annotate agda/similar.agda @ 31:33b386de3f56

Proof list-associative
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sat, 18 Oct 2014 13:38:29 +0900
parents c2f40b6d4027
children 71906644d206
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5ba82f107a95 Define Similar in Agda
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1 open import list
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6e6d646d7722 Split basic functions to file
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2 open import basic
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e0ba1bf564dd Apply level to some functions
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e0ba1bf564dd Apply level to some functions
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4 open import Level
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742e62fc63e4 Define Monad-law 1-4
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5 open import Relation.Binary.PropositionalEquality
742e62fc63e4 Define Monad-law 1-4
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6 open ≡-Reasoning
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5ba82f107a95 Define Similar in Agda
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8 module similar where
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10 data Similar {l : Level} (A : Set l) : (Set (suc l)) where
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11 similar : List String -> A -> List String -> A -> Similar A
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13 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Similar A) -> (Similar B)
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14 fmap f (similar xs x ys y) = similar xs (f x) ys (f y)
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16 mu : {l : Level} {A : Set l} -> Similar (Similar A) -> Similar A
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17 mu (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = similar (lx ++ llx) x (ly ++ lly) y
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19 return : {A : Set} -> A -> Similar A
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20 return x = similar [] x [] x
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22 returnS : {A : Set} -> A -> Similar A
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23 returnS x = similar [[ (show x) ]] x [[ (show x) ]] x
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25 returnSS : {A : Set} -> A -> A -> Similar A
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26 returnSS x y = similar [[ (show x) ]] x [[ (show y) ]] y
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28 --monad-law-1 : mu ∙ (fmap mu) ≡ mu ∙ mu
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30 {-
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31 monad-law-1 : {l : Level} {A : Set l} -> (s : Similar (Similar (Similar A))) -> ((mu ∙ (fmap mu)) s) ≡ ((mu ∙ mu) s)
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32 monad-law-1 s = begin
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33 ((mu ∙ (fmap mu)) s)
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34 ≡⟨⟩
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35 mu (fmap mu s)
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36 ≡⟨ cong mu {!!} ⟩
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37 mu (mu s)
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38 ≡⟨⟩
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39 ((mu ∙ mu) s)
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41 -}
c2f40b6d4027 Expand monad-law-1
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c2f40b6d4027 Expand monad-law-1
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43 monad-law-1-sub : {l : Level} {A : Set l} -> (s : Similar (Similar (Similar A))) -> fmap mu s ≡ mu s
c2f40b6d4027 Expand monad-law-1
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44 monad-law-1-sub (similar lx (similar llx (similar lllx x _ _) _ _) ly (similar _ _ lly (similar _ _ llly y))) = begin
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45 similar lx (mu (similar llx (similar lllx x _ _) _ _)) ly
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46 (mu (similar _ _ lly (similar _ _ llly y)))
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47 ≡⟨ {!!} ⟩
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48 similar (lx ++ llx) (similar lllx x _ _) (ly ++ lly)
c2f40b6d4027 Expand monad-law-1
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49 (similar _ _ llly y)
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56 {-
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57 --monad-law-2 : mu ∙ fmap return ≡ mu ∙ return ≡id
742e62fc63e4 Define Monad-law 1-4
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58 monad-law-2-1 : mu ∙ fmap return ≡ mu ∙ return
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59 monad-law-2-1 = {!!}
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61 monad-law-2-2 : mu ∙ return ≡ id
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62 monad-law-2-2 = {!!}
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64 monad-law-3 : ∀{f} -> return ∙ f ≡ fmap f ∙ return
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65 monad-law-3 = {!!}
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742e62fc63e4 Define Monad-law 1-4
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67 monad-law-4 : ∀{f} -> mu ∙ fmap (fmap f) ≡ fmap f ∙ mu
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68 monad-law-4 = {!!}
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69 -}