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annotate agda/delta/monad.agda @ 136:b6dcbe8617a9

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Delete unnecessary lines
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 15 Feb 2015 10:15:10 +0900
parents d205ff1e406f
children 2bf1fa6d2006
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1 open import Level
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2 open import Relation.Binary.PropositionalEquality
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3 open ≡-Reasoning
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5 open import basic
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6 open import delta
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7 open import delta.functor
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8 open import nat
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9 open import laws
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10
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11 module delta.monad where
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12
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14 tailDelta-to-tail-nt : {l : Level} {A B : Set l} (n m : Nat)
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15 (f : A -> B) (d : Delta (Delta A (S (S m))) (S n)) ->
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16 delta-fmap tailDelta (delta-fmap (delta-fmap f) d) ≡ delta-fmap (delta-fmap f) (delta-fmap tailDelta d)
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17 tailDelta-to-tail-nt O O f (mono (delta x d)) = refl
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18 tailDelta-to-tail-nt O (S m) f (mono (delta x d)) = refl
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19 tailDelta-to-tail-nt (S n) O f (delta (delta x (mono xx)) d) = begin
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20 delta (mono (f xx))
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21 (delta-fmap tailDelta (delta-fmap (delta-fmap f) d))
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22 ≡⟨ cong (\de -> delta (mono (f xx)) de) (tailDelta-to-tail-nt n O f d) ⟩
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23 delta (mono (f xx))
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24 (delta-fmap (delta-fmap f) (delta-fmap tailDelta d))
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25 ∎
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26 tailDelta-to-tail-nt (S n) (S m) f (delta (delta x (delta xx d)) ds) = begin
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27 delta (delta (f xx) (delta-fmap f d))
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28 (delta-fmap tailDelta (delta-fmap (delta-fmap f) ds))
107
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29 ≡⟨ cong (\de -> delta (delta (f xx) (delta-fmap f d)) de) (tailDelta-to-tail-nt n (S m) f ds) ⟩
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30 delta (delta (f xx) (delta-fmap f d))
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31 (delta-fmap (delta-fmap f) (delta-fmap tailDelta ds))
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32 ∎
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33
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34
105
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35 delta-eta-is-nt : {l : Level} {A B : Set l} -> {n : Nat}
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36 (f : A -> B) -> (x : A) -> (delta-eta {n = n} ∙ f) x ≡ delta-fmap f (delta-eta x)
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37 delta-eta-is-nt {n = O} f x = refl
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38 delta-eta-is-nt {n = S n} f x = begin
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39 (delta-eta ∙ f) x ≡⟨ refl ⟩
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40 delta-eta (f x) ≡⟨ refl ⟩
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41 delta (f x) (delta-eta (f x)) ≡⟨ cong (\de -> delta (f x) de) (delta-eta-is-nt f x) ⟩
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42 delta (f x) (delta-fmap f (delta-eta x)) ≡⟨ refl ⟩
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43 delta-fmap f (delta x (delta-eta x)) ≡⟨ refl ⟩
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44 delta-fmap f (delta-eta x) ∎
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45
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46
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47 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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48 -> delta-mu (delta-fmap (delta-fmap f) d) ≡ delta-fmap f (delta-mu d)
107
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49 delta-mu-is-nt {n = O} f (mono d) = refl
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50 delta-mu-is-nt {n = S n} f (delta (delta x d) ds) = begin
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51 delta (f x) (delta-mu (delta-fmap tailDelta (delta-fmap (delta-fmap f) ds)))
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52 ≡⟨ cong (\de -> delta (f x) (delta-mu de)) (tailDelta-to-tail-nt n n f ds ) ⟩
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53 delta (f x) (delta-mu (delta-fmap (delta-fmap f) (delta-fmap tailDelta ds)))
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54 ≡⟨ cong (\de -> delta (f x) de) (delta-mu-is-nt f (delta-fmap tailDelta ds)) ⟩
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55 delta (f x) (delta-fmap f (delta-mu (delta-fmap tailDelta ds)))
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56 ∎
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59 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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60 delta-fmap tailDelta (delta-fmap delta-mu d)
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61 ≡
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62 (delta-fmap delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta d)))
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63 delta-fmap-mu-to-tail O O (mono (delta d ds)) = refl
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64 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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65 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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66 delta (mono dxx) (delta-fmap tailDelta (delta-fmap delta-mu ds))
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67 ≡⟨ cong (\de -> delta (mono dxx) de) (delta-fmap-mu-to-tail n O ds) ⟩
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68 delta (mono dxx)
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69 (delta-fmap delta-mu
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70 (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds)))
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71 ∎
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72 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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73 delta (delta dxx (delta-mu (delta-fmap tailDelta (delta-fmap tailDelta ds))))
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74 (delta-fmap tailDelta (delta-fmap delta-mu dds))
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75 ≡⟨ 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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76 delta (delta dxx (delta-mu (delta-fmap tailDelta (delta-fmap tailDelta ds))))
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77 (delta-fmap delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta dds)))
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78 ∎
105
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81
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82 -- Monad-laws (Category)
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83 -- association-law : join . delta-fmap join = join . join
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84 delta-association-law : {l : Level} {A : Set l} {n : Nat} (d : Delta (Delta (Delta A (S n)) (S n)) (S n)) ->
105
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85 ((delta-mu ∙ (delta-fmap delta-mu)) d) ≡ ((delta-mu ∙ delta-mu) d)
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86 delta-association-law {n = O} (mono d) = refl
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87 delta-association-law {n = S n} (delta (delta (delta x d) dd) ds) = begin
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88 delta x (delta-mu (delta-fmap tailDelta (delta-fmap delta-mu ds)))
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89 ≡⟨ cong (\de -> delta x (delta-mu de)) (delta-fmap-mu-to-tail n n ds) ⟩
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90 delta x (delta-mu (delta-fmap delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds))))
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91 ≡⟨ cong (\de -> delta x de) (delta-association-law (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds))) ⟩
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92 delta x (delta-mu (delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta ds))))
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93 ≡⟨ cong (\de -> delta x (delta-mu de)) (delta-mu-is-nt tailDelta (delta-fmap tailDelta ds) ) ⟩
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94 delta x (delta-mu (delta-fmap tailDelta (delta-mu (delta-fmap tailDelta ds))))
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95 ∎
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98 delta-right-unity-law : {l : Level} {A : Set l} {n : Nat} (d : Delta A (S n)) -> (delta-mu ∙ delta-eta) d ≡ id d
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99 delta-right-unity-law (mono x) = refl
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100 delta-right-unity-law (delta x d) = begin
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101 (delta-mu ∙ delta-eta) (delta x d)
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102 ≡⟨ refl ⟩
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103 delta-mu (delta-eta (delta x d))
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parents: 104
diff changeset
104 ≡⟨ refl ⟩
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parents: 104
diff changeset
105 delta-mu (delta (delta x d) (delta-eta (delta x d)))
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
106 ≡⟨ refl ⟩
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parents: 104
diff changeset
107 delta (headDelta (delta x d)) (delta-mu (delta-fmap tailDelta (delta-eta (delta x d))))
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
108 ≡⟨ refl ⟩
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parents: 104
diff changeset
109 delta x (delta-mu (delta-fmap tailDelta (delta-eta (delta x d))))
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
110 ≡⟨ cong (\de -> delta x (delta-mu de)) (sym (delta-eta-is-nt tailDelta (delta x d))) ⟩
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
111 delta x (delta-mu (delta-eta (tailDelta (delta x d))))
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
112 ≡⟨ refl ⟩
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parents: 104
diff changeset
113 delta x (delta-mu (delta-eta d))
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
114 ≡⟨ cong (\de -> delta x de) (delta-right-unity-law d) ⟩
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
115 delta x d
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
116 ≡⟨ refl ⟩
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parents: 104
diff changeset
117 id (delta x d) ∎
88
526186c4f298 Split monad-proofs into delta.monad
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parents:
diff changeset
118
526186c4f298 Split monad-proofs into delta.monad
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parents:
diff changeset
119
105
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parents: 104
diff changeset
120 delta-left-unity-law : {l : Level} {A : Set l} {n : Nat} -> (d : Delta A (S n)) ->
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diff changeset
121 (delta-mu ∙ (delta-fmap delta-eta)) d ≡ id d
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parents: 104
diff changeset
122 delta-left-unity-law (mono x) = refl
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parents: 104
diff changeset
123 delta-left-unity-law {n = (S n)} (delta x d) = begin
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parents: 104
diff changeset
124 (delta-mu ∙ delta-fmap delta-eta) (delta x d) ≡⟨ refl ⟩
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parents: 104
diff changeset
125 delta-mu ( delta-fmap delta-eta (delta x d)) ≡⟨ refl ⟩
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parents: 104
diff changeset
126 delta-mu (delta (delta-eta x) (delta-fmap delta-eta d)) ≡⟨ refl ⟩
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parents: 104
diff changeset
127 delta (headDelta {n = S n} (delta-eta x)) (delta-mu (delta-fmap tailDelta (delta-fmap delta-eta d))) ≡⟨ refl ⟩
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parents: 104
