Heyting algebra morphisms #

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

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

Types of morphisms #

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

Typeclasses #

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

Type (max u_6 u_7)

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

toFun : α → β
map_sup' (a b : α) : self.toFun (a ⊔ b) = self.toFun a ⊔ self.toFun b
map_inf' (a b : α) : self.toFun (a ⊓ b) = self.toFun a ⊓ self.toFun b
map_bot' : self.toFun ⊥ = ⊥
The proposition that a Heyting homomorphism preserves the bottom element.
map_himp' (a b : α) : self.toFun (a ⇨ b) = self.toFun a ⇨ self.toFun b
The proposition that a Heyting homomorphism preserves the Heyting implication.

Instances For

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

Type (max u_6 u_7)

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

toFun : α → β
map_sup' (a b : α) : self.toFun (a ⊔ b) = self.toFun a ⊔ self.toFun b
map_inf' (a b : α) : self.toFun (a ⊓ b) = self.toFun a ⊓ self.toFun b
map_top' : self.toFun ⊤ = ⊤
The proposition that a co-Heyting homomorphism preserves the top element.
map_sdiff' (a b : α) : self.toFun (a \ b) = self.toFun a \ self.toFun b
The proposition that a co-Heyting homomorphism preserves the difference operation.

Instances For

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

Type (max u_6 u_7)

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

toFun : α → β
map_sup' (a b : α) : self.toFun (a ⊔ b) = self.toFun a ⊔ self.toFun b
map_inf' (a b : α) : self.toFun (a ⊓ b) = self.toFun a ⊓ self.toFun b
map_himp' (a b : α) : self.toFun (a ⇨ b) = self.toFun a ⇨ self.toFun b
The proposition that a bi-Heyting homomorphism preserves the Heyting implication.
map_sdiff' (a b : α) : self.toFun (a \ b) = self.toFun a \ self.toFun b
The proposition that a bi-Heyting homomorphism preserves the difference operation.

Instances For

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

Prop

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

You should extend this class when you extend HeytingHom.

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.

Instances

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

Prop

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

You should extend this class when you extend CoheytingHom.

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.

Instances

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

Prop

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

You should extend this class when you extend BiheytingHom.

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.

Instances

This section passes in some instances implicitly. See note [implicit instance arguments]

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@[instance 100]

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

BoundedLatticeHomClass F α β

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@[instance 100]

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

BoundedLatticeHomClass F α β

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@[instance 100]

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

HeytingHomClass F α β

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@[instance 100]

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

CoheytingHomClass F α β

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@[instance 100]

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

HeytingHomClass F α β

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@[instance 100]

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

CoheytingHomClass F α β

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@[instance 100]

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

BiheytingHomClass F α β

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

BiheytingHomClass F α β

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@[simp]

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

f aᶜ = (f a)ᶜ

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@[simp]

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

f (bihimp a b) = bihimp (f a) (f b)

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@[simp]

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

f (￢a) = ￢f a

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@[simp]

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

f (symmDiff a b) = symmDiff (f a) (f b)

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

CoeTC F (HeytingHom α β)

Equations

instCoeTCHeytingHomOfHeytingHomClass = { coe := fun (f : F) => { toFun := ⇑f, map_sup' := ⋯, map_inf' := ⋯, map_bot' := ⋯, map_himp' := ⋯ } }

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

CoeTC F (CoheytingHom α β)

Equations

instCoeTCCoheytingHomOfCoheytingHomClass = { coe := fun (f : F) => { toFun := ⇑f, map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_sdiff' := ⋯ } }

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

CoeTC F (BiheytingHom α β)

Equations

instCoeTCBiheytingHomOfBiheytingHomClass = { coe := fun (f : F) => { toFun := ⇑f, map_sup' := ⋯, map_inf' := ⋯, map_himp' := ⋯, map_sdiff' := ⋯ } }

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

FunLike (HeytingHom α β) α β

Equations

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

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

HeytingHomClass (HeytingHom α β) α β

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

f.toFun = ⇑f

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@[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

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

f = g

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

f = g ↔ ∀ (a : α), f a = g a

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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.

Equations

f.copy f' h = { toFun := f', map_sup' := ⋯, map_inf' := ⋯, map_bot' := ⋯, map_himp' := ⋯ }

Instances For

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@[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) :

⇑(f.copy f' h) = f'

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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) :

f.copy f' h = f

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

HeytingHom α α

id as a HeytingHom.

