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Prove right-unity-law for DeltaM
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Mon, 02 Feb 2015 14:09:30 +0900
parents 48b44bd85056
children 5902b2a24abf
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b7f0879e854e Trying Monad-laws for DeltaM
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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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9fe3d0bd1149 Prove right-unity-law on DeltaM
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8 open import delta.monad
98
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9 open import deltaM
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10 open import deltaM.functor
104
ebd0d6e2772c Trying redenition Delta with length constraints
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parents: 103
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11 open import nat
98
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12 open import laws
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13
b7f0879e854e Trying Monad-laws for DeltaM
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14 module deltaM.monad where
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15 open Functor
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16 open NaturalTransformation
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17 open Monad
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18
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19
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
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20 -- sub proofs
114
08403eb8db8b Prove natural transformation for deltaM-eta
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21
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
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22 deconstruct-id : {l : Level} {A : Set l} {n : Nat}
124
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23 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
48b44bd85056 Fix proof natural transformation for deltaM-eta
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24 (d : DeltaM M A (S n)) -> deltaM (unDeltaM d) ≡ d
48b44bd85056 Fix proof natural transformation for deltaM-eta
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parents: 118
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25 deconstruct-id {n = O} (deltaM x) = refl
117
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26 deconstruct-id {n = S n} (deltaM x) = refl
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27
6f86b55bf8b4 Temporary commit : Proving association-law ....
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28
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29
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08403eb8db8b Prove natural transformation for deltaM-eta
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30 fmap-headDeltaM-with-deltaM-eta : {l : Level} {A : Set l} {n : Nat}
124
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31 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
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32 (x : T A) -> (fmap F ((headDeltaM {n = n} {M = M}) ∙ deltaM-eta) x) ≡ fmap F (eta M) x
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
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parents: 118
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33 fmap-headDeltaM-with-deltaM-eta {n = O} x = refl
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34 fmap-headDeltaM-with-deltaM-eta {n = S n} x = refl
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35
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36
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37
08403eb8db8b Prove natural transformation for deltaM-eta
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38 fmap-tailDeltaM-with-deltaM-eta : {l : Level} {A : Set l} {n : Nat}
124
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39 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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parents: 118
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40 (d : DeltaM M A (S n)) ->
48b44bd85056 Fix proof natural transformation for deltaM-eta
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41 deltaM-fmap ((tailDeltaM {n = n} {M = M} ) ∙ deltaM-eta) d ≡ deltaM-fmap (deltaM-eta) d
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42 fmap-tailDeltaM-with-deltaM-eta {n = O} d = refl
114
08403eb8db8b Prove natural transformation for deltaM-eta
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43 fmap-tailDeltaM-with-deltaM-eta {n = S n} d = refl
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44
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45
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46
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47 fmap-headDeltaM-with-deltaM-mu : {l : Level} {A : Set l} {n : Nat}
124
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48 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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49 (x : T (DeltaM M (DeltaM M A (S n)) (S n))) ->
48b44bd85056 Fix proof natural transformation for deltaM-eta
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parents: 118
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50 fmap F (headDeltaM ∙ deltaM-mu) x ≡ fmap F (((mu M) ∙ (fmap F headDeltaM)) ∙ headDeltaM) x
118
53cb21845dea Prove association-law for DeltaM
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51 fmap-headDeltaM-with-deltaM-mu {n = O} x = refl
53cb21845dea Prove association-law for DeltaM
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parents: 117
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52 fmap-headDeltaM-with-deltaM-mu {n = S n} x = refl
114
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53
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53cb21845dea Prove association-law for DeltaM
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54
53cb21845dea Prove association-law for DeltaM
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55 fmap-tailDeltaM-with-deltaM-mu : {l : Level} {A : Set l} {n : Nat}
124
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56 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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parents: 118
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57 (d : DeltaM M (DeltaM M (DeltaM M A (S (S n))) (S (S n))) (S n)) ->
48b44bd85056 Fix proof natural transformation for deltaM-eta
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parents: 118
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58 deltaM-fmap (tailDeltaM ∙ deltaM-mu) d ≡ deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) d
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53cb21845dea Prove association-law for DeltaM
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parents: 117
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59 fmap-tailDeltaM-with-deltaM-mu {n = O} (deltaM (mono x)) = refl
53cb21845dea Prove association-law for DeltaM
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parents: 117
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60 fmap-tailDeltaM-with-deltaM-mu {n = S n} (deltaM d) = refl
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61
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6f86b55bf8b4 Temporary commit : Proving association-law ....
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65
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66 -- main proofs
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67
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68 deltaM-eta-is-nt : {l : Level} {A B : Set l} {n : Nat}
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69 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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parents: 112
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70 (f : A -> B) -> (x : A) ->
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71 ((deltaM-eta {l} {B} {n} {T} {F} {M} )∙ f) x ≡ deltaM-fmap f (deltaM-eta x)
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parents: 112
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72 deltaM-eta-is-nt {l} {A} {B} {O} {M} {fm} {mm} f x = begin
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73 deltaM-eta {n = O} (f x) ≡⟨ refl ⟩
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parents: 112
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74 deltaM (mono (eta mm (f x))) ≡⟨ cong (\de -> deltaM (mono de)) (eta-is-nt mm f x) ⟩
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75 deltaM (mono (fmap fm f (eta mm x))) ≡⟨ refl ⟩
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76 deltaM-fmap f (deltaM-eta {n = O} x) ∎
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77 deltaM-eta-is-nt {l} {A} {B} {S n} {M} {fm} {mm} f x = begin
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parents: 118
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78 deltaM-eta {n = S n} (f x)
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79 ≡⟨ refl ⟩
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parents: 118
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80 deltaM (delta-eta {n = S n} (eta mm (f x)))
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parents: 118
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81 ≡⟨ refl ⟩
114
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parents: 112
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82 deltaM (delta (eta mm (f x)) (delta-eta (eta mm (f x))))
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parents: 112
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83 ≡⟨ cong (\de -> deltaM (delta de (delta-eta de))) (eta-is-nt mm f x) ⟩
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parents: 112
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84 deltaM (delta (fmap fm f (eta mm x)) (delta-eta (fmap fm f (eta mm x))))
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parents: 112
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85 ≡⟨ cong (\de -> deltaM (delta (fmap fm f (eta mm x)) de)) (eta-is-nt delta-is-monad (fmap fm f) (eta mm x)) ⟩
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parents: 112
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86 deltaM (delta (fmap fm f (eta mm x)) (delta-fmap (fmap fm f) (delta-eta (eta mm x))))
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parents: 112
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87 ≡⟨ refl ⟩
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parents: 112
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88 deltaM-fmap f (deltaM-eta {n = S n} x)
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parents: 112
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89 ∎
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90
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parents: 118
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91
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6dcc68ef8f96 Prove right-unity-law for DeltaM
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92
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9c62373bd474 Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
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parents: 98
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93
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94 postulate deltaM-mu-is-nt : {l : Level} {A B : Set l} {n : Nat}
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95 {T : Set l -> Set l} {F : Functor T} {M : Monad T F}
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6dcc68ef8f96 Prove right-unity-law for DeltaM
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parents: 124
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96 (f : A -> B) -> (d : DeltaM M (DeltaM M A (S n)) (S n)) ->
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parents: 117
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97 deltaM-fmap f (deltaM-mu d) ≡ deltaM-mu (deltaM-fmap (deltaM-fmap f) d)
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6dcc68ef8f96 Prove right-unity-law for DeltaM
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parents: 124
diff changeset
98 {-
6dcc68ef8f96 Prove right-unity-law for DeltaM
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99 deltaM-mu-is-nt {l} {A} {B} {O} {T} {F} {M} f (deltaM (mono x)) = begin
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parents: 124
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100 deltaM-fmap f (deltaM-mu (deltaM (mono x)))
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parents: 124
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101 ≡⟨ refl ⟩
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parents: 124
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102 deltaM-fmap f (deltaM (mono (mu M (fmap F headDeltaM x))))
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parents: 124
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103 ≡⟨ refl ⟩
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parents: 124
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104 deltaM (mono (fmap F f (mu M (fmap F headDeltaM x))))
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parents: 124
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105 ≡⟨ cong (\de -> deltaM (mono de)) (sym (mu-is-nt M f (fmap F headDeltaM x))) ⟩
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parents: 124
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106 deltaM (mono (mu M (fmap F (fmap F f) (fmap F headDeltaM x))))
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parents: 124
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107 ≡⟨ cong (\de -> deltaM (mono (mu M de))) (sym (covariant F headDeltaM (fmap F f) x)) ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
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parents: 124
diff changeset
108 deltaM (mono (mu M (fmap F ((fmap F f) ∙ headDeltaM) x)))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
109 ≡⟨ {!!} ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
110 deltaM (mono (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x)))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
111 ≡⟨ {!!} ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
112 deltaM (mono (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x)))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
113 ≡⟨ cong (\de -> deltaM (mono (mu M de))) (covariant F (deltaM-fmap f) (headDeltaM) x) ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
114 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (fmap F (deltaM-fmap f) x))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
115 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
116 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {M = M} (deltaM (mono (fmap F (deltaM-fmap f) x)))))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
117 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
118 deltaM-mu (deltaM (mono (fmap F (deltaM-fmap f) x)))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
119 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
120 deltaM-mu (deltaM-fmap (deltaM-fmap f) (deltaM (mono x)))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
121 ∎
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
122 deltaM-mu-is-nt {n = S n} f (deltaM (delta x d)) = {!!}
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
123
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
124 -}
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
126
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
127 deltaM-right-unity-law : {l : Level} {A : Set l} {n : Nat}
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
128 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
129 (d : DeltaM M A (S n)) -> (deltaM-mu ∙ deltaM-eta) d ≡ id d
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
130 deltaM-right-unity-law {l} {A} {O} {M} {fm} {mm} (deltaM (mono x)) = begin
111
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
131 deltaM-mu (deltaM-eta (deltaM (mono x))) ≡⟨ refl ⟩
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
132 deltaM-mu (deltaM (mono (eta mm (deltaM (mono x))))) ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
133 deltaM (mono (mu mm (fmap fm (headDeltaM {M = mm})(eta mm (deltaM (mono x))))))
111
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
134 ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (sym (eta-is-nt mm headDeltaM (deltaM (mono x)) )) ⟩
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
135 deltaM (mono (mu mm (eta mm ((headDeltaM {l} {A} {O} {M} {fm} {mm}) (deltaM (mono x)))))) ≡⟨ refl ⟩
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
136 deltaM (mono (mu mm (eta mm x))) ≡⟨ cong (\de -> deltaM (mono de)) (sym (right-unity-law mm x)) ⟩
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
137 deltaM (mono x)
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
138 ∎
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
139 deltaM-right-unity-law {l} {A} {S n} {T} {F} {M} (deltaM (delta x d)) = begin
111
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
140 deltaM-mu (deltaM-eta (deltaM (delta x d)))
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
141 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
142 deltaM-mu (deltaM (delta (eta M (deltaM (delta x d))) (delta-eta (eta M (deltaM (delta x d))))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
143 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
144 deltaM (delta (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {M = M} (deltaM (delta {l} {T (DeltaM M A (S (S n)))} {n} (eta M (deltaM (delta x d))) (delta-eta (eta M (deltaM (delta x d)))))))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
145 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM (deltaM (delta (eta M (deltaM (delta x d))) (delta-eta (eta M (deltaM (delta x d)))))))))))
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
146 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
147 deltaM (delta (mu M (fmap F (headDeltaM {M = M}) (eta M (deltaM (delta x d)))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
148 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta M (deltaM (delta x d)))))))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
149 ≡⟨ cong (\de -> deltaM (delta (mu M de)
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
150 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta M (deltaM (delta x d))))))))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
151 (sym (eta-is-nt M headDeltaM (deltaM (delta x d)))) ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
152 deltaM (delta (mu M (eta M (headDeltaM {M = M} (deltaM (delta x d)))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
153 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta M (deltaM (delta x d)))))))))
111
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
154 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
155 deltaM (delta (mu M (eta M x))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
156 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta M (deltaM (delta x d)))))))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
157 ≡⟨ cong (\de -> deltaM (delta de (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta M (deltaM (delta x d))))))))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
158 (sym (right-unity-law M x)) ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
159 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta M (deltaM (delta x d)))))))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
160 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
161 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM (delta-fmap (fmap F tailDeltaM) (delta-eta (eta M (deltaM (delta x d)))))))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
162 ≡⟨ cong (\de -> deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM de)))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
163 (sym (delta-eta-is-nt (fmap F tailDeltaM) (eta M (deltaM (delta x d))))) ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
164 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM (delta-eta (fmap F tailDeltaM (eta M (deltaM (delta x d)))))))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
165 ≡⟨ cong (\de -> deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM (delta-eta de))))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
166 (sym (eta-is-nt M tailDeltaM (deltaM (delta x d)))) ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
167 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM (delta-eta (eta M (tailDeltaM (deltaM (delta x d)))))))))
103
a271f3ff1922 Delte type dependencie in Monad record for escape implicit type conflict
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 102
diff changeset
168 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
169 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM (delta-eta (eta M (deltaM d)))))))
111
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
170 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
171 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM-eta (deltaM d)))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
172 ≡⟨ cong (\de -> deltaM (delta x (unDeltaM {M = M} de))) (deltaM-right-unity-law (deltaM d)) ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
173 deltaM (delta x (unDeltaM {M = M} (deltaM d)))
111
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
174 ≡⟨ refl ⟩
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
175 deltaM (delta x d)
102
9c62373bd474 Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 98
diff changeset
176 ∎
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
177
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
178
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
179
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
180 {-
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
181
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
182
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
183 postulate deltaM-left-unity-law : {l : Level} {A : Set l}
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
184 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
185 {n : Nat}
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
186 (d : DeltaM M {functorM} {monadM} A (S n)) ->
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
187 (deltaM-mu ∙ (deltaM-fmap deltaM-eta)) d ≡ id d
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
188 {-
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
189 deltaM-left-unity-law {l} {A} {M} {fm} {mm} {O} (deltaM (mono x)) = begin
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
190 deltaM-mu (deltaM-fmap deltaM-eta (deltaM (mono x)))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
191 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
192 deltaM-mu (deltaM (delta-fmap (fmap fm deltaM-eta) (mono x)))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
193 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
194 deltaM-mu (deltaM (mono (fmap fm deltaM-eta x)))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
195 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
196 deltaM (mono (mu mm (fmap fm (headDeltaM {l} {A} {O} {M}) (fmap fm deltaM-eta x))))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
197 ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (sym (covariant fm deltaM-eta headDeltaM x)) ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
198 deltaM (mono (mu mm (fmap fm ((headDeltaM {l} {A} {O} {M} {fm} {mm}) ∙ deltaM-eta) x)))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
199 ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (fmap-headDeltaM-with-deltaM-eta {l} {A} {O} {M} {fm} {mm} x) ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
200 deltaM (mono (mu mm (fmap fm (eta mm) x)))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
201 ≡⟨ cong (\de -> deltaM (mono de)) (left-unity-law mm x) ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
202 deltaM (mono x)
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
203 ∎
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
204 deltaM-left-unity-law {l} {A} {M} {fm} {mm} {S n} (deltaM (delta x d)) = begin
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
205 deltaM-mu (deltaM-fmap deltaM-eta (deltaM (delta x d)))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
206 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
207 deltaM-mu (deltaM (delta-fmap (fmap fm deltaM-eta) (delta x d)))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
208 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
209 deltaM-mu (deltaM (delta (fmap fm deltaM-eta x) (delta-fmap (fmap fm deltaM-eta) d)))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
210 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
211 appendDeltaM (deltaM (mono (mu mm (fmap fm (headDeltaM {l} {A} {S n} {M} {fm} {mm}) (fmap fm deltaM-eta x)))))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
212 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
213 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm de)))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
214 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
215 (sym (covariant fm deltaM-eta headDeltaM x)) ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
216 appendDeltaM (deltaM (mono (mu mm (fmap fm ((headDeltaM {l} {A} {S n} {M} {fm} {mm}) ∙ deltaM-eta) x))))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
217 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
218 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm de)))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
219 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
220 (fmap-headDeltaM-with-deltaM-eta {l} {A} {S n} {M} {fm} {mm} x) ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
221 appendDeltaM (deltaM (mono (mu mm (fmap fm (eta mm) x))))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
222 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
223
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
224 ≡⟨ cong (\de -> (appendDeltaM (deltaM (mono de)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
225 (left-unity-law mm x) ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
226 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
227 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
228 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap (tailDeltaM {n = n})(deltaM-fmap deltaM-eta (deltaM d))))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
229 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu de)) (sym (covariant deltaM-is-functor deltaM-eta tailDeltaM (deltaM d))) ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
230 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap ((tailDeltaM {n = n}) ∙ deltaM-eta) (deltaM d)))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
231 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu de)) (fmap-tailDeltaM-with-deltaM-eta (deltaM d)) ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
232 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap deltaM-eta (deltaM d)))
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
233 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) de) (deltaM-left-unity-law (deltaM d)) ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
234 appendDeltaM (deltaM (mono x)) (deltaM d)
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
235 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
236 deltaM (delta x d)
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
237 ∎
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
238
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
239 -}
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
240 postulate nya : {l : Level} {A : Set l}
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
241 (M : Set l -> Set l) (fm : Functor M) (mm : Monad M fm)
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
242 (d : DeltaM M {fm} {mm} (DeltaM M {fm} {mm} (DeltaM M {fm} {mm} A (S O)) (S O)) (S O)) ->
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
243 deltaM-mu (deltaM-fmap deltaM-mu d) ≡ deltaM-mu (deltaM-mu d)
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
244
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
245
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
246
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
247
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
248
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
249
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
250
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
251 deltaM-association-law : {l : Level} {A : Set l} {n : Nat}
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
252 (M : Set l -> Set l) (fm : Functor M) (mm : Monad M fm)
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
253 (d : DeltaM M {fm} {mm} (DeltaM M {fm} {mm} (DeltaM M {fm} {mm} A (S n)) (S n)) (S n)) ->
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
254 deltaM-mu (deltaM-fmap deltaM-mu d) ≡ deltaM-mu (deltaM-mu d)
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
255 deltaM-association-law {l} {A} {O} M fm mm (deltaM (mono x)) = nya {l} {A} M fm mm (deltaM (mono x))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
256 {-
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
257 begin
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
258 deltaM-mu (deltaM-fmap deltaM-mu (deltaM (mono x))) ≡⟨ refl ⟩
116
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
259 deltaM-mu (deltaM (mono (fmap fm deltaM-mu x))) ≡⟨ refl ⟩
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
260 deltaM (mono (mu mm (fmap fm headDeltaM (headDeltaM {A = DeltaM M A (S O)} {monadM = mm} (deltaM (mono (fmap fm deltaM-mu x))))))) ≡⟨ refl ⟩
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
261 deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x)))) ≡⟨ refl ⟩
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
262 deltaM (mono (mu mm (fmap fm (headDeltaM {A = A} {monadM = mm}) (fmap fm
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
263 (\d -> (deltaM (mono (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d)))))) x))))
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
264 ≡⟨ cong (\de -> deltaM (mono (mu mm de)))
116
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
265 (sym (covariant fm (\d -> (deltaM (mono (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d)))))) headDeltaM x)) ⟩
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
266 deltaM (mono (mu mm (fmap fm ((headDeltaM {A = A} {monadM = mm}) ∙
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
267 (\d -> (deltaM (mono (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d))))))) x)))
116
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
268 ≡⟨ refl ⟩
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
269 deltaM (mono (mu mm (fmap fm (\d -> (headDeltaM {A = A} {monadM = mm} (deltaM (mono (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d))))))) x)))
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
270 ≡⟨ refl ⟩
116
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
271 deltaM (mono (mu mm (fmap fm (\d -> (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d)))) x)))
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
272 ≡⟨ refl ⟩
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
273 deltaM (mono (mu mm (fmap fm ((mu mm) ∙ (((fmap fm headDeltaM)) ∙ ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm})))) x)))
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
274 ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (covariant fm ((fmap fm headDeltaM) ∙ (headDeltaM)) (mu mm) x )⟩
116
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
275 deltaM (mono (mu mm (((fmap fm (mu mm)) ∙ (fmap fm ((fmap fm headDeltaM) ∙ (headDeltaM {l} {DeltaM M A (S O)} {monadM = mm})))) x)))
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
276 ≡⟨ refl ⟩
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
277 deltaM (mono (mu mm (fmap fm (mu mm) ((fmap fm ((fmap fm headDeltaM) ∙ (headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}))) x))))
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
278 ≡⟨ cong (\de -> deltaM (mono (mu mm (fmap fm (mu mm) de)))) (covariant fm headDeltaM (fmap fm headDeltaM) x) ⟩
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
279 deltaM (mono (mu mm (fmap fm (mu mm) (((fmap fm (fmap fm headDeltaM)) ∙ (fmap fm (headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}))) x))))
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
280 ≡⟨ refl ⟩
116
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
281 deltaM (mono (mu mm (fmap fm (mu mm) (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x)))))
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
282 ≡⟨ cong (\de -> deltaM (mono de)) (association-law mm (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x))) ⟩
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
283 deltaM (mono (mu mm (mu mm (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x)))))
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
284 ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (mu-is-nt mm headDeltaM (fmap fm headDeltaM x)) ⟩
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
285 deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))) ≡⟨ refl ⟩
116
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
286 deltaM (mono (mu mm (fmap fm headDeltaM (headDeltaM {A = DeltaM M A (S O)} {monadM = mm} (deltaM (mono (mu mm (fmap fm headDeltaM x)))))))) ≡⟨ refl ⟩
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
287 deltaM-mu (deltaM (mono (mu mm (fmap fm headDeltaM x)))) ≡⟨ refl ⟩
116
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
288 deltaM-mu (deltaM (mono (mu mm (fmap fm headDeltaM (headDeltaM {A = DeltaM M (DeltaM M A (S O)) (S O)} {monadM = mm} (deltaM (mono x))))))) ≡⟨ refl ⟩
f02c5ad4a327 Prove association-law for DeltaM by (S O) pattern with definition changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
289 deltaM-mu (deltaM-mu (deltaM (mono x))) ∎
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
290 -}
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
291 deltaM-association-law {l} {A} {S n} M fm mm (deltaM (delta x d)) = begin
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
292 deltaM-mu (deltaM-fmap deltaM-mu (deltaM (delta x d)))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
293 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
294 deltaM-mu (deltaM (delta (fmap fm deltaM-mu x) (delta-fmap (fmap fm deltaM-mu) d)))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
295 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
296 deltaM (delta (mu mm (fmap fm (headDeltaM {A = A} {monadM = mm}) (headDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM (delta (fmap fm deltaM-mu x) (delta-fmap (fmap fm deltaM-mu) d))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
297 (unDeltaM {A = A} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM (deltaM (delta (fmap fm deltaM-mu x) (delta-fmap (fmap fm deltaM-mu) d))))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
298
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
299 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
300 deltaM (delta (mu mm (fmap fm (headDeltaM {A = A} {monadM = mm}) (fmap fm deltaM-mu x)))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
301 (unDeltaM {A = A} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
302 ≡⟨ cong (\de -> deltaM (delta (mu mm de) (unDeltaM {A = A} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d)))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
303 (sym (covariant fm deltaM-mu headDeltaM x)) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
304 deltaM (delta (mu mm (fmap fm ((headDeltaM {A = A} {monadM = mm}) ∙ deltaM-mu) x))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
305 (unDeltaM {A = A} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
306 ≡⟨ cong (\de -> deltaM (delta (mu mm de)
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
307 (unDeltaM {A = A} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d)))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
308 (fmap-headDeltaM-with-deltaM-mu {A = A} {monadM = mm} x) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
309 deltaM (delta (mu mm (fmap fm (((mu mm) ∙ (fmap fm headDeltaM)) ∙ headDeltaM) x))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
310 (unDeltaM {A = A} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
311 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
312 deltaM (delta (mu mm (fmap fm (((mu mm) ∙ (fmap fm headDeltaM)) ∙ headDeltaM) x))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
313 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM-fmap deltaM-mu (deltaM d))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
314 ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm (((mu mm) ∙ (fmap fm headDeltaM)) ∙ headDeltaM) x))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
315 (unDeltaM {monadM = mm} (deltaM-mu de))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
316 (sym (deltaM-covariant fm tailDeltaM deltaM-mu (deltaM d))) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
317 deltaM (delta (mu mm (fmap fm (((mu mm) ∙ (fmap fm headDeltaM)) ∙ headDeltaM) x))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
318 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap (tailDeltaM ∙ deltaM-mu) (deltaM d)))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
319 ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm (((mu mm) ∙ (fmap fm headDeltaM)) ∙ headDeltaM) x))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
320 (unDeltaM {monadM = mm} (deltaM-mu de))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
321 (fmap-tailDeltaM-with-deltaM-mu (deltaM d)) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
322 deltaM (delta (mu mm (fmap fm (((mu mm) ∙ (fmap fm headDeltaM)) ∙ headDeltaM) x))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
323 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
324 ≡⟨ cong (\de -> deltaM (delta (mu mm de)
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
325 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
326 (covariant fm headDeltaM ((mu mm) ∙ (fmap fm headDeltaM)) x) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
327 deltaM (delta (mu mm (((fmap fm ((mu mm) ∙ (fmap fm headDeltaM))) ∙ (fmap fm headDeltaM)) x))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
328 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
329 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
330 deltaM (delta (mu mm (((fmap fm ((mu mm) ∙ (fmap fm headDeltaM))) (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
331 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
332 ≡⟨ cong (\de -> deltaM (delta (mu mm de)
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
333 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
334 (covariant fm (fmap fm headDeltaM) (mu mm) (fmap fm headDeltaM x)) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
335
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
336 deltaM (delta (mu mm ((((fmap fm (mu mm)) ∙ (fmap fm (fmap fm headDeltaM))) (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
337 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
338 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
339 deltaM (delta (mu mm (fmap fm (mu mm) (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
340 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
341 ≡⟨ cong (\de -> deltaM (delta de (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
342 (association-law mm (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x))) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
