annotate agda/similar.agda @ 27:742e62fc63e4

Define Monad-law 1-4
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Tue, 07 Oct 2014 14:53:56 +0900
parents 5ba82f107a95
children 6e6d646d7722
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
26
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import list
27
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
2 open import Relation.Binary.PropositionalEquality
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
3 open ≡-Reasoning
26
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5 module similar where
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6
27
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
7 id : {A : Set} -> A -> A
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
8 id x = x
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
9
26
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10 postulate String : Set
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11 postulate show : {A : Set} -> A -> String
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13 data Similar (A : Set) : Set where
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14 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
15
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
17 fmap : {A B : Set} -> (A -> B) -> (Similar A) -> (Similar B)
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
18 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
19
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
20
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
21 mu : {A : Set} -> Similar (Similar A) -> Similar A
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22 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
23
27
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
24 return : {A : Set} -> A -> Similar A
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
25 return x = similar [] x [] x
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
26
26
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
27 returnS : {A : Set} -> A -> Similar A
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
28 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
29
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 returnSS : {A : Set} -> A -> A -> Similar A
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
31 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
32
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
33
27
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
34 _∙_ : {A B C : Set} -> (A -> B) -> (B -> C) -> (A -> C)
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
35 f ∙ g = \x -> g (f x)
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
36
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
37 monad-law-1 : mu ∙ (fmap mu) ≡ mu ∙ mu
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
38 monad-law-1 = {!!}
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
39
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
40 --monad-law-2 : mu ∙ fmap return ≡ mu ∙ return ≡id
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
41 monad-law-2-1 : mu ∙ fmap return ≡ mu ∙ return
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
42 monad-law-2-1 = {!!}
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
43
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
44 monad-law-2-2 : mu ∙ return ≡ id
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
45 monad-law-2-2 = {!!}
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
46
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
47 monad-law-3 : ∀{f} -> return ∙ f ≡ fmap f ∙ return
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
48 monad-law-3 = {!!}
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
49
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
50 monad-law-4 : ∀{f} -> mu ∙ fmap (fmap f) ≡ fmap f ∙ mu
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
51 monad-law-4 = {!!}