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Proof Monad-law-3 (haskell)
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Fri, 24 Oct 2014 14:08:50 +0900
parents 23474bf242c6
children
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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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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
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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11 similar : List String -> A -> List String -> A -> Similar A
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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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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)
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19 -- Monad (Category)
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20 mu : {l : Level} {A : Set l} -> Similar (Similar A) -> Similar A
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21 mu (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = similar (lx ++ llx) x (ly ++ lly) y
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23 eta : {l : Level} {A : Set l} -> A -> Similar A
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24 eta x = similar [] x [] x
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742e62fc63e4 Define Monad-law 1-4
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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
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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
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0bc402f970b3 Proof Monad-law 1
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33 -- Monad (Haskell)
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34 return : {l : Level} {A : Set l} -> A -> Similar A
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35 return = eta
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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38 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
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39 (x : Similar A) -> (f : A -> (Similar B)) -> (Similar B)
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40 x >>= f = mu (fmap f x)
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6ce83b2c9e59 Proof Functor-laws
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44 -- 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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47 -- Functor-laws
6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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49 -- Functor-law-1 : T(id) = id'
6ce83b2c9e59 Proof Functor-laws
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50 functor-law-1 : {l : Level} {A : Set l} -> (s : Similar A) -> (fmap id) s ≡ id s
6ce83b2c9e59 Proof Functor-laws
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51 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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53 -- Functor-law-2 : T(f . g) = T(f) . T(g)
6ce83b2c9e59 Proof Functor-laws
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54 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
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55 (f : B -> C) -> (g : A -> B) -> (s : Similar A) ->
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56 (fmap (f ∙ g)) s ≡ ((fmap f) ∙ (fmap g)) s
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57 functor-law-2 f g (similar lx x ly y) = refl
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6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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61 -- Monad-laws (Category)
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63 -- monad-law-1 : join . fmap join = join . join
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64 monad-law-1 : {l : Level} {A : Set l} -> (s : Similar (Similar (Similar A))) -> ((mu ∙ (fmap mu)) s) ≡ ((mu ∙ mu) s)
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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65 monad-law-1 (similar lx (similar llx (similar lllx x _ _) _ (similar _ _ _ _))
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71906644d206 Expand monad-law 1
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66 ly (similar _ (similar _ _ _ _) lly (similar _ _ llly y))) = begin
71906644d206 Expand monad-law 1
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67 similar (lx ++ (llx ++ lllx)) x (ly ++ (lly ++ llly)) y
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0bc402f970b3 Proof Monad-law 1
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68 ≡⟨ cong (\left-list -> similar left-list x (ly ++ (lly ++ llly)) y) (list-associative lx llx lllx) ⟩
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69 similar (lx ++ llx ++ lllx) x (ly ++ (lly ++ llly)) y
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70 ≡⟨ 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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71 similar (lx ++ llx ++ lllx) x (ly ++ lly ++ llly) y
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72 ∎
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b7c4e6276bcf Proof Monad-law-2-1
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75 -- monad-law-2 : join . fmap return = join . return = id
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76 -- monad-law-2-1 join . fmap return = join . return
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b7c4e6276bcf Proof Monad-law-2-1
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77 monad-law-2-1 : {l : Level} {A : Set l} -> (s : Similar A) ->
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78 (mu ∙ fmap eta) s ≡ (mu ∙ eta) s
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b7c4e6276bcf Proof Monad-law-2-1
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79 monad-law-2-1 (similar lx x ly y) = begin
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80 similar (lx ++ []) x (ly ++ []) y
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81 ≡⟨ cong (\left-list -> similar left-list x (ly ++ []) y) (empty-append lx)⟩
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82 similar lx x (ly ++ []) y
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83 ≡⟨ cong (\right-list -> similar lx x right-list y) (empty-append ly) ⟩
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84 similar lx x ly y
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85 ∎
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86
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87 -- monad-law-2-2 : join . return = id
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88 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Similar A) -> (mu ∙ eta) s ≡ id s
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c5cdbedc68ad Proof Monad-law-2-2
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89 monad-law-2-2 (similar lx x ly y) = refl