343 deltaM (delta (mu mm (mu mm (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
344 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
345 ≡⟨ cong (\de -> deltaM (delta (mu mm de) (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
346 (mu-is-nt mm headDeltaM (fmap fm headDeltaM x)) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
347 deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
348 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
349 ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))) (unDeltaM {monadM = mm} (deltaM-mu de))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
350 (deltaM-covariant fm (deltaM-mu ∙ (deltaM-fmap tailDeltaM)) tailDeltaM (deltaM d)) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
351 deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
352 (unDeltaM {monadM = mm} (deltaM-mu (((deltaM-fmap (deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ (deltaM-fmap tailDeltaM)) (deltaM d))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
353 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
354 deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
355 (unDeltaM {monadM = mm} (deltaM-mu (((deltaM-fmap (deltaM-mu ∙ (deltaM-fmap tailDeltaM)) (deltaM-fmap tailDeltaM (deltaM d))))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
356 ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))) (unDeltaM {monadM = mm} (deltaM-mu de))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
357 (deltaM-covariant fm deltaM-mu (deltaM-fmap tailDeltaM) (deltaM-fmap tailDeltaM (deltaM d))) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
358 deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
359 (unDeltaM {monadM = mm} (deltaM-mu (((deltaM-fmap deltaM-mu) ∙ (deltaM-fmap (deltaM-fmap tailDeltaM))) (deltaM-fmap tailDeltaM (deltaM d))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
360 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
361 deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
362 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap deltaM-mu (deltaM-fmap (deltaM-fmap tailDeltaM) (deltaM-fmap tailDeltaM (deltaM d)))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
363 ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))) (unDeltaM {monadM = mm} de)))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
364 (deltaM-association-law M fm mm (deltaM-fmap (deltaM-fmap tailDeltaM) (deltaM-fmap tailDeltaM (deltaM d)))) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
365 deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
366 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-mu (deltaM-fmap (deltaM-fmap tailDeltaM) (deltaM-fmap tailDeltaM (deltaM d)))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
367
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
368 ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
369 (unDeltaM {monadM = mm} (deltaM-mu de))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
370 (sym (deltaM-mu-is-nt tailDeltaM (deltaM-fmap tailDeltaM (deltaM d)))) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
371 deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
372 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
373 ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
374 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM de)))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
375 (sym (deconstruct-id (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))) ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
376 deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
377 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
378 (deltaM (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
379
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
380
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
381 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
382 deltaM (delta (mu mm (fmap fm headDeltaM (headDeltaM {monadM = mm} ((deltaM (delta (mu mm (fmap fm headDeltaM x))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
383 (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
384 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM ((deltaM (delta (mu mm (fmap fm headDeltaM x))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
385 (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
386
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
387
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
388 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
389 deltaM-mu (deltaM (delta (mu mm (fmap fm headDeltaM x))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
390 (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
391 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
392 deltaM-mu (deltaM (delta (mu mm (fmap fm headDeltaM (headDeltaM {monadM = mm} (deltaM (delta x d)))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
393 (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM (deltaM (delta x d))))))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
394 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
395 deltaM-mu (deltaM-mu (deltaM (delta x d)))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
396 ∎
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
397 {-
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
398 deltaM-association-law {l} {A} {S n} M fm mm (deltaM (delta x d)) = begin
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
399 deltaM-mu (deltaM-fmap deltaM-mu (deltaM (delta x d))) ≡⟨ refl ⟩
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
400 deltaM-mu (deltaM (delta-fmap (fmap fm deltaM-mu) (delta x d))) ≡⟨ refl ⟩
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
401 deltaM-mu (deltaM (delta (fmap fm deltaM-mu x) (delta-fmap (fmap fm deltaM-mu) d))) ≡⟨ refl ⟩
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
402 appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x)))))
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
403 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d)))) ≡⟨ refl ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
404 appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x)))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
405 (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d))))
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
406 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x))))) de)
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
