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Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 25 Jan 2015 12:16:34 +0900
parents f26a954cd068
children a271f3ff1922
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1 open import Relation.Binary.PropositionalEquality
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2 open import Level
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3 open import basic
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4
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5 module laws where
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6
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7 record Functor {l : Level} (F : {l' : Level} -> Set l' -> Set l') : Set (suc l) where
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8 field
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9 fmap : {A B : Set l} -> (A -> B) -> (F A) -> (F B)
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10 field
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11 preserve-id : {A : Set l} (x : F A) → fmap id x ≡ id x
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12 covariant : {A B C : Set l} (f : A -> B) -> (g : B -> C) -> (x : F A)
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13 -> fmap (g ∙ f) x ≡ ((fmap g) ∙ (fmap f)) x
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15 open Functor
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16
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17 record NaturalTransformation {l : Level} (F G : {l' : Level} -> Set l' -> Set l')
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18 {fmapF : {A B : Set l} -> (A -> B) -> (F A) -> (F B)}
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19 {fmapG : {A B : Set l} -> (A -> B) -> (G A) -> (G B)}
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20 (natural-transformation : {A : Set l} -> F A -> G A)
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21 : Set (suc l) where
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22 field
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23 commute : {A B : Set l} -> (f : A -> B) -> (x : F A) ->
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24 natural-transformation (fmapF f x) ≡ fmapG f (natural-transformation x)
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25 open NaturalTransformation
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31 -- simple Monad definition. without NaturalTransformation (mu, eta) and monad-law with f.
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32 record Monad {l : Level} {A : Set l}
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33 (M : {ll : Level} -> Set ll -> Set ll)
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34 (functorM : Functor {l} M)
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35 : Set (suc l) where
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36 field -- category
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37 mu : {A : Set l} -> M (M A) -> M A
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38 eta : {A : Set l} -> A -> M A
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39 field -- haskell
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40 return : {A : Set l} -> A -> M A
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41 bind : {A B : Set l} -> M A -> (A -> (M B)) -> M B
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42 field -- category laws
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43 association-law : (x : (M (M (M A)))) -> (mu ∙ (fmap functorM mu)) x ≡ (mu ∙ mu) x
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44 left-unity-law : (x : M A) -> (mu ∙ (fmap functorM eta)) x ≡ id x
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45 right-unity-law : (x : M A) -> id x ≡ (mu ∙ eta) x
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46 field -- natural transformations
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47 eta-is-nt : {B : Set l} -> (f : A -> B) -> (x : A) -> (eta ∙ f) x ≡ fmap functorM f (eta x)
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50 open Monad