annotate agda/laws.agda @ 131:d205ff1e406f InfiniteDeltaWithMonad

Cleanup proofs
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Tue, 03 Feb 2015 12:57:13 +0900
parents 5902b2a24abf
children 575de2e38385
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1 open import Relation.Binary.PropositionalEquality
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2 open import Level
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4 open import basic
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5
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6 module laws where
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8 record Functor {l : Level} (F : Set l -> Set l) : Set (suc l) where
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9 field
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10 fmap : {A B : Set l} -> (A -> B) -> (F A) -> (F B)
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11 field -- laws
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12 preserve-id : {A : Set l} (x : F A) → fmap id x ≡ id x
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13 covariant : {A B C : Set l} (f : A -> B) -> (g : B -> C) -> (x : F A)
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14 -> fmap (g ∙ f) x ≡ ((fmap g) ∙ (fmap f)) x
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15 field -- proof assistant
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16 fmap-equiv : {A B : Set l} {f g : A -> B} -> ((x : A) -> f x ≡ g x) -> (fx : F A) -> fmap f fx ≡ fmap g fx
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17 open Functor
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19 record NaturalTransformation {l : Level} (F G : Set l -> Set l)
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20 {fmapF : {A B : Set l} -> (A -> B) -> (F A) -> (F B)}
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21 {fmapG : {A B : Set l} -> (A -> B) -> (G A) -> (G B)}
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22 (natural-transformation : {A : Set l} -> F A -> G A)
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23 : Set (suc l) where
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24 field
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25 commute : {A B : Set l} -> (f : A -> B) -> (x : F A) ->
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26 natural-transformation (fmapF f x) ≡ fmapG f (natural-transformation x)
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27 open NaturalTransformation
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33 -- Categorical Monad definition. without haskell-laws (bind)
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34 record Monad {l : Level} (T : Set l -> Set l) (F : Functor T) : Set (suc l) where
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35 field -- category
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36 mu : {A : Set l} -> T (T A) -> T A
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37 eta : {A : Set l} -> A -> T A
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38 field -- natural transformations
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39 eta-is-nt : {A B : Set l} -> (f : A -> B) -> (x : A) -> (eta ∙ f) x ≡ fmap F f (eta x)
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40 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)
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41 field -- category laws
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42 association-law : {A : Set l} -> (x : (T (T (T A)))) -> (mu ∙ (fmap F mu)) x ≡ (mu ∙ mu) x
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43 left-unity-law : {A : Set l} -> (x : T A) -> (mu ∙ (fmap F eta)) x ≡ id x
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44 right-unity-law : {A : Set l} -> (x : T A) -> id x ≡ (mu ∙ eta) x
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47 open Monad