHom = ↑f

source

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

f = g

source

def CoheytingHom.copy {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] (f : CoheytingHom α β) (f' : α → β) (h : f' = ↑f) :

CoheytingHom α β

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

Instances For

source

@[simp]

theorem CoheytingHom.coe_copy {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] (f : CoheytingHom α β) (f' : α → β) (h : f' = ↑f) :

↑(CoheytingHom.copy f f' h) = f'

source

theorem CoheytingHom.copy_eq {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] (f : CoheytingHom α β) (f' : α → β) (h : f' = ↑f) :

CoheytingHom.copy f f' h = f

source

def CoheytingHom.id (α : Type u_2) [CoheytingAlgebra α] :

CoheytingHom α α

id as a CoheytingHom.

Instances For

source

@[simp]

theorem CoheytingHom.coe_id (α : Type u_2) [CoheytingAlgebra α] :

↑(CoheytingHom.id α) = id

source

@[simp]

theorem CoheytingHom.id_apply {α : Type u_2} [CoheytingAlgebra α] (a : α) :

↑(CoheytingHom.id α) a = a

source

instance CoheytingHom.instInhabitedCoheytingHom {α : Type u_2} [CoheytingAlgebra α] :

Inhabited (CoheytingHom α α)

source

instance CoheytingHom.instPartialOrderCoheytingHom {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] :

PartialOrder (CoheytingHom α β)

source

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

CoheytingHom α γ

Composition of CoheytingHoms as a CoheytingHom.

Instances For

source

@[simp]

theorem CoheytingHom.coe_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingAlgebra γ] (f : CoheytingHom β γ) (g : CoheytingHom α β) :

↑(CoheytingHom.comp f g) = ↑f ∘ ↑g

source

@[simp]

theorem CoheytingHom.comp_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingAlgebra γ] (f : CoheytingHom β γ) (g : CoheytingHom α β) (a : α) :

↑(CoheytingHom.comp f g) a = ↑f (↑g a)

source

@[simp]

theorem CoheytingHom.comp_assoc {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingAlgebra γ] [CoheytingAlgebra δ] (f : CoheytingHom γ δ) (g : CoheytingHom β γ) (h : CoheytingHom α β) :

CoheytingHom.comp (CoheytingHom.comp f g) h = CoheytingHom.comp f (CoheytingHom.comp g h)

source

@[simp]

theorem CoheytingHom.comp_id {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] (f : CoheytingHom α β) :

CoheytingHom.comp f (CoheytingHom.id α) = f

source

@[simp]

theorem CoheytingHom.id_comp {α : Type u_2} {β : Type u_3} [CoheytingAlgebra α] [CoheytingAlgebra β] (f : CoheytingHom α β) :

CoheytingHom.comp (CoheytingHom.id β) f = f

source

@[simp]

theorem CoheytingHom.cancel_right {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingAlgebra γ] {f : CoheytingHom α β} {g₁ : CoheytingHom β γ} {g₂ : CoheytingHom β γ} (hf : Function.Surjective ↑f) :

CoheytingHom.comp g₁ f = CoheytingHom.comp g₂ f ↔ g₁ = g₂

source

@[simp]

theorem CoheytingHom.cancel_left {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingAlgebra γ] {f₁ : CoheytingHom α β} {f₂ : CoheytingHom α β} {g : CoheytingHom β γ} (hg : Function.Injective ↑g) :

CoheytingHom.comp g f₁ = CoheytingHom.comp g f₂ ↔ f₁ = f₂

source

instance BiheytingHom.instBiheytingHomClassBiheytingHom {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] :

BiheytingHomClass (BiheytingHom α β) α β

source

theorem BiheytingHom.toFun_eq_coe {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] {f : BiheytingHom α β} :

f.toFun = ↑f

source

@[simp]

theorem BiheytingHom.toFun_eq_coe_aux {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] {f : BiheytingHom α β} :

↑f.toLatticeHom = ↑f

source

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

f = g

source

def BiheytingHom.copy {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] (f : BiheytingHom α β) (f' : α → β) (h : f' = ↑f) :

BiheytingHom α β

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

Instances For

source

@[simp]

theorem BiheytingHom.coe_copy {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] (f : BiheytingHom α β) (f' : α → β) (h : f' = ↑f) :

↑(BiheytingHom.copy f f' h) = f'

source

theorem BiheytingHom.copy_eq {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] (f : BiheytingHom α β) (f' : α → β) (h : f' = ↑f) :

BiheytingHom.copy f f' h = f

source

def BiheytingHom.id (α : Type u_2) [BiheytingAlgebra α] :

BiheytingHom α α

id as a BiheytingHom.

Instances For

source

@[simp]

theorem BiheytingHom.coe_id (α : Type u_2) [BiheytingAlgebra α] :

↑(BiheytingHom.id α) = id

source

@[simp]

theorem BiheytingHom.id_apply {α : Type u_2} [BiheytingAlgebra α] (a : α) :

↑(BiheytingHom.id α) a = a

source

instance BiheytingHom.instInhabitedBiheytingHom {α : Type u_2} [BiheytingAlgebra α] :

Inhabited (BiheytingHom α α)

source

instance BiheytingHom.instPartialOrderBiheytingHom {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] :

PartialOrder (BiheytingHom α β)

source

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

BiheytingHom α γ

Composition of BiheytingHoms as a BiheytingHom.

Instances For

source

@[simp]

theorem BiheytingHom.coe_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [BiheytingAlgebra α] [BiheytingAlgebra β] [BiheytingAlgebra γ] (f : BiheytingHom β γ) (g : BiheytingHom α β) :

↑(BiheytingHom.comp f g) = ↑f ∘ ↑g

source

@[simp]

theorem BiheytingHom.comp_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [BiheytingAlgebra α] [BiheytingAlgebra β] [BiheytingAlgebra γ] (f : BiheytingHom β γ) (g : BiheytingHom α β) (a : α) :

↑(BiheytingHom.comp f g) a = ↑f (↑g a)

source

@[simp]

theorem BiheytingHom.comp_assoc {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [BiheytingAlgebra α] [BiheytingAlgebra β] [BiheytingAlgebra γ] [BiheytingAlgebra δ] (f : BiheytingHom γ δ) (g : BiheytingHom β γ) (h : BiheytingHom α β) :

BiheytingHom.comp (BiheytingHom.comp f g) h = BiheytingHom.comp f (BiheytingHom.comp g h)

source

@[simp]

theorem BiheytingHom.comp_id {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] (f : BiheytingHom α β) :

BiheytingHom.comp f (BiheytingHom.id α) = f

source

@[simp]

theorem BiheytingHom.id_comp {α : Type u_2} {β : Type u_3} [BiheytingAlgebra α] [BiheytingAlgebra β] (f : BiheytingHom α β) :

BiheytingHom.comp (BiheytingHom.id β) f = f

source

@[simp]

theorem BiheytingHom.cancel_right {α : Type u_2} {β : Type u_3} {γ : Type u_4} [BiheytingAlgebra α] [BiheytingAlgebra β] [BiheytingAlgebra γ] {f : BiheytingHom α β} {g₁ : BiheytingHom β γ} {g₂ : BiheytingHom β γ} (hf : Function.Surjective ↑f) :

BiheytingHom.comp g₁ f = BiheytingHom.comp g₂ f ↔ g₁ = g₂

source

@[simp]

theorem BiheytingHom.cancel_left {α : Type u_2} {β : Type u_3} {γ : Type u_4} [BiheytingAlgebra α] [BiheytingAlgebra β] [BiheytingAlgebra γ] {f₁ : BiheytingHom α β} {f₂ : BiheytingHom α β} {g : BiheytingHom β γ} (hg : Function.Injective ↑g) :

BiheytingHom.comp g f₁ = BiheytingHom.comp g f₂ ↔ f₁ = f₂