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Redefine Delta for infinite changes in Agda
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Wed, 19 Nov 2014 20:18:26 +0900
parents 9bb7c9bee94f
children bfb6be9a689d
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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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3
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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
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6 open ≡-Reasoning
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8 module delta where
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9
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10 DeltaLog : Set
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11 DeltaLog = List String
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12
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13 data Delta {l : Level} (A : Set l) : (Set (suc l)) where
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14 mono : DeltaLog -> A -> Delta A
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15 delta : DeltaLog -> A -> Delta A -> Delta A
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17 logAppend : {l : Level} {A : Set l} -> DeltaLog -> Delta A -> Delta A
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18 logAppend l (mono lx x) = mono (l ++ lx) x
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19 logAppend l (delta lx x d) = delta (l ++ lx) x (logAppend l d)
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21 deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A
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22 deltaAppend (mono lx x) d = delta lx x d
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23 deltaAppend (delta lx x d) ds = delta lx x (deltaAppend d ds)
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25 headDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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26 headDelta (mono lx x) = mono lx x
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27 headDelta (delta lx x _) = mono lx x
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29 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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30 tailDelta (mono lx x) = mono lx x
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31 tailDelta (delta _ _ d) = d
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33
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34 -- Functor
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35 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B)
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36 fmap f (mono lx x) = mono lx (f x)
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37 fmap f (delta lx x d) = delta lx (f x) (fmap f d)
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39
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40 {-# NO_TERMINATION_CHECK #-}
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41 -- Monad (Category)
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42 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A
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43 mu (mono ld d) = logAppend ld d
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44 mu (delta ld d ds) = deltaAppend (logAppend ld (headDelta d)) (mu (fmap tailDelta ds))
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46 eta : {l : Level} {A : Set l} -> A -> Delta A
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47 eta x = mono [] x
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48
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49 returnS : {l : Level} {A : Set l} -> A -> Delta A
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50 returnS x = mono [[ (show x) ]] x
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52 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A
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53 returnSS x y = delta [[ (show x) ]] x (mono [[ (show y) ]] y)
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0bc402f970b3 Proof Monad-law 1
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56 -- Monad (Haskell)
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57 return : {l : Level} {A : Set l} -> A -> Delta A
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58 return = eta
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59
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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60 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
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61 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
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62 x >>= f = mu (fmap f x)
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a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
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64
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66 -- proofs
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69 -- Functor-laws
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70
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71 -- Functor-law-1 : T(id) = id'
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72 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d
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73 functor-law-1 (mono lx x) = refl
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74 functor-law-1 (delta lx x d) = cong (delta lx x) (functor-law-1 d)
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76 -- Functor-law-2 : T(f . g) = T(f) . T(g)
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77 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
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78 (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
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79 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d
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80 functor-law-2 f g (mono lx x) = refl
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81 functor-law-2 f g (delta lx x d) = cong (delta lx (f (g x))) (functor-law-2 f g d)
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82
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83
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84 {-
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85 -- Monad-laws (Category)
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86
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87 -- monad-law-1 : join . fmap join = join . join
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88 monad-law-1 : {l : Level} {A : Set l} -> (s : Delta (Delta (Delta 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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89 monad-law-1 (similar lx (similar llx (similar lllx x _ _) _ (similar _ _ _ _))
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71906644d206 Expand monad-law 1
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90 ly (similar _ (similar _ _ _ _) lly (similar _ _ llly y))) = begin
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91 similar (lx ++ (llx ++ lllx)) x (ly ++ (lly ++ llly)) y
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0bc402f970b3 Proof Monad-law 1
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92 ≡⟨ cong (\left-list -> similar left-list x (ly ++ (lly ++ llly)) y) (list-associative lx llx lllx) ⟩
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93 similar (lx ++ llx ++ lllx) x (ly ++ (lly ++ llly)) y
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94 ≡⟨ 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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95 similar (lx ++ llx ++ lllx) x (ly ++ lly ++ llly) y
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e0ba1bf564dd Apply level to some functions
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96 ∎
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97
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b7c4e6276bcf Proof Monad-law-2-1
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98
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99 -- monad-law-2 : join . fmap return = join . return = id
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100 -- monad-law-2-1 join . fmap return = join . return
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101 monad-law-2-1 : {l : Level} {A : Set l} -> (s : Delta A) ->
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102 (mu ∙ fmap eta) s ≡ (mu ∙ eta) s
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b7c4e6276bcf Proof Monad-law-2-1
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103 monad-law-2-1 (similar lx x ly y) = begin
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104 similar (lx ++ []) x (ly ++ []) y
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105 ≡⟨ cong (\left-list -> similar left-list x (ly ++ []) y) (empty-append lx)⟩
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106 similar lx x (ly ++ []) y
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107 ≡⟨ cong (\right-list -> similar lx x right-list y) (empty-append ly) ⟩
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108 similar lx x ly y
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109 ∎
b7c4e6276bcf Proof Monad-law-2-1
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110
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111 -- monad-law-2-2 : join . return = id
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112 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Delta A) -> (mu ∙ eta) s ≡ id s
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c5cdbedc68ad Proof Monad-law-2-2
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parents: 34
diff changeset
113 monad-law-2-2 (similar lx x ly y) = refl
c5cdbedc68ad Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
114
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
115 -- monad-law-3 : return . f = fmap f . return
40
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-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x
36
169ec60fcd36 Proof Monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
117 monad-law-3 f x = refl
27
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
118
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
119 -- monad-law-4 : join . fmap (fmap f) = fmap f . join
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
120 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (s : Delta (Delta A)) ->
36
169ec60fcd36 Proof Monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
121 (mu ∙ fmap (fmap f)) s ≡ (fmap f ∙ mu) s
40
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-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = 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
123
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
124
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
125 -- Monad-laws (Haskell)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
126 -- monad-law-h-1 : return a >>= k = 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
127 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} ->
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
128 (a : A) -> (k : A -> (Delta B)) ->
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
129 (return a >>= k) ≡ (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
130 monad-law-h-1 a k = begin
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
131 return a >>= k
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
132 ≡⟨ 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
133 mu (fmap k (return 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
134 ≡⟨ 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
135 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
136 ≡⟨ 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
137 (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
138 ≡⟨ 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
139 (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
140 ≡⟨ (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
141 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
142 ≡⟨ 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
143 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
144 ∎
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
145
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
146 -- monad-law-h-2 : m >>= return = m
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
147 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (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
148 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
149
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
150 -- 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
151 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
152 (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) ->
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
153 (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
154 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
155 ((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
156 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
157 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
158 ≡⟨ refl ⟩
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 (\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
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 (\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
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 (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
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 (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
166 ≡⟨ refl ⟩
42
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
167 (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
168 ≡⟨ monad-law-1 (((fmap (\x -> fmap h (k x))) (similar lx x ly y))) ⟩
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
169 (mu ∙ mu) ((fmap (\x -> fmap h (k x))) (similar lx x ly y))
42
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
170 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
171 (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
172 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
173 (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
174 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
175 (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
176 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
177 (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
178 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
179 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
180 ≡⟨ 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
181 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
182 ≡⟨ 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
183 (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
184 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
185 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
186 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
187 (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
188 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
189 ((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
190 ∎
54
9bb7c9bee94f Trying redefine delta for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
191 -}