Mercurial > hg > Members > atton > delta_monad
annotate agda/deltaM/monad.agda @ 118:53cb21845dea
Prove association-law for DeltaM
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Mon, 02 Feb 2015 11:54:23 +0900 |
parents | 6f86b55bf8b4 |
children | 48b44bd85056 |
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Trying Monad-laws for DeltaM
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1 open import Level |
b7f0879e854e
Trying Monad-laws for DeltaM
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2 open import Relation.Binary.PropositionalEquality |
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Trying Monad-laws for DeltaM
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3 open ≡-Reasoning |
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Trying Monad-laws for DeltaM
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4 |
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Trying Monad-laws for DeltaM
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5 open import basic |
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Trying Monad-laws for DeltaM
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6 open import delta |
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7 open import delta.functor |
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Prove right-unity-law on DeltaM
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8 open import delta.monad |
98
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Trying Monad-laws for DeltaM
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9 open import deltaM |
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10 open import deltaM.functor |
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11 open import nat |
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Trying Monad-laws for DeltaM
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12 open import laws |
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13 |
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Trying Monad-laws for DeltaM
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14 module deltaM.monad where |
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Trying Monad-laws for DeltaM
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15 open Functor |
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16 open NaturalTransformation |
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Trying Monad-laws for DeltaM
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17 open Monad |
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Trying Monad-laws for DeltaM
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18 |
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Trying Monad-laws for DeltaM
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19 |
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Prove natural transformation for deltaM-eta
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20 -- sub proofs |
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21 |
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22 deconstruct-id : {l : Level} {A : Set l} {n : Nat} |
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23 {M : Set l -> Set l} {fm : Functor M} {mm : Monad M fm} |
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24 (d : DeltaM M {fm} {mm} A (S n)) -> deltaM (unDeltaM d) ≡ d |
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25 deconstruct-id {n = O} (deltaM x) = refl |
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26 deconstruct-id {n = S n} (deltaM x) = refl |
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27 |
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28 |
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29 headDeltaM-with-appendDeltaM : {l : Level} {A : Set l} {n m : Nat} |
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30 {M : Set l -> Set l} {fm : Functor M} {mm : Monad M fm} |
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31 (d : DeltaM M {fm} {mm} A (S n)) -> (ds : DeltaM M {fm} {mm} A (S m)) -> |
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32 headDeltaM (appendDeltaM d ds) ≡ headDeltaM d |
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33 headDeltaM-with-appendDeltaM {l} {A} {n = O} {O} (deltaM (mono _)) (deltaM _) = refl |
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34 headDeltaM-with-appendDeltaM {l} {A} {n = O} {S m} (deltaM (mono _)) (deltaM _) = refl |
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35 headDeltaM-with-appendDeltaM {l} {A} {n = S n} {O} (deltaM (delta _ _)) (deltaM _) = refl |
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36 headDeltaM-with-appendDeltaM {l} {A} {n = S n} {S m} (deltaM (delta _ _)) (deltaM _) = refl |
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Prove natural transformation for deltaM-eta
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38 fmap-headDeltaM-with-deltaM-eta : {l : Level} {A : Set l} {n : Nat} |
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39 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} |
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40 (x : M A) -> (fmap functorM ((headDeltaM {l} {A} {n} {M} {functorM} {monadM}) ∙ deltaM-eta) x) ≡ fmap functorM (eta monadM) x |
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Prove natural transformation for deltaM-eta
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41 fmap-headDeltaM-with-deltaM-eta {l} {A} {O} {M} {fm} {mm} x = refl |
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42 fmap-headDeltaM-with-deltaM-eta {l} {A} {S n} {M} {fm} {mm} x = refl |
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43 |
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Prove natural transformation for deltaM-eta
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44 |
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45 fmap-tailDeltaM-with-deltaM-eta : {l : Level} {A : Set l} {n : Nat} |
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46 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} |
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47 (d : DeltaM M {functorM} {monadM} A (S n)) -> |
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48 deltaM-fmap ((tailDeltaM {n = n} {monadM = monadM} ) ∙ deltaM-eta) d ≡ deltaM-fmap (deltaM-eta) d |
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49 fmap-tailDeltaM-with-deltaM-eta {n = O} d = refl |
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50 fmap-tailDeltaM-with-deltaM-eta {n = S n} d = refl |
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51 |
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52 fmap-headDeltaM-with-deltaM-mu : {l : Level} {A : Set l} {n : Nat} |
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53 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} |
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54 (x : M (DeltaM M (DeltaM M {functorM} {monadM} A (S n)) (S n))) -> |
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55 fmap functorM (headDeltaM ∙ deltaM-mu) x ≡ fmap functorM (((mu monadM) ∙ (fmap functorM headDeltaM)) ∙ headDeltaM) x |
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56 fmap-headDeltaM-with-deltaM-mu {n = O} x = refl |
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57 fmap-headDeltaM-with-deltaM-mu {n = S n} x = refl |
