Mercurial > hg > Members > atton > delta_monad
annotate agda/delta.agda @ 64:15eec529dfc4
Trying prove monad-law-1 ...
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Wed, 26 Nov 2014 16:17:53 +0900 |
parents | 474ed34e4f02 |
children | 6d0193011f89 |
rev | line source |
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Define Similar in Agda
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1 open import list |
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2 open import basic |
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3 |
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4 open import Level |
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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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7 |
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8 module delta where |
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9 |
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11 data Delta {l : Level} (A : Set l) : (Set (suc l)) where |
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12 mono : A -> Delta A |
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13 delta : A -> Delta A -> Delta A |
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14 |
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15 deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A |
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16 deltaAppend (mono x) d = delta x d |
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17 deltaAppend (delta x d) ds = delta x (deltaAppend d ds) |
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18 |
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19 headDelta : {l : Level} {A : Set l} -> Delta A -> Delta A |
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20 headDelta (mono x) = mono x |
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21 headDelta (delta x _) = mono x |
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22 |
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23 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A |
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24 tailDelta (mono x) = mono x |
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25 tailDelta (delta _ d) = d |
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26 |
38
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Proof Functor-laws
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27 |
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Proof Functor-laws
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28 -- Functor |
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29 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B) |
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30 fmap f (mono x) = mono (f x) |
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31 fmap f (delta x d) = delta (f x) (fmap f d) |
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32 |
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33 |
38
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34 |
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35 -- Monad (Category) |
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36 eta : {l : Level} {A : Set l} -> A -> Delta A |
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37 eta x = mono x |
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38 |
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39 bind : {l ll : Level} {A : Set l} {B : Set ll} -> (Delta A) -> (A -> Delta B) -> Delta B |
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40 bind (mono x) f = f x |
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41 bind (delta x d) f = deltaAppend (headDelta (f x)) (bind d (tailDelta ∙ f)) |
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42 |
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43 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A |
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44 mu d = bind d id |
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45 |
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46 |
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47 returnS : {l : Level} {A : Set l} -> A -> Delta A |
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48 returnS x = mono x |
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49 |
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50 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A |
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51 returnSS x y = deltaAppend (returnS x) (returnS y) |
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52 |
33 | 53 |
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54 -- Monad (Haskell) |
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55 return : {l : Level} {A : Set l} -> A -> Delta A |
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56 return = eta |
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57 |
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Proof monad-law-h-2, trying monad-law-h-3
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58 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} -> |
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59 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B) |
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60 (mono x) >>= f = f x |
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61 (delta x d) >>= f = deltaAppend (headDelta (f x)) (d >>= (tailDelta ∙ f)) |
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62 |
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63 |
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64 |
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65 -- proofs |
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66 |
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67 -- sub proofs |
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68 |
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69 head-delta-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll} -> |
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70 (f : A -> B) (d : Delta A) -> (headDelta (fmap f d)) ≡ fmap f (headDelta d) |
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71 head-delta-natural-transformation f (mono x) = refl |
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72 head-delta-natural-transformation f (delta x d) = refl |
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73 |
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74 tail-delta-natural-transfomation : {l ll : Level} {A : Set l} {B : Set ll} -> |
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75 (f : A -> B) (d : Delta A) -> (tailDelta (fmap f d)) ≡ fmap f (tailDelta d) |
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76 tail-delta-natural-transfomation f (mono x) = refl |
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77 tail-delta-natural-transfomation f (delta x d) = refl |
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78 |
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79 delta-append-natural-transfomation : {l ll : Level} {A : Set l} {B : Set ll} -> |
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80 (f : A -> B) (d : Delta A) (dd : Delta A) -> |
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81 deltaAppend (fmap f d) (fmap f dd) ≡ fmap f (deltaAppend d dd) |
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82 delta-append-natural-transfomation f (mono x) dd = refl |
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83 delta-append-natural-transfomation f (delta x d) dd = begin |
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84 deltaAppend (fmap f (delta x d)) (fmap f dd) |
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85 ≡⟨ refl ⟩ |
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86 deltaAppend (delta (f x) (fmap f d)) (fmap f dd) |
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87 ≡⟨ refl ⟩ |
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88 delta (f x) (deltaAppend (fmap f d) (fmap f dd)) |
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89 ≡⟨ cong (\d -> delta (f x) d) (delta-append-natural-transfomation f d dd) ⟩ |
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90 delta (f x) (fmap f (deltaAppend d dd)) |
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91 ≡⟨ refl ⟩ |
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92 fmap f (deltaAppend (delta x d) dd) |
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93 ∎ |
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94 {- |
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95 |
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96 mu-head-delta : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> mu (headDelta d) ≡ headDelta (mu d) |
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97 mu-head-delta (mono (mono x)) = refl |
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98 mu-head-delta (mono (delta x (mono xx))) = begin |
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99 mu (headDelta (mono (delta x (mono xx)))) |
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100 ≡⟨ refl ⟩ |
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101 bind (headDelta (mono (delta x (mono xx)))) id |
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Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
102 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
103 bind (delta x (mono xx)) return |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
104 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
105 deltaAppend (headDelta (return x)) (bind (mono xx) (tailDelta ∙ return)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
106 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
107 deltaAppend (headDelta (return x)) ((tailDelta ∙ return) xx) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
108 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
109 deltaAppend (headDelta (mono x)) (tailDelta (mono xx)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
110 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
111 deltaAppend (mono x) (mono xx) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
112 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
113 delta x (mono xx) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
114 ≡⟨ {!!} ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
115 headDelta (delta x (mono xx)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
116 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
117 headDelta (bind (mono (delta x (mono xx))) id) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
118 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
119 headDelta (mu (mono (delta x (mono xx)))) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
120 ∎ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
121 mu-head-delta (mono (delta x (delta x₁ d))) = {!!} |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
122 mu-head-delta (delta d dd) = {!!} |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
123 -} |
38
6ce83b2c9e59
Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
36
diff
changeset
|
124 -- Functor-laws |
6ce83b2c9e59
Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
36
diff
changeset
|
125 |
6ce83b2c9e59
Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
36
diff
changeset
|
126 -- Functor-law-1 : T(id) = id' |
55
9c8c09334e32
Redefine Delta for infinite changes in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
54
diff
changeset
|
127 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d |
57
dfcd72dc697e
ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
56
diff
changeset
|
128 functor-law-1 (mono x) = refl |
dfcd72dc697e
ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
56
diff
changeset
|
129 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d) |
38
6ce83b2c9e59
Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
36
diff
changeset
|
130 |
6ce83b2c9e59
Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
36
diff
changeset
|
131 -- Functor-law-2 : T(f . g) = T(f) . T(g) |
6ce83b2c9e59
Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
36
diff
changeset
|
132 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> |
55
9c8c09334e32
Redefine Delta for infinite changes in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
54
diff
changeset
|
133 (f : B -> C) -> (g : A -> B) -> (d : Delta A) -> |
9c8c09334e32
Redefine Delta for infinite changes in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
54
diff
changeset
|
134 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d |
57
dfcd72dc697e
ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
56
diff
changeset
|
135 functor-law-2 f g (mono x) = refl |
dfcd72dc697e
ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
56
diff
changeset
|
136 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d) |
38
6ce83b2c9e59
Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
36
diff
changeset
|
137 |
39 | 138 -- Monad-laws (Category) |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
139 |
64
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
140 monad-law-1-5 : {l : Level} {A : Set l} -> (ds : Delta (Delta A)) -> |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
141 (tailDelta ∙ tailDelta) (bind ds tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
142 ≡ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
143 bind ((tailDelta ∙ tailDelta) ds) ((tailDelta ∙ tailDelta) ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
144 monad-law-1-5 (mono ds) = refl |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
145 monad-law-1-5 (delta (mono x) ds) = {!!} |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
146 monad-law-1-5 (delta (delta x d) ds) = {!!} |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
147 |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
148 monad-law-1-4 : {l : Level} {A : Set l} -> (x : A) -> (ds : Delta (Delta (Delta A))) -> |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
149 delta x (bind (fmap mu ds) (tailDelta ∙ (tailDelta ∙ tailDelta))) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
150 ≡ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
151 bind (delta (mono x) (bind ds ((tailDelta ∙ tailDelta ) ∙ tailDelta))) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
152 monad-law-1-4 x (mono (mono ds)) = refl |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
153 monad-law-1-4 x (mono (delta (mono xx) ds)) = begin |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
154 delta x (bind (fmap mu (mono (delta (mono xx) ds))) (tailDelta ∙ (tailDelta ∙ tailDelta))) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
155 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
156 delta x (bind (mono (mu (delta (mono xx) ds))) (tailDelta ∙ (tailDelta ∙ tailDelta))) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
157 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
158 delta x (bind (mono (bind (delta (mono xx) ds) id)) (tailDelta ∙ (tailDelta ∙ tailDelta))) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
159 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
160 delta x (bind (mono (deltaAppend (headDelta (mono xx)) (bind ds tailDelta))) (tailDelta ∙ (tailDelta ∙ tailDelta))) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
161 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
162 delta x (bind (mono (deltaAppend (mono xx) (bind ds tailDelta))) (tailDelta ∙ (tailDelta ∙ tailDelta))) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
163 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
164 delta x (bind (mono (delta xx (bind ds tailDelta))) (tailDelta ∙ (tailDelta ∙ tailDelta))) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
165 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
166 delta x ((tailDelta ∙ (tailDelta ∙ tailDelta)) (delta xx (bind ds tailDelta))) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
167 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
168 delta x ((tailDelta ∙ tailDelta) (bind ds tailDelta)) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
169 ≡⟨ cong (\d -> delta x d) (monad-law-1-5 ds) ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
170 delta x (bind ((tailDelta ∙ tailDelta) ds) ((tailDelta ∙ tailDelta) ∙ tailDelta)) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
171 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
172 deltaAppend (mono x) (bind ((tailDelta ∙ tailDelta) ds) ((tailDelta ∙ tailDelta) ∙ tailDelta)) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
173 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
174 deltaAppend (headDelta ((tailDelta ∙ tailDelta) (mono x))) (bind ((tailDelta ∙ tailDelta) ds) ((tailDelta ∙ tailDelta) ∙ tailDelta)) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
175 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
176 bind (delta (mono x) ((tailDelta ∙ tailDelta) ds)) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
177 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
178 bind (delta (mono x) (bind (mono (delta (mono xx) ds)) ((tailDelta ∙ tailDelta) ∙ tailDelta))) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
179 ∎ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
180 monad-law-1-4 x (mono (delta (delta x₁ ds) ds₁)) = {!!} |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
181 monad-law-1-4 x (delta ds ds₁) = {!!} |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
182 |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
183 monad-law-1-3 : {l : Level} {A : Set l} -> (ds : Delta (Delta A)) -> |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
184 tailDelta (bind ds tailDelta) ≡ bind (tailDelta ds) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
185 monad-law-1-3 (mono ds) = refl |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
186 monad-law-1-3 (delta (mono x) ds) = refl |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
187 monad-law-1-3 (delta (delta x (mono x₁)) ds) = refl |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
188 monad-law-1-3 (delta (delta x (delta x₁ d)) ds) = refl |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
189 |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
190 monad-law-1-sub-sub : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> |
64
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
191 bind (fmap mu d) (tailDelta ∙ tailDelta) ≡ bind (bind d (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
192 monad-law-1-sub-sub (mono (mono d)) = refl |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
193 monad-law-1-sub-sub (mono (delta (mono x) ds)) = begin |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
194 bind (fmap mu (mono (delta (mono x) ds))) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
195 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
196 bind (mono (mu (delta (mono x) ds))) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
197 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
198 bind (mono (bind (delta (mono x) ds) id)) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
199 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
200 bind (mono (deltaAppend (headDelta (mono x)) (bind ds tailDelta))) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
201 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
202 bind (mono (deltaAppend (mono x) (bind ds tailDelta))) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
203 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
204 bind (mono (delta x (bind ds tailDelta))) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
205 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
206 (tailDelta ∙ tailDelta) (delta x (bind ds tailDelta)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
207 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
208 tailDelta (bind ds tailDelta) |
64
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
209 ≡⟨ monad-law-1-3 ds ⟩ |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
210 bind (tailDelta ds) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
211 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
212 bind ((tailDelta ∙ tailDelta) (delta (mono x) ds)) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
213 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
214 bind (bind (mono (delta (mono x) ds)) (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
215 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
216 bind (bind (headDelta (tailDelta (mono (delta (mono x) ds)))) (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
217 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
218 bind (bind (mono (delta (mono x) ds)) (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
219 ∎ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
220 monad-law-1-sub-sub (mono (delta (delta x (mono x₁)) ds)) = begin |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
221 bind (fmap mu (mono (delta (delta x (mono x₁)) ds))) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
222 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
223 bind (mono (mu (delta (delta x (mono x₁)) ds))) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
224 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
225 (tailDelta ∙ tailDelta) (mu (delta (delta x (mono x₁)) ds)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
226 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
227 (tailDelta ∙ tailDelta) (bind (delta (delta x (mono x₁)) ds) id) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
228 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
229 (tailDelta ∙ tailDelta) (deltaAppend (headDelta (delta x (mono x₁))) (bind ds (tailDelta ∙ id))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
230 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
231 (tailDelta ∙ tailDelta) (deltaAppend (mono x) (bind ds (tailDelta ∙ id))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
232 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
233 (tailDelta ∙ tailDelta) (delta x (bind ds (tailDelta ∙ id))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
234 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
235 tailDelta (bind ds (tailDelta ∙ id)) |
64
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
236 ≡⟨ monad-law-1-3 ds ⟩ |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
237 bind (tailDelta ds) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
238 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
239 bind ((tailDelta ∙ tailDelta) (delta (delta x (mono x₁)) ds)) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
240 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
241 bind (bind (mono (delta (delta x (mono x₁)) ds)) (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
242 ∎ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
243 monad-law-1-sub-sub (mono (delta (delta x (delta xx d)) ds)) = begin |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
244 bind (fmap mu (mono (delta (delta x (delta xx d)) ds))) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
245 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
246 bind (mono (mu (delta (delta x (delta xx d)) ds))) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
247 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
248 (tailDelta ∙ tailDelta) (mu (delta (delta x (delta xx d)) ds)) |
64
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
249 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
250 (tailDelta ∙ tailDelta) (bind (delta (delta x (delta xx d)) ds) id) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
251 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
252 (tailDelta ∙ tailDelta) (deltaAppend (headDelta (delta x (delta xx d))) (bind ds tailDelta)) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
253 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
254 (tailDelta ∙ tailDelta) (deltaAppend (mono x) (bind ds tailDelta)) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
255 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
256 (tailDelta ∙ tailDelta) (delta x (bind ds tailDelta)) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
257 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
258 tailDelta (bind ds tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
259 ≡⟨ monad-law-1-3 ds ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
260 bind (tailDelta ds) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
261 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
262 bind ((tailDelta ∙ tailDelta) (delta (delta x (delta xx d)) ds)) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
263 ≡⟨ refl ⟩ |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
264 bind (bind (mono (delta (delta x (delta xx d)) ds)) (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
265 ∎ |
64
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
266 monad-law-1-sub-sub (delta (mono (mono x)) ds) = begin |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
267 bind (fmap mu (delta (mono (mono x)) ds)) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
268 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
269 bind (delta (mu (mono (mono x))) (fmap mu ds)) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
270 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
271 bind (delta (mono x) (fmap mu ds)) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
272 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
273 deltaAppend (headDelta ((tailDelta ∙ tailDelta) (mono x))) (bind (fmap mu ds) (tailDelta ∙ (tailDelta ∙ tailDelta))) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
274 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
275 deltaAppend ((tailDelta ∙ tailDelta) (mono x)) (bind (fmap mu ds) (tailDelta ∙ (tailDelta ∙ tailDelta))) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
276 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
277 deltaAppend (mono x) (bind (fmap mu ds) (tailDelta ∙ (tailDelta ∙ tailDelta))) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
278 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
279 delta x (bind (fmap mu ds) (tailDelta ∙ (tailDelta ∙ tailDelta))) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
280 ≡⟨ monad-law-1-4 x ds ⟩ -- ? |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
281 bind (delta (mono x) (bind ds ((tailDelta ∙ tailDelta ) ∙ tailDelta))) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
282 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
283 bind (delta (tailDelta (mono x)) (bind ds (tailDelta ∙ (tailDelta ∙ tailDelta)))) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
284 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
285 bind (deltaAppend (mono (mono x)) (bind ds (tailDelta ∙ (tailDelta ∙ tailDelta)))) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
286 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
287 bind (deltaAppend (headDelta ((tailDelta ∙ tailDelta) (mono (mono x)))) (bind ds (tailDelta ∙ (tailDelta ∙ tailDelta)))) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
288 ≡⟨ refl ⟩ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
289 bind (bind (delta (mono (mono x)) ds) (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta) |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
290 ∎ |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
291 monad-law-1-sub-sub (delta (mono (delta x d)) ds) = {!!} |
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
292 monad-law-1-sub-sub (delta (delta d d₁) ds) = {!!} |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
293 |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
294 |
64
15eec529dfc4
Trying prove monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
63
diff
changeset
|
295 monad-law-1-sub : {l : Level} {A : Set l} -> (x : Delta (Delta A)) -> (d : Delta (Delta (Delta A))) -> |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
296 deltaAppend (headDelta (mu x)) (bind (fmap mu d) tailDelta) ≡ mu (deltaAppend (headDelta x) (bind d tailDelta)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
297 monad-law-1-sub (mono (mono _)) (mono (mono _)) = refl |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
298 monad-law-1-sub (mono (mono _)) (mono (delta (mono _) _)) = refl |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
299 monad-law-1-sub (mono (mono _)) (mono (delta (delta _ _) _)) = refl |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
300 monad-law-1-sub (mono (mono x)) (delta (mono (mono xx)) d) = begin |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
301 deltaAppend (headDelta (mu (mono (mono x)))) (bind (fmap mu (delta (mono (mono xx)) d)) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
302 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
303 deltaAppend (headDelta (mu (mono (mono x)))) (bind (delta (mu (mono (mono xx))) (fmap mu d)) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
304 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
305 deltaAppend (headDelta (bind (mono (mono x)) id)) (bind (delta (mu (mono (mono xx))) (fmap mu d)) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
306 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
307 deltaAppend (headDelta (mono x)) (bind (delta (mu (mono (mono xx))) (fmap mu d)) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
308 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
309 deltaAppend (headDelta (mono x)) (bind (delta (mono xx) (fmap mu d)) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
310 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
311 deltaAppend (mono x) (bind (delta (mono xx) (fmap mu d)) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
312 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
313 deltaAppend (mono x) (bind (delta (mono xx) (fmap mu d)) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
314 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
315 deltaAppend (mono x) (deltaAppend (tailDelta (mono xx)) (bind (fmap mu d) (tailDelta ∙ tailDelta))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
316 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
317 deltaAppend (mono x) (deltaAppend (mono xx) (bind (fmap mu d) (tailDelta ∙ tailDelta))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
318 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
319 deltaAppend (mono x) (deltaAppend (mu (mono (mono xx))) (bind (fmap mu d) (tailDelta ∙ tailDelta))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
320 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
321 deltaAppend (mono x) (deltaAppend (mono xx) (bind (fmap mu d) (tailDelta ∙ tailDelta))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
322 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
323 delta x (deltaAppend (mono xx) (bind (fmap mu d) (tailDelta ∙ tailDelta))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
324 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
325 delta x (delta xx (bind (fmap mu d) (tailDelta ∙ tailDelta))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
326 ≡⟨ cong (\d -> (delta x (delta xx d))) (monad-law-1-sub-sub d) ⟩ -- ??? |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
327 delta x (delta xx (bind (bind d (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
328 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
329 delta x ((deltaAppend (mono xx) (bind (bind d (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta)))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
330 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
331 delta x ((deltaAppend (tailDelta (mono xx)) (bind (bind d (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta)))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
332 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
333 delta x (bind (delta (mono xx) (bind d (tailDelta ∙ tailDelta))) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
334 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
335 delta x (bind (deltaAppend (mono (mono xx)) (bind d (tailDelta ∙ tailDelta))) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
336 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
337 delta x (bind (deltaAppend (headDelta (tailDelta (mono (mono xx)))) (bind d (tailDelta ∙ tailDelta))) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
338 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
339 delta x (bind (bind (delta (mono (mono xx)) d) tailDelta) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
340 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
341 deltaAppend (mono x) (bind (bind (delta (mono (mono xx)) d) tailDelta) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
342 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
343 bind (delta (mono x) (bind (delta (mono (mono xx)) d) tailDelta)) id |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
344 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
345 mu (delta (mono x) (bind (delta (mono (mono xx)) d) tailDelta)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
346 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
347 mu (deltaAppend (mono (mono x)) (bind (delta (mono (mono xx)) d) tailDelta)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
348 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
349 mu (deltaAppend (headDelta (mono (mono x))) (bind (delta (mono (mono xx)) d) tailDelta)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
350 ∎ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
351 monad-law-1-sub (mono (mono x)) (delta (mono (delta x₁ d)) d₁) = {!!} |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
352 monad-law-1-sub (mono (mono x)) (delta (delta d d₁) d₂) = {!!} |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
353 monad-law-1-sub (mono (delta x x₁)) d = {!!} |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
354 monad-law-1-sub (delta x x₁) d = {!!} |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
355 |
39 | 356 -- 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
|
357 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) |
46b15f368905
Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
57
diff
changeset
|
358 monad-law-1 (mono d) = refl |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
359 monad-law-1 (delta x d) = begin |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
360 (mu ∙ (fmap mu)) (delta x d) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
361 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
362 mu (fmap mu (delta x d)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
363 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
364 mu (delta (mu x) (fmap mu d)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
365 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
366 bind (delta (mu x) (fmap mu d)) id |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
367 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
368 deltaAppend (headDelta (mu x)) (bind (fmap mu d) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
369 ≡⟨ monad-law-1-sub x d ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
370 mu (deltaAppend (headDelta x) (bind d tailDelta)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
371 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
372 mu (bind (delta x d) id) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
373 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
374 mu (mu (delta x d)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
375 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
376 (mu ∙ mu) (delta x d) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
377 ∎ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
378 |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
379 -- split d |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
380 {- |
62
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
381 monad-law-1 (delta x (mono d)) = begin |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
382 |
62
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
383 (mu ∙ fmap mu) (delta x (mono d)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
384 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
385 mu (fmap mu (delta x (mono d))) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
386 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
387 mu (delta (mu x) (mono (mu d))) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
388 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
389 bind (delta (mu x) (mono (mu d))) id |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
390 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
391 deltaAppend (headDelta (mu x)) (bind (mono (mu d)) tailDelta) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
392 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
393 deltaAppend (headDelta (mu x)) (tailDelta (mu d)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
394 ≡⟨ {!!} ⟩ |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
395 mu (deltaAppend (headDelta x) (tailDelta d)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
396 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
397 mu (deltaAppend (headDelta x) (tailDelta (id d))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
398 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
399 mu (deltaAppend (headDelta x) ((tailDelta ∙ id) d)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
400 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
401 mu (deltaAppend (headDelta x) (bind (mono d) (tailDelta ∙ id))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
402 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
403 mu (bind (delta x (mono d)) id) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
404 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
405 mu (mu (delta x (mono d))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
406 ≡⟨ refl ⟩ |
62
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
407 (mu ∙ mu) (delta x (mono d)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
408 ∎ |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
409 monad-law-1 (delta x (delta d ds)) = begin |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
410 (mu ∙ fmap mu) (delta x (delta d ds)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
411 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
412 mu (fmap mu (delta x (delta d ds))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
413 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
414 mu (delta (mu x) (delta (mu d) (fmap mu ds))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
415 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
416 bind (delta (mu x) (delta (mu d) (fmap mu ds))) id |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
417 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
418 deltaAppend (headDelta (mu x)) (bind (delta (mu d) (fmap mu ds)) tailDelta) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
419 ≡⟨ refl ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
420 deltaAppend (headDelta (mu x)) (deltaAppend (headDelta (tailDelta (mu d))) (bind (fmap mu ds) (tailDelta ∙ tailDelta))) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
421 |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
422 ≡⟨ {!!} ⟩ |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
423 (mu ∙ mu) (delta x (delta d ds)) |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
424 ∎ |
62
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
425 -} |
59
46b15f368905
Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
57
diff
changeset
|
426 |
29
e0ba1bf564dd
Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
427 |
62
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
428 {- |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
429 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
430 monad-law-1 (mono d) = refl |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
431 monad-law-1 (delta x (mono d)) = begin |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
432 (mu ∙ fmap mu) (delta x (mono d)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
433 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
434 mu ((fmap mu) (delta x (mono d))) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
435 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
436 mu (delta (mu x) (fmap mu (mono d))) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
437 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
438 mu (delta (mu x) (fmap mu (mono d))) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
439 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
440 mu (delta (mu x) (mono (mu d))) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
441 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
442 bind (delta (mu x) (mono (mu d))) id |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
443 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
444 deltaAppend (headDelta (mu x)) (bind (mono (mu d)) (tailDelta ∙ id)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
445 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
446 deltaAppend (headDelta (mu x)) (bind (mono (mu d)) (tailDelta)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
447 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
448 deltaAppend (headDelta (mu x)) (tailDelta (mu d)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
449 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
450 deltaAppend (headDelta (mu x)) ((tailDelta ∙ mu) d) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
451 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
452 deltaAppend (headDelta (mu x)) (bind (mono d) (tailDelta ∙ mu)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
453 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
454 bind (delta x (mono d)) mu |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
455 ≡⟨ {!!} ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
456 mu (deltaAppend (headDelta x) (tailDelta d)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
457 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
458 mu (deltaAppend (headDelta x) (bind (mono d) tailDelta)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
459 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
460 mu (deltaAppend (headDelta (id x)) (bind (mono d) (tailDelta ∙ id))) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
461 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
462 mu (deltaAppend (headDelta x) (bind (mono d) (tailDelta ∙ id))) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
463 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
464 mu (bind (delta x (mono d)) id) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
465 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
466 mu (deltaAppend (headDelta (id x)) (bind (mono d) (tailDelta ∙ id))) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
467 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
468 mu (mu (delta x (mono d))) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
469 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
470 (mu ∙ mu) (delta x (mono d)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
471 ∎ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
472 monad-law-1 (delta x (delta xx d)) = {!!} |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
473 |
62
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
474 monad-law-1 (delta x d) = begin |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
475 (mu ∙ fmap mu) (delta x d) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
476 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
477 mu ((fmap mu) (delta x d)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
478 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
479 mu (delta (mu x) (fmap mu d)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
480 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
481 bind (delta (mu x) (fmap mu d)) id |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
482 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
483 deltaAppend (headDelta (mu x)) (bind (fmap mu d) (tailDelta ∙ id)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
484 ≡⟨ refl ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
485 deltaAppend (headDelta (mu x)) (bind (fmap mu d) (tailDelta ∙ id)) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
486 ≡⟨ {!!} ⟩ |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
487 (mu ∙ mu) (delta x d) |
0f308ddd6136
Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
60
diff
changeset
|
488 ∎ |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
489 |
34
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
490 |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
491 |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
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492 |
39 | 493 -- monad-law-2-2 : join . return = id |
43
90b171e3a73e
Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
42
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494 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
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495 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
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496 |
39 | 497 -- 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
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498 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x |
36 | 499 monad-law-3 f x = refl |
27
742e62fc63e4
Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
26
diff
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500 |
39 | 501 -- monad-law-4 : join . fmap (fmap f) = fmap f . join |
43
90b171e3a73e
Rename to Delta from Similar
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parents:
42
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502 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (s : Delta (Delta A)) -> |
36 | 503 (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
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504 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
|
505 |
a7cd7740f33e
Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
39
diff
changeset
|
506 |
a7cd7740f33e
Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
39
diff
changeset
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507 -- 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
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508 -- 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
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509 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
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changeset
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510 (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
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511 (return a >>= k) ≡ (k a) |
59
46b15f368905
Define bind and mu for Infinite Delta
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512 monad-law-h-1 a k = refl |
46b15f368905
Define bind and mu for Infinite Delta
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|
513 |
46b15f368905
Define bind and mu for Infinite Delta
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|
514 |
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
|
515 |
a7cd7740f33e
Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
39
diff
changeset
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516 -- monad-law-h-2 : m >>= return = m |
43
90b171e3a73e
Rename to Delta from Similar
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parents:
42
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517 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m |
59
46b15f368905
Define bind and mu for Infinite Delta
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518 monad-law-h-2 (mono x) = refl |
46b15f368905
Define bind and mu for Infinite Delta
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519 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d) |
46b15f368905
Define bind and mu for Infinite Delta
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parents:
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|
520 |
46b15f368905
Define bind and mu for Infinite Delta
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|
521 |
46b15f368905
Define bind and mu for Infinite Delta
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57
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changeset
|
522 |
41
23474bf242c6
Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
40
diff
changeset
|
523 |
23474bf242c6
Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
40
diff
changeset
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524 -- 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
|
525 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
|
526 (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
|
527 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h) |
59
46b15f368905
Define bind and mu for Infinite Delta
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parents:
57
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changeset
|
528 monad-law-h-3 (mono x) k h = refl |
46b15f368905
Define bind and mu for Infinite Delta
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parents:
57
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|
529 monad-law-h-3 (delta x d) k h = begin |
46b15f368905
Define bind and mu for Infinite Delta
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parents:
57
diff
changeset
|
530 (delta x d) >>= (\x -> k x >>= h) |
42
1df4f9d88025
Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
41
diff
changeset
|
531 ≡⟨ refl ⟩ |
59
46b15f368905
Define bind and mu for Infinite Delta
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parents:
57
diff
changeset
|
532 -- (delta x d) >>= f = deltaAppend (headDelta (f x)) (d >>= (tailDelta ∙ f)) |
46b15f368905
Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
57
diff
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|
533 deltaAppend (headDelta ((\x -> k x >>= h) x)) (d >>= (tailDelta ∙ (\x -> k x >>= h))) |
42
1df4f9d88025
Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
41
diff
changeset
|
534 ≡⟨ refl ⟩ |
59
46b15f368905
Define bind and mu for Infinite Delta
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parents:
57
diff
changeset
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535 deltaAppend (headDelta (k x >>= h)) (d >>= (tailDelta ∙ (\x -> k x >>= h))) |
46b15f368905
Define bind and mu for Infinite Delta
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parents:
57
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changeset
|
536 ≡⟨ {!!} ⟩ |
46b15f368905
Define bind and mu for Infinite Delta
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parents:
57
diff
changeset
|
537 ((delta x d) >>= k) >>= h |
41
23474bf242c6
Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
40
diff
changeset
|
538 ∎ |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
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|
539 -} |