Mercurial > hg > Members > atton > delta_monad
annotate agda/similar.agda @ 36:169ec60fcd36
Proof Monad-law-4
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
---|---|
date | Sat, 18 Oct 2014 14:22:34 +0900 |
parents | c5cdbedc68ad |
children | 6ce83b2c9e59 |
rev | line source |
---|---|
26
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
1 open import list |
28
6e6d646d7722
Split basic functions to file
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
27
diff
changeset
|
2 open import basic |
29
e0ba1bf564dd
Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
3 |
e0ba1bf564dd
Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
4 open import Level |
27
742e62fc63e4
Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
26
diff
changeset
|
5 open import Relation.Binary.PropositionalEquality |
742e62fc63e4
Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
26
diff
changeset
|
6 open ≡-Reasoning |
26
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
7 |
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
8 module similar where |
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
9 |
29
e0ba1bf564dd
Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
10 data Similar {l : Level} (A : Set l) : (Set (suc l)) where |
26
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
11 similar : List String -> A -> List String -> A -> Similar A |
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
12 |
29
e0ba1bf564dd
Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
13 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Similar A) -> (Similar B) |
26
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
14 fmap f (similar xs x ys y) = similar xs (f x) ys (f y) |
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
15 |
29
e0ba1bf564dd
Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
16 mu : {l : Level} {A : Set l} -> Similar (Similar A) -> Similar A |
26
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
17 mu (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = similar (lx ++ llx) x (ly ++ lly) y |
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
18 |
34
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
19 return : {l : Level} {A : Set l} -> A -> Similar A |
27
742e62fc63e4
Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
26
diff
changeset
|
20 return x = similar [] x [] x |
742e62fc63e4
Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
26
diff
changeset
|
21 |
26
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
22 returnS : {A : Set} -> A -> Similar A |
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
23 returnS x = similar [[ (show x) ]] x [[ (show x) ]] x |
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
24 |
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
25 returnSS : {A : Set} -> A -> A -> Similar A |
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
26 returnSS x y = similar [[ (show x) ]] x [[ (show y) ]] y |
5ba82f107a95
Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
27 |
33 | 28 |
29
e0ba1bf564dd
Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
29 --monad-law-1 : mu ∙ (fmap mu) ≡ mu ∙ mu |
e0ba1bf564dd
Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
30 monad-law-1 : {l : Level} {A : Set l} -> (s : Similar (Similar (Similar A))) -> ((mu ∙ (fmap mu)) s) ≡ ((mu ∙ mu) s) |
32
71906644d206
Expand monad-law 1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
30
diff
changeset
|
31 monad-law-1 (similar lx (similar llx (similar lllx x _ _) _ (similar _ _ _ _)) |
71906644d206
Expand monad-law 1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
30
diff
changeset
|
32 ly (similar _ (similar _ _ _ _) lly (similar _ _ llly y))) = begin |
71906644d206
Expand monad-law 1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
30
diff
changeset
|
33 similar (lx ++ (llx ++ lllx)) x (ly ++ (lly ++ llly)) y |
33 | 34 ≡⟨ cong (\left-list -> similar left-list x (ly ++ (lly ++ llly)) y) (list-associative lx llx lllx) ⟩ |
35 similar (lx ++ llx ++ lllx) x (ly ++ (lly ++ llly)) y | |
36 ≡⟨ cong (\right-list -> similar (lx ++ llx ++ lllx) x right-list y ) (list-associative ly lly llly) ⟩ | |
32
71906644d206
Expand monad-law 1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
30
diff
changeset
|
37 similar (lx ++ llx ++ lllx) x (ly ++ lly ++ llly) y |
29
e0ba1bf564dd
Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
38 ∎ |
e0ba1bf564dd
Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
39 |
34
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
40 |
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
41 --monad-law-2 : mu ∙ fmap return ≡ mu ∙ return ≡ id |
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
42 monad-law-2-1 : {l : Level} {A : Set l} -> (s : Similar A) -> |
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
43 (mu ∙ fmap return) s ≡ (mu ∙ return) s |
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
44 monad-law-2-1 (similar lx x ly y) = begin |
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
45 similar (lx ++ []) x (ly ++ []) y |
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
46 ≡⟨ cong (\left-list -> similar left-list x (ly ++ []) y) (empty-append lx)⟩ |
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
47 similar lx x (ly ++ []) y |
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
48 ≡⟨ cong (\right-list -> similar lx x right-list y) (empty-append ly) ⟩ |
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
49 similar lx x ly y |
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
50 ∎ |
b7c4e6276bcf
Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
33
diff
changeset
|
51 |
35
c5cdbedc68ad
Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
34
diff
changeset
|
52 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Similar A) -> (mu ∙ return) s ≡ id s |
c5cdbedc68ad
Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
34
diff
changeset
|
53 monad-law-2-2 (similar lx x ly y) = refl |
c5cdbedc68ad
Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
34
diff
changeset
|
54 |
36 | 55 |
56 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (return ∙ f) x ≡ (fmap f ∙ return) x | |
57 monad-law-3 f x = refl | |
27
742e62fc63e4
Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
26
diff
changeset
|
58 |
36 | 59 |
60 monad-law-4 : {l : Level} {A B : Set l} (f : A -> B) (s : Similar (Similar A)) -> | |
61 (mu ∙ fmap (fmap f)) s ≡ (fmap f ∙ mu) s | |
62 monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl |