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Prove mu-is-nt for DeltaM with fmap-equiv
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Tue, 03 Feb 2015 11:45:33 +0900
parents 6dcc68ef8f96
children d56596e4e784
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b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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1 open import Level
b7f0879e854e Trying Monad-laws for DeltaM
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2 open import Relation.Binary.PropositionalEquality
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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3 open ≡-Reasoning
b7f0879e854e Trying Monad-laws for DeltaM
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4
b7f0879e854e Trying Monad-laws for DeltaM
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5 open import basic
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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6 open import delta
b7f0879e854e Trying Monad-laws for DeltaM
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7 open import delta.functor
111
9fe3d0bd1149 Prove right-unity-law on DeltaM
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parents: 104
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8 open import delta.monad
98
b7f0879e854e Trying Monad-laws for DeltaM
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9 open import deltaM
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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10 open import deltaM.functor
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
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11 open import nat
98
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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12 open import laws
b7f0879e854e Trying Monad-laws for DeltaM
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13
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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14 module deltaM.monad where
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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15 open Functor
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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16 open NaturalTransformation
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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17 open Monad
b7f0879e854e Trying Monad-laws for DeltaM
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18
b7f0879e854e Trying Monad-laws for DeltaM
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19
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
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20 -- sub proofs
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
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21
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
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22 deconstruct-id : {l : Level} {A : Set l} {n : Nat}
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
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parents: 118
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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
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
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24 (d : DeltaM M A (S n)) -> deltaM (unDeltaM d) ≡ d
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
25 deconstruct-id {n = O} (deltaM x) = refl
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
26 deconstruct-id {n = S n} (deltaM x) = refl
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
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27
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
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28 headDeltaM-with-f : {l : Level} {A B : Set l} {n : Nat}
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
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parents: 125
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29 {T : Set l -> Set l} {F : Functor T} {M : Monad T F}
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
30 (f : A -> B) -> (x : (DeltaM M A (S n))) ->
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
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parents: 125
diff changeset
31 ((fmap F f) ∙ headDeltaM) x ≡ (headDeltaM ∙ (deltaM-fmap f)) x
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
32 headDeltaM-with-f {n = O} f (deltaM (mono x)) = refl
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
33 headDeltaM-with-f {n = S n} f (deltaM (delta x d)) = refl
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
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parents: 125
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34
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
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parents: 125
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35 tailDeltaM-with-f : {l : Level} {A B : Set l} {n : Nat}
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
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parents: 125
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36 {T : Set l -> Set l} {F : Functor T} {M : Monad T F}
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
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parents: 125
diff changeset
37 (f : A -> B) -> (d : (DeltaM M A (S (S n)))) ->
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
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38 (tailDeltaM ∙ (deltaM-fmap f)) d ≡ ((deltaM-fmap f) ∙ tailDeltaM) d
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
39 tailDeltaM-with-f {n = O} f (deltaM (delta x d)) = refl
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
40 tailDeltaM-with-f {n = S n} f (deltaM (delta x d)) = refl
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
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41
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
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42
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08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
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43 fmap-headDeltaM-with-deltaM-eta : {l : Level} {A : Set l} {n : Nat}
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
44 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
45 (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
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
46 fmap-headDeltaM-with-deltaM-eta {n = O} x = refl
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
47 fmap-headDeltaM-with-deltaM-eta {n = S n} x = refl
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
48
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08403eb8db8b Prove natural transformation for deltaM-eta
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parents: 112
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49
08403eb8db8b Prove natural transformation for deltaM-eta
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parents: 112
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50
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
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51 fmap-tailDeltaM-with-deltaM-eta : {l : Level} {A : Set l} {n : Nat}
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
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parents: 118
diff changeset
52 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
53 (d : DeltaM M A (S n)) ->
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
54 deltaM-fmap ((tailDeltaM {n = n} {M = M} ) ∙ deltaM-eta) d ≡ deltaM-fmap (deltaM-eta) d
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
55 fmap-tailDeltaM-with-deltaM-eta {n = O} d = refl
114
08403eb8db8b Prove natural transformation for deltaM-eta
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parents: 112
diff changeset
56 fmap-tailDeltaM-with-deltaM-eta {n = S n} d = refl
08403eb8db8b Prove natural transformation for deltaM-eta
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parents: 112
diff changeset
57
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48b44bd85056 Fix proof natural transformation for deltaM-eta
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parents: 118
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58
48b44bd85056 Fix proof natural transformation for deltaM-eta
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parents: 118
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59
118
53cb21845dea Prove association-law for DeltaM
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parents: 117
diff changeset
60 fmap-headDeltaM-with-deltaM-mu : {l : Level} {A : Set l} {n : Nat}
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
61 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
62 (x : T (DeltaM M (DeltaM M A (S n)) (S n))) ->
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
63 fmap F (headDeltaM ∙ deltaM-mu) x ≡ fmap F (((mu M) ∙ (fmap F headDeltaM)) ∙ headDeltaM) x
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
64 fmap-headDeltaM-with-deltaM-mu {n = O} x = refl
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
65 fmap-headDeltaM-with-deltaM-mu {n = S n} x = refl
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08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
66
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53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
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67
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
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68 fmap-tailDeltaM-with-deltaM-mu : {l : Level} {A : Set l} {n : Nat}
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48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
69 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
70 (d : DeltaM M (DeltaM M (DeltaM M A (S (S n))) (S (S n))) (S n)) ->
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
71 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
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
72 fmap-tailDeltaM-with-deltaM-mu {n = O} (deltaM (mono x)) = refl
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
73 fmap-tailDeltaM-with-deltaM-mu {n = S n} (deltaM d) = refl
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
74
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
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75
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6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
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6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
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6f86b55bf8b4 Temporary commit : Proving association-law ....
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parents: 116
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78
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08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
79 -- main proofs
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
80
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
81 deltaM-eta-is-nt : {l : Level} {A B : Set l} {n : Nat}
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
82 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
83 (f : A -> B) -> (x : A) ->
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48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
84 ((deltaM-eta {l} {B} {n} {T} {F} {M} )∙ f) x ≡ deltaM-fmap f (deltaM-eta x)
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08403eb8db8b Prove natural transformation for deltaM-eta
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parents: 112
diff changeset
85 deltaM-eta-is-nt {l} {A} {B} {O} {M} {fm} {mm} f x = begin
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
86 deltaM-eta {n = O} (f x) ≡⟨ refl ⟩
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
87 deltaM (mono (eta mm (f x))) ≡⟨ cong (\de -> deltaM (mono de)) (eta-is-nt mm f x) ⟩
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
88 deltaM (mono (fmap fm f (eta mm x))) ≡⟨ refl ⟩
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
89 deltaM-fmap f (deltaM-eta {n = O} x) ∎
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
90 deltaM-eta-is-nt {l} {A} {B} {S n} {M} {fm} {mm} f x = begin
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
91 deltaM-eta {n = S n} (f x)
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
92 ≡⟨ refl ⟩
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
93 deltaM (delta-eta {n = S n} (eta mm (f x)))
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
94 ≡⟨ refl ⟩
114
08403eb8db8b Prove natural transformation for deltaM-eta
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parents: 112
diff changeset
95 deltaM (delta (eta mm (f x)) (delta-eta (eta mm (f x))))
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
96 ≡⟨ cong (\de -> deltaM (delta de (delta-eta de))) (eta-is-nt mm f x) ⟩
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
97 deltaM (delta (fmap fm f (eta mm x)) (delta-eta (fmap fm f (eta mm x))))
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
98 ≡⟨ cong (\de -> deltaM (delta (fmap fm f (eta mm x)) de)) (eta-is-nt delta-is-monad (fmap fm f) (eta mm x)) ⟩
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
99 deltaM (delta (fmap fm f (eta mm x)) (delta-fmap (fmap fm f) (delta-eta (eta mm x))))
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
100 ≡⟨ refl ⟩
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
101 deltaM-fmap f (deltaM-eta {n = S n} x)
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
102 ∎
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
103
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
104
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6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
105
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
106
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
107 deltaM-mu-is-nt : {l : Level} {A B : Set l} {n : Nat}
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
108 {T : Set l -> Set l} {F : Functor T} {M : Monad T F}
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
109 (f : A -> B) -> (d : DeltaM M (DeltaM M A (S n)) (S n)) ->
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
110 deltaM-fmap f (deltaM-mu d) ≡ deltaM-mu (deltaM-fmap (deltaM-fmap f) d)
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
111 deltaM-mu-is-nt {l} {A} {B} {O} {T} {F} {M} f (deltaM (mono x)) =
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
112 {-
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
113 deltaM-mu-is-nt {l} {A} {B} {O} {T} {F} {M} f (deltaM (mono x)) = begin
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
114 deltaM-fmap f (deltaM-mu (deltaM (mono x))) ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
115 deltaM-fmap f (deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {M = M} (deltaM (mono x))))))) ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
116 deltaM-fmap f (deltaM (mono (mu M (fmap F (headDeltaM {M = M}) x)))) ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
117 deltaM (mono (fmap F f (mu M (fmap F (headDeltaM {M = M}) x)))) ≡⟨ {!!} ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
118 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (fmap F (deltaM-fmap f) x)))) ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
119 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {M = M} (deltaM (mono (fmap F (deltaM-fmap f) x))))))) ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
120 deltaM-mu (deltaM (mono (fmap F (deltaM-fmap f) x))) ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
121 deltaM-mu (deltaM (mono (fmap F (deltaM-fmap f) x))) ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
122 deltaM-mu (deltaM-fmap (deltaM-fmap f) (deltaM (mono x)))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
123 ∎
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
124 -}
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
125
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
126 begin
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
127 deltaM-fmap f (deltaM-mu (deltaM (mono x)))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
128 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
129 deltaM-fmap f (deltaM (mono (mu M (fmap F headDeltaM x))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
130 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
131 deltaM (mono (fmap F f (mu M (fmap F headDeltaM x))))
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
132 ≡⟨ cong (\de -> deltaM (mono de)) (sym (mu-is-nt M f (fmap F headDeltaM x))) ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
133 deltaM (mono (mu M (fmap F (fmap F f) (fmap F headDeltaM x))))
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
134 ≡⟨ cong (\de -> deltaM (mono (mu M de))) (sym (covariant F headDeltaM (fmap F f) x)) ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
135 deltaM (mono (mu M (fmap F ((fmap F f) ∙ headDeltaM) x)))
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
136 ≡⟨ cong (\de -> deltaM (mono (mu M de))) (fmap-equiv F (headDeltaM-with-f f) x) ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
137 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
138 ≡⟨ cong (\de -> deltaM (mono (mu M de))) (covariant F (deltaM-fmap f) (headDeltaM) x) ⟩
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
139 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (fmap F (deltaM-fmap f) x))))
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
140 ≡⟨ refl ⟩
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
141 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {M = M} (deltaM (mono (fmap F (deltaM-fmap f) x)))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
142 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
143 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
144 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
145 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
146 ∎
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
147
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
148 deltaM-mu-is-nt {l} {A} {B} {S n} {T} {F} {M} f (deltaM (delta x d)) = begin
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
149 deltaM-fmap f (deltaM-mu (deltaM (delta x d))) ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
150 deltaM-fmap f (deltaM (delta (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {M = M} (deltaM (delta x d)))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
151 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM (deltaM (delta x d))))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
152 ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
153 deltaM-fmap f (deltaM (delta (mu M (fmap F (headDeltaM {M = M}) x))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
154 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
155 ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
156 deltaM (delta (fmap F f (mu M (fmap F (headDeltaM {M = M}) x)))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
157 (delta-fmap (fmap F f) (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
158 ≡⟨ cong (\de -> deltaM (delta de
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
159 (delta-fmap (fmap F f) (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
160 (sym (mu-is-nt M f (fmap F headDeltaM x))) ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
161 deltaM (delta (mu M (fmap F (fmap F f) (fmap F (headDeltaM {M = M}) x)))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
162 (delta-fmap (fmap F f) (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
163 ≡⟨ cong (\de -> deltaM (delta (mu M de) (delta-fmap (fmap F f) (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
164 (sym (covariant F headDeltaM (fmap F f) x)) ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
165 deltaM (delta (mu M (fmap F ((fmap F f) ∙ (headDeltaM {M = M})) x))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
166 (delta-fmap (fmap F f) (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
167 ≡⟨ cong (\de -> deltaM (delta (mu M de) (delta-fmap (fmap F f) (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
168 (fmap-equiv F (headDeltaM-with-f f) x) ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
169 deltaM (delta (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
170 (delta-fmap (fmap F f) (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
171 ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
172 deltaM (delta (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
173 (unDeltaM {M = M} (deltaM-fmap f (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
174 ≡⟨ cong (\de -> deltaM (delta (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x)) (unDeltaM de)))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
175 (deltaM-mu-is-nt {l} {A} {B} {n} {T} {F} {M} f (deltaM-fmap tailDeltaM (deltaM d))) ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
176 deltaM (delta (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
177 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap (deltaM-fmap {n = n} f) (deltaM-fmap {n = n} (tailDeltaM {n = n}) (deltaM d))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
178 ≡⟨ cong (\de -> deltaM (delta (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x)) (unDeltaM {M = M} (deltaM-mu de))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
179 (sym (deltaM-covariant (deltaM-fmap f) tailDeltaM (deltaM d))) ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
180 deltaM (delta (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
181 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap {n = n} ((deltaM-fmap {n = n} f) ∙ (tailDeltaM {n = n})) (deltaM d)))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
182
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
183 ≡⟨ cong (\de -> deltaM (delta (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x)) (unDeltaM {M = M} (deltaM-mu de))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
184 (sym (deltaM-fmap-equiv (tailDeltaM-with-f f) (deltaM d))) ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
185 deltaM (delta (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
186 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap (tailDeltaM ∙ (deltaM-fmap f)) (deltaM d)))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
187 ≡⟨ cong (\de -> deltaM (delta (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x)) (unDeltaM {M = M} (deltaM-mu de))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
188 (deltaM-covariant tailDeltaM (deltaM-fmap f) (deltaM d)) ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
189 deltaM (delta (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
190 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM-fmap (deltaM-fmap f) (deltaM d))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
191 ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
192 deltaM (delta (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
193 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F (deltaM-fmap f)) d))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
194 ≡⟨ cong (\de -> deltaM (delta (mu M de) (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F (deltaM-fmap f)) d)))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
195 (covariant F (deltaM-fmap f) headDeltaM x) ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
196 deltaM (delta (mu M (fmap F (headDeltaM {M = M}) (fmap F (deltaM-fmap f) x)))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
197 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F (deltaM-fmap f)) d))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
198 ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
199 deltaM (delta (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {M = M} (deltaM (delta (fmap F (deltaM-fmap f) x) (delta-fmap (fmap F (deltaM-fmap f)) d))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
200 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM (deltaM (delta (fmap F (deltaM-fmap f) x) (delta-fmap (fmap F (deltaM-fmap f)) d))))))))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
201 ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
202 deltaM-mu (deltaM (delta (fmap F (deltaM-fmap f) x) (delta-fmap (fmap F (deltaM-fmap f)) d)))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
203 ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
204 deltaM-mu (deltaM-fmap (deltaM-fmap f) (deltaM (delta x d)))
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
205 ∎
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
206
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
207
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
208
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
209
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
210 {-
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
211 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
212 {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
213 (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
214 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
215 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
216 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
217 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
218 ≡⟨ 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
219 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
220 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
221 deltaM (mono x)
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
222 ∎
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
223 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
224 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
225 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
226 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
227 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
228 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
229 (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
230 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
231 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
232 (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
233 ≡⟨ 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
234 (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
235 (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
236 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
237 (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
238 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
239 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
240 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta M (deltaM (delta x d)))))))))
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
241 ≡⟨ cong (\de -> deltaM (delta de (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta M (deltaM (delta x d))))))))))
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
242 (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
243 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta M (deltaM (delta x d)))))))))
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
244 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
245 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
246 ≡⟨ 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
247 (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
248 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
249 ≡⟨ 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
