annotate agda/delta.agda @ 87:6789c65a75bc

Split functor-proofs into delta.functor
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Mon, 19 Jan 2015 11:00:34 +0900
parents fc5cd8c50312
children 526186c4f298
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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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3 open import nat
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6789c65a75bc Split functor-proofs into delta.functor
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4 open import laws
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e0ba1bf564dd Apply level to some functions
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5
e0ba1bf564dd Apply level to some functions
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6 open import Level
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742e62fc63e4 Define Monad-law 1-4
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7 open import Relation.Binary.PropositionalEquality
742e62fc63e4 Define Monad-law 1-4
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8 open ≡-Reasoning
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9
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10 module delta where
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6789c65a75bc Split functor-proofs into delta.functor
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13 data Delta {l : Level} (A : Set l) : (Set l) where
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14 mono : A -> Delta A
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15 delta : A -> Delta A -> Delta A
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17 deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A
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18 deltaAppend (mono x) d = delta x d
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19 deltaAppend (delta x d) ds = delta x (deltaAppend d ds)
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20
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21 headDelta : {l : Level} {A : Set l} -> Delta A -> A
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22 headDelta (mono x) = x
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23 headDelta (delta x _) = x
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25 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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26 tailDelta (mono x) = mono x
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27 tailDelta (delta _ d) = d
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29 n-tail : {l : Level} {A : Set l} -> Nat -> ((Delta A) -> (Delta A))
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30 n-tail O = id
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31 n-tail (S n) = tailDelta ∙ (n-tail n)
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33
6ce83b2c9e59 Proof Functor-laws
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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 x) = mono (f x)
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37 fmap f (delta x d) = delta (f x) (fmap f d)
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41 -- Monad (Category)
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42 eta : {l : Level} {A : Set l} -> A -> Delta A
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43 eta x = mono x
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742e62fc63e4 Define Monad-law 1-4
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44
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45 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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46 bind (mono x) f = f x
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47 bind (delta x d) f = delta (headDelta (f x)) (bind d (tailDelta ∙ f))
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49 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A
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50 mu d = bind d id
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51
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52 returnS : {l : Level} {A : Set l} -> A -> Delta A
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53 returnS x = mono x
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55 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A
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56 returnSS x y = deltaAppend (returnS x) (returnS y)
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0bc402f970b3 Proof Monad-law 1
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59 -- Monad (Haskell)
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60 return : {l : Level} {A : Set l} -> A -> Delta A
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61 return = eta
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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63 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
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64 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
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65 (mono x) >>= f = f x
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66 (delta x d) >>= f = delta (headDelta (f x)) (d >>= (tailDelta ∙ f))
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70 -- proofs
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72 -- sub-proofs
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73
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74 n-tail-plus : {l : Level} {A : Set l} -> (n : Nat) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n)
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75 n-tail-plus O = refl
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76 n-tail-plus (S n) = begin
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77 n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩
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78 (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩
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79 tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩
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80 n-tail (S (S n))
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81
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82
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83 n-tail-add : {l : Level} {A : Set l} {d : Delta A} -> (n m : Nat) -> (n-tail {l} {A} n) ∙ (n-tail m) ≡ n-tail (n + m)
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84 n-tail-add O m = refl
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85 n-tail-add (S n) O = begin
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86 n-tail (S n) ∙ n-tail O ≡⟨ refl ⟩
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4b16b485a4b2 Split nat definition
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87 n-tail (S n) ≡⟨ cong (\n -> n-tail n) (nat-add-right-zero (S n))⟩
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88 n-tail (S n + O)
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89
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90 n-tail-add {l} {A} {d} (S n) (S m) = begin
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91 n-tail (S n) ∙ n-tail (S m) ≡⟨ refl ⟩
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92 (tailDelta ∙ (n-tail n)) ∙ n-tail (S m) ≡⟨ refl ⟩
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93 tailDelta ∙ ((n-tail n) ∙ n-tail (S m)) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-add {l} {A} {d} n (S m)) ⟩
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94 tailDelta ∙ (n-tail (n + (S m))) ≡⟨ refl ⟩
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95 n-tail (S (n + S m)) ≡⟨ refl ⟩
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96 n-tail (S n + S m) ∎
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97
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98 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Nat) -> (x : A) ->
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99 (n-tail n) (mono x) ≡ (mono x)
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100 tail-delta-to-mono O x = refl
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101 tail-delta-to-mono (S n) x = begin
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102 n-tail (S n) (mono x) ≡⟨ refl ⟩
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103 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩
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104 tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩
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105 tailDelta (mono x) ≡⟨ refl ⟩
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106 mono x ∎
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107
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108 head-delta-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll}
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109 -> (f : A -> B) -> (d : Delta A) -> headDelta (fmap f d) ≡ f (headDelta d)
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110 head-delta-natural-transformation f (mono x) = refl
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111 head-delta-natural-transformation f (delta x d) = refl
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112
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113 n-tail-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll}
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114 -> (n : Nat) -> (f : A -> B) -> (d : Delta A) -> n-tail n (fmap f d) ≡ fmap f (n-tail n d)
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115 n-tail-natural-transformation O f d = refl
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parents: 78
diff changeset
116 n-tail-natural-transformation (S n) f (mono x) = begin
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
117 n-tail (S n) (fmap f (mono x)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
