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annotate agda/similar.agda @ 38:6ce83b2c9e59

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Proof Functor-laws
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 19 Oct 2014 15:53:55 +0900
parents 169ec60fcd36
children b9b26b470cc2
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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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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9
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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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6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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14 -- Functor
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e0ba1bf564dd Apply level to some functions
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15 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Similar A) -> (Similar B)
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16 fmap f (similar xs x ys y) = similar xs (f x) ys (f y)
5ba82f107a95 Define Similar in Agda
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6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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19 -- Monad
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20 mu : {l : Level} {A : Set l} -> Similar (Similar A) -> Similar A
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5ba82f107a95 Define Similar in Agda
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21 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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23 return : {l : Level} {A : Set l} -> A -> Similar A
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742e62fc63e4 Define Monad-law 1-4
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24 return x = similar [] x [] x
742e62fc63e4 Define Monad-law 1-4
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6ce83b2c9e59 Proof Functor-laws
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26 returnS : {l : Level} {A : Set l} -> A -> Similar A
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27 returnS x = similar [[ (show x) ]] x [[ (show x) ]] x
5ba82f107a95 Define Similar in Agda
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6ce83b2c9e59 Proof Functor-laws
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29 returnSS : {l : Level} {A : Set l} -> A -> A -> Similar A
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30 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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6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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34 -- proofs
6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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37 -- Functor-laws
6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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39 -- Functor-law-1 : T(id) = id'
6ce83b2c9e59 Proof Functor-laws
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40 functor-law-1 : {l : Level} {A : Set l} -> (s : Similar A) -> (fmap id) s ≡ id s
6ce83b2c9e59 Proof Functor-laws
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41 functor-law-1 (similar lx x ly y) = refl
6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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43 -- Functor-law-2 : T(f . g) = T(f) . T(g)
6ce83b2c9e59 Proof Functor-laws
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44 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
6ce83b2c9e59 Proof Functor-laws
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45 (f : B -> C) -> (g : A -> B) -> (s : Similar A) ->
6ce83b2c9e59 Proof Functor-laws
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46 (fmap (f ∙ g)) s ≡ ((fmap f) ∙ (fmap g)) s
6ce83b2c9e59 Proof Functor-laws
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47 functor-law-2 f g (similar lx x ly y) = refl
6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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51 -- Monad-laws
6ce83b2c9e59 Proof Functor-laws
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52
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53 --monad-law-1 : mu ∙ (fmap mu) ≡ mu ∙ mu
e0ba1bf564dd Apply level to some functions
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54 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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55 monad-law-1 (similar lx (similar llx (similar lllx x _ _) _ (similar _ _ _ _))
71906644d206 Expand monad-law 1
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56 ly (similar _ (similar _ _ _ _) lly (similar _ _ llly y))) = begin
71906644d206 Expand monad-law 1
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57 similar (lx ++ (llx ++ lllx)) x (ly ++ (lly ++ llly)) y
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0bc402f970b3 Proof Monad-law 1
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58 ≡⟨ 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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59 similar (lx ++ llx ++ lllx) x (ly ++ (lly ++ llly)) y
0bc402f970b3 Proof Monad-law 1
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60 ≡⟨ 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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61 similar (lx ++ llx ++ lllx) x (ly ++ lly ++ llly) y
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62 ∎
e0ba1bf564dd Apply level to some functions
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b7c4e6276bcf Proof Monad-law-2-1
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b7c4e6276bcf Proof Monad-law-2-1
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65 --monad-law-2 : mu ∙ fmap return ≡ mu ∙ return ≡ id
b7c4e6276bcf Proof Monad-law-2-1
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66 monad-law-2-1 : {l : Level} {A : Set l} -> (s : Similar A) ->
b7c4e6276bcf Proof Monad-law-2-1
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67 (mu ∙ fmap return) s ≡ (mu ∙ return) s
b7c4e6276bcf Proof Monad-law-2-1
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68 monad-law-2-1 (similar lx x ly y) = begin
b7c4e6276bcf Proof Monad-law-2-1
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69 similar (lx ++ []) x (ly ++ []) y
b7c4e6276bcf Proof Monad-law-2-1
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70 ≡⟨ cong (\left-list -> similar left-list x (ly ++ []) y) (empty-append lx)⟩
b7c4e6276bcf Proof Monad-law-2-1
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71 similar lx x (ly ++ []) y
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72 ≡⟨ cong (\right-list -> similar lx x right-list y) (empty-append ly) ⟩
b7c4e6276bcf Proof Monad-law-2-1
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73 similar lx x ly y
b7c4e6276bcf Proof Monad-law-2-1
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74 ∎
b7c4e6276bcf Proof Monad-law-2-1
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c5cdbedc68ad Proof Monad-law-2-2
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76 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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77 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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80 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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81 monad-law-3 f x = refl
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742e62fc63e4 Define Monad-law 1-4
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82
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169ec60fcd36 Proof Monad-law-4
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169ec60fcd36 Proof Monad-law-4
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84 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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85 (mu ∙ fmap (fmap f)) s ≡ (fmap f ∙ mu) s
169ec60fcd36 Proof Monad-law-4
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86 monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl