### view agda/laws.agda @ 90:55d11ce7e223

Unify levels on data type. only use suc to proofs
author Yasutaka Higa Mon, 19 Jan 2015 12:11:38 +0900 6789c65a75bc bcd4fe52a504
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open import Relation.Binary.PropositionalEquality
open import Level
open import basic

module laws where

record Functor {l : Level} (F : 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

record NaturalTransformation {l : Level} (F G : Set l -> Set l)
(functorF : Functor F)
(functorG : Functor G) : Set (suc l) where
field
natural-transformation : {A : Set l}  -> F A -> G A
field
commute : ∀ {A B} -> (f : A -> B) -> (x : F A) ->
natural-transformation (fmap functorF f x) ≡  fmap functorG f (natural-transformation x)
open NaturalTransformation

-- simple Monad definition. without NaturalTransformation (mu, eta) and monad-law with f.
record Monad {l : Level} {A : Set l}
(M : {ll : Level} -> Set ll -> Set ll)
(functorM : Functor M)
: Set (suc l)  where
field
mu  : {A : Set l} -> M (M A) -> M A
eta : {A : Set l} -> A -> M A
field
association-law : (x : (M (M (M A)))) -> (mu ∙ (fmap functorM mu)) x ≡ (mu ∙ mu) x
left-unity-law  : (x : M A) -> (mu  ∙ (fmap functorM eta)) x ≡ id x
right-unity-law : (x : M A) -> id x ≡ (mu ∙ eta) x