c5cdbedc68ad Proof Monad-law-2-2
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90
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91 -- monad-law-3 : return . f = fmap f . return
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92 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x
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169ec60fcd36 Proof Monad-law-4
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93 monad-law-3 f x = refl
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742e62fc63e4 Define Monad-law 1-4
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95 -- monad-law-4 : join . fmap (fmap f) = fmap f . join
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1df4f9d88025 Proof Monad-law-3 (haskell)
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96 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (s : Similar (Similar A)) ->
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169ec60fcd36 Proof Monad-law-4
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97 (mu ∙ fmap (fmap f)) s ≡ (fmap f ∙ mu) s
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98 monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl
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101 -- Monad-laws (Haskell)
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102 -- monad-law-h-1 : return a >>= k = k a
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103 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} ->
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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104 (a : A) -> (k : A -> (Similar B)) ->
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105 (return a >>= k) ≡ (k a)
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106 monad-law-h-1 a k = begin
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107 return a >>= k
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108 ≡⟨ refl ⟩
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109 mu (fmap k (return a))
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110 ≡⟨ refl ⟩
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Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
111 mu (return (k a))
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
112 ≡⟨ refl ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
113 (mu ∙ return) (k a)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
114 ≡⟨ refl ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
115 (mu ∙ eta) (k a)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
116 ≡⟨ (monad-law-2-2 (k a)) ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
117 id (k a)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
118 ≡⟨ refl ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
119 k a
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
120 ∎
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
121
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
122 -- monad-law-h-2 : m >>= return = m
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
123 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Similar A) -> (m >>= return) ≡ m
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
124 monad-law-h-2 (similar lx x ly y) = monad-law-2-1 (similar lx x ly y)
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
125
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
126 -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
127 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
128 (m : Similar A) -> (k : A -> (Similar B)) -> (h : B -> (Similar C)) ->
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
129 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h)
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
130 monad-law-h-3 (similar lx x ly y) k h = begin
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
131 ((similar lx x ly y) >>= (\x -> (k x) >>= h))
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
132 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
133 mu (fmap (\x -> k x >>= h) (similar lx x ly y))
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
134 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
135 (mu ∙ fmap (\x -> k x >>= h)) (similar lx x ly y)
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
136 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
137 (mu ∙ fmap (\x -> mu (fmap h (k x)))) (similar lx x ly y)
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
138 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
139 (mu ∙ fmap (mu ∙ (\x -> fmap h (k x)))) (similar lx x ly y)
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
140 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
141 (mu ∙ (fmap mu ∙ (fmap (\x -> fmap h (k x))))) (similar lx x ly y)
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
142 ≡⟨ refl ⟩
42
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
143 (mu ∙ (fmap mu)) ((fmap (\x -> fmap h (k x))) (similar lx x ly y))
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
144 ≡⟨ monad-law-1 (((fmap (\x -> fmap h (k x))) (similar lx x ly y))) ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
145 (mu ∙ mu) ((fmap (\x -> fmap h (k x))) (similar lx x ly y))
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
146 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
147 (mu ∙ (mu ∙ (fmap (\x -> fmap h (k x))))) (similar lx x ly y)
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
148 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
149 (mu ∙ (mu ∙ (fmap ((fmap h) ∙ k)))) (similar lx x ly y)
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
150 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
151 (mu ∙ (mu ∙ ((fmap (fmap h)) ∙ (fmap k)))) (similar lx x ly y)
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
152 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
153 (mu ∙ (mu ∙ (fmap (fmap h)))) (fmap k (similar lx x ly y))
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
154 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
155 mu ((mu ∙ (fmap (fmap h))) (fmap k (similar lx x ly y)))
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
156 ≡⟨ cong (\fx -> mu fx) (monad-law-4 h (fmap k (similar lx x ly y))) ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
157 mu (fmap h (mu (similar lx (k x) ly (k y))))
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
158 ≡⟨ refl ⟩
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
159 (mu ∙ fmap h) (mu (fmap k (similar lx x ly y)))
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
160 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
161 mu (fmap h (mu (fmap k (similar lx x ly y))))
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
162 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
163 (mu (fmap k (similar lx x ly y))) >>= h
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
164 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
165 ((similar lx x ly y) >>= k) >>= h
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
166 ∎