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Trying Monad-laws for DeltaM
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Wed, 21 Jan 2015 17:43:53 +0900
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children 9c62373bd474
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b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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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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4
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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 deltaM
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9 open import deltaM.functor
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10 open import laws
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11
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12 module deltaM.monad where
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13 open Functor
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14 open NaturalTransformation
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15 open Monad
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16
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17
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18 postulate deltaM-mu-is-natural-transformation : {l : Level} {A : Set l}
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19 {M : {l' : Level} -> Set l' -> Set l'} ->
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20 {functorM : {l' : Level} -> Functor {l'} M}
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21 {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M (functorM ) } ->
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22 NaturalTransformation (\A -> DeltaM M (DeltaM M A)) (\A -> DeltaM M A)
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23 {deltaM-fmap ∙ deltaM-fmap} {deltaM-fmap {l}}
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24 (deltaM-mu {_} {_} {M} {functorM} {monadM})
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25
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26 headDeltaM-commute : {l : Level} {A B : Set l}
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27 {M : {l' : Level} -> Set l' -> Set l'} ->
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28 {functorM : {l' : Level} -> Functor {l'} M} ->
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29 {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M (functorM ) } ->
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30 (f : A -> B) -> (x : DeltaM M {functorM} {monadM} A) ->
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31 headDeltaM (deltaM-fmap f x) ≡ fmap functorM f (headDeltaM x)
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32 headDeltaM-commute f (deltaM (mono x)) = refl
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33 headDeltaM-commute f (deltaM (delta x d)) = refl
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34
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35
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36 headDeltaM-is-natural-transformation : {l : Level} {A : Set l}
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37 {M : {l' : Level} -> Set l' -> Set l'} ->
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38 {functorM : {l' : Level} -> Functor {l'} M}
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39 {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM } ->
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40 NaturalTransformation {l} (\A -> DeltaM M {functorM} {monadM} A) M
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41 {\f d → deltaM (mono (headDeltaM (deltaM-fmap f d)))} {fmap functorM} headDeltaM
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42 -- {deltaM-fmap} {fmap (functorM {l} {A})} headDeltaM
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43 headDeltaM-is-natural-transformation = record { commute = headDeltaM-commute }
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44
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45
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46
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47 deltaM-association-law : {l : Level} {A : Set l}
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48 {M : {l' : Level} -> Set l' -> Set l'}
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49 (functorM : {l' : Level} -> Functor {l'} M)
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50 (monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM)
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51 -> (d : DeltaM M (DeltaM M (DeltaM M {functorM} {monadM} A)))
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52 -> ((deltaM-mu ∙ (deltaM-fmap deltaM-mu)) d) ≡ ((deltaM-mu ∙ deltaM-mu) d)
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53 deltaM-association-law functorM monadM (deltaM x) = {!!}
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54 {-
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55 begin
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56 (deltaM-mu ∙ deltaM-fmap deltaM-mu) d ≡⟨ refl ⟩
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57 deltaM-mu (deltaM-fmap deltaM-mu d) ≡⟨ {!!} ⟩
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58 deltaM-mu (deltaM-mu d) ≡⟨ refl ⟩
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59 (deltaM-mu ∙ deltaM-mu) d
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60 ∎
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61 -}
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62 {-
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63 deltaM-association-law functorM monadM (deltaM (mono x)) = begin
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64 (deltaM-mu ∙ deltaM-fmap deltaM-mu) (deltaM (mono x)) ≡⟨ refl ⟩
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65 deltaM-mu (deltaM-fmap deltaM-mu (deltaM (mono x))) ≡⟨ refl ⟩
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66 deltaM-mu (deltaM (delta-fmap (fmap functorM deltaM-mu) (mono x))) ≡⟨ refl ⟩
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67 deltaM-mu (deltaM (mono (fmap functorM deltaM-mu x))) ≡⟨ refl ⟩
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68 deltaM-bind (deltaM (mono (fmap functorM deltaM-mu x))) id ≡⟨ refl ⟩