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58 |
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59 |
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60 fmap-tailDeltaM-with-deltaM-mu : {l : Level} {A : Set l} {n : Nat} |
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61 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} |
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62 (d : DeltaM M {functorM} {monadM} (DeltaM M {functorM} {monadM} (DeltaM M A (S (S n))) (S (S n))) (S n)) -> |
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63 deltaM-fmap (tailDeltaM ∙ deltaM-mu) d ≡ deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) d |
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64 fmap-tailDeltaM-with-deltaM-mu {n = O} (deltaM (mono x)) = refl |
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65 fmap-tailDeltaM-with-deltaM-mu {n = S n} (deltaM d) = refl |
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66 |
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67 |
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68 |
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69 |
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71 -- main proofs |
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73 postulate deltaM-eta-is-nt : {l : Level} {A B : Set l} {n : Nat} |
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74 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} |
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75 (f : A -> B) -> (x : A) -> |
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76 ((deltaM-eta {l} {B} {n} {M} {functorM} {monadM} )∙ f) x ≡ deltaM-fmap f (deltaM-eta x) |
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77 {- |
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78 deltaM-eta-is-nt {l} {A} {B} {O} {M} {fm} {mm} f x = begin |
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79 deltaM-eta {n = O} (f x) ≡⟨ refl ⟩ |
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80 deltaM (mono (eta mm (f x))) ≡⟨ cong (\de -> deltaM (mono de)) (eta-is-nt mm f x) ⟩ |
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81 deltaM (mono (fmap fm f (eta mm x))) ≡⟨ refl ⟩ |
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82 deltaM-fmap f (deltaM-eta {n = O} x) ∎ |
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83 deltaM-eta-is-nt {l} {A} {B} {S n} {M} {fm} {mm} f x = begin |
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84 deltaM-eta {n = S n} (f x) ≡⟨ refl ⟩ |
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85 deltaM (delta-eta {n = S n} (eta mm (f x))) ≡⟨ refl ⟩ |
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86 deltaM (delta (eta mm (f x)) (delta-eta (eta mm (f x)))) |
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87 ≡⟨ cong (\de -> deltaM (delta de (delta-eta de))) (eta-is-nt mm f x) ⟩ |
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88 deltaM (delta (fmap fm f (eta mm x)) (delta-eta (fmap fm f (eta mm x)))) |
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89 ≡⟨ 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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90 deltaM (delta (fmap fm f (eta mm x)) (delta-fmap (fmap fm f) (delta-eta (eta mm x)))) |
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91 ≡⟨ refl ⟩ |
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92 deltaM-fmap f (deltaM-eta {n = S n} x) |
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93 ∎ |
115
e6bcc7467335
Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
114
diff
changeset
|
94 -} |
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
|
95 |
118
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
96 postulate deltaM-mu-is-nt : {l : Level} {A B : Set l} {n : Nat} |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
97 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
98 (f : A -> B) -> |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
99 (d : DeltaM T {F} {M} (DeltaM T A (S n)) (S n)) -> |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
100 deltaM-fmap f (deltaM-mu d) ≡ deltaM-mu (deltaM-fmap (deltaM-fmap f) d) |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
101 |
112
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
102 postulate deltaM-right-unity-law : {l : Level} {A : Set l} |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
103 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} {n : Nat} |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
104 (d : DeltaM M {functorM} {monadM} A (S n)) -> (deltaM-mu ∙ deltaM-eta) d ≡ id d |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
105 {- |
111
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
106 deltaM-right-unity-law {l} {A} {M} {fm} {mm} {O} (deltaM (mono x)) = begin |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
107 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
|
108 deltaM-mu (deltaM (mono (eta mm (deltaM (mono x))))) ≡⟨ refl ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
109 deltaM (mono (mu mm (fmap fm (headDeltaM {M = M})(eta mm (deltaM (mono x)))))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
110 ≡⟨ 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
|
111 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
|
112 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
|
113 deltaM (mono x) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
114 ∎ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
115 deltaM-right-unity-law {l} {A} {M} {fm} {mm} {S n} (deltaM (delta x d)) = begin |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
116 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
|
117 ≡⟨ refl ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
118 deltaM-mu (deltaM (delta (eta mm (deltaM (delta x d))) (delta-eta (eta mm (deltaM (delta x d)))))) |
104
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
119 ≡⟨ refl ⟩ |
112
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
120 appendDeltaM (deltaM (mono (mu mm (fmap fm (headDeltaM {monadM = mm}) (eta mm (deltaM (delta x d))))))) |
111
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
121 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d))))))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
122 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm de))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
123 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d)))))))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
124 (sym (eta-is-nt mm headDeltaM (deltaM (delta x d)))) ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
125 appendDeltaM (deltaM (mono (mu mm (eta mm ((headDeltaM {monadM = mm}) (deltaM (delta x d))))))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
126 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d))))))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
127 ≡⟨ refl ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
128 appendDeltaM (deltaM (mono (mu mm (eta mm x)))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
129 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d))))))) |
112
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
130 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono de)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d)))))))) |
111
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
131 (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
|
132 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (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
|
133 ≡⟨ refl ⟩ |
111
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
134 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-eta (eta mm (deltaM (delta x d))))))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
135 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM de))) (sym (eta-is-nt delta-is-monad (fmap fm tailDeltaM) (eta mm (deltaM (delta x d))))) ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
136 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-eta (fmap fm tailDeltaM (eta mm (deltaM (delta x d))))))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
137 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-eta de)))) (sym (eta-is-nt mm tailDeltaM (deltaM (delta x d)))) ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
138 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-eta (eta mm (tailDeltaM (deltaM (delta x d))))))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
139 ≡⟨ refl ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
140 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-eta (eta mm (deltaM d))))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
141 ≡⟨ refl ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
142 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-eta (deltaM d))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
143 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) de) (deltaM-right-unity-law (deltaM d)) ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
144 appendDeltaM (deltaM (mono x)) (deltaM d) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
145 ≡⟨ refl ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
146 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
|
147 ∎ |
112
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
148 -} |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
149 |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
150 |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
151 |
104
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
152 |
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
153 |
114
08403eb8db8b
Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
112
diff
changeset
|
154 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
|
155 {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
|
156 {n : Nat} |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
157 (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
|
158 (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
|
159 {- |
112
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
160 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
|
161 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
|
162 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
163 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
|
164 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
165 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
|
166 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
167 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
|
168 ≡⟨ 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
|
169 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
|
170 ≡⟨ 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
|
171 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
|
172 ≡⟨ 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
|
173 deltaM (mono x) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
174 ∎ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
175 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
|
176 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
|
177 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
178 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
|
179 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
180 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
|
181 ≡⟨ refl ⟩ |
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Prove left-unity-law for DeltaM
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182 appendDeltaM (deltaM (mono (mu mm (fmap fm (headDeltaM {l} {A} {S n} {M} {fm} {mm}) (fmap fm deltaM-eta x))))) |
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Prove left-unity-law for DeltaM
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183 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))) |
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Prove left-unity-law for DeltaM
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184 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm de))) |
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Prove left-unity-law for DeltaM
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185 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))) |
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Prove left-unity-law for DeltaM
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186 (sym (covariant fm deltaM-eta headDeltaM x)) ⟩ |
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Prove left-unity-law for DeltaM
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187 appendDeltaM (deltaM (mono (mu mm (fmap fm ((headDeltaM {l} {A} {S n} {M} {fm} {mm}) ∙ deltaM-eta) x)))) |
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Prove left-unity-law for DeltaM
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188 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))) |
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Prove left-unity-law for DeltaM
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189 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm de))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
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190 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))) |
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Prove left-unity-law for DeltaM
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191 (fmap-headDeltaM-with-deltaM-eta {l} {A} {S n} {M} {fm} {mm} x) ⟩ |
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Prove left-unity-law for DeltaM
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192 appendDeltaM (deltaM (mono (mu mm (fmap fm (eta mm) x)))) |
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Prove left-unity-law for DeltaM
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193 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))) |
104
ebd0d6e2772c
Trying redenition Delta with length constraints
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103
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194 |
112
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Prove left-unity-law for DeltaM
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195 ≡⟨ cong (\de -> (appendDeltaM (deltaM (mono de)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))))) |
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Prove left-unity-law for DeltaM
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196 (left-unity-law mm x) ⟩ |
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Prove left-unity-law for DeltaM
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197 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))) |
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Prove left-unity-law for DeltaM
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198 ≡⟨ refl ⟩ |
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199 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap (tailDeltaM {n = n})(deltaM-fmap deltaM-eta (deltaM d)))) |
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Prove left-unity-law for DeltaM
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111
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200 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu de)) (sym (covariant deltaM-is-functor deltaM-eta tailDeltaM (deltaM d))) ⟩ |
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Prove left-unity-law for DeltaM
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201 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap ((tailDeltaM {n = n}) ∙ deltaM-eta) (deltaM d))) |
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Prove left-unity-law for DeltaM
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202 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu de)) (fmap-tailDeltaM-with-deltaM-eta (deltaM d)) ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
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203 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap deltaM-eta (deltaM d))) |
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Prove left-unity-law for DeltaM
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204 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) de) (deltaM-left-unity-law (deltaM d)) ⟩ |
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Prove left-unity-law for DeltaM
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205 appendDeltaM (deltaM (mono x)) (deltaM d) |
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Prove left-unity-law for DeltaM
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206 ≡⟨ refl ⟩ |
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Prove left-unity-law for DeltaM
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207 deltaM (delta x d) |
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208 ∎ |
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Prove left-unity-law for DeltaM
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209 |
114
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Prove natural transformation for deltaM-eta
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112
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210 -} |
117
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211 postulate nya : {l : Level} {A : Set l} |
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212 (M : Set l -> Set l) (fm : Functor M) (mm : Monad M fm) |
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213 (d : DeltaM M {fm} {mm} (DeltaM M {fm} {mm} (DeltaM M {fm} {mm} A (S O)) (S O)) (S O)) -> |
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214 deltaM-mu (deltaM-fmap deltaM-mu d) ≡ deltaM-mu (deltaM-mu d) |
114
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Prove natural transformation for deltaM-eta
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112
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215 |
112
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Prove left-unity-law for DeltaM
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216 |
117
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217 |
115
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218 |
117
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219 |
115
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220 |
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221 |
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222 deltaM-association-law : {l : Level} {A : Set l} {n : Nat} |
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223 (M : Set l -> Set l) (fm : Functor M) (mm : Monad M fm) |
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224 (d : DeltaM M {fm} {mm} (DeltaM M {fm} {mm} (DeltaM M {fm} {mm} A (S n)) (S n)) (S n)) -> |
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225 deltaM-mu (deltaM-fmap deltaM-mu d) ≡ deltaM-mu (deltaM-mu d) |
117
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226 deltaM-association-law {l} {A} {O} M fm mm (deltaM (mono x)) = nya {l} {A} M fm mm (deltaM (mono x)) |
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227 {- |
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228 begin |
115
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229 deltaM-mu (deltaM-fmap deltaM-mu (deltaM (mono x))) ≡⟨ refl ⟩ |
116
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Prove association-law for DeltaM by (S O) pattern with definition changes
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230 deltaM-mu (deltaM (mono (fmap fm deltaM-mu x))) ≡⟨ refl ⟩ |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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231 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
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232 deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x)))) ≡⟨ refl ⟩ |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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233 deltaM (mono (mu mm (fmap fm (headDeltaM {A = A} {monadM = mm}) (fmap fm |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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234 (\d -> (deltaM (mono (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d)))))) x)))) |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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235 ≡⟨ cong (\de -> deltaM (mono (mu mm de))) |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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236 (sym (covariant fm (\d -> (deltaM (mono (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d)))))) headDeltaM x)) ⟩ |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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237 deltaM (mono (mu mm (fmap fm ((headDeltaM {A = A} {monadM = mm}) ∙ |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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238 (\d -> (deltaM (mono (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
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239 ≡⟨ refl ⟩ |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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240 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
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241 ≡⟨ refl ⟩ |
116
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Prove association-law for DeltaM by (S O) pattern with definition changes
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242 deltaM (mono (mu mm (fmap fm (\d -> (mu mm (fmap fm headDeltaM ((headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}) d)))) x))) |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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243 ≡⟨ refl ⟩ |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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244 deltaM (mono (mu mm (fmap fm ((mu mm) ∙ (((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
