annotate agda/similar.agda @ 36:169ec60fcd36

Proof Monad-law-4
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sat, 18 Oct 2014 14:22:34 +0900
parents c5cdbedc68ad
children 6ce83b2c9e59
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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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5ba82f107a95 Define Similar in Agda
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8 module similar where
5ba82f107a95 Define Similar in Agda
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10 data Similar {l : Level} (A : Set l) : (Set (suc l)) where
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5ba82f107a95 Define Similar in Agda
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11 similar : List String -> A -> List String -> A -> Similar A
5ba82f107a95 Define Similar in Agda
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12
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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)
5ba82f107a95 Define Similar in Agda
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16 mu : {l : Level} {A : Set l} -> Similar (Similar A) -> Similar A
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5ba82f107a95 Define Similar in Agda
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17 mu (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = similar (lx ++ llx) x (ly ++ lly) y
5ba82f107a95 Define Similar in Agda
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b7c4e6276bcf Proof Monad-law-2-1
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19 return : {l : Level} {A : Set l} -> A -> Similar A
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742e62fc63e4 Define Monad-law 1-4
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20 return x = similar [] x [] x
742e62fc63e4 Define Monad-law 1-4
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5ba82f107a95 Define Similar in Agda
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22 returnS : {A : Set} -> A -> Similar A
5ba82f107a95 Define Similar in Agda
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23 returnS x = similar [[ (show x) ]] x [[ (show x) ]] x
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5ba82f107a95 Define Similar in Agda
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25 returnSS : {A : Set} -> A -> A -> Similar A
5ba82f107a95 Define Similar in Agda
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26 returnSS x y = similar [[ (show x) ]] x [[ (show y) ]] y
5ba82f107a95 Define Similar in Agda
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0bc402f970b3 Proof Monad-law 1
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e0ba1bf564dd Apply level to some functions
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29 --monad-law-1 : mu ∙ (fmap mu) ≡ mu ∙ mu
e0ba1bf564dd Apply level to some functions
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30 monad-law-1 : {l : Level} {A : Set l} -> (s : Similar (Similar (Similar A))) -> ((mu ∙ (fmap mu)) s) ≡ ((mu ∙ mu) s)
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71906644d206 Expand monad-law 1
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31 monad-law-1 (similar lx (similar llx (similar lllx x _ _) _ (similar _ _ _ _))
71906644d206 Expand monad-law 1
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32 ly (similar _ (similar _ _ _ _) lly (similar _ _ llly y))) = begin
71906644d206 Expand monad-law 1
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33 similar (lx ++ (llx ++ lllx)) x (ly ++ (lly ++ llly)) y
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0bc402f970b3 Proof Monad-law 1
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34 ≡⟨ cong (\left-list -> similar left-list x (ly ++ (lly ++ llly)) y) (list-associative lx llx lllx) ⟩
0bc402f970b3 Proof Monad-law 1
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35 similar (lx ++ llx ++ lllx) x (ly ++ (lly ++ llly)) y
0bc402f970b3 Proof Monad-law 1
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36 ≡⟨ cong (\right-list -> similar (lx ++ llx ++ lllx) x right-list y ) (list-associative ly lly llly) ⟩
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71906644d206 Expand monad-law 1
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37 similar (lx ++ llx ++ lllx) x (ly ++ lly ++ llly) y
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e0ba1bf564dd Apply level to some functions
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38
e0ba1bf564dd Apply level to some functions
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b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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b7c4e6276bcf Proof Monad-law-2-1
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41 --monad-law-2 : mu ∙ fmap return ≡ mu ∙ return ≡ id
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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42 monad-law-2-1 : {l : Level} {A : Set l} -> (s : Similar A) ->
b7c4e6276bcf Proof Monad-law-2-1
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43 (mu ∙ fmap return) s ≡ (mu ∙ return) s
b7c4e6276bcf Proof Monad-law-2-1
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44 monad-law-2-1 (similar lx x ly y) = begin
b7c4e6276bcf Proof Monad-law-2-1
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45 similar (lx ++ []) x (ly ++ []) y
b7c4e6276bcf Proof Monad-law-2-1
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46 ≡⟨ cong (\left-list -> similar left-list x (ly ++ []) y) (empty-append lx)⟩
b7c4e6276bcf Proof Monad-law-2-1
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47 similar lx x (ly ++ []) y
b7c4e6276bcf Proof Monad-law-2-1
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48 ≡⟨ cong (\right-list -> similar lx x right-list y) (empty-append ly) ⟩
b7c4e6276bcf Proof Monad-law-2-1
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49 similar lx x ly y
b7c4e6276bcf Proof Monad-law-2-1
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b7c4e6276bcf Proof Monad-law-2-1
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c5cdbedc68ad Proof Monad-law-2-2
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52 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Similar A) -> (mu ∙ return) s ≡ id s
c5cdbedc68ad Proof Monad-law-2-2
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53 monad-law-2-2 (similar lx x ly y) = refl
c5cdbedc68ad Proof Monad-law-2-2
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169ec60fcd36 Proof Monad-law-4
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169ec60fcd36 Proof Monad-law-4
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56 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (return ∙ f) x ≡ (fmap f ∙ return) x
169ec60fcd36 Proof Monad-law-4
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57 monad-law-3 f x = refl
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742e62fc63e4 Define Monad-law 1-4
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169ec60fcd36 Proof Monad-law-4
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169ec60fcd36 Proof Monad-law-4
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60 monad-law-4 : {l : Level} {A B : Set l} (f : A -> B) (s : Similar (Similar A)) ->
169ec60fcd36 Proof Monad-law-4
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61 (mu ∙ fmap (fmap f)) s ≡ (fmap f ∙ mu) s
169ec60fcd36 Proof Monad-law-4
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62 monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl