Mercurial > hg > Members > atton > similar_monad

comparison agda/similar.agda @ 39:b9b26b470cc2

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author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 19 Oct 2014 16:00:52 +0900
parents 6ce83b2c9e59
children a7cd7740f33e
comparison
equal deleted inserted replaced
38:6ce83b2c9e59 39:b9b26b470cc2
46 (fmap (f ∙ g)) s ≡ ((fmap f) ∙ (fmap g)) s 46 (fmap (f ∙ g)) s ≡ ((fmap f) ∙ (fmap g)) s
47 functor-law-2 f g (similar lx x ly y) = refl 47 functor-law-2 f g (similar lx x ly y) = refl
48 48
49 49
50 50
51 -- Monad-laws 51 -- Monad-laws (Category)
52 52
53 --monad-law-1 : mu ∙ (fmap mu) ≡ mu ∙ mu 53 -- monad-law-1 : join . fmap join = join . join
54 monad-law-1 : {l : Level} {A : Set l} -> (s : Similar (Similar (Similar A))) -> ((mu ∙ (fmap mu)) s) ≡ ((mu ∙ mu) s) 54 monad-law-1 : {l : Level} {A : Set l} -> (s : Similar (Similar (Similar A))) -> ((mu ∙ (fmap mu)) s) ≡ ((mu ∙ mu) s)
55 monad-law-1 (similar lx (similar llx (similar lllx x _ _) _ (similar _ _ _ _)) 55 monad-law-1 (similar lx (similar llx (similar lllx x _ _) _ (similar _ _ _ _))
56 ly (similar _ (similar _ _ _ _) lly (similar _ _ llly y))) = begin 56 ly (similar _ (similar _ _ _ _) lly (similar _ _ llly y))) = begin
57 similar (lx ++ (llx ++ lllx)) x (ly ++ (lly ++ llly)) y 57 similar (lx ++ (llx ++ lllx)) x (ly ++ (lly ++ llly)) y
58 ≡⟨ cong (\left-list -> similar left-list x (ly ++ (lly ++ llly)) y) (list-associative lx llx lllx) ⟩ 58 ≡⟨ cong (\left-list -> similar left-list x (ly ++ (lly ++ llly)) y) (list-associative lx llx lllx) ⟩
60 ≡⟨ cong (\right-list -> similar (lx ++ llx ++ lllx) x right-list y ) (list-associative ly lly llly) ⟩ 60 ≡⟨ cong (\right-list -> similar (lx ++ llx ++ lllx) x right-list y ) (list-associative ly lly llly) ⟩
61 similar (lx ++ llx ++ lllx) x (ly ++ lly ++ llly) y 61 similar (lx ++ llx ++ lllx) x (ly ++ lly ++ llly) y
62 ∎ 62 ∎
63 63
64 64
65 --monad-law-2 : mu ∙ fmap return ≡ mu ∙ return ≡ id 65 -- monad-law-2 : join . fmap return = join . return = id
66 -- monad-law-2-1 join . fmap return = join . return
66 monad-law-2-1 : {l : Level} {A : Set l} -> (s : Similar A) -> 67 monad-law-2-1 : {l : Level} {A : Set l} -> (s : Similar A) ->
67 (mu ∙ fmap return) s ≡ (mu ∙ return) s 68 (mu ∙ fmap return) s ≡ (mu ∙ return) s
68 monad-law-2-1 (similar lx x ly y) = begin 69 monad-law-2-1 (similar lx x ly y) = begin
69 similar (lx ++ []) x (ly ++ []) y 70 similar (lx ++ []) x (ly ++ []) y
70 ≡⟨ cong (\left-list -> similar left-list x (ly ++ []) y) (empty-append lx)⟩ 71 ≡⟨ cong (\left-list -> similar left-list x (ly ++ []) y) (empty-append lx)⟩
71 similar lx x (ly ++ []) y 72 similar lx x (ly ++ []) y
72 ≡⟨ cong (\right-list -> similar lx x right-list y) (empty-append ly) ⟩ 73 ≡⟨ cong (\right-list -> similar lx x right-list y) (empty-append ly) ⟩
73 similar lx x ly y 74 similar lx x ly y
74 ∎ 75 ∎
75 76
77 -- monad-law-2-2 : join . return = id
76 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Similar A) -> (mu ∙ return) s ≡ id s 78 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Similar A) -> (mu ∙ return) s ≡ id s
77 monad-law-2-2 (similar lx x ly y) = refl 79 monad-law-2-2 (similar lx x ly y) = refl
78 80
79 81 -- monad-law-3 : return . f = fmap f . return
80 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (return ∙ f) x ≡ (fmap f ∙ return) x 82 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (return ∙ f) x ≡ (fmap f ∙ return) x
81 monad-law-3 f x = refl 83 monad-law-3 f x = refl
82 84
83 85 -- monad-law-4 : join . fmap (fmap f) = fmap f . join
84 monad-law-4 : {l : Level} {A B : Set l} (f : A -> B) (s : Similar (Similar A)) -> 86 monad-law-4 : {l : Level} {A B : Set l} (f : A -> B) (s : Similar (Similar A)) ->
85 (mu ∙ fmap (fmap f)) s ≡ (fmap f ∙ mu) s 87 (mu ∙ fmap (fmap f)) s ≡ (fmap f ∙ mu) s
86 monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl 88 monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl