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Fix names left/right unity law
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Wed, 25 Feb 2015 14:49:50 +0900
parents d205ff1e406f
children
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
98
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import Level
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 open import Relation.Binary.PropositionalEquality
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3 open ≡-Reasoning
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
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4
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5 open import basic
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6 open import delta
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7 open import delta.functor
111
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
8 open import delta.monad
98
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9 open import deltaM
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10 open import deltaM.functor
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
11 open import nat
98
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12 open import laws
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14 module deltaM.monad where
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15 open Functor
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16 open NaturalTransformation
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
17 open Monad
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
18
b7f0879e854e Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
19
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
20 -- sub proofs
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
21
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
22 deconstruct-id : {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
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
diff changeset
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
diff changeset
27
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
28 headDeltaM-with-f : {l : Level} {A B : Set l} {n : Nat}
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
29 {T : Set l -> Set l} {F : Functor T} {M : Monad T F}
130
ac45d065cbf2 Prove all Monad-laws for Delta with Monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
30 (f : A -> B) -> (x : (DeltaM M A (S n))) ->
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
34
131
d205ff1e406f Cleanup proofs
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 130
diff changeset
35
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
36 tailDeltaM-with-f : {l : Level} {A B : Set l} {n : Nat}
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
37 {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
38 (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
diff changeset
39 (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
40 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
41 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
diff changeset
42
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
43
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
44
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
45 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
46 {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
47 (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
48 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
49 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
50 fmap-headDeltaM-with-deltaM-mu {n = S n} x = refl
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
51
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
52
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
53 fmap-tailDeltaM-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
54 {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
55 (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
56 deltaM-fmap (tailDeltaM ∙ deltaM-mu) d ≡ deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) d
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
57 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
58 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
59
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
60
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
61
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
62
130
ac45d065cbf2 Prove all Monad-laws for Delta with Monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
63
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
64 -- main proofs
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
65
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
66 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
67 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
68 (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
69 ((deltaM-eta {l} {B} {n} {T} {F} {M} )∙ f) x ≡ deltaM-fmap f (deltaM-eta x)
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
70 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
71 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
72 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
73 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
74 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
75 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
76 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
77 ≡⟨ refl ⟩
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
78 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
79 ≡⟨ refl ⟩
114
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
80 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
81 ≡⟨ 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
82 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
83 ≡⟨ 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
84 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
85 ≡⟨ refl ⟩
08403eb8db8b Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
86 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
87 ∎
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
88
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
89
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
90
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
91
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
92 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
93 {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
94 (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
95 deltaM-fmap f (deltaM-mu d) ≡ deltaM-mu (deltaM-fmap (deltaM-fmap f) d)
127
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
96 deltaM-mu-is-nt {l} {A} {B} {O} {T} {F} {M} f (deltaM (mono x)) = begin
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
97 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
98 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
99 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
100 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
101 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
102 ≡⟨ 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
103 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
104 ≡⟨ 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
105 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
106 ≡⟨ 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
107 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
108 ≡⟨ 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
109 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
110 ≡⟨ refl ⟩
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 (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
112 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
113 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
114 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
115 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
116 ∎
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
117 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
118 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
119 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
120 (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
121 ≡⟨ 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-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
123 (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
124 ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
125 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
126 (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
127 ≡⟨ 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
128 (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
129 (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
130 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
131 (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
132 ≡⟨ 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
133 (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
134 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
135 (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
136 ≡⟨ 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
137 (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
138 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
139 (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
140 ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
141 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
142 (unDeltaM {M = M} (deltaM-fmap f (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))
130
ac45d065cbf2 Prove all Monad-laws for Delta with Monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
143 ≡⟨ cong (\de -> deltaM (delta (mu M (fmap F ((headDeltaM {M = M}) ∙ (deltaM-fmap f)) x)) (unDeltaM de)))
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
144 (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
145 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
146 (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
147 ≡⟨ 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
148 (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
149 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
150 (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
151
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
152 ≡⟨ 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
153 (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
154 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
155 (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
156 ≡⟨ 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
157 (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
158 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
159 (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
160 ≡⟨ refl ⟩
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 ((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
162 (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
163 ≡⟨ 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
164 (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
165 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
166 (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
167 ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
168 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
169 (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
170 ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
171 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
172 ≡⟨ refl ⟩
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
173 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
174 ∎
129
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
175
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
176
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
177
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
178
127
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
179
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
180 deltaM-left-unity-law : {l : Level} {A : Set l} {n : Nat}
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
181 {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
182 (d : DeltaM M A (S n)) -> (deltaM-mu ∙ deltaM-eta) d ≡ id d
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
183 deltaM-left-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
184 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
185 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
186 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
187 ≡⟨ 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
188 deltaM (mono (mu mm (eta mm ((headDeltaM {l} {A} {O} {M} {fm} {mm}) (deltaM (mono x)))))) ≡⟨ refl ⟩
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
189 deltaM (mono (mu mm (eta mm x))) ≡⟨ cong (\de -> deltaM (mono de)) (sym (left-unity-law mm x)) ⟩
111
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
190 deltaM (mono x)
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
191 ∎
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
192 deltaM-left-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
193 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
194 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
195 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
196 ≡⟨ refl ⟩
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
197 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
198 (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
199 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
200 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
201 (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
202 ≡⟨ 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
203 (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
204 (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
205 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
206 (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
207 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
208 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
209 (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
210 ≡⟨ cong (\de -> deltaM (delta de (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta M (deltaM (delta x d))))))))))
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
211 (sym (left-unity-law M x)) ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
212 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
213 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
214 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
215 ≡⟨ 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
216 (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
217 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
218 ≡⟨ 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
219 (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
220 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
221 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
222 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
223 ≡⟨ refl ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
224 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM-eta (deltaM d)))))
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
225 ≡⟨ cong (\de -> deltaM (delta x (unDeltaM {M = M} de))) (deltaM-left-unity-law (deltaM d)) ⟩
125
6dcc68ef8f96 Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 124
diff changeset
226 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
227 ≡⟨ refl ⟩
9fe3d0bd1149 Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 104
diff changeset
228 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
229 ∎
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
230
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
231
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
232
126
5902b2a24abf Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
233
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
234
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
235
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
236 deltaM-right-unity-law : {l : Level} {A : Set l} {n : Nat}
127
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
237 {T : Set l -> Set l} {F : Functor T} {M : Monad T F}
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
238 (d : DeltaM M A (S n)) -> (deltaM-mu ∙ (deltaM-fmap deltaM-eta)) d ≡ id d
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
239 deltaM-right-unity-law {l} {A} {O} {T} {F} {M} (deltaM (mono x)) = begin
127
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
240 deltaM-mu (deltaM-fmap deltaM-eta (deltaM (mono x))) ≡⟨ refl ⟩
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
241 deltaM-mu (deltaM (mono (fmap F deltaM-eta x))) ≡⟨ refl ⟩
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
242 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {M = M} (deltaM (mono (fmap F deltaM-eta x))))))) ≡⟨ refl ⟩
130
ac45d065cbf2 Prove all Monad-laws for Delta with Monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
243 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (fmap F deltaM-eta x))))
127
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
244 ≡⟨ cong (\de -> deltaM (mono (mu M de))) (sym (covariant F deltaM-eta headDeltaM x)) ⟩
130
ac45d065cbf2 Prove all Monad-laws for Delta with Monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
245 deltaM (mono (mu M (fmap F ((headDeltaM {n = O} {M = M}) ∙ deltaM-eta) x)))
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
246 ≡⟨ refl ⟩
127
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
247 deltaM (mono (mu M (fmap F (eta M) x)))
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
248 ≡⟨ cong (\de -> deltaM (mono de)) (right-unity-law M x) ⟩
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
249 deltaM (mono x)
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
250 ∎
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
251 deltaM-right-unity-law {l} {A} {S n} {T} {F} {M} (deltaM (delta x d)) = begin
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
252 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
253 ≡⟨ refl ⟩
127
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
254 deltaM-mu (deltaM (delta (fmap F deltaM-eta x) (delta-fmap (fmap F deltaM-eta) d)))
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
255 ≡⟨ refl ⟩
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
256 deltaM (delta (mu M (fmap F (headDeltaM {n = S n} {M = M}) (headDeltaM {n = S n} {M = M} (deltaM (delta (fmap F deltaM-eta x) (delta-fmap (fmap F deltaM-eta) d))))))
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
257 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM (deltaM (delta (fmap F deltaM-eta x) (delta-fmap (fmap F deltaM-eta) d))))))))
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
258 ≡⟨ refl ⟩
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
259 deltaM (delta (mu M (fmap F (headDeltaM {n = S n} {M = M}) (fmap F deltaM-eta x)))
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
260 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F deltaM-eta) d))))))
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
261 ≡⟨ cong (\de -> deltaM (delta (mu M de) (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F deltaM-eta) d)))))))
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
262 (sym (covariant F deltaM-eta headDeltaM x)) ⟩
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
263 deltaM (delta (mu M (fmap F ((headDeltaM {n = S n} {M = M}) ∙ deltaM-eta) x))
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
264 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F deltaM-eta) d))))))
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
265 ≡⟨ refl ⟩
127
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
266 deltaM (delta (mu M (fmap F (eta M) x))
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
267 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F deltaM-eta) d))))))
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
268 ≡⟨ cong (\de -> deltaM (delta de (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F deltaM-eta) d)))))))
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
269 (right-unity-law M x) ⟩
127
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
270 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F deltaM-eta) d))))))
112
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
271 ≡⟨ refl ⟩
127
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
272 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM-fmap (tailDeltaM {n = n})(deltaM-fmap (deltaM-eta {n = S n})(deltaM d))))))
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
273 ≡⟨ cong (\de -> deltaM (delta x (unDeltaM {M = M} (deltaM-mu de)))) (sym (deltaM-covariant tailDeltaM deltaM-eta (deltaM d))) ⟩
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
274 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM-fmap ((tailDeltaM {n = n}) ∙ (deltaM-eta {n = S n})) (deltaM d)))))
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
275 ≡⟨ refl ⟩
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
276 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM-fmap deltaM-eta (deltaM d)))))
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
277 ≡⟨ cong (\de -> deltaM (delta x (unDeltaM {M = M} de))) (deltaM-right-unity-law (deltaM d)) ⟩
127
d56596e4e784 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
278 deltaM (delta x (unDeltaM {M = M} (deltaM d)))
112
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 (delta x d)
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
281 ∎
0a3b6cb91a05 Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
282
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
283 deltaM-association-law : {l : Level} {A : Set l} {n : Nat}
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
284 {T : Set l -> Set l} {F : Functor T} {M : Monad T F}
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
285 (d : DeltaM M (DeltaM M (DeltaM M A (S n)) (S n)) (S n)) ->
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
286 deltaM-mu (deltaM-fmap deltaM-mu d) ≡ deltaM-mu (deltaM-mu d)
129
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
287 deltaM-association-law {l} {A} {O} {T} {F} {M} (deltaM (mono x)) = begin
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
288 deltaM-mu (deltaM-fmap deltaM-mu (deltaM (mono x)))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
289 ≡⟨ refl ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
290 deltaM-mu (deltaM (mono (fmap F deltaM-mu x)))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
291 ≡⟨ refl ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
292 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {M = M} (deltaM (mono (fmap F deltaM-mu x)))))))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
293 ≡⟨ refl ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
294 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (fmap F deltaM-mu x))))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
295 ≡⟨ cong (\de -> deltaM (mono (mu M de))) (sym (covariant F deltaM-mu headDeltaM x)) ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
296 deltaM (mono (mu M (fmap F ((headDeltaM {M = M}) ∙ deltaM-mu) x)))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
297 ≡⟨ refl ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
298 deltaM (mono (mu M (fmap F (((mu M) ∙ (fmap F headDeltaM)) ∙ headDeltaM) x)))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
299 ≡⟨ cong (\de -> deltaM (mono (mu M de))) (covariant F headDeltaM ((mu M) ∙ (fmap F headDeltaM)) x) ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
300 deltaM (mono (mu M ((((fmap F (((mu M) ∙ (fmap F headDeltaM)))) ∙ (fmap F headDeltaM)) x))))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
301 ≡⟨ refl ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
302 deltaM (mono (mu M ((((fmap F (((mu M) ∙ (fmap F headDeltaM)))) (fmap F headDeltaM x))))))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
303 ≡⟨ cong (\de -> deltaM (mono (mu M de))) (covariant F (fmap F headDeltaM) (mu M) (fmap F headDeltaM x)) ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
304 deltaM (mono (mu M (((fmap F (mu M)) ∙ (fmap F (fmap F headDeltaM))) (fmap F headDeltaM x))))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
305 ≡⟨ refl ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
306 deltaM (mono (mu M (fmap F (mu M) ((fmap F (fmap F headDeltaM) (fmap F headDeltaM x))))))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
307 ≡⟨ cong (\de -> deltaM (mono de)) (association-law M (fmap F (fmap F headDeltaM) (fmap F headDeltaM x))) ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
308 deltaM (mono (mu M (mu M (fmap F (fmap F headDeltaM) (fmap F headDeltaM x)))))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
309 ≡⟨ cong (\de -> deltaM (mono (mu M de))) (mu-is-nt M headDeltaM (fmap F headDeltaM x)) ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
310 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (mu M (fmap F (headDeltaM {M = M}) x)))))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
311 ≡⟨ refl ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
312 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {M = M} (deltaM (mono (mu M (fmap F (headDeltaM {M = M}) x))))))))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
313 ≡⟨ refl ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
314 deltaM-mu (deltaM (mono (mu M (fmap F (headDeltaM {M = M}) x))))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
315 ≡⟨ refl ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
316 deltaM-mu (deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {M = M} (deltaM (mono x)))))))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
317 ≡⟨ refl ⟩
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
318 deltaM-mu (deltaM-mu (deltaM (mono x)))
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
319
d57c88171f38 Prove assciation-law for DeltaM on (S O)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
320 ∎
130
ac45d065cbf2 Prove all Monad-laws for Delta with Monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
321
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
322 deltaM-association-law {l} {A} {S n} {T} {F} {M} (deltaM (delta x d)) = begin
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
323 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
324 ≡⟨ refl ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
325 deltaM-mu (deltaM (delta (fmap F deltaM-mu x) (delta-fmap (fmap F deltaM-mu) d)))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
326 ≡⟨ refl ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
327 deltaM (delta (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {A = DeltaM M A (S (S n))} {M = M} (deltaM (delta (fmap F deltaM-mu x) (delta-fmap (fmap F deltaM-mu) d))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
328 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM (deltaM (delta (fmap F deltaM-mu x) (delta-fmap (fmap F deltaM-mu) d))))))))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
329
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
330 ≡⟨ refl ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
331 deltaM (delta (mu M (fmap F (headDeltaM {A = A} {M = M}) (fmap F deltaM-mu x)))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
332 (unDeltaM {A = A} {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F deltaM-mu) d))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
333 ≡⟨ cong (\de -> deltaM (delta (mu M de) (unDeltaM {A = A} {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F deltaM-mu) d)))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
334 (sym (covariant F deltaM-mu headDeltaM x)) ⟩
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
335 deltaM (delta (mu M (fmap F ((headDeltaM {A = A} {M = M}) ∙ deltaM-mu) x))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
336 (unDeltaM {A = A} {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F deltaM-mu) d))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
337 ≡⟨ cong (\de -> deltaM (delta (mu M de)
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
338 (unDeltaM {A = A} {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F deltaM-mu) d)))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
339 (fmap-headDeltaM-with-deltaM-mu {A = A} {M = M} x) ⟩
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
340 deltaM (delta (mu M (fmap F (((mu M) ∙ (fmap F headDeltaM)) ∙ headDeltaM) x))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
341 (unDeltaM {A = A} {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap F deltaM-mu) d))))))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
342 ≡⟨ refl ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
343 deltaM (delta (mu M (fmap F (((mu M) ∙ (fmap F headDeltaM)) ∙ headDeltaM) x))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
344 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM-fmap deltaM-mu (deltaM d))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
345 ≡⟨ cong (\de -> deltaM (delta (mu M (fmap F (((mu M) ∙ (fmap F headDeltaM)) ∙ headDeltaM) x))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
346 (unDeltaM {M = M} (deltaM-mu de))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
347 (sym (deltaM-covariant tailDeltaM deltaM-mu (deltaM d))) ⟩
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
348 deltaM (delta (mu M (fmap F (((mu M) ∙ (fmap F headDeltaM)) ∙ headDeltaM) x))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
349 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap (tailDeltaM ∙ deltaM-mu) (deltaM d)))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
350 ≡⟨ cong (\de -> deltaM (delta (mu M (fmap F (((mu M) ∙ (fmap F headDeltaM)) ∙ headDeltaM) x))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
351 (unDeltaM {M = M} (deltaM-mu de))))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
352 (fmap-tailDeltaM-with-deltaM-mu (deltaM d)) ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
353 deltaM (delta (mu M (fmap F (((mu M) ∙ (fmap F headDeltaM)) ∙ headDeltaM) x))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
354 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
355 ≡⟨ cong (\de -> deltaM (delta (mu M de)
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
356 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
357 (covariant F headDeltaM ((mu M) ∙ (fmap F headDeltaM)) x) ⟩
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
358 deltaM (delta (mu M (((fmap F ((mu M) ∙ (fmap F headDeltaM))) ∙ (fmap F headDeltaM)) x))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
359 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
360 ≡⟨ refl ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
361 deltaM (delta (mu M (((fmap F ((mu M) ∙ (fmap F headDeltaM))) (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
362 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
363 ≡⟨ cong (\de -> deltaM (delta (mu M de)
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
364 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
365 (covariant F (fmap F headDeltaM) (mu M) (fmap F headDeltaM x)) ⟩
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
366
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
367 deltaM (delta (mu M ((((fmap F (mu M)) ∙ (fmap F (fmap F headDeltaM))) (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
368 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
369 ≡⟨ refl ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
370 deltaM (delta (mu M (fmap F (mu M) (fmap F (fmap F headDeltaM) (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
371 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
372 ≡⟨ cong (\de -> deltaM (delta de (unDeltaM {M = M} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
373 (association-law M (fmap F (fmap F headDeltaM) (fmap F headDeltaM x))) ⟩
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
374 deltaM (delta (mu M (mu M (fmap F (fmap F headDeltaM) (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
375 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
376 ≡⟨ cong (\de -> deltaM (delta (mu M de) (unDeltaM {M = M} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
377 (mu-is-nt M headDeltaM (fmap F headDeltaM x)) ⟩
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
378 deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
379 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap ((deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ tailDeltaM) (deltaM d)))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
380 ≡⟨ cong (\de -> deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x)))) (unDeltaM {M = M} (deltaM-mu de))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
381 (deltaM-covariant (deltaM-mu ∙ (deltaM-fmap tailDeltaM)) tailDeltaM (deltaM d)) ⟩
d9a30f696933 Fix association-law for DeltaM in (S n)
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parents: 127
diff changeset
382 deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
383 (unDeltaM {M = M} (deltaM-mu (((deltaM-fmap (deltaM-mu ∙ (deltaM-fmap tailDeltaM)) ∙ (deltaM-fmap tailDeltaM)) (deltaM d))))))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
384 ≡⟨ refl ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
385 deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
386 (unDeltaM {M = M} (deltaM-mu (((deltaM-fmap (deltaM-mu ∙ (deltaM-fmap tailDeltaM)) (deltaM-fmap tailDeltaM (deltaM d))))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
387 ≡⟨ cong (\de -> deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x)))) (unDeltaM {M = M} (deltaM-mu de))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
388 (deltaM-covariant deltaM-mu (deltaM-fmap tailDeltaM) (deltaM-fmap tailDeltaM (deltaM d))) ⟩
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
389 deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
390 (unDeltaM {M = M} (deltaM-mu (((deltaM-fmap deltaM-mu) ∙ (deltaM-fmap (deltaM-fmap tailDeltaM))) (deltaM-fmap tailDeltaM (deltaM d))))))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
391 ≡⟨ refl ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
392 deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
393 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap deltaM-mu (deltaM-fmap (deltaM-fmap tailDeltaM) (deltaM-fmap tailDeltaM (deltaM d)))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
394 ≡⟨ cong (\de -> deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x)))) (unDeltaM {M = M} de)))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
395 (deltaM-association-law (deltaM-fmap (deltaM-fmap tailDeltaM) (deltaM-fmap tailDeltaM (deltaM d)))) ⟩
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
396 deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
397 (unDeltaM {M = M} (deltaM-mu (deltaM-mu (deltaM-fmap (deltaM-fmap tailDeltaM) (deltaM-fmap tailDeltaM (deltaM d)))))))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
398
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
399 ≡⟨ cong (\de -> deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
400 (unDeltaM {M = M} (deltaM-mu de))))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
401 (sym (deltaM-mu-is-nt tailDeltaM (deltaM-fmap tailDeltaM (deltaM d)))) ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
402 deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
403 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
404 ≡⟨ cong (\de -> deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
405 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM de)))))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
406 (sym (deconstruct-id (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))) ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
407 deltaM (delta (mu M (fmap F headDeltaM (mu M (fmap F headDeltaM x))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
408 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
409 (deltaM (unDeltaM {A = DeltaM M A (S (S n))} {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))))))))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
410
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
411
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
412 ≡⟨ refl ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
413 deltaM (delta (mu M (fmap F headDeltaM (headDeltaM {M = M} ((deltaM (delta (mu M (fmap F headDeltaM x))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
414 (unDeltaM {A = DeltaM M A (S (S n))} {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
415 (unDeltaM {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM ((deltaM (delta (mu M (fmap F headDeltaM x))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
416 (unDeltaM {A = DeltaM M A (S (S n))} {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))))))))
118
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
417
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
418
53cb21845dea Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
419 ≡⟨ refl ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
420 deltaM-mu (deltaM (delta (mu M (fmap F headDeltaM x))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
421 (unDeltaM {A = DeltaM M A (S (S n))} {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d))))))
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
422 ≡⟨ refl ⟩
128
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
423 deltaM-mu (deltaM (delta (mu M (fmap F headDeltaM (headDeltaM {M = M} (deltaM (delta x d)))))
d9a30f696933 Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
424 (unDeltaM {A = DeltaM M A (S (S n))} {M = M} (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM (deltaM (delta x d))))))))
117
6f86b55bf8b4 Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
425 ≡⟨ refl ⟩
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
426 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
427 ∎
130
ac45d065cbf2 Prove all Monad-laws for Delta with Monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
428
115
e6bcc7467335 Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
429
130
ac45d065cbf2 Prove all Monad-laws for Delta with Monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
430
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
431
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
432 deltaM-is-monad : {l : Level} {A : Set l} {n : Nat}
130
ac45d065cbf2 Prove all Monad-laws for Delta with Monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
433 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
ac45d065cbf2 Prove all Monad-laws for Delta with Monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
434 Monad {l} (\A -> DeltaM M A (S n)) (deltaM-is-functor {l} {n})
ac45d065cbf2 Prove all Monad-laws for Delta with Monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
435 deltaM-is-monad {l} {A} {n} {T} {F} {M} =
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
436 record { mu = deltaM-mu
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
437 ; eta = deltaM-eta
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
438 ; 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
439 ; mu-is-nt = (\f x -> (sym (deltaM-mu-is-nt f x)))
130
ac45d065cbf2 Prove all Monad-laws for Delta with Monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
440 ; association-law = deltaM-association-law
144
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
441 ; right-unity-law = deltaM-right-unity-law
575de2e38385 Fix names left/right unity law
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 131
diff changeset
442 ; left-unity-law = (\x -> (sym (deltaM-left-unity-law x)))
124
48b44bd85056 Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
443 }