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245 ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (covariant fm ((fmap fm headDeltaM) ∙ (headDeltaM)) (mu mm) x )⟩ |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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246 deltaM (mono (mu mm (((fmap fm (mu mm)) ∙ (fmap fm ((fmap fm headDeltaM) ∙ (headDeltaM {l} {DeltaM M A (S O)} {monadM = mm})))) x))) |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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247 ≡⟨ refl ⟩ |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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248 deltaM (mono (mu mm (fmap fm (mu mm) ((fmap fm ((fmap fm headDeltaM) ∙ (headDeltaM {l} {DeltaM M A (S O)} {monadM = mm}))) x)))) |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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249 ≡⟨ cong (\de -> deltaM (mono (mu mm (fmap fm (mu mm) de)))) (covariant fm headDeltaM (fmap fm headDeltaM) x) ⟩ |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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250 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)))) |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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251 ≡⟨ refl ⟩ |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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252 deltaM (mono (mu mm (fmap fm (mu mm) (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x))))) |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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253 ≡⟨ cong (\de -> deltaM (mono de)) (association-law mm (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x))) ⟩ |
115
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254 deltaM (mono (mu mm (mu mm (fmap fm (fmap fm headDeltaM) (fmap fm headDeltaM x))))) |
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255 ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (mu-is-nt mm headDeltaM (fmap fm headDeltaM x)) ⟩ |
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256 deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x))))) ≡⟨ refl ⟩ |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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257 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 ⟩ |
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258 deltaM-mu (deltaM (mono (mu mm (fmap fm headDeltaM x)))) ≡⟨ refl ⟩ |
116
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Prove association-law for DeltaM by (S O) pattern with definition changes
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259 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 ⟩ |
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Prove association-law for DeltaM by (S O) pattern with definition changes
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260 deltaM-mu (deltaM-mu (deltaM (mono x))) ∎ |
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|
261 -} |
118
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
262 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
|
263 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
|
264 ≡⟨ refl ⟩ |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
265 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
|
266 ≡⟨ refl ⟩ |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
267 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
|
268 (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
|
269 |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
270 ≡⟨ refl ⟩ |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
271 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
|
272 (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
|
273 ≡⟨ 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
|
274 (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
|
275 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
|
276 (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
|
277 ≡⟨ 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
|
278 (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
|
279 (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
|
280 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
|
281 (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
|
282 ≡⟨ refl ⟩ |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
283 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
|
284 (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
|
285 ≡⟨ 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
|
286 (unDeltaM {monadM = mm} (deltaM-mu de)))) |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
287 (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
|
288 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
|
289 (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
|
290 ≡⟨ 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
|
291 (unDeltaM {monadM = mm} (deltaM-mu de)))) |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
292 (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
|
293 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
|
294 (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
|
295 ≡⟨ 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
|
296 (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
|
297 (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
|
298 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
|
299 (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
|
300 ≡⟨ refl ⟩ |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
301 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
|
302 (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
|
303 ≡⟨ 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
|
304 (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
|
305 (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
|
306 |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
307 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
|
308 (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
|
309 ≡⟨ refl ⟩ |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
310 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
|
311 (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
|
312 ≡⟨ 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
|
313 (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
|
314 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
|
315 (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
|
316 ≡⟨ 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
|
317 (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
|
318 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
|
319 (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
|
320 ≡⟨ 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
|
321 (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
|
322 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
|
323 (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
|
324 ≡⟨ refl ⟩ |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
325 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
|
326 (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
|
327 ≡⟨ 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
|
328 (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
|
329 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
|
330 (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
|
331 ≡⟨ refl ⟩ |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
332 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
|
