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Fix Monad-proof for Delta
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Mon, 02 Feb 2015 12:12:14 +0900
parents e6bcc7467335
children d205ff1e406f
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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1 open import basic
526186c4f298 Split monad-proofs into delta.monad
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2 open import delta
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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3 open import delta.functor
526186c4f298 Split monad-proofs into delta.monad
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4 open import nat
526186c4f298 Split monad-proofs into delta.monad
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5 open import laws
526186c4f298 Split monad-proofs into delta.monad
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6
526186c4f298 Split monad-proofs into delta.monad
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7
526186c4f298 Split monad-proofs into delta.monad
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8 open import Level
526186c4f298 Split monad-proofs into delta.monad
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9 open import Relation.Binary.PropositionalEquality
526186c4f298 Split monad-proofs into delta.monad
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10 open ≡-Reasoning
526186c4f298 Split monad-proofs into delta.monad
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11
526186c4f298 Split monad-proofs into delta.monad
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12 module delta.monad where
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13
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d47aea3f9246 Delete comment outed temporary code
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14 tailDelta-is-nt : {l : Level} {A B : Set l} {n : Nat}
107
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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15 (f : A -> B) -> (d : Delta A (S (S n))) ->
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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16 (tailDelta {n = n} ∙ (delta-fmap f)) d ≡ ((delta-fmap f) ∙ tailDelta {n = n}) d
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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17 tailDelta-is-nt f (delta x d) = refl
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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18
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d47aea3f9246 Delete comment outed temporary code
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19
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caaf364f45ac Prove monad-laws for length fixed infinite Delta
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20 tailDelta-to-tail-nt : {l : Level} {A B : Set l} (n m : Nat)
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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21 (f : A -> B) (d : Delta (Delta A (S (S m))) (S n)) ->
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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22 delta-fmap tailDelta (delta-fmap (delta-fmap f) d) ≡ delta-fmap (delta-fmap f) (delta-fmap tailDelta d)
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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23 tailDelta-to-tail-nt O O f (mono (delta x d)) = refl
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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24 tailDelta-to-tail-nt O (S m) f (mono (delta x d)) = refl
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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25 tailDelta-to-tail-nt (S n) O f (delta (delta x (mono xx)) d) = begin
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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26 delta (mono (f xx))
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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27 (delta-fmap tailDelta (delta-fmap (delta-fmap f) d))
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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28 ≡⟨ cong (\de -> delta (mono (f xx)) de) (tailDelta-to-tail-nt n O f d) ⟩
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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29 delta (mono (f xx))
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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30 (delta-fmap (delta-fmap f) (delta-fmap tailDelta d))
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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31 ∎
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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32 tailDelta-to-tail-nt (S n) (S m) f (delta (delta x (delta xx d)) ds) = begin
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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33 delta (delta (f xx) (delta-fmap f d))
108
d47aea3f9246 Delete comment outed temporary code
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34 (delta-fmap tailDelta (delta-fmap (delta-fmap f) ds))
107
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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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) ⟩
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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36 delta (delta (f xx) (delta-fmap f d))
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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diff changeset
37 (delta-fmap (delta-fmap f) (delta-fmap tailDelta ds))
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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38 ∎
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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39
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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40
105
e6499a50ccbd Retrying prove monad-laws for delta
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41 delta-eta-is-nt : {l : Level} {A B : Set l} -> {n : Nat}
e6499a50ccbd Retrying prove monad-laws for delta
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42 (f : A -> B) -> (x : A) -> (delta-eta {n = n} ∙ f) x ≡ delta-fmap f (delta-eta x)
e6499a50ccbd Retrying prove monad-laws for delta
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43 delta-eta-is-nt {n = O} f x = refl
e6499a50ccbd Retrying prove monad-laws for delta
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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 ⟩
e6499a50ccbd Retrying prove monad-laws for delta
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48 delta (f x) (delta-eta (f x)) ≡⟨ cong (\de -> delta (f x) de) (delta-eta-is-nt f x) ⟩
e6499a50ccbd Retrying prove monad-laws for delta
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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 ⟩
e6499a50ccbd Retrying prove monad-laws for delta
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51 delta-fmap f (delta-eta x) ∎
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52
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caaf364f45ac Prove monad-laws for length fixed infinite Delta
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53
105
e6499a50ccbd Retrying prove monad-laws for delta
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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))
e6499a50ccbd Retrying prove monad-laws for delta
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55 -> delta-mu (delta-fmap (delta-fmap f) d) ≡ delta-fmap f (delta-mu d)
107
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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56 delta-mu-is-nt {n = O} f (mono d) = refl
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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57 delta-mu-is-nt {n = S n} f (delta (delta x d) ds) = begin
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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58 delta (f x) (delta-mu (delta-fmap tailDelta (delta-fmap (delta-fmap f) ds)))
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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59 ≡⟨ cong (\de -> delta (f x) (delta-mu de)) (tailDelta-to-tail-nt n n f ds ) ⟩
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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60 delta (f x) (delta-mu (delta-fmap (delta-fmap f) (delta-fmap tailDelta ds)))
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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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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65
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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)) ->
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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67 delta-fmap tailDelta (delta-fmap delta-mu d)
105
e6499a50ccbd Retrying prove monad-laws for delta
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68 ≡
107
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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
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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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
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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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
108
d47aea3f9246 Delete comment outed temporary code
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73 delta (mono dxx) (delta-fmap tailDelta (delta-fmap delta-mu ds))
107
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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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 ∎
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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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
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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80 delta (delta dxx (delta-mu (delta-fmap tailDelta (delta-fmap tailDelta ds))))
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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81 (delta-fmap tailDelta (delta-fmap delta-mu dds))
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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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) ⟩
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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83 delta (delta dxx (delta-mu (delta-fmap tailDelta (delta-fmap tailDelta ds))))
caaf364f45ac Prove monad-laws for length fixed infinite Delta
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84 (delta-fmap delta-mu (delta-fmap (delta-fmap tailDelta) (delta-fmap tailDelta dds)))
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85 ∎
105
e6499a50ccbd Retrying prove monad-laws for delta
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86
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87
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526186c4f298 Split monad-proofs into delta.monad
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88
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89 -- Monad-laws (Category)
105
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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)) ->
e6499a50ccbd Retrying prove monad-laws for delta
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parents: 104
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92 ((delta-mu ∙ (delta-fmap delta-mu)) d) ≡ ((delta-mu ∙ delta-mu) d)
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caaf364f45ac Prove monad-laws for length fixed infinite Delta
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93 monad-law-1 {n = O} (mono d) = refl
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parents: 105
diff changeset
94 monad-law-1 {n = S n} (delta (delta (delta x d) dd) ds) = begin
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d47aea3f9246 Delete comment outed temporary code
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parents: 107
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95 delta x (delta-mu (delta-fmap tailDelta (delta-fmap delta-mu ds)))
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caaf364f45ac Prove monad-laws for length fixed infinite Delta
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parents: 105
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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
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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
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parents: 105
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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
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parents: 105
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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
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parents: 105
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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
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parents: 105
diff changeset
101 delta x (delta-mu (delta-fmap tailDelta (delta-mu (delta-fmap tailDelta ds))))
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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
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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
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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;
105
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
147 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
148 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
149 association-law = monad-law-1;
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
150 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
151 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
152
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
153
88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
154
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
155
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
156
88
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 -- Monad-laws (Haskell)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
160 -- 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
161 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
162 (a : A) -> (k : A -> (Delta B)) ->
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
163 (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
164 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
165
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
166
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 -- 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
169 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
170 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
171 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
172
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
173
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
174
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
175 -- 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
176 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
177 (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
178 (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
179 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
180 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
181 (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
182 (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
183
96
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
184
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
185
dfe8c67390bd Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
186
105
e6499a50ccbd Retrying prove monad-laws for delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
187 -}