diff changeset
128 delta x (delta-mu (delta-fmap tailDelta (delta-fmap delta-eta d)))
131
d205ff1e406f Cleanup proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
129 ≡⟨ cong (\de -> delta x (delta-mu de)) (sym (delta-covariant tailDelta delta-eta d)) ⟩
105
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
130 delta x (delta-mu (delta-fmap (tailDelta ∙ delta-eta {n = S n}) d)) ≡⟨ refl ⟩
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parents: 104
diff changeset
131 delta x (delta-mu (delta-fmap (delta-eta {n = n}) d)) ≡⟨ cong (\de -> delta x de) (delta-left-unity-law d) ⟩
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parents: 104
diff changeset
132 delta x d ≡⟨ refl ⟩
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parents: 104
diff changeset
133 id (delta x d) ∎
88
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parents:
diff changeset
134
105
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parents: 104
diff changeset
135
104
ebd0d6e2772c Trying redenition Delta with length constraints
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parents: 103
diff changeset
136
105
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
137 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
138 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
139 mu = delta-mu;
105
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
140 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
141 mu-is-nt = delta-mu-is-nt;
131
d205ff1e406f Cleanup proofs
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parents: 122
diff changeset
142 association-law = delta-association-law;
105
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
143 left-unity-law = delta-left-unity-law ;
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
144 right-unity-law = \x -> (sym (delta-right-unity-law x)) }
104
ebd0d6e2772c Trying redenition Delta with length constraints
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parents: 103
diff changeset
145
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parents: 103
diff changeset
146
88
526186c4f298 Split monad-proofs into delta.monad
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parents:
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147
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parents:
diff changeset
148
96
dfe8c67390bd Unify Levels in delta
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parents: 94
diff changeset
149
88
526186c4f298 Split monad-proofs into delta.monad
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parents:
diff changeset
150 {-
96
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parents: 94
diff changeset
151
88
526186c4f298 Split monad-proofs into delta.monad
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parents:
diff changeset
152 -- Monad-laws (Haskell)
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parents:
diff changeset
153 -- monad-law-h-1 : return a >>= k = k a
105
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parents: 104
diff changeset
154 monad-law-h-1 : {l : Level} {A B : Set l} ->
88
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parents:
diff changeset
155 (a : A) -> (k : A -> (Delta B)) ->
96
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parents: 94
diff changeset
156 (delta-return a >>= k) ≡ (k a)
88
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parents:
diff changeset
157 monad-law-h-1 a k = refl
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parents:
diff changeset
158
526186c4f298 Split monad-proofs into delta.monad
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parents:
diff changeset
159
526186c4f298 Split monad-proofs into delta.monad
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parents:
diff changeset
160
526186c4f298 Split monad-proofs into delta.monad
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parents:
diff changeset
161 -- monad-law-h-2 : m >>= return = m
96
dfe8c67390bd Unify Levels in delta
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parents: 94
diff changeset
162 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= delta-return) ≡ m
88
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parents:
diff changeset
163 monad-law-h-2 (mono x) = refl
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parents:
diff changeset
164 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d)
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parents:
diff changeset
165
526186c4f298 Split monad-proofs into delta.monad
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parents:
diff changeset
166
526186c4f298 Split monad-proofs into delta.monad
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parents:
diff changeset
167
96
dfe8c67390bd Unify Levels in delta
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parents: 94
diff changeset
168 -- monad-law-h-3 : m >>= (\x -> f x >>= g) = (m >>= f) >>= g
105
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
169 monad-law-h-3 : {l : Level} {A B C : Set l} ->
96
dfe8c67390bd Unify Levels in delta
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parents: 94
diff changeset
170 (m : Delta A) -> (f : A -> (Delta B)) -> (g : B -> (Delta C)) ->
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parents: 94
diff changeset
171 (delta-bind m (\x -> delta-bind (f x) g)) ≡ (delta-bind (delta-bind m f) g)
dfe8c67390bd Unify Levels in delta
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parents: 94
diff changeset
172 monad-law-h-3 (mono x) f g = refl
dfe8c67390bd Unify Levels in delta
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parents: 94
diff changeset
173 monad-law-h-3 (delta x d) f g = begin
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parents: 94
diff changeset
174 (delta-bind (delta x d) (\x -> delta-bind (f x) g)) ≡⟨ {!!} ⟩
dfe8c67390bd Unify Levels in delta
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parents: 94
diff changeset
175 (delta-bind (delta-bind (delta x d) f) g) ∎
88
526186c4f298 Split monad-proofs into delta.monad
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parents:
diff changeset
176
96
dfe8c67390bd Unify Levels in delta
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parents: 94
diff changeset
177
dfe8c67390bd Unify Levels in delta
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parents: 94
diff changeset
178
dfe8c67390bd Unify Levels in delta
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parents: 94
diff changeset
179
105
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
diff changeset
180 -}