Mercurial > hg > Members > atton > delta_monad
annotate agda/delta/monad.agda @ 115:e6bcc7467335
Temporary commit : Proving association-law ...
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Sun, 01 Feb 2015 17:06:55 +0900 |
parents | d47aea3f9246 |
children | e1900c89dea9 |
rev | line source |
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1 open import basic |
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2 open import delta |
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3 open import delta.functor |
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4 open import nat |
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5 open import laws |
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6 |
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7 |
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8 open import Level |
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9 open import Relation.Binary.PropositionalEquality |
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10 open ≡-Reasoning |
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11 |
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12 module delta.monad where |
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108
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14 tailDelta-is-nt : {l : Level} {A B : Set l} {n : Nat} |
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15 (f : A -> B) -> (d : Delta A (S (S n))) -> |
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16 (tailDelta {n = n} ∙ (delta-fmap f)) d ≡ ((delta-fmap f) ∙ tailDelta {n = n}) d |
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17 tailDelta-is-nt f (delta x d) = refl |
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108
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19 |
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20 tailDelta-to-tail-nt : {l : Level} {A B : Set l} (n m : Nat) |
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21 (f : A -> B) (d : Delta (Delta A (S (S m))) (S n)) -> |
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22 delta-fmap tailDelta (delta-fmap (delta-fmap f) d) ≡ delta-fmap (delta-fmap f) (delta-fmap tailDelta d) |
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23 tailDelta-to-tail-nt O O f (mono (delta x d)) = refl |
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24 tailDelta-to-tail-nt O (S m) f (mono (delta x d)) = refl |
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25 tailDelta-to-tail-nt (S n) O f (delta (delta x (mono xx)) d) = begin |
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26 delta (mono (f xx)) |
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27 (delta-fmap tailDelta (delta-fmap (delta-fmap f) d)) |
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28 ≡⟨ cong (\de -> delta (mono (f xx)) de) (tailDelta-to-tail-nt n O f d) ⟩ |
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29 delta (mono (f xx)) |
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30 (delta-fmap (delta-fmap f) (delta-fmap tailDelta d)) |
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31 ∎ |
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32 tailDelta-to-tail-nt (S n) (S m) f (delta (delta x (delta xx d)) ds) = begin |
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33 delta (delta (f xx) (delta-fmap f d)) |
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34 (delta-fmap tailDelta (delta-fmap (delta-fmap f) ds)) |
107
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35 ≡⟨ cong (\de -> delta (delta (f xx) (delta-fmap f d)) de) (tailDelta-to-tail-nt n (S m) f ds) ⟩ |
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36 delta (delta (f xx) (delta-fmap f d)) |
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37 (delta-fmap (delta-fmap f) (delta-fmap tailDelta ds)) |
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38 ∎ |
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39 |
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40 |
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41 delta-eta-is-nt : {l : Level} {A B : Set l} -> {n : Nat} |
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42 (f : A -> B) -> (x : A) -> (delta-eta {n = n} ∙ f) x ≡ delta-fmap f (delta-eta x) |
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43 delta-eta-is-nt {n = O} f x = refl |
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44 delta-eta-is-nt {n = S O} f x = refl |
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45 delta-eta-is-nt {n = S n} f x = begin |
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46 (delta-eta ∙ f) x ≡⟨ refl ⟩ |
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47 delta-eta (f x) ≡⟨ refl ⟩ |
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48 delta (f x) (delta-eta (f x)) ≡⟨ cong (\de -> delta (f x) de) (delta-eta-is-nt f x) ⟩ |
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49 delta (f x) (delta-fmap f (delta-eta x)) ≡⟨ refl ⟩ |
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50 delta-fmap f (delta x (delta-eta x)) ≡⟨ refl ⟩ |
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51 delta-fmap f (delta-eta x) ∎ |
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52 |
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53 |
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54 delta-mu-is-nt : {l : Level} {A B : Set l} {n : Nat} -> (f : A -> B) -> (d : Delta (Delta A (S n)) (S n)) |
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55 -> delta-mu (delta-fmap (delta-fmap f) d) ≡ delta-fmap f (delta-mu d) |
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56 delta-mu-is-nt {n = O} f (mono d) = refl |
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57 delta-mu-is-nt {n = S n} f (delta (delta x d) ds) = begin |
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58 delta (f x) (delta-mu (delta-fmap tailDelta (delta-fmap (delta-fmap f) ds))) |
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59 ≡⟨ cong (\de -> delta (f x) (delta-mu de)) (tailDelta-to-tail-nt n n f ds ) ⟩ |
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60 delta (f x) (delta-mu (delta-fmap (delta-fmap f) (delta-fmap tailDelta ds))) |
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61 ≡⟨ cong (\de -> delta (f x) de) (delta-mu-is-nt f (delta-fmap tailDelta ds)) ⟩ |
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62 delta (f x) (delta-fmap f (delta-mu (delta-fmap tailDelta ds))) |
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63 ∎ |
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64 |
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66 delta-fmap-mu-to-tail : {l : Level} {A : Set l} (n m : Nat) (d : Delta (Delta (Delta A (S (S m))) (S (S m))) (S n)) -> |
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67 delta-fmap tailDelta (delta-fmap delta-mu d) |
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68 ≡ |
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69 (delta-fmap delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta d))) |
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70 delta-fmap-mu-to-tail O O (mono (delta d ds)) = refl |
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71 delta-fmap-mu-to-tail O (S n) (mono (delta (delta x (delta xx d)) (delta (delta dx (delta dxx dd)) ds))) = refl |
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72 delta-fmap-mu-to-tail (S n) O (delta (delta (delta x (mono xx)) (mono (delta dx (mono dxx)))) ds) = begin |
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73 delta (mono dxx) (delta-fmap tailDelta (delta-fmap delta-mu ds)) |
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74 ≡⟨ cong (\de -> delta (mono dxx) de) (delta-fmap-mu-to-tail n O ds) ⟩ |
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75 delta (mono dxx) |
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76 (delta-fmap delta-mu |
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77 (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds))) |
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78 ∎ |
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79 delta-fmap-mu-to-tail (S n) (S n₁) (delta (delta (delta x (delta xx d)) (delta (delta dx (delta dxx dd)) ds)) dds) = begin |
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80 delta (delta dxx (delta-mu (delta-fmap tailDelta (delta-fmap tailDelta ds)))) |
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81 (delta-fmap tailDelta (delta-fmap delta-mu dds)) |
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82 ≡⟨ cong (\de -> delta (delta dxx (delta-mu (delta-fmap tailDelta (delta-fmap tailDelta ds)))) de) (delta-fmap-mu-to-tail n (S n₁) dds) ⟩ |
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83 delta (delta dxx (delta-mu (delta-fmap tailDelta (delta-fmap tailDelta ds)))) |
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84 (delta-fmap delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta dds))) |
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85 ∎ |
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86 |
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87 |
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88 |
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89 -- Monad-laws (Category) |
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90 -- monad-law-1 : join . delta-fmap join = join . join |
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91 monad-law-1 : {l : Level} {A : Set l} {n : Nat} (d : Delta (Delta (Delta A (S n)) (S n)) (S n)) -> |
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92 ((delta-mu ∙ (delta-fmap delta-mu)) d) ≡ ((delta-mu ∙ delta-mu) d) |
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105
diff
changeset
|
93 monad-law-1 {n = O} (mono d) = refl |
caaf364f45ac
Prove monad-laws for length fixed infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
105
diff
changeset
|
94 monad-law-1 {n = S n} (delta (delta (delta x d) dd) ds) = begin |
108
d47aea3f9246
Delete comment outed temporary code
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
107
diff
changeset
|
95 delta x (delta-mu (delta-fmap tailDelta (delta-fmap delta-mu ds))) |
107
caaf364f45ac
Prove monad-laws for length fixed infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
105
diff
changeset
|
96 ≡⟨ cong (\de -> delta x (delta-mu de)) (delta-fmap-mu-to-tail n n ds) ⟩ |
caaf364f45ac
Prove monad-laws for length fixed infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
105
diff
changeset
|
97 delta x (delta-mu (delta-fmap delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds)))) |
caaf364f45ac
Prove monad-laws for length fixed infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
105
diff
changeset
|
98 ≡⟨ cong (\de -> delta x de) (monad-law-1 (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds))) ⟩ |
caaf364f45ac
Prove monad-laws for length fixed infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
105
diff
changeset
|
99 delta x (delta-mu (delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds)))) |
caaf364f45ac
Prove monad-laws for length fixed infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
105
diff
changeset
|
100 ≡⟨ cong (\de -> delta x (delta-mu de)) (delta-mu-is-nt tailDelta (delta-fmap tailDelta ds) ) ⟩ |
caaf364f45ac
Prove monad-laws for length fixed infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
105
diff
changeset
|
101 delta x (delta-mu (delta-fmap tailDelta (delta-mu (delta-fmap tailDelta ds)))) |
caaf364f45ac
Prove monad-laws for length fixed infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
105
diff
changeset
|
102 ∎ |
caaf364f45ac
Prove monad-laws for length fixed infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
105
diff
changeset
|
103 |
caaf364f45ac
Prove monad-laws for length fixed infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
105
diff
changeset
|
104 |
105
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
105 delta-right-unity-law : {l : Level} {A : Set l} {n : Nat} (d : Delta A (S n)) -> (delta-mu ∙ delta-eta) d ≡ id d |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
106 delta-right-unity-law (mono x) = refl |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
107 delta-right-unity-law (delta x d) = begin |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
108 (delta-mu ∙ delta-eta) (delta x d) |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
109 ≡⟨ refl ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
110 delta-mu (delta-eta (delta x d)) |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
111 ≡⟨ refl ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
112 delta-mu (delta (delta x d) (delta-eta (delta x d))) |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
113 ≡⟨ refl ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
114 delta (headDelta (delta x d)) (delta-mu (delta-fmap tailDelta (delta-eta (delta x d)))) |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
115 ≡⟨ refl ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
116 delta x (delta-mu (delta-fmap tailDelta (delta-eta (delta x d)))) |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
117 ≡⟨ cong (\de -> delta x (delta-mu de)) (sym (delta-eta-is-nt tailDelta (delta x d))) ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
118 delta x (delta-mu (delta-eta (tailDelta (delta x d)))) |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
119 ≡⟨ refl ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
120 delta x (delta-mu (delta-eta d)) |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
121 ≡⟨ cong (\de -> delta x de) (delta-right-unity-law d) ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
122 delta x d |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
123 ≡⟨ refl ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
124 id (delta x d) ∎ |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
125 |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
126 |
105
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
127 delta-left-unity-law : {l : Level} {A : Set l} {n : Nat} -> (d : Delta A (S n)) -> |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
128 (delta-mu ∙ (delta-fmap delta-eta)) d ≡ id d |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
129 delta-left-unity-law (mono x) = refl |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
130 delta-left-unity-law {n = (S n)} (delta x d) = begin |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
131 (delta-mu ∙ delta-fmap delta-eta) (delta x d) ≡⟨ refl ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
132 delta-mu ( delta-fmap delta-eta (delta x d)) ≡⟨ refl ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
133 delta-mu (delta (delta-eta x) (delta-fmap delta-eta d)) ≡⟨ refl ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
134 delta (headDelta {n = S n} (delta-eta x)) (delta-mu (delta-fmap tailDelta (delta-fmap delta-eta d))) ≡⟨ refl ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
135 delta x (delta-mu (delta-fmap tailDelta (delta-fmap delta-eta d))) |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
136 ≡⟨ cong (\de -> delta x (delta-mu de)) (sym (functor-law-2 tailDelta delta-eta d)) ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
137 delta x (delta-mu (delta-fmap (tailDelta ∙ delta-eta {n = S n}) d)) ≡⟨ refl ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
138 delta x (delta-mu (delta-fmap (delta-eta {n = n}) d)) ≡⟨ cong (\de -> delta x de) (delta-left-unity-law d) ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
139 delta x d ≡⟨ refl ⟩ |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
140 id (delta x d) ∎ |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
141 |
105
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
142 |
104
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
143 |
105
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
144 delta-is-monad : {l : Level} {n : Nat} -> Monad {l} (\A -> Delta A (S n)) delta-is-functor |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
145 delta-is-monad = record { eta = delta-eta; |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
146 mu = delta-mu; |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
147 return = delta-eta; |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
148 bind = delta-bind; |
105
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
149 eta-is-nt = delta-eta-is-nt; |
115
e6bcc7467335
Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
108
diff
changeset
|
150 mu-is-nt = delta-mu-is-nt; |
105
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
151 association-law = monad-law-1; |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
152 left-unity-law = delta-left-unity-law ; |
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
153 right-unity-law = \x -> (sym (delta-right-unity-law x)) } |
104
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
154 |
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
155 |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
156 |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
157 |
96
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
158 |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
159 {- |
96
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
160 |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
161 -- Monad-laws (Haskell) |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
162 -- monad-law-h-1 : return a >>= k = k a |
105
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
163 monad-law-h-1 : {l : Level} {A B : Set l} -> |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
164 (a : A) -> (k : A -> (Delta B)) -> |
96
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
165 (delta-return a >>= k) ≡ (k a) |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
166 monad-law-h-1 a k = refl |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
167 |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
168 |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
169 |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
170 -- monad-law-h-2 : m >>= return = m |
96
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
171 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= delta-return) ≡ m |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
172 monad-law-h-2 (mono x) = refl |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
173 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d) |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
174 |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
175 |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
176 |
96
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
177 -- monad-law-h-3 : m >>= (\x -> f x >>= g) = (m >>= f) >>= g |
105
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
178 monad-law-h-3 : {l : Level} {A B C : Set l} -> |
96
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
179 (m : Delta A) -> (f : A -> (Delta B)) -> (g : B -> (Delta C)) -> |
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
180 (delta-bind m (\x -> delta-bind (f x) g)) ≡ (delta-bind (delta-bind m f) g) |
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
181 monad-law-h-3 (mono x) f g = refl |
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
182 monad-law-h-3 (delta x d) f g = begin |
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
183 (delta-bind (delta x d) (\x -> delta-bind (f x) g)) ≡⟨ {!!} ⟩ |
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
184 (delta-bind (delta-bind (delta x d) f) g) ∎ |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
185 |
96
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
186 |
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
187 |
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
188 |
105
e6499a50ccbd
Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
189 -} |