Functors #

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This module provides additional lemmas, definitions, and instances for functors.

Main definitions #

const α is the functor that sends all types to α.
add_const α is const α but for when α has an additive structure.
comp F G for functors F and G is the functor composition of F and G.
liftp and liftr respectively lift predicates and relations on a type α to F α. Terms of F α are considered to, in some sense, contain values of type α.

Tags #

functor, applicative

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theorem functor.map_id {F : Type u → Type v} {α : Type u} [functor F] [is_lawful_functor F] :

functor.map id = id

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theorem functor.map_comp_map {F : Type u → Type v} {α β γ : Type u} [functor F] [is_lawful_functor F] (f : α → β) (g : β → γ) :

functor.map g ∘ functor.map f = functor.map (g ∘ f)

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theorem functor.ext {F : Type u_1 → Type u_2} {F1 F2 : functor F} [is_lawful_functor F] [is_lawful_functor F] (H : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x) :

F1 = F2

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def id.mk {α : Sort u} :

α → id α

Introduce the id functor. Incidentally, this is pure for id as a monad and as an applicative functor.

Equations

id.mk = id

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@[nolint]

def functor.const (α : Type u_1) (β : Type u_2) :

Type u_1

const α is the constant functor, mapping every type to α. When α has a monoid structure, const α has an applicative instance. (If α has an additive monoid structure, see functor.add_const.)

Equations

functor.const α β = α

Instances for functor.const

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def functor.const.mk {α : Type u_1} {β : Type u_2} (x : α) :

functor.const α β

const.mk is the canonical map α → const α β (the identity), and it can be used as a pattern to extract this value.

Equations

functor.const.mk x = x

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def functor.const.mk' {α : Type u_1} (x : α) :

functor.const α punit

const.mk' is const.mk but specialized to map α to const α punit, where punit is the terminal object in Type*.

Equations

functor.const.mk' x = x

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def functor.const.run {α : Type u_1} {β : Type u_2} (x : functor.const α β) :

Extract the element of α from the const functor.

Equations

x.run = x

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@[protected]

theorem functor.const.ext {α : Type u_1} {β : Type u_2} {x y : functor.const α β} (h : x.run = y.run) :

x = y

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@[protected, nolint]

def functor.const.map {γ : Type u_1} {α : Type u_2} {β : Type u_3} (f : α → β) (x : functor.const γ β) :

functor.const γ α

The map operation of the const γ functor.

Equations

functor.const.map f x = x

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@[protected, instance]

def functor.const.functor {γ : Type u_1} :

functor (functor.const γ)

Equations

functor.const.functor = {map := functor.const.map γ, map_const := λ (α β : Type u_2), functor.const.map ∘ function.const β}

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@[protected, instance]

def functor.const.is_lawful_functor {γ : Type u_1} :

is_lawful_functor (functor.const γ)

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@[protected, instance]

def functor.const.inhabited {α : Type u_1} {β : Type u_2} [inhabited α] :

inhabited (functor.const α β)

Equations

functor.const.inhabited = {default := inhabited.default _inst_1}

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def functor.add_const (α : Type u_1) (β : Type u_2) :

Type u_1

add_const α is a synonym for constant functor const α, mapping every type to α. When α has a additive monoid structure, add_const α has an applicative instance. (If α has a multiplicative monoid structure, see functor.const.)

Equations

functor.add_const α = functor.const α

Instances for functor.add_const

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def functor.add_const.mk {α : Type u_1} {β : Type u_2} (x : α) :

functor.add_const α β

add_const.mk is the canonical map α → add_const α β, which is the identity, where add_const α β = const α β. It can be used as a pattern to extract this value.

Equations

functor.add_const.mk x = x

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def functor.add_const.run {α : Type u_1} {β : Type u_2} :

functor.add_const α β → α

Extract the element of α from the constant functor.

Equations

functor.add_const.run = id

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@[protected, instance]

def functor.add_const.functor {γ : Type u_1} :

functor (functor.add_const γ)

Equations

functor.add_const.functor = functor.const.functor

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@[protected, instance]

def functor.add_const.is_lawful_functor {γ : Type u_1} :

is_lawful_functor (functor.add_const γ)

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@[protected, instance]

def functor.add_const.inhabited {α : Type u_1} {β : Type u_2} [inhabited α] :

inhabited (functor.add_const α β)

Equations

functor.add_const.inhabited = {default := inhabited.default _inst_1}

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def functor.comp (F : Type u → Type w) (G : Type v → Type u) (α : Type v) :

Type w

functor.comp is a wrapper around function.comp for types. It prevents Lean's type class resolution mechanism from trying a functor (comp F id) when functor F would do.

Equations

functor.comp F G α = F (G α)

Instances for functor.comp

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def functor.comp.mk {F : Type u → Type w} {G : Type v → Type u} {α : Type v} (x : F (G α)) :

functor.comp F G α

Construct a term of comp F G α from a term of F (G α), which is the same type. Can be used as a pattern to extract a term of F (G α).

Equations

functor.comp.mk x = x

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def functor.comp.run {F : Type u → Type w} {G : Type v → Type u} {α : Type v} (x : functor.comp F G α) :

F (G α)

Extract a term of F (G α) from a term of comp F G α, which is the same type.

Equations

x.run = x

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@[protected]

theorem functor.comp.ext {F : Type u → Type w} {G : Type v → Type u} {α : Type v} {x y : functor.comp F G α} :

x.run = y.run → x = y

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@[protected, instance]

def functor.comp.inhabited {F : Type u → Type w} {G : Type v → Type u} {α : Type v} [inhabited (F (G α))] :

inhabited (functor.comp F G α)

Equations

functor.comp.inhabited = {default := inhabited.default _inst_1}

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@[protected]

def functor.comp.map {F : Type u → Type w} {G : Type v → Type u} [functor F] [functor G] {α β : Type v} (h : α → β) :

functor.comp F G α → functor.comp F G β

The map operation for the composition comp F G of functors F and G.

Equations

functor.comp.map h (functor.comp.mk x) = functor.comp.mk (functor.map h <$> x)

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@[protected, instance]

def functor.comp.functor {F : Type u → Type w} {G : Type v → Type u} [functor F] [functor G] :

functor (functor.comp F G)

Equations

functor.comp.functor = {map := functor.comp.map _inst_2, map_const := λ (α β : Type v), functor.comp.map ∘ function.const β}

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theorem functor.comp.map_mk {F : Type u → Type w} {G : Type v → Type u} [functor F] [functor G] {α β : Type v} (h : α → β) (x : F (G α)) :

h <$> functor.comp.mk x = functor.comp.mk (functor.map h <$> x)

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@[protected, simp]

theorem functor.comp.run_map {F : Type u → Type w} {G : Type v → Type u} [functor F] [functor G] {α β : Type v} (h : α → β) (x : functor.comp F G α) :

functor.comp.run (h <$> x) = functor.map h <$> x.run

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@[protected]

theorem functor.comp.id_map {F : Type u → Type w} {G : Type v → Type u} [functor F] [functor G] [is_lawful_functor F] [is_lawful_functor G] {α : Type v} (x : functor.comp F G α) :

functor.comp.map id x = x

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@[protected]

theorem functor.comp.comp_map {F : Type u → Type w} {G : Type v → Type u} [functor F] [functor G] [is_lawful_functor F] [is_lawful_functor G] {α β γ : Type v} (g' : α → β) (h : β → γ) (x : functor.comp F G α) :

functor.comp.map (h ∘ g') x = functor.comp.map h (functor.comp.map g' x)

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@[protected, instance]

def functor.comp.is_lawful_functor {F : Type u → Type w} {G : Type v → Type u} [functor F] [functor G] [is_lawful_functor F] [is_lawful_functor G] :

is_lawful_functor (functor.comp F G)

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theorem functor.comp.functor_comp_id {F : Type u_1 → Type u_2} [AF : functor F] [is_lawful_functor F] :

functor.comp.functor = AF

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theorem functor.comp.functor_id_comp {F : Type u_1 → Type u_2} [AF : functor F] [is_lawful_functor F] :

functor.comp.functor = AF

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@[protected]

def functor.comp.seq {F : Type u → Type w} {G : Type v → Type u} [applicative F] [applicative G] {α β : Type v} :

functor.comp F G (α → β) → functor.comp F G α → functor.comp F G β

The <*> operation for the composition of applicative functors.

Equations

(functor.comp.mk f).seq (functor.comp.mk x) = functor.comp.mk (has_seq.seq <$> f <*> x)

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@[protected, instance]

def functor.comp.has_pure {F : Type u → Type w} {G : Type v → Type u} [applicative F] [applicative G] :

has_pure (functor.comp F G)

Equations

functor.comp.has_pure = {pure := λ (_x : Type v) (x : _x), functor.comp.mk (has_pure.pure (has_pure.pure x))}

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@[protected, instance]

def functor.comp.has_seq {F : Type u → Type w} {G : Type v → Type u} [applicative F] [applicative G] :

has_seq (functor.comp F G)

Equations

functor.comp.has_seq = {seq := λ (_x _x_1 : Type v) (f : functor.comp F G (_x → _x_1)) (x : functor.comp F G _x), f.seq x}

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@[protected, simp]

theorem functor.comp.run_pure {F : Type u → Type w} {G : Type v → Type u} [applicative F] [applicative G] {α : Type v} (x : α) :

(has_pure.pure x).run = has_pure.pure (has_pure.pure x)

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@[protected, simp]

theorem functor.comp.run_seq {F : Type u → Type w} {G : Type v → Type u} [applicative F] [applicative G] {α β : Type v} (f : functor.comp F G (α → β)) (x : functor.comp F G α) :

(f <*> x).run = has_seq.seq <$> f.run <*> x.run

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@[protected, instance]

def functor.comp.applicative {F : Type u → Type w} {G : Type v → Type u} [applicative F] [applicative G] :

applicative (functor.comp F G)

Equations

functor.comp.applicative = applicative.mk

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def functor.liftp {F : Type u → Type u} [functor F] {α : Type u} (p : α → Prop) (x : F α) :

Prop

If we consider x : F α to, in some sense, contain values of type α, predicate liftp p x holds iff every value contained by x satisfies p.

Equations

functor.liftp p x = ∃ (u : F (subtype p)), subtype.val <$> u = x

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def functor.liftr {F : Type u → Type u} [functor F] {α : Type u} (r : α → α → Prop) (x y : F α) :

Prop

If we consider x : F α to, in some sense, contain values of type α, then liftr r x y relates x and y iff (1) x and y have the same shape and (2) we can pair values a from x and b from y so that r a b holds.

Equations

functor.liftr r x y = ∃ (u : F {p // r p.fst p.snd}), (λ (t : {p // r p.fst p.snd}), t.val.fst) <$> u = x ∧ (λ (t : {p // r p.fst p.snd}), t.val.snd) <$> u = y

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def functor.supp {F : Type u → Type u} [functor F] {α : Type u} (x : F α) :

set α

If we consider x : F α to, in some sense, contain values of type α, then supp x is the set of values of type α that x contains.

Equations

functor.supp x = {y : α | ∀ ⦃p : α → Prop⦄, functor.liftp p x → p y}

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theorem functor.of_mem_supp {F : Type u → Type u} [functor F] {α : Type u} {x : F α} {p : α → Prop} (h : functor.liftp p x) (y : α) (H : y ∈ functor.supp x) :

p y