Equations

HeytingHom.id α = { toLatticeHom := LatticeHom.id α, map_bot' := ⋯, map_himp' := ⋯ }

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@[simp]

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

⇑(HeytingHom.id α) = id

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@[simp]

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

(HeytingHom.id α) a = a

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

Inhabited (HeytingHom α α)

Equations

HeytingHom.instInhabited = { default := HeytingHom.id α }

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

PartialOrder (HeytingHom α β)

Equations

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

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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.

Equations

f.comp g = { toFun := ⇑f ∘ ⇑g, map_sup' := ⋯, map_inf' := ⋯, map_bot' := ⋯, map_himp' := ⋯ }

Instances For

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@[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) :

⇑(f.comp g) = ⇑f ∘ ⇑g

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@[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) :

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

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@[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) :

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

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@[simp]

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

f.comp (HeytingHom.id α) = f

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@[simp]

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

(HeytingHom.id β).comp f = f

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@[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 g₂ : HeytingHom β γ} (hf : Function.Surjective ⇑f) :

g₁.comp f = g₂.comp f ↔ g₁ = g₂

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@[simp]

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

g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

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

FunLike (CoheytingHom α β) α β

Equations

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

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

CoheytingHomClass (CoheytingHom α β) α β

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

f.toFun = ⇑f

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@[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

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

f = g

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

f = g ↔ ∀ (a : α), f a = g a

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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.

Equations

f.copy f' h = { toFun := f', map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_sdiff' := ⋯ }

Instances For

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@[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) :

⇑(f.copy f' h) = f'

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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) :

f.copy f' h = f

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

CoheytingHom α α

id as a CoheytingHom.

Equations

CoheytingHom.id α = { toLatticeHom := LatticeHom.id α, map_top' := ⋯, map_sdiff' := ⋯ }

Instances For

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@[simp]

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

⇑(CoheytingHom.id α) = id

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@[simp]

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

(CoheytingHom.id α) a = a

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

Inhabited (CoheytingHom α α)

Equations

CoheytingHom.instInhabited = { default := CoheytingHom.id α }

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

PartialOrder (CoheytingHom α β)

Equations

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

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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.

Equations

f.comp g = { toFun := ⇑f ∘ ⇑g, map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_sdiff' := ⋯ }

Instances For

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@[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) :

⇑(f.comp g) = ⇑f ∘ ⇑g

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@[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) :

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

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@[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) :

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

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@[simp]

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

f.comp (CoheytingHom.id α) = f

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@[simp]

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

(CoheytingHom.id β).comp f = f

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@[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 g₂ : CoheytingHom β γ} (hf : Function.Surjective ⇑f) :

g₁.comp f = g₂.comp f ↔ g₁ = g₂

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@[simp]

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

g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

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

FunLike (BiheytingHom α β) α β

Equations

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

source

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

BiheytingHomClass (BiheytingHom α β) α β

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

f.toFun = ⇑f

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@[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

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

f = g

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

f = g ↔ ∀ (a : α), f a = g a

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.

Equations

f.copy f' h = { toFun := f', map_sup' := ⋯, map_inf' := ⋯, map_himp' := ⋯, map_sdiff' := ⋯ }

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) :

⇑(f.copy 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) :

f.copy f' h = f

source

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

BiheytingHom α α

id as a BiheytingHom.

Equations

BiheytingHom.id α = { toLatticeHom := LatticeHom.id α, map_himp' := ⋯, map_sdiff' := ⋯ }

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.instInhabited {α : Type u_2} [BiheytingAlgebra α] :

Inhabited (BiheytingHom α α)

Equations

BiheytingHom.instInhabited = { default := BiheytingHom.id α }

source

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

PartialOrder (BiheytingHom α β)

Equations

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

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.

Equations

f.comp g = { toFun := ⇑f ∘ ⇑g, map_sup' := ⋯, map_inf' := ⋯, map_himp' := ⋯, map_sdiff' := ⋯ }

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) :

⇑(f.comp 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) :

(f.comp 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) :

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

source

@[simp]

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

f.comp (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.id β).comp 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 g₂ : BiheytingHom β γ} (hf : Function.Surjective ⇑f) :

g₁.comp f = g₂.comp 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 f₂ : BiheytingHom α β} {g : BiheytingHom β γ} (hg : Function.Injective ⇑g) :

g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

Documentation

Mathlib.Order.Heyting.Hom

Heyting algebra morphisms #

Types of morphisms #

Typeclasses #