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Change headDelta definition. return non-delta value
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Thu, 27 Nov 2014 19:12:44 +0900
parents f9c9207c40b7
children 18a20a14c4b2
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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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diff changeset
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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9
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10
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90b171e3a73e Rename to Delta from Similar
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11 data Delta {l : Level} (A : Set l) : (Set (suc l)) where
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dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
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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
9bb7c9bee94f Trying redefine delta for infinite changes
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19 headDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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20 headDelta (mono x) = mono x
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21 headDelta (delta x _) = mono 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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38
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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a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
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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 = deltaAppend (headDelta (f x)) (bind d (tailDelta ∙ f))
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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44 mu d = bind d id
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46b15f368905 Define bind and mu for Infinite Delta
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47 returnS : {l : Level} {A : Set l} -> A -> Delta A
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48 returnS x = mono x
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49
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50 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A
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51 returnSS x y = deltaAppend (returnS x) (returnS y)
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52
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0bc402f970b3 Proof Monad-law 1
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53
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54 -- Monad (Haskell)
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55 return : {l : Level} {A : Set l} -> A -> Delta A
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56 return = eta
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57
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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58 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
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59 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
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60 (mono x) >>= f = f x
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61 (delta x d) >>= f = deltaAppend (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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63
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64
6ce83b2c9e59 Proof Functor-laws
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65 -- proofs
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67 -- sub proofs
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68
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69 head-delta-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll} ->
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70 (f : A -> B) (d : Delta A) -> (headDelta (fmap f d)) ≡ fmap f (headDelta d)
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71 head-delta-natural-transformation f (mono x) = refl
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72 head-delta-natural-transformation f (delta x d) = refl
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73
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74 tail-delta-natural-transfomation : {l ll : Level} {A : Set l} {B : Set ll} ->
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75 (f : A -> B) (d : Delta A) -> (tailDelta (fmap f d)) ≡ fmap f (tailDelta d)
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76 tail-delta-natural-transfomation f (mono x) = refl
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77 tail-delta-natural-transfomation f (delta x d) = refl
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78
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79 delta-append-natural-transfomation : {l ll : Level} {A : Set l} {B : Set ll} ->
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80 (f : A -> B) (d : Delta A) (dd : Delta A) ->
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81 deltaAppend (fmap f d) (fmap f dd) ≡ fmap f (deltaAppend d dd)
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82 delta-append-natural-transfomation f (mono x) dd = refl
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83 delta-append-natural-transfomation f (delta x d) dd = begin
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84 deltaAppend (fmap f (delta x d)) (fmap f dd)
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85 ≡⟨ refl ⟩
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86 deltaAppend (delta (f x) (fmap f d)) (fmap f dd)
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87 ≡⟨ refl ⟩
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88 delta (f x) (deltaAppend (fmap f d) (fmap f dd))
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89 ≡⟨ cong (\d -> delta (f x) d) (delta-append-natural-transfomation f d dd) ⟩
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90 delta (f x) (fmap f (deltaAppend d dd))
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91 ≡⟨ refl ⟩
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92 fmap f (deltaAppend (delta x d) dd)
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93 ∎
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6ce83b2c9e59 Proof Functor-laws
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94 -- Functor-laws
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95
6ce83b2c9e59 Proof Functor-laws
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96 -- Functor-law-1 : T(id) = id'
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97 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d
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98 functor-law-1 (mono x) = refl
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99 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)
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100
6ce83b2c9e59 Proof Functor-laws
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101 -- Functor-law-2 : T(f . g) = T(f) . T(g)
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102 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
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103 (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
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104 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d
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105 functor-law-2 f g (mono x) = refl
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106 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)
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6ce83b2c9e59 Proof Functor-laws
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107
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295e8ed39c0c Change headDelta definition. return non-delta value
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108 {-
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b9b26b470cc2 Add Comments
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109 -- Monad-laws (Category)
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474ed34e4f02 proving monad-law-1 ...
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110
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b9b26b470cc2 Add Comments
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111 -- monad-law-1 : join . fmap join = join . join
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46b15f368905 Define bind and mu for Infinite Delta
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112 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
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295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
113 monad-law-1 d = ?
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
114
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
115
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
116 -- monad-law-2-2 : join . return = id
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
117 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Delta A) -> (mu ∙ eta) s ≡ id s
35
c5cdbedc68ad Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
118 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
119
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
120 -- 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
121 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
122 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
123
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
124 -- 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
125 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
126 (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
127 monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl
69
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
128 -}
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
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-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
131 -- 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
132 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
133 (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
134 (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
135 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
136
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
137
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
138
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 -- 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
140 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
141 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
142 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
143
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
144
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
145 -- 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
146 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
147 (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
148 (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
149 monad-law-h-3 (mono x) k h = refl
69
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
150 monad-law-h-3 (delta x (mono xx)) k h = begin
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
151 delta x (mono xx) >>= (\x → k x >>= h)
42
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
152 ≡⟨ refl ⟩
69
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
153 deltaAppend (headDelta ((\x -> k x >>= h) x)) ((mono xx) >>= (tailDelta ∙ ((\x → k x >>= h))))
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
154 ≡⟨ refl ⟩
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
155 deltaAppend (headDelta ((\x -> k x >>= h) x)) ((tailDelta ∙ (\x → k x >>= h)) xx)
42
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
156 ≡⟨ refl ⟩
69
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
157 deltaAppend (headDelta (k x >>= h)) (tailDelta (k xx >>= h))
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
158 ≡⟨ {!!} ⟩ -- ?
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
159 deltaAppend (headDelta (k x)) (tailDelta (k xx)) >>= h
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
160 ≡⟨ refl ⟩
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
161 (delta x (mono xx) >>= k) >>= h
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
162 ∎
69
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
163 monad-law-h-3 (delta x (delta xx d)) k h = {!!}
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
164