407 (sym (deconstruct-id (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d)))))) ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
408
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
409 appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x)))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
410 (deltaM (deconstruct {A = A} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d))))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
411 ≡⟨ refl ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
412 deltaM (deltaAppend (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
413 (deconstruct {A = A} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d))))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
414 ≡⟨ refl ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
415 deltaM (delta (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x)))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
416 (deconstruct {A = A} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d))))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
417 ≡⟨ {!!} ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
418 appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
419 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d)))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
420 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))))
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
421 (deltaM-mu (deltaM-fmap tailDeltaM de)))
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
422 (sym (deconstruct-id (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))) ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
423 appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
424 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d)))))))
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
425
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
426 ≡⟨ refl ⟩
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
427 appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))))
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
428 (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM ( (deltaM (delta (mu mm (fmap fm headDeltaM x))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
429 (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))))))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
430 ≡⟨ refl ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
431 appendDeltaM (deltaM (mono (mu mm (fmap fm (headDeltaM {monadM = mm}) (headDeltaM {monadM = mm} ((deltaM (delta (mu mm (fmap fm headDeltaM x))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
432 (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))))))))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
433 (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM ( (deltaM (delta (mu mm (fmap fm headDeltaM x))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
434 (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))))))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
435 ≡⟨ refl ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
436 deltaM-mu (deltaM (delta (mu mm (fmap fm headDeltaM x))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
437 (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
438 ≡⟨ refl ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
439 deltaM-mu (deltaM (deltaAppend (mono (mu mm (fmap fm headDeltaM x)))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
440 (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
441 ≡⟨ refl ⟩
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
442 deltaM-mu (appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x))))
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
443 (deltaM (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
444 ≡⟨ cong (\de -> deltaM-mu (appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x)))) de))
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
445 (deconstruct-id (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d)))) ⟩
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
446 deltaM-mu (appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x))))
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
447 (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
448 ≡⟨ refl ⟩
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
449 deltaM-mu (appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x))))
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
450 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))≡⟨ refl ⟩
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
451 deltaM-mu (deltaM-mu (deltaM (delta x d)))
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
452 ∎
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
453 -}
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
454
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
455
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
456
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
457 deltaM-is-monad : {l : Level} {A : Set l} {n : Nat}
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
458 {M : Set l -> Set l}
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
459 (functorM : Functor M)
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
460 (monadM : Monad M functorM) ->
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
461 Monad {l} (\A -> DeltaM M {functorM} {monadM} A (S n)) (deltaM-is-functor {l} {n})
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
462 deltaM-is-monad {l} {A} {n} {M} functorM monadM =
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
463 record { mu = deltaM-mu
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
464 ; eta = deltaM-eta
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
465 ; eta-is-nt = deltaM-eta-is-nt
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
466 ; mu-is-nt = (\f x -> (sym (deltaM-mu-is-nt f x)))
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
467 ; association-law = (deltaM-association-law M functorM monadM)
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
468 ; left-unity-law = deltaM-left-unity-law
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
469 ; right-unity-law = (\x -> (sym (deltaM-right-unity-law x)))
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
470 }
102
9c62373bd474 Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 98
diff changeset
471
9c62373bd474 Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 98
diff changeset
472
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
473 -}