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annotate agda/delta.agda @ 70:18a20a14c4b2

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Change prove method. use Int ...
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Thu, 27 Nov 2014 22:44:57 +0900
parents 295e8ed39c0c
children 56da62d57c95
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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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3
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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8 module delta where
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11 data Delta {l : Level} (A : Set l) : (Set (suc l)) where
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12 mono : A -> Delta A
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13 delta : A -> Delta A -> Delta A
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14
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15 deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A
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16 deltaAppend (mono x) d = delta x d
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17 deltaAppend (delta x d) ds = delta x (deltaAppend d ds)
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18
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19 headDelta : {l : Level} {A : Set l} -> Delta A -> A
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20 headDelta (mono x) = x
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21 headDelta (delta x _) = x
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22
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23 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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24 tailDelta (mono x) = mono x
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25 tailDelta (delta _ d) = d
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5ba82f107a95 Define Similar in Agda
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6ce83b2c9e59 Proof Functor-laws
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27
6ce83b2c9e59 Proof Functor-laws
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28 -- Functor
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29 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B)
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30 fmap f (mono x) = mono (f x)
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31 fmap f (delta x d) = delta (f x) (fmap f d)
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35 -- Monad (Category)
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36 eta : {l : Level} {A : Set l} -> A -> Delta A
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37 eta x = mono x
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742e62fc63e4 Define Monad-law 1-4
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38
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46b15f368905 Define bind and mu for Infinite Delta
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39 bind : {l ll : Level} {A : Set l} {B : Set ll} -> (Delta A) -> (A -> Delta B) -> Delta B
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40 bind (mono x) f = f x
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41 bind (delta x d) f = delta (headDelta (f x)) (bind d (tailDelta ∙ f))
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46b15f368905 Define bind and mu for Infinite Delta
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46b15f368905 Define bind and mu for Infinite Delta
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43 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A
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6d0193011f89 Trying prove monad-law-1 by another pattern
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44 mu d = bind d id
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45
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46 returnS : {l : Level} {A : Set l} -> A -> Delta A
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47 returnS x = mono x
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49 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A
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50 returnSS x y = deltaAppend (returnS x) (returnS y)
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51
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0bc402f970b3 Proof Monad-law 1
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53 -- Monad (Haskell)
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54 return : {l : Level} {A : Set l} -> A -> Delta A
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55 return = eta
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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57 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
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58 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
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59 (mono x) >>= f = f x
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60 (delta x d) >>= f = delta (headDelta (f x)) (d >>= (tailDelta ∙ f))
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a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
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6ce83b2c9e59 Proof Functor-laws
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6ce83b2c9e59 Proof Functor-laws
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64 -- proofs
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65
6ce83b2c9e59 Proof Functor-laws
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66 -- Functor-laws
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67
6ce83b2c9e59 Proof Functor-laws
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68 -- Functor-law-1 : T(id) = id'
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69 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d
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70 functor-law-1 (mono x) = refl
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71 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)
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72
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73 -- Functor-law-2 : T(f . g) = T(f) . T(g)
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74 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
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75 (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
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76 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d
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77 functor-law-2 f g (mono x) = refl
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78 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)
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b9b26b470cc2 Add Comments
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81 -- Monad-laws (Category)
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83
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84 data Int : Set where
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85 one : Int
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86 succ : Int -> Int
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87
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88 n-times-tail-delta : {l : Level} {A : Set l} -> Int -> ((Delta A) -> (Delta A))
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89 n-times-tail-delta one = tailDelta
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90 n-times-tail-delta (succ n) = (n-times-tail-delta n) ∙ tailDelta
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91