250 (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
251 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
252 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
253 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
254 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
255 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
256 ≡⟨ 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
257 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
258 ≡⟨ refl ⟩
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
259 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
260 ∎
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
261
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
262
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
263
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
264
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
265
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
266
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
267 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
268 {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
269 {n : Nat}
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
270 (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
271 (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
272 {-
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
273 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
274 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
275 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
276 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
277 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
278 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
279 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
280 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
281 ≡⟨ 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
282 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
283 ≡⟨ 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
284 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
285 ≡⟨ 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
286 deltaM (mono x)
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
287 ∎
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
288 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
289 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
290 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
291 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
292 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
293 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
294 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
295 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
296 (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
297 ≡⟨ 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
298 (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
299 (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
300 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
301 (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
302 ≡⟨ 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
303 (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
304 (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
305 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
306 (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
307
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
308 ≡⟨ 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
309 (left-unity-law mm x) ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
310 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
311 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
312 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
313 ≡⟨ 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
314 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
315 ≡⟨ 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
316 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
317 ≡⟨ 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
318 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
319 ≡⟨ refl ⟩
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
320 deltaM (delta x d)
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
321 ∎
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
322
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
323 -}
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
324 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
325 (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
326 (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
327 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
328
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
329
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
330
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
331
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
332
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
333
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
334
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
335 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
336 (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
337 (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
338 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
339 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
340 {-
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
341 begin
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
342 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
343 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
344 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
345 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
346 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
347 (\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
348 ≡⟨ 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
349 (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
350 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
351 (\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
352 ≡⟨ 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
353 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
354 ≡⟨ 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
355 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
356 ≡⟨ 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
357 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
358 ≡⟨ 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
359 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
360 ≡⟨ 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
361 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
362 ≡⟨ 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
363 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
364 ≡⟨ 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
365 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
366 ≡⟨ 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
367 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
368 ≡⟨ 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
369 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
370 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
371 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
372 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
373 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
374 -}
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
375 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
376 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
377 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
378 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
379 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
380 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
381 (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
382
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
383 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
384 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
385 (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
386 ≡⟨ 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
387 (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
388 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
389 (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
390 ≡⟨ 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
391 (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
392 (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
393 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
394 (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
395 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
396 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
397 (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
398 ≡⟨ 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
399 (unDeltaM {monadM = mm} (deltaM-mu de))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
400 (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
401 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
402 (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
403 ≡⟨ 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
404 (unDeltaM {monadM = mm} (deltaM-mu de))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
