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ReDefine Delta used non-empty-list for infinite changes
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sat, 22 Nov 2014 12:29:32 +0900
parents bfb6be9a689d
children 46b15f368905
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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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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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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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 -> 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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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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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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dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
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37 -- TODO: mu
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
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38 -- TODO: bind
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41 eta : {l : Level} {A : Set l} -> A -> Delta A
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42 eta x = mono x
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742e62fc63e4 Define Monad-law 1-4
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43
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44 returnS : {l : Level} {A : Set l} -> A -> Delta A
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45 returnS x = mono x
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47 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A
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48 returnSS x y = deltaAppend (returnS x) (returnS y)
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49
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0bc402f970b3 Proof Monad-law 1
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51 -- Monad (Haskell)
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52 return : {l : Level} {A : Set l} -> A -> Delta A
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53 return = eta
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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55 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
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56 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
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57 (mono x) >>= f = f x
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58 (delta x d) >>= f = deltaAppend (headDelta (f x)) (d >>= (tailDelta ∙ f))
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6ce83b2c9e59 Proof Functor-laws
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62 -- proofs
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6ce83b2c9e59 Proof Functor-laws
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65 -- Functor-laws
6ce83b2c9e59 Proof Functor-laws
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66
6ce83b2c9e59 Proof Functor-laws
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67 -- Functor-law-1 : T(id) = id'
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68 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d
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69 functor-law-1 (mono x) = refl
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70 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)
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71
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72 -- Functor-law-2 : T(f . g) = T(f) . T(g)
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73 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
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74 (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
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75 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d
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76 functor-law-2 f g (mono x) = refl
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77 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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bfb6be9a689d Trying redefine monad-laws-1
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b9b26b470cc2 Add Comments
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81 -- Monad-laws (Category)
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82
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83 -- monad-law-1 : join . fmap join = join . join
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84 --monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
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85
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b7c4e6276bcf Proof Monad-law-2-1
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87 {-
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88 -- monad-law-2-2 : join . return = id
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89 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Delta A) -> (mu ∙ eta) s ≡ id s
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c5cdbedc68ad Proof Monad-law-2-2
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90 monad-law-2-2 (similar lx x ly y) = refl
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91
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92 -- monad-law-3 : return . f = fmap f . return
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93 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x
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169ec60fcd36 Proof Monad-law-4
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94 monad-law-3 f x = refl
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95
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96 -- monad-law-4 : join . fmap (fmap f) = fmap f . join
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97 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (s : Delta (Delta A)) ->
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169ec60fcd36 Proof Monad-law-4
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98 (mu ∙ fmap (fmap f)) s ≡ (fmap f ∙ mu) s
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99 monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl
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102 -- Monad-laws (Haskell)
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103 -- monad-law-h-1 : return a >>= k = k a
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104 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} ->
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105 (a : A) -> (k : A -> (Delta B)) ->
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106 (return a >>= k) ≡ (k a)
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107 monad-law-h-1 a k = begin
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108 return a >>= k
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109 ≡⟨ refl ⟩
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110 mu (fmap k (return a))
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111 ≡⟨ refl ⟩
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112 mu (return (k a))
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113 ≡⟨ refl ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
114 (mu ∙ return) (k a)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
115 ≡⟨ refl ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
116 (mu ∙ eta) (k a)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
117 ≡⟨ (monad-law-2-2 (k a)) ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
118 id (k a)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
119 ≡⟨ refl ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
120 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
121 ∎
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
122
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
123 -- 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
124 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
125 monad-law-h-2 (similar lx x ly y) = monad-law-2-1 (similar lx x ly y)
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
126
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
127 -- monad-law-h-3 : 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
128 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
129 (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
130 (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
131 monad-law-h-3 (similar lx x ly y) k h = begin
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
132 ((similar lx x ly y) >>= (\x -> (k x) >>= h))
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
133 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
134 mu (fmap (\x -> k x >>= h) (similar lx x ly y))
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
135 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
136 (mu ∙ fmap (\x -> k x >>= h)) (similar lx x ly y)
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
137 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
138 (mu ∙ fmap (\x -> mu (fmap h (k x)))) (similar lx x ly y)
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
139 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
140 (mu ∙ fmap (mu ∙ (\x -> fmap h (k x)))) (similar lx x ly y)
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
141 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
142 (mu ∙ (fmap mu ∙ (fmap (\x -> fmap h (k x))))) (similar lx x ly y)
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
143 ≡⟨ refl ⟩
42
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
144 (mu ∙ (fmap mu)) ((fmap (\x -> fmap h (k x))) (similar lx x ly y))
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
145 ≡⟨ monad-law-1 (((fmap (\x -> fmap h (k x))) (similar lx x ly y))) ⟩
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
146 (mu ∙ mu) ((fmap (\x -> fmap h (k x))) (similar lx x ly y))
42
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
147 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
148 (mu ∙ (mu ∙ (fmap (\x -> fmap h (k x))))) (similar lx x ly y)
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
149 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
150 (mu ∙ (mu ∙ (fmap ((fmap h) ∙ k)))) (similar lx x ly y)
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
151 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
152 (mu ∙ (mu ∙ ((fmap (fmap h)) ∙ (fmap k)))) (similar lx x ly y)
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
153 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
154 (mu ∙ (mu ∙ (fmap (fmap h)))) (fmap k (similar lx x ly y))
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
155 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
156 mu ((mu ∙ (fmap (fmap h))) (fmap k (similar lx x ly y)))
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
157 ≡⟨ cong (\fx -> mu fx) (monad-law-4 h (fmap k (similar lx x ly y))) ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
158 mu (fmap h (mu (similar lx (k x) ly (k y))))
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
159 ≡⟨ refl ⟩
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
160 (mu ∙ fmap h) (mu (fmap k (similar lx x ly y)))
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
161 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
162 mu (fmap h (mu (fmap k (similar lx x ly y))))
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
163 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
164 (mu (fmap k (similar lx x ly y))) >>= h
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
165 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
166 ((similar lx x ly y) >>= k) >>= h
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
167 ∎
57
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
168
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
169 -}