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92 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Int) -> (x : A) ->
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93 (n-times-tail-delta n) (mono x) ≡ (mono x)
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94 tail-delta-to-mono one x = refl
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95 tail-delta-to-mono (succ n) x = begin
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96 n-times-tail-delta (succ n) (mono x)
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97 ≡⟨ refl ⟩
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98 n-times-tail-delta n (mono x)
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99 ≡⟨ tail-delta-to-mono n x ⟩
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100 mono x
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101 ∎
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102
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103 monad-law-1-4 : {l : Level} {A : Set l} -> (n : Int) (d : Delta (Delta A)) ->
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104 (headDelta ((n-times-tail-delta n) (headDelta ((n-times-tail-delta n) d)))) ≡
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105 (headDelta ((n-times-tail-delta n) (mu d)))
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106 monad-law-1-4 one (mono d) = refl
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107 monad-law-1-4 one (delta d (mono ds)) = refl
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108 monad-law-1-4 one (delta d (delta ds ds₁)) = refl
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109 monad-law-1-4 (succ n) (mono d) = begin
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110 headDelta (n-times-tail-delta (succ n) (headDelta (n-times-tail-delta (succ n) (mono d))))
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111 ≡⟨ refl ⟩
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112 headDelta (n-times-tail-delta (succ n) (headDelta ((n-times-tail-delta n) (mono d))))
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113 ≡⟨ cong (\d -> headDelta (n-times-tail-delta (succ n) (headDelta d))) (tail-delta-to-mono n d) ⟩
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diff changeset
114 headDelta (n-times-tail-delta (succ n) (headDelta (mono d)))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
115 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
116 headDelta (n-times-tail-delta (succ n) d)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
117 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
118 headDelta (n-times-tail-delta (succ n) (mu (mono d)))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
119 ∎
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
120 monad-law-1-4 (succ n) (delta d (mono ds)) = begin
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
121 headDelta (n-times-tail-delta (succ n) (headDelta (n-times-tail-delta (succ n) (delta d (mono ds)))))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
122 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
123 headDelta (n-times-tail-delta (succ n) (headDelta (n-times-tail-delta n (mono ds))))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
124 ≡⟨ cong (\d -> headDelta (n-times-tail-delta (succ n) (headDelta d))) (tail-delta-to-mono n ds) ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
125 headDelta (n-times-tail-delta (succ n) (headDelta (mono ds)))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
126 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
127 headDelta (n-times-tail-delta (succ n) ds)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
128 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
129 headDelta (n-times-tail-delta n (tailDelta ds))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
130 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
131 headDelta (n-times-tail-delta n ((bind (mono ds) tailDelta)))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
132 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
133 headDelta (n-times-tail-delta (succ n) (delta (headDelta d) (bind (mono ds) tailDelta)))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
134 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
135 headDelta (n-times-tail-delta (succ n) (mu (delta d (mono ds))))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
136 ∎
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
137 monad-law-1-4 (succ n) (delta d (delta dd ds)) = begin
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
138 headDelta (n-times-tail-delta (succ n) (headDelta (n-times-tail-delta (succ n) (delta d (delta dd ds)))))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
139 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
140 headDelta (n-times-tail-delta (succ n) (headDelta (n-times-tail-delta n (delta dd ds))))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
141 ≡⟨ {!!} ⟩ -- ?
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
142
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
143 headDelta (n-times-tail-delta n (delta (headDelta (tailDelta dd)) (bind ds (tailDelta ∙ tailDelta))))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
144 ≡⟨ {!!} ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
145 headDelta (n-times-tail-delta n (delta (headDelta (tailDelta dd)) (bind ds (tailDelta ∙ tailDelta ))))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
146 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
147 headDelta (n-times-tail-delta n (bind (delta dd ds) (tailDelta)))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
148 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
149 headDelta (n-times-tail-delta (succ n) (delta (headDelta d) (bind (delta dd ds) (tailDelta))))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
150 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
151 headDelta (n-times-tail-delta (succ n) (mu (delta d (delta dd ds))))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
152 ∎
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
153
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
154
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
155
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
156
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
157 monad-law-1-3 : {l : Level} {A : Set l} -> (i : Int) -> (d : Delta (Delta (Delta A))) ->
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
158 (bind (fmap mu d) (n-times-tail-delta i) ≡ (bind (bind d (n-times-tail-delta i)) (n-times-tail-delta i)))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
159 monad-law-1-3 one (mono (mono d)) = refl
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
160 monad-law-1-3 one (mono (delta d d₁)) = refl
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
161 monad-law-1-3 one (delta d ds) = begin
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
162 bind (fmap mu (delta d ds)) (n-times-tail-delta one)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
163 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
164 bind (delta (mu d) (fmap mu ds)) (n-times-tail-delta one)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
165 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
166 delta (headDelta ((n-times-tail-delta one) (mu d))) (bind (fmap mu ds) ((n-times-tail-delta one) ∙ tailDelta))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
167 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
168 delta (headDelta ((n-times-tail-delta one) (mu d))) (bind (fmap mu ds) (n-times-tail-delta (succ one)))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
169 ≡⟨ cong (\dx -> delta (headDelta ((n-times-tail-delta one) (mu d))) dx) (monad-law-1-3 (succ one) ds) ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
