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Trying prove infinite-delta. but I think this definition was missed.
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 30 Nov 2014 19:00:32 +0900
parents 56da62d57c95
children 0ad0ae7a3cbe
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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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5ba82f107a95 Define Similar in Agda
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7
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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
9bb7c9bee94f Trying redefine delta for infinite changes
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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
9bb7c9bee94f Trying redefine delta for infinite changes
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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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32
26
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6ce83b2c9e59 Proof Functor-laws
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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
46b15f368905 Define bind and mu for Infinite Delta
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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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42
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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48
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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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56
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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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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
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68 -- Functor-law-1 : T(id) = id'
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9c8c09334e32 Redefine Delta for infinite changes in Agda
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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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81 -- Monad-laws (Category)
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84 data Int : Set where
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85 O : Int
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86 S : Int -> Int
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87
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88 _+_ : Int -> Int -> Int
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89 O + n = n
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90 (S m) + n = S (m + n)
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91
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92 n-tail : {l : Level} {A : Set l} -> Int -> ((Delta A) -> (Delta A))
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93 n-tail O = id
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94 n-tail (S n) = (n-tail n) ∙ tailDelta
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95
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96 postulate n-tail-plus : (n : Int) -> (tailDelta ∙ (n-tail n)) ≡ n-tail (S n)
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98
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99
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101
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102 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Int) -> (x : A) ->
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103 (n-tail n) (mono x) ≡ (mono x)
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104 tail-delta-to-mono O x = refl
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105 tail-delta-to-mono (S n) x = begin
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106 n-tail (S n) (mono x) ≡⟨ refl ⟩
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107 ((n-tail n) ∙ tailDelta) (mono x) ≡⟨ refl ⟩
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108 (n-tail n) (tailDelta (mono x)) ≡⟨ refl ⟩
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109 (n-tail n) (mono x) ≡⟨ tail-delta-to-mono n x ⟩
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110 mono x
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111 ∎
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112 {- begin
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113 n-tail (S n) (mono x) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
114 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
115 tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
116 tailDelta (mono x) ≡⟨ refl ⟩
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
117 mono x
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
118 ∎
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
119 -}
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
120 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
121 monad-law-1-2 (mono _) = refl
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
122 monad-law-1-2 (delta _ _) = refl
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
123
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
124 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Int) -> (d : Delta (Delta (Delta A))) ->
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
125 bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n)
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
126 monad-law-1-3 O (mono d) = refl
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
127 monad-law-1-3 O (delta d ds) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
128 bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
129 bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
130 delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
131 delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
132 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
133 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
134 bind (bind (delta d ds) (n-tail O)) (n-tail O)
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
135 ∎
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
136 monad-law-1-3 (S n) (mono (mono d)) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
137 bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
138 bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
139 (n-tail (S n)) d ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
140 bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
141 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
142 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
143 bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
144 ∎
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
145 monad-law-1-3 (S n) (mono (delta d ds)) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
146 bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
147 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
148 n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
149 n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
150 n-tail n (bind ds tailDelta) ≡⟨ {!!} ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
151 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
152 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
153 bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
154 ∎
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
155 monad-law-1-3 (S n) (delta (mono d) ds) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
156 bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
157 bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
158 bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
159 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ {!!} ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
160 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ {!!} ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
161 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
162 delta (headDelta ((n-tail (S n)) (headDelta (mono d)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) (headDelta de))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (sym (tail-delta-to-mono (S n) d)) ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
163 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (mono d))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
164 bind (delta (headDelta ((n-tail (S n)) (mono d))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
165 bind (bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
166 ∎
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
167 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
168 bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
169 bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
170 delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
171 delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
172 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ {!!} ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
173
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
174 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
175 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
176 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
177 bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
178 ∎
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
179
71
56da62d57c95 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 70
diff changeset
