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Trying redefine monad-laws-1
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Wed, 19 Nov 2014 21:09:45 +0900
parents 9c8c09334e32
children dfcd72dc697e
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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
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6 open ≡-Reasoning
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8 module delta where
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9
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10 DeltaLog : Set
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11 DeltaLog = List String
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12
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13 data Delta {l : Level} (A : Set l) : (Set (suc l)) where
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14 mono : DeltaLog -> A -> Delta A
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15 delta : DeltaLog -> A -> Delta A -> Delta A
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16
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17 logAppend : {l : Level} {A : Set l} -> DeltaLog -> Delta A -> Delta A
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18 logAppend l (mono lx x) = mono (l ++ lx) x
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19 logAppend l (delta lx x d) = delta (l ++ lx) x (logAppend l d)
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21 deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A
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22 deltaAppend (mono lx x) d = delta lx x d
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23 deltaAppend (delta lx x d) ds = delta lx x (deltaAppend d ds)
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25 headDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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26 headDelta (mono lx x) = mono lx x
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27 headDelta (delta lx x _) = mono lx x
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29 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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30 tailDelta (mono lx x) = mono lx x
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31 tailDelta (delta _ _ d) = d
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33
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34 -- Functor
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35 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B)
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36 fmap f (mono lx x) = mono lx (f x)
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37 fmap f (delta lx x d) = delta lx (f x) (fmap f d)
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40 {-# NO_TERMINATION_CHECK #-}
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41 -- Monad (Category)
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42 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A
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43 mu (mono ld d) = logAppend ld d
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44 mu (delta ld d ds) = deltaAppend (logAppend ld (headDelta d)) (mu (fmap tailDelta ds))
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45
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46 eta : {l : Level} {A : Set l} -> A -> Delta A
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47 eta x = mono [] x
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742e62fc63e4 Define Monad-law 1-4
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48
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49 returnS : {l : Level} {A : Set l} -> A -> Delta A
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50 returnS x = mono [[ (show x) ]] x
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52 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A
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53 returnSS x y = delta [[ (show x) ]] x (mono [[ (show y) ]] y)
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0bc402f970b3 Proof Monad-law 1
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56 -- Monad (Haskell)
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57 return : {l : Level} {A : Set l} -> A -> Delta A
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58 return = eta
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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60 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
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61 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
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62 x >>= f = mu (fmap f x)
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64
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6ce83b2c9e59 Proof Functor-laws
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66 -- proofs
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68 -- sub proofs
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69 twice-log-append : {l : Level} {A : Set l} -> (l : List String) -> (ll : List String) -> (d : Delta A) ->
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70 logAppend l (logAppend ll d) ≡ logAppend (l ++ ll) d
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71 twice-log-append l ll (mono lx x) = begin
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72 mono (l ++ (ll ++ lx)) x
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73 ≡⟨ cong (\l -> mono l x) (list-associative l ll lx) ⟩
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74 mono (l ++ ll ++ lx) x
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75 ∎
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76 twice-log-append l ll (delta lx x d) = begin
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77 delta (l ++ (ll ++ lx)) x (logAppend l (logAppend ll d))
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78 ≡⟨ cong (\lx -> delta lx x (logAppend l (logAppend ll d))) (list-associative l ll lx) ⟩
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79 delta (l ++ ll ++ lx) x (logAppend l (logAppend ll d))
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80 ≡⟨ cong (delta (l ++ ll ++ lx) x) (twice-log-append l ll d) ⟩
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81 delta (l ++ ll ++ lx) x (logAppend (l ++ ll) d)
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82 ∎
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83
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84
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85 -- Functor-laws
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86
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87 -- Functor-law-1 : T(id) = id'
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88 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d
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89 functor-law-1 (mono lx x) = refl
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90 functor-law-1 (delta lx x d) = cong (delta lx x) (functor-law-1 d)
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91
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92 -- Functor-law-2 : T(f . g) = T(f) . T(g)
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93 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
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94 (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
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95 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d
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96 functor-law-2 f g (mono lx x) = refl
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97 functor-law-2 f g (delta lx x d) = cong (delta lx (f (g x))) (functor-law-2 f g d)
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98
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100
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101 -- Monad-laws (Category)
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102
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103 -- monad-law-1 : join . fmap join = join . join
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104 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
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105 monad-law-1 (mono lx (mono llx (mono lllx x))) = begin
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106 mono (lx ++ (llx ++ lllx)) x
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107 ≡⟨ cong (\l -> mono l x) (list-associative lx llx lllx) ⟩
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108 mono (lx ++ llx ++ lllx) x
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109 ∎
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110 monad-law-1 (mono lx (mono llx (delta lllx x d))) = begin
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111 delta (lx ++ (llx ++ lllx)) x (logAppend lx (logAppend llx d))
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112 ≡⟨ cong (\l -> delta l x (logAppend lx (logAppend llx d))) (list-associative lx llx lllx) ⟩
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diff changeset
113 delta (lx ++ llx ++ lllx) x (logAppend lx (logAppend llx d))
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
114 ≡⟨ cong (\d -> delta (lx ++ llx ++ lllx) x d) (twice-log-append lx llx d) ⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
115 delta (lx ++ llx ++ lllx) x (logAppend (lx ++ llx) d)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
116 ∎
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
117 monad-law-1 (mono lx (delta ld (mono x x₁) (mono x₂ (mono x₃ x₄)))) = begin
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
118 delta (lx ++ (ld ++ x)) x₁ (mono (lx ++ (x₂ ++ x₃)) x₄)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
119 ≡⟨ cong (\l -> delta l x₁(mono (lx ++ (x₂ ++ x₃)) x₄)) (list-associative lx ld x) ⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
120 delta (lx ++ ld ++ x) x₁ (mono (lx ++ (x₂ ++ x₃)) x₄)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
121 ≡⟨ cong (\l -> delta (lx ++ ld ++ x) x₁ (mono l x₄)) (list-associative lx x₂ x₃) ⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
122 delta (lx ++ ld ++ x) x₁ (mono (lx ++ x₂ ++ x₃) x₄)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
123 ∎
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
124 monad-law-1 (mono lx (delta ld (mono x x₁) (mono x₂ (delta x₃ x₄ ds)))) = begin
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
125 delta (lx ++ (ld ++ x)) x₁ (logAppend lx (logAppend x₂ ds))
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
126 ≡⟨ cong (\l -> delta l x₁ (logAppend lx (logAppend x₂ ds))) (list-associative lx ld x) ⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
127 delta (lx ++ ld ++ x) x₁ (logAppend lx (logAppend x₂ ds))
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
128 ≡⟨ cong (\d -> delta (lx ++ ld ++ x) x₁ d) (twice-log-append lx x₂ ds) ⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
129 delta (lx ++ ld ++ x) x₁ (logAppend (lx ++ x₂) ds)
29
e0ba1bf564dd Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
130 ∎
56
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
131 monad-law-1 (mono lx (delta ld (delta x x₁ (mono x₂ x₃)) (mono x₄ (mono x₅ x₆)))) = begin
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
132 delta (lx ++ (ld ++ x)) x₁ (mono (lx ++ (x₄ ++ x₅)) x₆)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
133 ≡⟨ cong (\l -> delta l x₁ (mono (lx ++ (x₄ ++ x₅)) x₆)) (list-associative lx ld x ) ⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
134 delta (lx ++ ld ++ x) x₁ (mono (lx ++ (x₄ ++ x₅)) x₆)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
135 ≡⟨ cong (\l -> delta (lx ++ ld ++ x) x₁ (mono l x₆)) (list-associative lx x₄ x₅)⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
136 delta (lx ++ ld ++ x) x₁ (mono (lx ++ x₄ ++ x₅) x₆)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
137 ∎
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
138 monad-law-1 (mono lx (delta ld (delta x x₁ (mono x₂ x₃)) (mono x₄ (delta x₅ x₆ ds)))) = begin
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
139 delta (lx ++ (ld ++ x)) x₁ (logAppend lx (logAppend x₄ ds))
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
140 ≡⟨ cong (\l -> delta l x₁(logAppend lx (logAppend x₄ ds))) (list-associative lx ld x ) ⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
141 delta (lx ++ ld ++ x) x₁ (logAppend lx (logAppend x₄ ds))
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
142 ≡⟨ cong (\d -> delta (lx ++ ld ++ x) x₁ d) (twice-log-append lx x₄ ds ) ⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
143 delta (lx ++ ld ++ x) x₁ (logAppend (lx ++ x₄) ds)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
144 ∎
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
145
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
146 monad-law-1 (mono lx (delta ld (delta x x₁ (delta ly y (mono x₂ x₃))) (mono x₄ (mono x₅ x₆)))) = begin
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
147 delta (lx ++ (ld ++ x)) x₁ (mono (lx ++ (x₄ ++ x₅)) x₆)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
148 ≡⟨ {!!} ⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
149 delta (lx ++ ld ++ x) x₁ (mono (lx ++ x₄ ++ x₅) x₆)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
150 ∎
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
151 monad-law-1 (mono lx (delta ld (delta x x₁ (delta ly y (mono x₂ x₃))) (mono x₄ (delta x₅ x₆ ds)))) = begin
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
152 delta (lx ++ (ld ++ x)) x₁ (logAppend lx (logAppend x₄ ds))
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
153 ≡⟨ {!!} ⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
154 delta (lx ++ ld ++ x) x₁ (logAppend (lx ++ x₄) ds)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
155 ∎
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
156 monad-law-1 (mono lx (delta ld (delta x x₁ (delta ly y (delta x₂ x₃ d))) (mono x₄ (mono x₅ x₆)))) = begin
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
157 delta (lx ++ (ld ++ x)) x₁ (mono (lx ++ (x₄ ++ x₅)) x₆)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
158 ≡⟨ {!!} ⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
159 delta (lx ++ ld ++ x) x₁ (mono (lx ++ x₄ ++ x₅) x₆)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
160 ∎
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
