Mercurial > hg > Members > atton > delta_monad
annotate agda/deltaM/monad.agda @ 128:d9a30f696933
Fix association-law for DeltaM in (S n)
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
---|---|
date | Tue, 03 Feb 2015 12:24:26 +0900 |
parents | d56596e4e784 |
children | d57c88171f38 |
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:
diff
changeset
|
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} |
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
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 |
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
35 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
|
36 {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
|
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
diff
changeset
|
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
diff
changeset
|
41 |
127
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
42 headDeltaM-with-deltaM-eta : {l : Level} {A : Set l} {n : Nat} |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
43 {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
|
44 (x : DeltaM M A (S n)) -> |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
45 ((headDeltaM {n = n} {M = M}) ∙ deltaM-eta) x ≡ eta M x |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
46 headDeltaM-with-deltaM-eta {n = O} (deltaM (mono x)) = refl |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
47 headDeltaM-with-deltaM-eta {n = S n} (deltaM (delta x d)) = refl |
114
08403eb8db8b
Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
112
diff
changeset
|
48 |
08403eb8db8b
Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
112
diff
changeset
|
49 |
08403eb8db8b
Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
112
diff
changeset
|
50 fmap-tailDeltaM-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
|
51 {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
|
52 (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
|
53 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
|
54 fmap-tailDeltaM-with-deltaM-eta {n = O} d = refl |
114
08403eb8db8b
Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
112
diff
changeset
|
55 fmap-tailDeltaM-with-deltaM-eta {n = S n} d = refl |
08403eb8db8b
Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
112
diff
changeset
|
56 |
124
48b44bd85056
Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
118
diff
changeset
|
57 |
48b44bd85056
Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
118
diff
changeset
|
58 |
118
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
59 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
|
60 {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
|
61 (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
|
62 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
|
63 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
|
64 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
|
65 |
118
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
66 |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
67 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
|
68 {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
|
69 (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
|
70 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
|
71 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
|
72 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
|
73 |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
74 |
117
6f86b55bf8b4
Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
116
diff
changeset
|
75 |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
76 |
117
6f86b55bf8b4
Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
116
diff
changeset
|
77 |
114
08403eb8db8b
Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
112
diff
changeset
|
78 -- main proofs |
08403eb8db8b
Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
112
diff
changeset
|
79 |
124
48b44bd85056
Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
118
diff
changeset
|
80 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
|
81 {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
|
82 (f : A -> B) -> (x : A) -> |
124
48b44bd85056
Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
118
diff
changeset
|
83 ((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
|
84 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
|
85 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
|
86 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
|
87 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
|
88 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
|
89 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
|
90 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
|
91 ≡⟨ refl ⟩ |
48b44bd85056
Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
118
diff
changeset
|
92 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
|
93 ≡⟨ refl ⟩ |
114
08403eb8db8b
Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
112
diff
changeset
|
94 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
|
95 ≡⟨ 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
|
96 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
|
97 ≡⟨ 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
|
98 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
|
99 ≡⟨ refl ⟩ |
08403eb8db8b
Prove natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
112
diff
changeset
|
100 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
|
101 ∎ |
124
48b44bd85056
Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
118
diff
changeset
|
102 |
48b44bd85056
Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
118
diff
changeset
|
103 |
125
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
104 |
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
|
105 |
126
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
106 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
|
107 {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
|
108 (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
|
109 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
|
110 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
|
111 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
|
112 ≡⟨ refl ⟩ |
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
113 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
|
114 ≡⟨ refl ⟩ |
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
115 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
|
116 ≡⟨ 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
|
117 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
|
118 ≡⟨ 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
|
119 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
|
120 ≡⟨ 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
|
121 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
|
122 ≡⟨ 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
|
123 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
|
124 ≡⟨ refl ⟩ |
126
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
125 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
|
126 ≡⟨ refl ⟩ |
125
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
127 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
|
128 ≡⟨ refl ⟩ |
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
129 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
|
130 ∎ |
126
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
131 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
|
132 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
|
133 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
|
134 (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
|
135 ≡⟨ refl ⟩ |
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
136 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
|
137 (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
|
138 ≡⟨ refl ⟩ |
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
139 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
|
140 (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
|
141 ≡⟨ 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
|
142 (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
|
143 (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
|
144 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
|
145 (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
|
146 ≡⟨ 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
|
147 (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
|
148 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
|
149 (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
|
150 ≡⟨ 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
|
151 (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
|
152 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
|
153 (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
|
154 ≡⟨ refl ⟩ |
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
155 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
|
156 (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
|
157 ≡⟨ 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
|
158 (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
|
159 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
|
160 (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
|
161 ≡⟨ 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
|
162 (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
|
163 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
|
164 (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
|
165 |
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
166 ≡⟨ 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
|
167 (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
|
168 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
|
169 (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
|
170 ≡⟨ 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
|
171 (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
|
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-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
|
174 ≡⟨ refl ⟩ |
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
175 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
|
176 (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
|
177 ≡⟨ 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
|
178 (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
|
179 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
|
180 (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
|
181 ≡⟨ refl ⟩ |
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
182 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
|
183 (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
|
184 ≡⟨ refl ⟩ |
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
185 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
|
186 ≡⟨ refl ⟩ |
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
187 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
|
188 ∎ |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
189 {- |
125
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
190 |
126
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
191 |
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
192 |
127
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
193 |
125
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
194 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
|
195 {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
|
196 (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
|
197 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
|
198 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
|
199 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
|
200 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
|
201 ≡⟨ 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
|
202 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
|
203 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
|
204 deltaM (mono x) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
205 ∎ |
125
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
206 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
|
207 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
|
208 ≡⟨ refl ⟩ |
125
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
209 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
|
210 ≡⟨ refl ⟩ |
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
211 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
|
212 (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
|
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 (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
|
215 (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
|
216 ≡⟨ 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
|
217 (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
|
218 (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
|
219 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
|
220 (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
|
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 (mu M (eta M x)) |
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
223 (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
|
224 ≡⟨ 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
|
225 (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
|
226 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
|
227 ≡⟨ refl ⟩ |
125
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
228 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
|
229 ≡⟨ 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
|
230 (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
|
231 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
|
232 ≡⟨ 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
|
233 (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
|
234 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
|
235 ≡⟨ refl ⟩ |
125
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
236 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
|
237 ≡⟨ refl ⟩ |
125
6dcc68ef8f96
Prove right-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
124
diff
changeset
|
238 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
|
239 ≡⟨ 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
|
240 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
|
241 ≡⟨ refl ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
242 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
|
243 ∎ |
112
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
244 |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
245 |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
246 |
126
5902b2a24abf
Prove mu-is-nt for DeltaM with fmap-equiv
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
125
diff
changeset
|
247 |
104
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
248 |
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
249 |
127
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
250 deltaM-left-unity-law : {l : Level} {A : Set l} {n : Nat} |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
251 {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
|
252 (d : DeltaM M A (S n)) -> (deltaM-mu ∙ (deltaM-fmap deltaM-eta)) d ≡ id d |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
253 deltaM-left-unity-law {l} {A} {O} {T} {F} {M} (deltaM (mono x)) = begin |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
254 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
|
255 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
|
256 deltaM (mono (mu M (fmap F (headDeltaM {M = M}) (headDeltaM {M = M} (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
|
257 deltaM (mono (mu M (fmap F (headDeltaM {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
|
258 ≡⟨ cong (\de -> deltaM (mono (mu M de))) (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
|
259 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
|
260 ≡⟨ refl ⟩ |
127
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
261 deltaM (mono (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
|
262 ≡⟨ cong (\de -> deltaM (mono de)) (left-unity-law M x) ⟩ |
112
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
263 deltaM (mono x) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
264 ∎ |
127
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
265 deltaM-left-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
|
266 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
|
267 ≡⟨ refl ⟩ |
127
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
268 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
|
269 ≡⟨ refl ⟩ |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
270 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
|
271 (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
|
272 ≡⟨ refl ⟩ |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
273 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
|
274 (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
|
275 ≡⟨ 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
|
276 (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
|
277 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
|
278 (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
|
279 ≡⟨ refl ⟩ |
127
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
280 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
|
281 (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
|
282 ≡⟨ cong (\de -> deltaM (delta 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
|
283 (left-unity-law M x) ⟩ |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
284 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
|
285 ≡⟨ refl ⟩ |
127
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
286 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
|
287 ≡⟨ 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
|
288 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
|
289 ≡⟨ refl ⟩ |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
290 deltaM (delta x (unDeltaM {M = M} (deltaM-mu (deltaM-fmap deltaM-eta (deltaM d))))) |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
291 ≡⟨ cong (\de -> deltaM (delta x (unDeltaM {M = M} de))) (deltaM-left-unity-law (deltaM d)) ⟩ |
d56596e4e784
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
126
diff
changeset
|
292 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
|
293 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
294 deltaM (delta x d) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
295 ∎ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
296 |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
297 -} |
115
e6bcc7467335
Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
114
diff
changeset
|
298 |
e6bcc7467335
Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
114
diff
changeset
|
299 |
e6bcc7467335
Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
114
diff
changeset
|
300 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
|
301 {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
|
302 (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
|
303 deltaM-mu (deltaM-fmap deltaM-mu d) ≡ deltaM-mu (deltaM-mu d) |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
304 deltaM-association-law {l} {A} {O} {T} {F} {M} (deltaM (mono x)) = {!!} |
117
6f86b55bf8b4
Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
116
diff
changeset
|
305 {- |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
306 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
|
307 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
|
308 ≡⟨ refl ⟩ |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
309 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
|
310 ≡⟨ refl ⟩ |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
311 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
|
312 (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
|
313 |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
314 ≡⟨ refl ⟩ |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
315 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
|
316 (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
|
317 ≡⟨ 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
|
318 (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
|
319 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
|
320 (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
|
321 ≡⟨ 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
|
322 (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
|
323 (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
|
324 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
|
325 (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
|
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 (((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
|
328 (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
|
329 ≡⟨ 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
|
330 (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
|
331 (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
|
332 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
|
333 (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
|
334 ≡⟨ 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
|
335 (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
|
336 (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
|
337 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
|
338 (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
|
339 ≡⟨ 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
|
340 (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
|
341 (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
|
342 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
|
343 (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
|
344 ≡⟨ refl ⟩ |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
345 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
|
346 (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
|
347 ≡⟨ 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
|
348 (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
|
349 (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
|
350 |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
351 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
|
352 (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
|
353 ≡⟨ refl ⟩ |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
354 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
|
355 (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
|
356 ≡⟨ 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
|
357 (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
|
358 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
|
359 (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
|
360 ≡⟨ 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
|
361 (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
|
362 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
|
363 (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
|
364 ≡⟨ 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
|
365 (deltaM-covariant (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
|
366 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
|
367 (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
|
368 ≡⟨ refl ⟩ |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
369 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
|
370 (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
|
371 ≡⟨ 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
|
372 (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
|
373 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
|
374 (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
|
375 ≡⟨ refl ⟩ |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
376 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
|
377 (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
|
378 ≡⟨ 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
|
379 (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
|
380 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
|
381 (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
|
382 |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
383 ≡⟨ 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
|
384 (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
|
385 (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
|
386 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
|
387 (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
|
388 ≡⟨ 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
|
389 (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
|
390 (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
|
391 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
|
392 (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
|
393 (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
|
394 |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
395 |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
396 ≡⟨ refl ⟩ |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
397 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
|
398 (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
|
399 (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
|
400 (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
|
401 |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
402 |
53cb21845dea
Prove association-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
117
diff
changeset
|
403 ≡⟨ refl ⟩ |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
404 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
|
405 (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
|
406 ≡⟨ refl ⟩ |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
407 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
|
408 (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
|
409 ≡⟨ refl ⟩ |
115
e6bcc7467335
Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
114
diff
changeset
|
410 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
|
411 ∎ |
117
6f86b55bf8b4
Temporary commit : Proving association-law ....
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
116
diff
changeset
|
412 -} |
115
e6bcc7467335
Temporary commit : Proving association-law ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
114
diff
changeset
|
413 |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
414 {- |
104
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
415 |
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
416 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
|
417 {M : Set l -> Set l} |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
418 (functorM : Functor M) |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
419 (M : Monad M functorM) -> |
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
420 Monad {l} (\A -> DeltaM M {functorM} {M} A (S n)) (deltaM-is-functor {l} {n}) |
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
421 deltaM-is-monad {l} {A} {n} {M} functorM M = |
124
48b44bd85056
Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
118
diff
changeset
|
422 record { mu = deltaM-mu |
48b44bd85056
Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
118
diff
changeset
|
423 ; eta = deltaM-eta |
48b44bd85056
Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
118
diff
changeset
|
424 ; 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
|
425 ; mu-is-nt = (\f x -> (sym (deltaM-mu-is-nt f x))) |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
426 ; association-law = (deltaM-association-law M functorM M) |
124
48b44bd85056
Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
118
diff
changeset
|
427 ; 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
|
428 ; 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
|
429 } |
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
|
430 |
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
|
431 |
128
d9a30f696933
Fix association-law for DeltaM in (S n)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
127
diff
changeset
|
432 |
124
48b44bd85056
Fix proof natural transformation for deltaM-eta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
118
diff
changeset
|
433 -} |