118 n-tail (S n) (mono (f x)) ≡⟨ tail-delta-to-mono (S n) (f x) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
119 (mono (f x)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
120 fmap f (mono x) ≡⟨ cong (\d -> fmap f d) (sym (tail-delta-to-mono (S n) x)) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
121 fmap f (n-tail (S n) (mono x)) ∎
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
122 n-tail-natural-transformation (S n) f (delta x d) = begin
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
123 n-tail (S n) (fmap f (delta x d)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
124 n-tail (S n) (delta (f x) (fmap f d)) ≡⟨ cong (\t -> t (delta (f x) (fmap f d))) (sym (n-tail-plus n)) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
125 ((n-tail n) ∙ tailDelta) (delta (f x) (fmap f d)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
126 n-tail n (fmap f d) ≡⟨ n-tail-natural-transformation n f d ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
127 fmap f (n-tail n d) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
128 fmap f (((n-tail n) ∙ tailDelta) (delta x d)) ≡⟨ cong (\t -> fmap f (t (delta x d))) (n-tail-plus n) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
129 fmap f (n-tail (S n) (delta x d)) ∎
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
130
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
131
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
132
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
133
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
134
87
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 80
diff changeset
135 {-
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
136 -- Monad-laws (Category)
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
137
76
c7076f9bbaed Refactors
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
138 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) ->
c7076f9bbaed Refactors
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
139 n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n))
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
140 monad-law-1-5 O O ds = refl
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
141 monad-law-1-5 O (S n) (mono ds) = begin
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
142 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
143 n-tail (S n) ds ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
144 bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
145 bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
146 bind (n-tail (S n) (mono ds)) (n-tail (O + S n))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
147
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
148 monad-law-1-5 O (S n) (delta d ds) = begin
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
149 n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
150 n-tail (S n) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
151 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
152 (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
153 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
154 bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S n))) (n-tail-plus n) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
155 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
156 bind (n-tail (S n) (delta d ds)) (n-tail (O + S n))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
157
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
158 monad-law-1-5 (S m) n (mono (mono x)) = begin
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
159 n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
160 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
161 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
162 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
163 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
164 bind (mono (mono x)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (mono x))) ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
165 bind (n-tail n (mono (mono x))) (n-tail (S m + n))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
166
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
167 monad-law-1-5 (S m) n (mono (delta x ds)) = begin
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
168 n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
169 n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
170 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
171 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
172 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
173 n-tail (m + n) ds ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
174 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
175 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
176 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
177 bind (mono (delta x ds)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (delta x ds))) ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
178 bind (n-tail n (mono (delta x ds))) (n-tail (S m + n))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
179
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
180 monad-law-1-5 (S m) O (delta d ds) = begin
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
181 n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
182 (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
183 delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
184 bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
185 bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> bind (n-tail O (delta d ds)) (n-tail n)) (nat-add-right-zero (S m)) ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
186 bind (n-tail O (delta d ds)) (n-tail (S m + O))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
187
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
188 monad-law-1-5 (S m) (S n) (delta d ds) = begin
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
189 n-tail (S n) (bind (delta d ds) (n-tail (S m))) ≡⟨ cong (\t -> t ((bind (delta d ds) (n-tail (S m))))) (sym (n-tail-plus n)) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
190 ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
191 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
192 (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
193 (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
194 bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> bind ((n-tail n) ds) (n-tail nm)) (sym (nat-right-increment (S m) n)) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
195 bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
196 bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
197 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
198
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
199
76
c7076f9bbaed Refactors
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
200 monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) ->
c7076f9bbaed Refactors
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
201 headDelta ((n-tail n) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd)))
74
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
202 monad-law-1-4 O O (mono dd) = refl
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
203 monad-law-1-4 O O (delta dd dd₁) = refl
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
204 monad-law-1-4 O (S n) (mono dd) = begin
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
205 headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
206 headDelta (n-tail (S n) dd) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
207 headDelta (n-tail (S n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S n) (headDelta de))) (sym (tail-delta-to-mono (S n) dd)) ⟩
74
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
208 headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
209 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd))))
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
210
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
211 monad-law-1-4 O (S n) (delta d ds) = begin
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
212 headDelta (n-tail (S n) (bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
213 headDelta (n-tail (S n) (bind (delta d ds) id)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
214 headDelta (n-tail (S n) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta d) (bind ds tailDelta)))) (sym (n-tail-plus n)) ⟩
74
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
215 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ refl ⟩
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
216 headDelta (n-tail n (bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
217 headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
218 headDelta (n-tail (S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
219 headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩
74
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
220 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds))))
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
221
74
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
222 monad-law-1-4 (S m) n (mono dd) = begin
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
223 headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
224 headDelta (n-tail n ((n-tail (S m)) dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
225 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
226 headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
227 headDelta (n-tail (S m + n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S m + n) (headDelta de))) (sym (tail-delta-to-mono n dd)) ⟩
74
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
228 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd))))
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
229
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
230 monad-law-1-4 (S m) O (delta d ds) = begin
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
231 headDelta (n-tail O (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
232 headDelta (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
74
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
233 headDelta (delta (headDelta ((n-tail (S m) d))) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
234 headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
235 headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
236 headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩
74
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
237 headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds))))
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
238
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
239 monad-law-1-4 (S m) (S n) (delta d ds) = begin
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
240 headDelta (n-tail (S n) (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
241 headDelta (n-tail (S n) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) (sym (n-tail-plus n)) ⟩
74
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
242 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
243 headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
244 headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
245 headDelta (n-tail ((S (S m) + n)) (headDelta (n-tail n ds))) ≡⟨ cong (\nm -> headDelta ((n-tail nm) (headDelta (n-tail n ds)))) (sym (nat-right-increment (S m) n)) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
246 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
247 headDelta (n-tail (S m + S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S m + S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩
74
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
248 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds))))
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
249
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
250
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
251 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
252 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
253 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
254
76
c7076f9bbaed Refactors
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
255 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) ->
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
256 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
257 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
258 monad-law-1-3 O (delta d ds) = begin
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
259 bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
260 bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
261 delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
262 delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩
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
263 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
264 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩
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
265 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
266
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
267 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
268 bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
269 bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
270 (n-tail (S n)) d ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
271 bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
272 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
273 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
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
274 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
275
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
276 monad-law-1-3 (S n) (mono (delta d ds)) = begin
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
277 bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
278 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
279 n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
280 n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta))) (sym (n-tail-plus n)) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
281 (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
282 n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
283 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
284 bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> (bind (t (delta d ds)) (n-tail (S n)))) (n-tail-plus n) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
285 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
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
286 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
287
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
288 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
289 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
290 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
291 bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
292 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
293 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) d)) de) (monad-law-1-3 (S (S n)) ds) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
294 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
295 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
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
296 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
297 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
298 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
299 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
300 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
301
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
302 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
303 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
304 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
305 delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
306 delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
307 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
308 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
309 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail n) (bind dd tailDelta))) de) (monad-law-1-3 (S (S n)) ds) ⟩
74
1f4ea5cb153d Prove monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
310 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de ( (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))))) (monad-law-1-4 (S O) n dd) ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
311 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
312 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
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
313 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 ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
314 delta (headDelta ((n-tail (S n)) (headDelta (((n-tail n) ∙ tailDelta) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta ((n-tail (S n)) (headDelta (t (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (n-tail-plus n) ⟩
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
315 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
316 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
317 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
318
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
319
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
320
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
321 -- 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
322 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
323 monad-law-1 (mono d) = refl
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
324 monad-law-1 (delta x d) = begin
80
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
325 (mu ∙ fmap mu) (delta x d) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
326 mu (fmap mu (delta x d)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
327 mu (delta (mu x) (fmap mu d)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
328 delta (headDelta (mu x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
329 delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
330 delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
331 mu (delta (headDelta x) (bind d tailDelta)) ≡⟨ refl ⟩
fc5cd8c50312 Adjust proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
332 mu (mu (delta x d)) ≡⟨ refl ⟩
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
333 (mu ∙ mu) (delta x d)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
334
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
335
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
336
78
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
337 monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (fmap eta d) (n-tail n)) ≡ d
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
338 monad-law-2-1 O (mono x) = refl
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
339 monad-law-2-1 O (delta x d) = begin
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
340 bind (fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
341 bind (delta (eta x) (fmap eta d)) id ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
342 delta (headDelta (eta x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
343 delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\de -> delta x de) (monad-law-2-1 (S O) d) ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
344 delta x d ∎
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
345 monad-law-2-1 (S n) (mono x) = begin
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
346 bind (fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
347 bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
348 n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
349 mono x ∎
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
350 monad-law-2-1 (S n) (delta x d) = begin
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
351 bind (fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
352 bind (delta (eta x) (fmap eta d)) (n-tail (S n)) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
353 delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
354 delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta (de)) (bind (fmap eta d) (n-tail (S (S n))))) (tail-delta-to-mono (S n) x) ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
355 delta (headDelta (eta x)) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
356 delta x (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S (S n)) d) ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
357 delta x d
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
358
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
359
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
360
78
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
361 -- monad-law-2 : join . fmap return = join . return = id
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
362 -- monad-law-2 join . fmap return = join . return
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
363 monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) ->
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
364 (mu ∙ fmap eta) d ≡ (mu ∙ eta) d
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
365 monad-law-2 (mono x) = refl
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
366 monad-law-2 (delta x d) = begin
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
367 (mu ∙ fmap eta) (delta x d) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
368 mu (fmap eta (delta x d)) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
369 mu (delta (mono x) (fmap eta d)) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
370 delta (headDelta (mono x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
371 delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
372 (delta x d) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
373 mu (mono (delta x d)) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
374 mu (eta (delta x d)) ≡⟨ refl ⟩
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
375 (mu ∙ eta) (delta x d)
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
376
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
377
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
378
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
379 -- monad-law-2' : join . return = id
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
380 monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (mu ∙ eta) d ≡ id d
f02391a7402f Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
381 monad-law-2' d = refl
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
382
35
c5cdbedc68ad Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
383
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
384 -- 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
385 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
386 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
387
79
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
388
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
389 monad-law-4-1 : {l ll : Level} {A : Set l} {B : Set ll} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) ->
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
390 bind (fmap (fmap f) ds) (n-tail n) ≡ fmap f (bind ds (n-tail n))
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
391 monad-law-4-1 O f (mono d) = refl
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
392 monad-law-4-1 O f (delta d ds) = begin
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
393 bind (fmap (fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
394 bind (delta (fmap f d) (fmap (fmap f) ds)) (n-tail O) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
395 delta (headDelta (fmap f d)) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta de (bind (fmap (fmap f) ds) tailDelta)) (head-delta-natural-transformation f d) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
396 delta (f (headDelta d)) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f (headDelta d)) de) (monad-law-4-1 (S O) f ds) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
397 delta (f (headDelta d)) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
398 fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
399 fmap f (bind (delta d ds) (n-tail O)) ∎
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
400 monad-law-4-1 (S n) f (mono d) = begin
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
401 bind (fmap (fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
402 bind (mono (fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
403 n-tail (S n) (fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
404 fmap f (n-tail (S n) d) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
405 fmap f (bind (mono d) (n-tail (S n)))
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
406
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
407 monad-law-4-1 (S n) f (delta d ds) = begin
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
408 bind (fmap (fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
409 delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
410 delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta de) (bind (fmap (fmap f) ds) (n-tail (S (S n))))) (n-tail-natural-transformation (S n) f d) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
411 delta (headDelta (fmap f ((n-tail (S n) d)))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de (bind (fmap (fmap f) ds) (n-tail (S (S n))))) (head-delta-natural-transformation f (n-tail (S n) d)) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
412 delta (f (headDelta (n-tail (S n) d))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (f (headDelta (n-tail (S n) d))) de) (monad-law-4-1 (S (S n)) f ds) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
413 delta (f (headDelta (n-tail (S n) d))) (fmap f (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
414 fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
415 fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
416 fmap f (bind (delta d ds) (n-tail (S n))) ∎
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
417
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
418
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
419 -- 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
420 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
421 (mu ∙ fmap (fmap f)) d ≡ (fmap f ∙ mu) d
79
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
422 monad-law-4 f (mono d) = refl
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
423 monad-law-4 f (delta (mono x) ds) = begin
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
424 (mu ∙ fmap (fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
425 mu ( fmap (fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
426 mu (delta (mono (f x)) (fmap (fmap f) ds)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
427 delta (headDelta (mono (f x))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
428 delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
429 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
430 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
431 fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
432 fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
433 (fmap f ∙ mu) (delta (mono x) ds) ∎
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
434 monad-law-4 f (delta (delta x d) ds) = begin
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
435 (mu ∙ fmap (fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
436 mu (fmap (fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
437 mu (delta (delta (f x) (fmap f d)) (fmap (fmap f) ds)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
438 delta (headDelta (delta (f x) (fmap f d))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
439 delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
440 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
441 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
442 fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
443 fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
444 (fmap f ∙ mu) (delta (delta x d) ds) ∎
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
445
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
446
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
447
79
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
448 {-
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
449 -- 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
450 -- 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
451 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
452 (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
453 (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
454 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
455
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
456
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
457
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
458 -- 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
459 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
460 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
461 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
462
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
463
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
464 -- 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
465 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
466 (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
467 (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
468 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
469 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
470
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
471 -}
87
6789c65a75bc Split functor-proofs into delta.functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 80
diff changeset
472 -}