170 delta (headDelta ((n-times-tail-delta one) (mu d))) (bind (bind ds (n-times-tail-delta (succ one))) (n-times-tail-delta (succ one)))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
171 ≡⟨ cong (\dx -> delta dx (bind (bind ds (n-times-tail-delta (succ one))) (n-times-tail-delta (succ one )))) (sym (monad-law-1-4 one d)) ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
172 delta (headDelta ((n-times-tail-delta one) (headDelta ((n-times-tail-delta one) d)))) (bind (bind ds (n-times-tail-delta (succ one))) (n-times-tail-delta (succ one)))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
173 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
174 delta (headDelta ((n-times-tail-delta one) (headDelta ((n-times-tail-delta one) d)))) ((bind (bind ds (n-times-tail-delta (succ one)))) ((n-times-tail-delta one) ∙ tailDelta))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
175 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
176 bind (delta (headDelta ((n-times-tail-delta one) d)) (bind ds (n-times-tail-delta (succ one)))) (n-times-tail-delta one)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
177 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
178 bind (delta (headDelta ((n-times-tail-delta one) d)) (bind ds ((n-times-tail-delta one) ∙ tailDelta))) (n-times-tail-delta one)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
179 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
180 bind (bind (delta d ds) (n-times-tail-delta one)) (n-times-tail-delta one)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
181 ∎
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
182 monad-law-1-3 (succ i) d = {!!}
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
183
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
184
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
185 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
186 monad-law-1-2 (mono _) = refl
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
187 monad-law-1-2 (delta _ _) = refl
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
188
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
189 -- monad-law-1 : join . fmap join = join . join
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
190 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
191 monad-law-1 (mono d) = refl
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
192 monad-law-1 (delta x d) = begin
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
193 (mu ∙ fmap mu) (delta x d)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
194 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
195 mu (fmap mu (delta x d))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
196 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
197 mu (delta (mu x) (fmap mu d))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
198 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
199 delta (headDelta (mu x)) (bind (fmap mu d) tailDelta)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
200 ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
201 delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
202 ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 one d) ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
203 delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
204 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
205 mu (delta (headDelta x) (bind d tailDelta))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
206 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
207 mu (mu (delta x d))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
208 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
209 (mu ∙ mu) (delta x d)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
210 ∎
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
211
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
212
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
213
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
214 {-
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
215 -- monad-law-2 : join . fmap return = join . return = id
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
216 -- monad-law-2-1 join . fmap return = join . return
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
217 monad-law-2-1 : {l : Level} {A : Set l} -> (d : Delta A) ->
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
218 (mu ∙ fmap eta) d ≡ (mu ∙ eta) d
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
219 monad-law-2-1 (mono x) = refl
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
220 monad-law-2-1 (delta x d) = {!!}
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
221
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
222
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
223 -- monad-law-2-2 : join . return = id
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
224 monad-law-2-2 : {l : Level} {A : Set l } -> (d : Delta A) -> (mu ∙ eta) d ≡ id d
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
225 monad-law-2-2 d = refl
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
226
35
c5cdbedc68ad Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
227
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
228 -- 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
229 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
230 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
231
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
232
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
233 -- monad-law-4 : join . fmap (fmap f) = fmap f . join
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
234 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (d : Delta (Delta A)) ->
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
235 (mu ∙ fmap (fmap f)) d ≡ (fmap f ∙ mu) d
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
236 monad-law-4 f d = {!!}
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
237
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
238
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
239
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
240
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
241 -- 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
242 -- 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
243 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
244 (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
245 (return a >>= k) ≡ (k a)
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
246 monad-law-h-1 a k = refl
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
247
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
248
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
249
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
250 -- 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
251 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
252 monad-law-h-2 (mono x) = refl
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
253 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d)
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
254
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
255
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
256 -- 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
257 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
258 (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
259 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h)
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
260 monad-law-h-3 (mono x) k h = refl
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
261 monad-law-h-3 (delta x d) k h = {!!}
69
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
262
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
263 -}