405 (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
406 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
407 (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
408 ≡⟨ 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
409 (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
410 (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
411 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
412 (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
413 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
414 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
415 (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
416 ≡⟨ 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
417 (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
418 (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
419
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
420 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
421 (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
422 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
423 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
424 (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
425 ≡⟨ 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
426 (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
427 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
428 (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
429 ≡⟨ 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
430 (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
431 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
432 (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
433 ≡⟨ 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
434 (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
435 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
436 (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
437 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
438 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
439 (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
440 ≡⟨ 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
441 (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
442 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
443 (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
444 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
445 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
446 (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
447 ≡⟨ 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
448 (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
449 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
450 (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
451
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
452 ≡⟨ 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
453 (unDeltaM {monadM = mm} (deltaM-mu de))))
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
454 (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
455 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
456 (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
457 ≡⟨ 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
458 (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
459 (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
460 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
461 (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
462 (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
463
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
464
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
465 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
466 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
467 (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
468 (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
469 (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
470
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
471
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
472 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
473 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
474 (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
475 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
476 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
477 (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
478 ≡⟨ refl ⟩
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
479 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
480 ∎
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
481 {-
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
482 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
483 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
484 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
485 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
486 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
487 (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
488 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
489 (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
490 ≡⟨ 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
491 (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
492
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
493 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
494 (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
495 ≡⟨ refl ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
496 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
497 (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
498 ≡⟨ refl ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
499 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
500 (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
501 ≡⟨ {!!} ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
502 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
503 (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
504 ≡⟨ 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
505 (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
506 (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
507 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
508 (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
509
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
510 ≡⟨ refl ⟩
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
511 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
512 (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
513 (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
514 ≡⟨ refl ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
515 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
516 (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
517 (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
518 (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
519 ≡⟨ refl ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
520 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
521 (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
522 ≡⟨ refl ⟩
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
523 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
524 (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
525 ≡⟨ refl ⟩
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
526 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
527 (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
528 ≡⟨ 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
529 (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
530 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
531 (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
532 ≡⟨ refl ⟩
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
533 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
534 (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
535 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
536 ∎
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
537 -}
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
538
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
539
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
540
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
541 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
542 {M : Set l -> Set l}
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
543 (functorM : Functor M)
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
544 (monadM : Monad M functorM) ->
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
545 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
546 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
547 record { mu = deltaM-mu
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
548 ; eta = deltaM-eta
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
549 ; 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
550 ; 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
551 ; 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
552 ; 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
553 ; 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
554 }
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
555
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
556
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
557 -}