Mercurial > hg > Members > atton > agda-proofs

annotate sandbox/FunctorExample.agda @ 7:c11c259916b7

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Example for natural transformation
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sat, 17 Jan 2015 22:13:47 +0900
parents 90abb3f53c03
children a3509dbb9e49
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
6
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import Level
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 open import Relation.Binary.PropositionalEquality
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3 open ≡-Reasoning
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6 module FunctorExample where
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8 id : {l : Level} {A : Set l} -> A -> A
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9 id x = x
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11 _∙_ : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> (B -> C) -> (A -> B) -> (A -> C)
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12 f ∙ g = \x -> f (g x)
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16 record Functor {l : Level} (F : Set l -> Set (suc l)) : (Set (suc l)) where
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
17 field
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
18 fmap : ∀{A B} -> (A -> B) -> (F A) -> (F B)
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
19 field
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
20 preserve-id : ∀{A} (Fa : F A) → fmap id Fa ≡ id Fa
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
21 covariant : ∀{A B C} (f : A → B) → (g : B → C) → (x : F A)
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22 → fmap (g ∙ f) x ≡ fmap g (fmap f x)
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
23
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
24 data List {l : Level} (A : Set l) : (Set (suc l)) where
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
25 nil : List A
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
26 cons : A -> List A -> List A
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
27
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
28 list-fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> List A -> List B
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
29 list-fmap f nil = nil
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 list-fmap f (cons x xs) = cons (f x) (list-fmap f xs)
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
31
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
32 list-preserve-id : {l : Level} {A : Set l} -> (xs : List A) -> list-fmap id xs ≡ id xs
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
33 list-preserve-id nil = refl
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
34 list-preserve-id (cons x xs) = cong (\li -> cons x li) (list-preserve-id xs)
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
35
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
36 list-covariant : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
37 (f : A -> B) → (g : B -> C) → (x : List A) → list-fmap (g ∙ f) x ≡ list-fmap g (list-fmap f x)
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
38 list-covariant f g nil = refl
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
39 list-covariant f g (cons x xs) = cong (\li -> cons (g (f x)) li) (list-covariant f g xs)
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
40
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
41
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
42 list-is-functor : {l : Level} -> Functor List
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
43 list-is-functor {l} = record { fmap = list-fmap ;
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
44 preserve-id = list-preserve-id ;
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
45 covariant = list-covariant {l}}
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
46
7
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
47
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
48 data Identity {l : Level} (A : Set l) : Set (suc l) where
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
49 identity : A -> Identity A
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
50
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
51 identity-fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> Identity A -> Identity B
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
52 identity-fmap f (identity a) = identity (f a)
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
53
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
54 identity-preserve-id : {l : Level} {A : Set l} -> (x : Identity A) -> identity-fmap id x ≡ id x
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
55 identity-preserve-id (identity x) = refl
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
56
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
57 identity-covariant : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
58 (f : A -> B) → (g : B -> C) → (x : Identity A) → identity-fmap (g ∙ f) x ≡ identity-fmap g (identity-fmap f x)
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
59 identity-covariant f g (identity x) = refl
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
60
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
61 identity-is-functor : {l : Level} -> Functor Identity
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
62 identity-is-functor {l} = record { fmap = identity-fmap {l};
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
63 preserve-id = identity-preserve-id ;
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
64 covariant = identity-covariant }
6
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
65
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
66
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
67
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
68
7
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
69 record NaturalTransformation {l ll : Level} (F G : Set l -> Set (suc l))
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
70 (functorF : Functor F)
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
71 (functorG : Functor G) : Set (suc (l ⊔ ll)) where
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
72 field
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
73 natural-transformation : {A : Set l} -> F A -> G A
6
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
74 field
7
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
75 commute : ∀ {A B} -> (function : A -> B) -> (x : F A) ->
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
76 natural-transformation (Functor.fmap functorF function x) ≡ Functor.fmap functorG function (natural-transformation x)
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
77
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
78 tail : {l : Level} {A : Set l} -> List A -> List A
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
79 tail nil = nil
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
80 tail (cons _ xs) = xs
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
81
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
82 tail-commute : {l ll : Level} {A : Set l} {B : Set ll} -> (f : A -> B) -> (xs : List A) ->
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
83 tail (list-fmap f xs) ≡ list-fmap f (tail xs)
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
84 tail-commute f nil = refl
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
85 tail-commute f (cons x xs) = refl
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
86
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
87 tail-is-natural-transformation : {l ll : Level} -> NaturalTransformation {l} {ll} List List list-is-functor list-is-functor
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
88 tail-is-natural-transformation = record { natural-transformation = tail;
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
89 commute = tail-commute}