333 (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
|
334 ≡⟨ 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
|
335 (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
|
336 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
|
337 (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
|
338 |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
339 ≡⟨ 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
|
340 (unDeltaM {monadM = mm} (deltaM-mu de)))) |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
341 (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
|
342 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
|
343 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))) |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
diff
changeset
|
344 ≡⟨ cong (\de -> deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))) |
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Prove association-law for DeltaM
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parents:
117
diff
changeset
|
345 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM de))))) |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
diff
changeset
|
346 (sym (deconstruct-id (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))) ⟩ |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
diff
changeset
|
347 deltaM (delta (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))) |
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Prove association-law for DeltaM
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parents:
117
diff
changeset
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348 (unDeltaM {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM |
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Prove association-law for DeltaM
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parents:
117
diff
changeset
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349 (deltaM (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))))) |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
diff
changeset
|
350 |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
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changeset
|
351 |
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Prove association-law for DeltaM
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117
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|
352 ≡⟨ refl ⟩ |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
diff
changeset
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353 deltaM (delta (mu mm (fmap fm headDeltaM (headDeltaM {monadM = mm} ((deltaM (delta (mu mm (fmap fm headDeltaM x)) |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
diff
changeset
|
354 (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
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355 (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
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356 (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))))))))))) |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
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changeset
|
357 |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
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changeset
|
358 |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
diff
changeset
|
359 ≡⟨ refl ⟩ |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
diff
changeset
|
360 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
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changeset
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361 (unDeltaM {A = DeltaM M A (S (S n))} {monadM = mm} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))))) |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
diff
changeset
|
362 ≡⟨ refl ⟩ |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
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363 deltaM-mu (deltaM (delta (mu mm (fmap fm headDeltaM (headDeltaM {monadM = mm} (deltaM (delta x d))))) |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
diff
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364 (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
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parents:
117
diff
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365 ≡⟨ refl ⟩ |
53cb21845dea
Prove association-law for DeltaM
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parents:
117
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366 deltaM-mu (deltaM-mu (deltaM (delta x d))) |
53cb21845dea
Prove association-law for DeltaM
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117
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367 ∎ |
117
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368 {- |
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369 deltaM-association-law {l} {A} {S n} M fm mm (deltaM (delta x d)) = begin |
115
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370 deltaM-mu (deltaM-fmap deltaM-mu (deltaM (delta x d))) ≡⟨ refl ⟩ |
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371 deltaM-mu (deltaM (delta-fmap (fmap fm deltaM-mu) (delta x d))) ≡⟨ refl ⟩ |
e6bcc7467335
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372 deltaM-mu (deltaM (delta (fmap fm deltaM-mu x) (delta-fmap (fmap fm deltaM-mu) d))) ≡⟨ refl ⟩ |
e6bcc7467335
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114
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changeset
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373 appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x))))) |
117
6f86b55bf8b4
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374 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-mu) d)))) ≡⟨ refl ⟩ |
6f86b55bf8b4
Temporary commit : Proving association-law ....
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116
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changeset
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375 appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x))))) |
6f86b55bf8b4
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376 (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d)))) |
6f86b55bf8b4
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377 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x))))) de) |
6f86b55bf8b4
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378 (sym (deconstruct-id (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d)))))) ⟩ |
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changeset
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379 |
6f86b55bf8b4
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380 appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x))))) |
6f86b55bf8b4
Temporary commit : Proving association-law ....
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381 (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 ....
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parents:
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changeset
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382 ≡⟨ refl ⟩ |
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383 deltaM (deltaAppend (mono (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x)))) |
6f86b55bf8b4
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384 (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 ....