161 monad-law-1 (mono lx (delta ld (delta x x₁ (delta ly y (delta x₂ x₃ d))) (mono x₄ (delta x₅ x₆ ds)))) = begin
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
162 delta (lx ++ (ld ++ x)) x₁ (logAppend lx (logAppend x₄ ds))
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
163 ≡⟨ {!!} ⟩
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
164 delta (lx ++ ld ++ x) x₁ (logAppend (lx ++ x₄) ds)
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
165 ∎
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
166
29
e0ba1bf564dd Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
167
34
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
168
56
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
169
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
170 monad-law-1 (mono lx (delta ld d (delta x ds ds₁))) = {!!}
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
171
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
172
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
173
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
174 monad-law-1 (delta lx x d) = {!!}
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
175
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
176 {-
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
177 -- monad-law-2 : join . fmap return = join . return = id
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
178 -- monad-law-2-1 join . fmap return = join . return
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
179 monad-law-2-1 : {l : Level} {A : Set l} -> (s : Delta A) ->
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
180 (mu ∙ fmap eta) s ≡ (mu ∙ eta) s
34
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
181 monad-law-2-1 (similar lx x ly y) = begin
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
182 similar (lx ++ []) x (ly ++ []) y
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
183 ≡⟨ cong (\left-list -> similar left-list x (ly ++ []) y) (empty-append lx)⟩
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
184 similar lx x (ly ++ []) y
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
185 ≡⟨ cong (\right-list -> similar lx x right-list y) (empty-append ly) ⟩
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
186 similar lx x ly y
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
187 ∎
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
188
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
189 -- 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
190 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
191 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
192
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
193 -- 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
194 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
195 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
196
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
197 -- 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
198 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
199 (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
200 monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
201
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
202
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
203 -- 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
204 -- 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
205 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
206 (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
207 (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
208 monad-law-h-1 a k = begin
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
209 return a >>= k
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
210 ≡⟨ 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
211 mu (fmap k (return a))
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
212 ≡⟨ 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
213 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
214 ≡⟨ 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
215 (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
216 ≡⟨ 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
217 (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
218 ≡⟨ (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
219 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
220 ≡⟨ 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
221 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
222 ∎
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
223
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
224 -- 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
225 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
226 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
227
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
228 -- 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
229 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
230 (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
231 (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
232 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
233 ((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
234 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
235 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
236 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
237 (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
238 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
239 (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
240 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
241 (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
242 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
243 (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
244 ≡⟨ refl ⟩
42
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
245 (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
246 ≡⟨ 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
247 (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
248 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
249 (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
250 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
251 (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
252 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
253 (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
254 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
255 (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
256 ≡⟨ refl ⟩
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
257 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
258 ≡⟨ 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
259 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
260 ≡⟨ 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
261 (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
262 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
263 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
264 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
265 (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
266 ≡⟨ refl ⟩
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
267 ((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
268 ∎
54
9bb7c9bee94f Trying redefine delta for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
269 -}