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Split functor-proofs into delta.functor
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Mon, 19 Jan 2015 11:00:34 +0900
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children 5411ce26d525
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86:5c083ddd73ed 87:6789c65a75bc
1 open import delta
2 open import basic
3 open import laws
4
5 open import Level
6 open import Relation.Binary.PropositionalEquality
7
8
9 module delta.functor where
10
11 -- Functor-laws
12
13 -- Functor-law-1 : T(id) = id'
14 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d
15 functor-law-1 (mono x) = refl
16 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)
17
18 -- Functor-law-2 : T(f . g) = T(f) . T(g)
19 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
20 (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
21 (fmap (f ∙ g)) d ≡ (fmap f) (fmap g d)
22 functor-law-2 f g (mono x) = refl
23 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)
24
25 delta-is-functor : {l : Level} -> Functor (Delta {l})
26 delta-is-functor = record { fmap = fmap ;
27 preserve-id = functor-law-1;
28 covariant = \f g -> functor-law-2 g f}