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parents:
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changeset
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385 ≡⟨ refl ⟩ |
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386 deltaM (delta (mu mm (fmap fm headDeltaM (fmap fm deltaM-mu x))) |
6f86b55bf8b4
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387 (deconstruct {A = A} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-fmap (fmap fm deltaM-mu) d)))))) |
6f86b55bf8b4
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388 ≡⟨ {!!} ⟩ |
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389 appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))))) |
6f86b55bf8b4
Temporary commit : Proving association-law ....
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390 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))) |
6f86b55bf8b4
Temporary commit : Proving association-law ....
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parents:
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391 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))))) |
6f86b55bf8b4
Temporary commit : Proving association-law ....
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392 (deltaM-mu (deltaM-fmap tailDeltaM de))) |
6f86b55bf8b4
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393 (sym (deconstruct-id (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d))))) ⟩ |
6f86b55bf8b4
Temporary commit : Proving association-law ....
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394 appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))))) |
6f86b55bf8b4
Temporary commit : Proving association-law ....
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parents:
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395 (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
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parents:
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diff
changeset
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396 |
117
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parents:
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397 ≡⟨ refl ⟩ |
115
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parents:
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398 appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM (mu mm (fmap fm headDeltaM x)))))) |
117
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parents:
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399 (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM ( (deltaM (delta (mu mm (fmap fm headDeltaM x)) |
6f86b55bf8b4
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parents:
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400 (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 ....
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401 ≡⟨ refl ⟩ |
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402 appendDeltaM (deltaM (mono (mu mm (fmap fm (headDeltaM {monadM = mm}) (headDeltaM {monadM = mm} ((deltaM (delta (mu mm (fmap fm headDeltaM x)) |
6f86b55bf8b4
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parents:
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403 (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 ....
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parents:
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404 (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM ( (deltaM (delta (mu mm (fmap fm headDeltaM x)) |
6f86b55bf8b4
Temporary commit : Proving association-law ....
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parents:
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405 (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 ....
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406 ≡⟨ refl ⟩ |
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407 deltaM-mu (deltaM (delta (mu mm (fmap fm headDeltaM x)) |
6f86b55bf8b4
Temporary commit : Proving association-law ....
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408 (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 ....
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409 ≡⟨ refl ⟩ |
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410 deltaM-mu (deltaM (deltaAppend (mono (mu mm (fmap fm headDeltaM x))) |
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411 (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d)))))) |
6f86b55bf8b4
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412 ≡⟨ refl ⟩ |
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413 deltaM-mu (appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x)))) |
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414 (deltaM (deconstruct {A = DeltaM M A (S (S n))} {mm = mm} (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d)))))) |
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Temporary commit : Proving association-law ....
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415 ≡⟨ cong (\de -> deltaM-mu (appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x)))) de)) |
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Temporary commit : Proving association-law ....
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416 (deconstruct-id (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d)))) ⟩ |
6f86b55bf8b4
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417 deltaM-mu (appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x)))) |
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418 (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) d)))) |
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419 ≡⟨ refl ⟩ |
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420 deltaM-mu (appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x)))) |
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421 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))≡⟨ refl ⟩ |
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422 deltaM-mu (deltaM-mu (deltaM (delta x d))) |
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423 ∎ |
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424 -} |
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425 |
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426 |
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Trying redenition Delta with length constraints
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427 |
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428 deltaM-is-monad : {l : Level} {A : Set l} {n : Nat} |
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429 {M : Set l -> Set l} |
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430 (functorM : Functor M) |
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431 (monadM : Monad M functorM) -> |
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432 Monad {l} (\A -> DeltaM M {functorM} {monadM} A (S n)) (deltaM-is-functor {l} {n}) |
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433 deltaM-is-monad {l} {A} {n} {M} functorM monadM = |
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434 record { mu = deltaM-mu; |
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435 eta = deltaM-eta; |
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436 return = deltaM-eta; |
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437 bind = deltaM-bind; |
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438 association-law = (deltaM-association-law M functorM monadM) ; |
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439 left-unity-law = deltaM-left-unity-law; |
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440 right-unity-law = (\x -> (sym (deltaM-right-unity-law x))) ; |
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441 eta-is-nt = deltaM-eta-is-nt; |
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442 mu-is-nt = (\f x -> (sym (deltaM-mu-is-nt f x)))} |
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443 |
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444 |