### view sandbox/FunctorExample.agda @ 11:26e64661b969

author Yasutaka Higa Tue, 20 Jan 2015 13:35:12 +0900 7c7659d4521d
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open import Level
open import Relation.Binary.PropositionalEquality
open ≡-Reasoning

module FunctorExample where

id : {l : Level} {A : Set l} -> A -> A
id x = x

_∙_ : {l : Level} {A B C : Set l} -> (B -> C) -> (A -> B) -> (A -> C)
f ∙ g = \x -> f (g x)

record Functor {l : Level} {A : Set l} (F : {l' : Level} -> Set l' -> Set l') : Set (suc l) where
field
fmap : {A B : Set l} -> (A -> B) -> (F A) -> (F B)
field
preserve-id : {A : Set l} (x : F A) → fmap id x ≡ id x
covariant   : {A B C : Set l} (f : A -> B) -> (g : B -> C) -> (x : F A)
-> fmap (g ∙ f) x ≡ ((fmap g) ∙ (fmap f)) x
open Functor

data List {l : Level} (A : Set l) : (Set l) where
nil  : List A
cons : A -> List A -> List A

list-fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> List A -> List B
list-fmap f nil         = nil
list-fmap f (cons x xs) = cons (f x) (list-fmap f xs)

list-preserve-id : {l : Level} {A : Set l} -> (xs : List A) -> list-fmap id xs ≡ id xs
list-preserve-id nil = refl
list-preserve-id (cons x xs) = cong (\li -> cons x li) (list-preserve-id xs)

list-covariant : {l : Level} {A B C : Set l} ->
(f : A -> B) → (g : B -> C) → (x : List A) → list-fmap (g ∙ f) x ≡ list-fmap g (list-fmap f x)
list-covariant f g nil         = refl
list-covariant f g (cons x xs) = cong (\li -> cons (g (f x)) li) (list-covariant f g xs)

list-is-functor : {l : Level} {A : Set l}-> Functor {l} {A} List
list-is-functor = record { fmap        = list-fmap ;
preserve-id = list-preserve-id ;
covariant   = list-covariant}

fmap-to-nest-list : {l : Level} {A B : Set l}
-> (A -> B) -> (List (List A)) -> (List (List B))
fmap-to-nest-list {l} {A} {B} f xs = Functor.fmap (list-is-functor {l} {List A}) (Functor.fmap {l} {A} list-is-functor f)  xs

data Identity {l : Level} (A : Set l) : Set l where
identity : A -> Identity A

identity-fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> Identity A -> Identity B
identity-fmap f (identity a) = identity (f a)

identity-preserve-id : {l : Level} {A : Set l} -> (x : Identity A) -> identity-fmap id x ≡ id x
identity-preserve-id (identity x) = refl

identity-covariant : {l : Level} {A B C : Set l}
(f : A -> B) → (g : B -> C) → (x : Identity A) → identity-fmap (g ∙ f) x ≡ identity-fmap g (identity-fmap f x)
identity-covariant f g (identity x) = refl

identity-is-functor : {l : Level} {A : Set l} -> Functor {l} {A} Identity
identity-is-functor {l} = record { fmap        = identity-fmap {l};
preserve-id = identity-preserve-id ;
covariant   = identity-covariant }

record NaturalTransformation {l : Level} (F G : {l' : Level} -> Set l' -> Set l')
{fmapF : {A B : Set l} -> (A -> B) -> (F A) -> (F B)}
{fmapG : {A B : Set l} -> (A -> B) -> (G A) -> (G B)}
(natural-transformation : {A : Set l}  -> F A -> G A)
: Set (suc l) where
field
commute : {A B : Set l} -> (f : A -> B) -> (x : F A) ->
natural-transformation (fmapF f x) ≡  fmapG f (natural-transformation x)
open NaturalTransformation

tail : {l : Level} {A : Set l} -> List A -> List A
tail nil         = nil
tail (cons _ xs) = xs

tail-commute : {l : Level} {A B : Set l} -> (f : A -> B) -> (xs : List A) ->
tail (list-fmap f xs) ≡ list-fmap f (tail xs)
tail-commute f nil = refl
tail-commute f (cons x xs) = refl

tail-is-natural-transformation : {l : Level} -> NaturalTransformation List List {list-fmap} {list-fmap {l}} tail
tail-is-natural-transformation = record { commute                = tail-commute}

append : {l : Level} {A : Set l} -> List A -> List A -> List A
append nil ys = ys
append (cons x xs) ys = cons x (append xs ys)

append-with-fmap : {l : Level} {A B : Set l} -> (f : A -> B) -> (xs : List A) -> (ys : List A) ->
append (list-fmap f xs) (list-fmap f ys) ≡  list-fmap f (append xs ys)
append-with-fmap f nil ys         = refl
append-with-fmap f (cons x xs) ys = begin
append (list-fmap f (cons x xs)) (list-fmap f ys)     ≡⟨ refl ⟩
append (cons (f x) (list-fmap f xs)) (list-fmap f ys) ≡⟨ refl ⟩
cons (f x) (append (list-fmap f xs) (list-fmap f ys)) ≡⟨ cong (\li -> cons (f x) li) (append-with-fmap f xs ys) ⟩
list-fmap f (append (cons x xs) ys)                   ∎

concat : {l : Level} {A : Set l} -> List (List A) -> List A
concat nil         = nil
concat (cons x xs) = append x (concat xs)

concat-commute :  {l : Level} {A B : Set l} -> (f : A -> B) -> (xs : List (List A)) ->
concat ((list-fmap ∙ list-fmap) f xs) ≡ list-fmap  f (concat xs)
concat-commute f nil         = refl
concat-commute f (cons x xs) = begin
concat ((list-fmap ∙ list-fmap) f (cons x xs))                 ≡⟨ refl ⟩
concat (cons (list-fmap f x) ((list-fmap ∙ list-fmap) f xs))   ≡⟨ refl ⟩
append (list-fmap f x) (concat ((list-fmap ∙ list-fmap) f xs)) ≡⟨ cong (\li -> append (list-fmap f x) li) (concat-commute f xs) ⟩
append (list-fmap f x) (list-fmap f (concat xs))               ≡⟨ append-with-fmap f x (concat xs) ⟩
list-fmap f (append x (concat xs)) ≡⟨ refl ⟩
list-fmap f (concat (cons x xs))
∎

concat-is-natural-transformation : {l : Level} -> NaturalTransformation (\A -> List (List A)) List
{list-fmap ∙ list-fmap} {list-fmap {l}} concat
concat-is-natural-transformation = record {commute = concat-commute}

data NonEmptyList {l : Level} (A : Set l)  : Set l where
val : A -> NonEmptyList A
con : A -> NonEmptyList A -> NonEmptyList A

head : {l : Level} {A : Set l} -> NonEmptyList A -> A