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comparison agda/laws.agda @ 121:673e1ca0d1a9

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Refactor monad definition
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Mon, 02 Feb 2015 12:11:24 +0900
parents e6bcc7467335
children 5902b2a24abf
comparison
equal deleted inserted replaced
120:0f9ecd118a03 121:673e1ca0d1a9
11 preserve-id : {A : Set l} (x : F A) → fmap id x ≡ id x 11 preserve-id : {A : Set l} (x : F A) → fmap id x ≡ id x
12 covariant : {A B C : Set l} (f : A -> B) -> (g : B -> C) -> (x : F A) 12 covariant : {A B C : Set l} (f : A -> B) -> (g : B -> C) -> (x : F A)
13 -> fmap (g ∙ f) x ≡ ((fmap g) ∙ (fmap f)) x 13 -> fmap (g ∙ f) x ≡ ((fmap g) ∙ (fmap f)) x
14 open Functor 14 open Functor
15 15
16 record NaturalTransformation {l : Level} (F G : {l' : Level} -> Set l' -> Set l') 16 record NaturalTransformation {l : Level} (F G : Set l -> Set l)
17 {fmapF : {A B : Set l} -> (A -> B) -> (F A) -> (F B)} 17 {fmapF : {A B : Set l} -> (A -> B) -> (F A) -> (F B)}
18 {fmapG : {A B : Set l} -> (A -> B) -> (G A) -> (G B)} 18 {fmapG : {A B : Set l} -> (A -> B) -> (G A) -> (G B)}
19 (natural-transformation : {A : Set l} -> F A -> G A) 19 (natural-transformation : {A : Set l} -> F A -> G A)
20 : Set (suc l) where 20 : Set (suc l) where
21 field 21 field
25 25
26 26
27 27
28 28
29 29
30 -- simple Monad definition. without NaturalTransformation (mu, eta) and monad-law with f. 30 -- Categorical Monad definition. without haskell-laws (bind)
31 record Monad {l : Level} (M : Set l -> Set l) 31 record Monad {l : Level} (T : Set l -> Set l) (F : Functor T) : Set (suc l) where
32 (functorM : Functor M)
33 : Set (suc l) where
34 field -- category 32 field -- category
35 mu : {A : Set l} -> M (M A) -> M A 33 mu : {A : Set l} -> T (T A) -> T A
36 eta : {A : Set l} -> A -> M A 34 eta : {A : Set l} -> A -> T A
37 field -- haskell 35 field -- natural transformations
38 return : {A : Set l} -> A -> M A 36 eta-is-nt : {A B : Set l} -> (f : A -> B) -> (x : A) -> (eta ∙ f) x ≡ fmap F f (eta x)
39 bind : {A B : Set l} -> M A -> (A -> (M B)) -> M B 37 mu-is-nt : {A B : Set l} -> (f : A -> B) -> (x : T (T A)) -> mu (fmap F (fmap F f) x) ≡ fmap F f (mu x)
40 field -- category laws 38 field -- category laws
41 association-law : {A : Set l} -> (x : (M (M (M A)))) -> (mu ∙ (fmap functorM mu)) x ≡ (mu ∙ mu) x 39 association-law : {A : Set l} -> (x : (T (T (T A)))) -> (mu ∙ (fmap F mu)) x ≡ (mu ∙ mu) x
42 left-unity-law : {A : Set l} -> (x : M A) -> (mu ∙ (fmap functorM eta)) x ≡ id x 40 left-unity-law : {A : Set l} -> (x : T A) -> (mu ∙ (fmap F eta)) x ≡ id x
43 right-unity-law : {A : Set l} -> (x : M A) -> id x ≡ (mu ∙ eta) x 41 right-unity-law : {A : Set l} -> (x : T A) -> id x ≡ (mu ∙ eta) x
44 field -- natural transformations
45 eta-is-nt : {A B : Set l} -> (f : A -> B) -> (x : A) -> (eta ∙ f) x ≡ fmap functorM f (eta x)
46 mu-is-nt : {A B : Set l} -> (f : A -> B) -> (x : M (M A)) -> mu (fmap functorM (fmap functorM f) x) ≡ fmap functorM f (mu x)
47 42
48 43
49 open Monad 44 open Monad