Functors with two arguments

This file defines bifunctors.

A bifunctor is a function F : Type* → Type* → Type* along with a bimap which turns F α β into F α' β' given two functions α → α' and β → β'. It further

• respects the identity: bimap id id = id
• composes in the obvious way: (bimap f' g') ∘ (bimap f g) = bimap (f' ∘ f) (g' ∘ g)

Main declarations

• Bifunctor: A typeclass for the bare bimap of a bifunctor.
• LawfulBifunctor: A typeclass asserting this bimap respects the bifunctor laws.

class Bifunctor (F : Type u₀ → Type u₁ → Type u₂) : Type (max (max (u₀ + 1) (u₁ + 1)) u₂)

Lawless bifunctor. This typeclass only holds the data for the bimap.

• bimap {α α' : Type u₀} {β β' : Type u₁} : (α → α') → (β → β') → F α β → F α' β'

class LawfulBifunctor (F : Type u₀ → Type u₁ → Type u₂) [Bifunctor F] : Prop

Bifunctor. This typeclass asserts that a lawless Bifunctor is lawful.

• id_bimap {α : Type u₀} {β : Type u₁} (x : F α β) : bimap id id x = x
• bimap_bimap {α₀ α₁ α₂ : Type u₀} {β₀ β₁ β₂ : Type u₁} (f : α₀ → α₁) (f' : α₁ → α₂) (g : β₀ → β₁) (g' : β₁ → β₂) (x : F α₀ β₀) : bimap f' g' (bimap f g x) = bimap (f' ∘ f) (g' ∘ g) x

theorem LawfulBifunctor.bimap_id_id {F : Type u₀ → Type u₁ → Type u₂} {inst✝ : Bifunctor F} [self : LawfulBifunctor F] {α : Type u₀} {β : Type u₁} : bimap id id = id

theorem LawfulBifunctor.bimap_comp_bimap {F : Type u₀ → Type u₁ → Type u₂} {inst✝ : Bifunctor F} [self : LawfulBifunctor F] {α₀ α₁ α₂ : Type u₀} {β₀ β₁ β₂ : Type u₁} (f : α₀ → α₁) (f' : α₁ → α₂) (g : β₀ → β₁) (g' : β₁ → β₂) : bimap f' g' ∘ bimap f g = bimap (f' ∘ f) (g' ∘ g)

@[reducible, inline]

abbrev Bifunctor.fst {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] {α α' : Type u₀} {β : Type u₁} (f : α → α') : F α β → F α' β

Left map of a bifunctor.

@[reducible, inline]

abbrev Bifunctor.snd {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] {α : Type u₀} {β β' : Type u₁} (f : β → β') : F α β → F α β'

Right map of a bifunctor.

theorem Bifunctor.id_fst {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α : Type u₀} {β : Type u₁} (x : F α β) : fst id x = x

theorem Bifunctor.fst_id {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α : Type u₀} {β : Type u₁} : fst id = id

theorem Bifunctor.id_snd {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α : Type u₀} {β : Type u₁} (x : F α β) : snd id x = x

theorem Bifunctor.snd_id {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α : Type u₀} {β : Type u₁} : snd id = id

theorem Bifunctor.comp_fst {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α₀ α₁ α₂ : Type u₀} {β : Type u₁} (f : α₀ → α₁) (f' : α₁ → α₂) (x : F α₀ β) : fst f' (fst f x) = fst (f' ∘ f) x

theorem Bifunctor.fst_comp_fst {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α₀ α₁ α₂ : Type u₀} {β : Type u₁} (f : α₀ → α₁) (f' : α₁ → α₂) : fst f' ∘ fst f = fst (f' ∘ f)

theorem Bifunctor.fst_snd {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α₀ α₁ : Type u₀} {β₀ β₁ : Type u₁} (f : α₀ → α₁) (f' : β₀ → β₁) (x : F α₀ β₀) : fst f (snd f' x) = bimap f f' x

theorem Bifunctor.fst_comp_snd {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α₀ α₁ : Type u₀} {β₀ β₁ : Type u₁} (f : α₀ → α₁) (f' : β₀ → β₁) : fst f ∘ snd f' = bimap f f'

theorem Bifunctor.snd_fst {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α₀ α₁ : Type u₀} {β₀ β₁ : Type u₁} (f : α₀ → α₁) (f' : β₀ → β₁) (x : F α₀ β₀) : snd f' (fst f x) = bimap f f' x

theorem Bifunctor.snd_comp_fst {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α₀ α₁ : Type u₀} {β₀ β₁ : Type u₁} (f : α₀ → α₁) (f' : β₀ → β₁) : snd f' ∘ fst f = bimap f f'

theorem Bifunctor.comp_snd {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α : Type u₀} {β₀ β₁ β₂ : Type u₁} (g : β₀ → β₁) (g' : β₁ → β₂) (x : F α β₀) : snd g' (snd g x) = snd (g' ∘ g) x

theorem Bifunctor.snd_comp_snd {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α : Type u₀} {β₀ β₁ β₂ : Type u₁} (g : β₀ → β₁) (g' : β₁ → β₂) : snd g' ∘ snd g = snd (g' ∘ g)

@[instance_reducible]

instance Prod.bifunctor : Bifunctor Prod

instance Prod.lawfulBifunctor : LawfulBifunctor Prod

@[instance_reducible]

instance Bifunctor.const : Bifunctor Functor.Const

instance LawfulBifunctor.const : LawfulBifunctor Functor.Const

@[instance_reducible]

instance Bifunctor.flip {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] : Bifunctor (_root_.flip F)

instance LawfulBifunctor.flip {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] : LawfulBifunctor (_root_.flip F)

@[instance_reducible]

instance Sum.bifunctor : Bifunctor Sum

instance Sum.lawfulBifunctor : LawfulBifunctor Sum

@[instance_reducible, instance 10]

instance Bifunctor.functor {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] {α : Type u₀} : Functor (F α)

@[instance 10]

instance Bifunctor.lawfulFunctor {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] [LawfulBifunctor F] {α : Type u₀} : LawfulFunctor (F α)

@[instance_reducible]

instance Function.bicompl.bifunctor {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] (G : Type u_1 → Type u₀) (H : Type u_2 → Type u₁) [Functor G] [Functor H] : Bifunctor (bicompl F G H)

instance Function.bicompl.lawfulBifunctor {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] (G : Type u_1 → Type u₀) (H : Type u_2 → Type u₁) [Functor G] [Functor H] [LawfulFunctor G] [LawfulFunctor H] [LawfulBifunctor F] : LawfulBifunctor (bicompl F G H)

@[instance_reducible]

instance Function.bicompr.bifunctor {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] (G : Type u₂ → Type u_1) [Functor G] : Bifunctor (bicompr G F)

instance Function.bicompr.lawfulBifunctor {F : Type u₀ → Type u₁ → Type u₂} [Bifunctor F] (G : Type u₂ → Type u_1) [Functor G] [LawfulFunctor G] [LawfulBifunctor F] : LawfulBifunctor (bicompr G F)