view agda/delta/functor.agda @ 126:5902b2a24abf

Prove mu-is-nt for DeltaM with fmap-equiv
author Yasutaka Higa Tue, 03 Feb 2015 11:45:33 +0900 47f144540d51 d205ff1e406f
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```
open import Level
open import Relation.Binary.PropositionalEquality
open ≡-Reasoning

open import basic
open import delta
open import laws
open import nat

module delta.functor where

-- Functor-laws

-- Functor-law-1 : T(id) = id'
functor-law-1 :  {l : Level} {A : Set l} {n : Nat} ->  (d : Delta A (S n)) -> (delta-fmap id) d ≡ id d
functor-law-1 (mono x)    = refl
functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)

-- Functor-law-2 : T(f . g) = T(f) . T(g)
functor-law-2 : {l : Level} {n : Nat} {A B C : Set l} ->
(f : B -> C) -> (g : A -> B) -> (d : Delta A (S n)) ->
(delta-fmap (f ∙ g)) d ≡ ((delta-fmap f) ∙ (delta-fmap g)) d
functor-law-2 f g (mono x)    = refl
functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)

delta-fmap-equiv : {l : Level} {A B : Set l} {n : Nat} {f g : A -> B}
(eq : (x : A) -> f x ≡ g x) -> (d : Delta A (S n)) ->
delta-fmap f d ≡ delta-fmap g d
delta-fmap-equiv {l} {A} {B} {O} {f} {g} eq (mono x) = begin
mono (f x) ≡⟨ cong mono (eq x) ⟩
mono (g x)
∎
delta-fmap-equiv {l} {A} {B} {S n} {f} {g} eq (delta x d) = begin
delta (f x) (delta-fmap f d) ≡⟨ cong (\de -> delta de (delta-fmap f d)) (eq x) ⟩
delta (g x) (delta-fmap f d) ≡⟨ cong (\de -> delta (g x) de) (delta-fmap-equiv eq d) ⟩
delta (g x) (delta-fmap g d)
∎

delta-is-functor : {l : Level} {n : Nat} -> Functor {l} (\A -> Delta A (S n))
delta-is-functor = record { fmap       = delta-fmap
;preserve-id = functor-law-1
; covariant  = \f g -> functor-law-2 g f
; fmap-equiv = delta-fmap-equiv
}
```