180 {-
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
181 monad-law-1-3 (S n) (mono d) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
182 bind (fmap mu (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
183 bind (mono (mu d)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
184 n-tail (S n) (mu d) ≡⟨ {!!} ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
185 bind (n-tail (S n) d) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
186 bind (bind (mono d) (n-tail (S n))) (n-tail (S n))
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
187 ∎
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
188 monad-law-1-3 (S n) (delta d ds) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
189 bind (fmap mu (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
190 bind (delta (mu d) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
191 delta (headDelta ((n-tail (S n)) (mu d))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
192 delta (headDelta ((n-tail (S n)) (mu d))) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ {!!} ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
193 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) d)))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
194 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) d)))) (bind (bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
195 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) d)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
196 bind (delta (headDelta ((n-tail (S n)) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
197 bind (bind (delta d ds) (n-tail (S n))) (n-tail (S n))
71
56da62d57c95 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 70
diff changeset
198 ∎
56da62d57c95 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 70
diff changeset
199 -}
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
200
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
201 -- 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
202 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
203 monad-law-1 (mono d) = refl
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
204 {-
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
205 monad-law-1 (delta x (mono d)) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
206 (mu ∙ fmap mu) (delta x (mono d)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
207 mu (fmap mu (delta x (mono d))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
208 mu (delta (mu x) (mono (mu d))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
209 delta (headDelta (mu x)) (bind (mono (mu d)) tailDelta) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
210 delta (headDelta (mu x)) (tailDelta (mu d)) ≡⟨ cong (\dx -> delta dx (tailDelta (mu d))) (monad-law-1-2 x) ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
211 delta (headDelta (headDelta x)) (tailDelta (mu d)) ≡⟨ {!!} ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
212 delta (headDelta (headDelta x)) (bind (tailDelta d) tailDelta) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
213 mu (delta (headDelta x) (tailDelta d)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
214 mu (delta (headDelta x) (bind (mono d) tailDelta)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
215 mu (mu (delta x (mono d))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
216 (mu ∙ mu) (delta x (mono d))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
217 ∎
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
218 monad-law-1 (delta x (delta d ds)) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
219 (mu ∙ fmap mu) (delta x (delta d ds)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
220 mu (fmap mu (delta x (delta d ds))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
221 mu (delta (mu x) (delta (mu d) (fmap mu ds))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
222 delta (headDelta (mu x)) (bind (delta (mu d) (fmap mu ds)) tailDelta) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
223 delta (headDelta (mu x)) (delta (headDelta (tailDelta (mu d))) (bind (fmap mu ds) (tailDelta ∙ tailDelta))) ≡⟨ {!!} ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
224 delta (headDelta (headDelta x)) (delta (headDelta (tailDelta (headDelta (tailDelta d)))) (bind (bind ds (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
225 delta (headDelta (headDelta x)) (bind (delta (headDelta (tailDelta d)) (bind ds (tailDelta ∙ tailDelta))) tailDelta) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
226 delta (headDelta (headDelta x)) (bind (bind (delta d ds) tailDelta) tailDelta) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
227 mu (delta (headDelta x) (bind (delta d ds) tailDelta)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
228 mu (mu (delta x (delta d ds))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
229 (mu ∙ mu) (delta x (delta d ds))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
230 ∎
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
231 -}
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
232
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
233 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
234 (mu ∙ fmap mu) (delta x d)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
235 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
236 mu (fmap mu (delta x d))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
237 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
238 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
239 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
240 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
241 ≡⟨ 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
242 delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta)
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
243 ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
244 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
245 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
246 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
247 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
248 mu (mu (delta x d))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
249 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
250 (mu ∙ mu) (delta x d)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
251 ∎
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
252
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
253
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
254
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
255 {-
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
256 -- 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
257 -- 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
258 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
259 (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
260 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
261 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
262
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
263
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
264 -- 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
265 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
266 monad-law-2-2 d = refl
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
267
35
c5cdbedc68ad Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
268
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
269 -- 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
270 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
271 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
272
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
273
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
274 -- 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
275 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
276 (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
277 monad-law-4 f d = {!!}
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
278
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
279
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
280
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
281
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
282 -- 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
283 -- 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
284 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
285 (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
286 (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
287 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
288
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
289
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
290
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
291 -- 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
292 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
293 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
294 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
295
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
296
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
297 -- 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
298 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
299 (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
300 (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
301 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
302 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
303
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
304 -}