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69 deltaM (mono (bind monadM (fmap functorM deltaM-mu x) (headDeltaM ∙ id))) ≡⟨ refl ⟩
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70 deltaM (mono (bind monadM (fmap functorM deltaM-mu x) headDeltaM)) ≡⟨ {!!} ⟩
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71 deltaM (mono (bind monadM (bind monadM x headDeltaM) headDeltaM)) ≡⟨ refl ⟩
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72 deltaM (mono (bind monadM (bind monadM x (headDeltaM ∙ id)) (headDeltaM ∙ id))) ≡⟨ refl ⟩
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73 deltaM-mu (deltaM (mono (bind monadM x (headDeltaM ∙ id)))) ≡⟨ refl ⟩
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74 deltaM-mu (deltaM-mu (deltaM (mono x))) ≡⟨ refl ⟩
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75 (deltaM-mu ∙ deltaM-mu) (deltaM (mono x)) ∎
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76 deltaM-association-law functorM monadM (deltaM (delta x d)) = {!!}
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77 -}
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78
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79 {-
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80
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81 nya : {l : Level} {A B C : Set l} ->
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82 {M : {l' : Level} -> Set l' -> Set l'}
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83 {functorM : {l' : Level} -> Functor {l'} M }
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84 {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM}
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85 (m : DeltaM M {functorM} {monadM} A) -> (f : A -> (DeltaM M {functorM} {monadM} B)) -> (g : B -> (DeltaM M C)) ->
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86 (x : M A) ->
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87 (deltaM (fmap delta-is-functor (\x -> bind {l} {A} monadM x (headDeltaM ∙ (\x -> deltaM-bind (f x) g))) (mono x))) ≡
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88 (deltaM-bind (deltaM (fmap delta-is-functor (\x -> (bind {l} {A} monadM x (headDeltaM ∙ f))) (mono x))) g)
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89 nya = {!!}
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90
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91 deltaM-monad-law-h-3 : {l : Level} {A B C : Set l} ->
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92 {M : {l' : Level} -> Set l' -> Set l'}
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93 {functorM : {l' : Level} -> Functor {l'} M }
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94 {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM}
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95 (m : DeltaM M {functorM} {monadM} A) -> (f : A -> (DeltaM M B)) -> (g : B -> (DeltaM M C)) ->
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96 (deltaM-bind m (\x -> deltaM-bind (f x) g)) ≡ (deltaM-bind (deltaM-bind m f) g)
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97 {-
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98 deltaM-monad-law-h-3 {l} {A} {B} {C} {M} {functorM} {monadM} (deltaM (mono x)) f g = begin
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99 (deltaM-bind (deltaM (mono x)) (\x -> deltaM-bind (f x) g)) ≡⟨ refl ⟩
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100
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101 (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ (\x -> deltaM-bind (f x) g))))) ≡⟨ {!!} ⟩
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102 (deltaM-bind (deltaM (fmap delta-is-functor (\x -> (bind {l} {A} monadM x (headDeltaM ∙ f))) (mono x))) g) ≡⟨ refl ⟩
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103 (deltaM-bind (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ f)))) g) ≡⟨ refl ⟩
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104 (deltaM-bind (deltaM-bind (deltaM (mono x)) f) g) ∎
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105 -}
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106
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107 deltaM-monad-law-h-3 {l} {A} {B} {C} {M} {functorM} {monadM} (deltaM (mono x)) f g = begin
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Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
108 (deltaM-bind (deltaM (mono x)) (\x -> deltaM-bind (f x) g)) ≡⟨ refl ⟩
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Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
109 (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ (\x -> deltaM-bind (f x) g))))) ≡⟨ {!!} ⟩
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Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
110 -- (deltaM (fmap delta-is-functor (\x -> bind {l} {A} monadM x (headDeltaM ∙ (\x -> deltaM-bind (f x) g))) (mono x))) ≡⟨ {!!} ⟩
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
111 deltaM (mono (bind {l} {B} monadM (bind {_} {A} monadM x (headDeltaM ∙ f)) (headDeltaM ∙ g))) ≡⟨ {!!} ⟩
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
112 deltaM (mono (bind {l} {B} monadM (bind {_} {A} monadM x (headDeltaM ∙ f)) (headDeltaM ∙ g))) ≡⟨ {!!} ⟩
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
113 (deltaM-bind (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ f)))) g) ≡⟨ refl ⟩
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
114 (deltaM-bind (deltaM-bind (deltaM (mono x)) f) g)
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Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
115 ∎
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
116 deltaM-monad-law-h-3 (deltaM (delta x d)) f g = {!!}
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
117 {-
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
118 begin
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
119 (deltaM-bind m (\x -> deltaM-bind (f x) g)) ≡⟨ {!!} ⟩
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
120 (deltaM-bind (deltaM-bind m f) g)
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Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
121 ∎
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
122 -}
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
123 -}