annotate sandbox/FunctorExample.agda @ 10:7c7659d4521d

Improve NaturalTransformation definition
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Tue, 20 Jan 2015 10:51:33 +0900
parents 4a0841123810
children 26e64661b969
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
10
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
11 _∙_ : {l : Level} {A B C : Set l} -> (B -> C) -> (A -> B) -> (A -> C)
6
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
10
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
14 record Functor {l : Level} {A : Set l} (F : {l' : Level} -> Set l' -> Set l') : Set (suc l) where
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
15 field
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
16 fmap : {A B : Set l} -> (A -> B) -> (F A) -> (F B)
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
17 field
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
18 preserve-id : {A : Set l} (x : F A) → fmap id x ≡ id x
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
19 covariant : {A B C : Set l} (f : A -> B) -> (g : B -> C) -> (x : F A)
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
20 -> fmap (g ∙ f) x ≡ ((fmap g) ∙ (fmap f)) x
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
21 open Functor
6
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22
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
9
4a0841123810 fmap for nested functor without implicit level
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
25 data List {l : Level} (A : Set l) : (Set l) where
6
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
26 nil : List A
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
27 cons : A -> List A -> List A
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
28
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
29 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
30 list-fmap f nil = nil
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
31 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
32
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
33 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
34 list-preserve-id nil = refl
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
35 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
36
10
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
37 list-covariant : {l : Level} {A B C : Set l} ->
6
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
38 (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
39 list-covariant f g nil = refl
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
40 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
41
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
42
10
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
43 list-is-functor : {l : Level} {A : Set l}-> Functor {l} {A} List
9
4a0841123810 fmap for nested functor without implicit level
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
44 list-is-functor = record { fmap = list-fmap ;
4a0841123810 fmap for nested functor without implicit level
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
45 preserve-id = list-preserve-id ;
4a0841123810 fmap for nested functor without implicit level
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
46 covariant = list-covariant}
6
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
47
10
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
48 fmap-to-nest-list : {l : Level} {A B : Set l}
8
a3509dbb9e49 Example for implicit-level functor
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
49 -> (A -> B) -> (List (List A)) -> (List (List B))
10
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
50 fmap-to-nest-list {l} {A} {B} f xs = Functor.fmap (list-is-functor {l} {List A}) (Functor.fmap {l} {A} list-is-functor f) xs
7
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
51
9
4a0841123810 fmap for nested functor without implicit level
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
52 data Identity {l : Level} (A : Set l) : Set l where
7
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
53 identity : A -> Identity A
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
54
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
55 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
56 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
57
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
58 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
59 identity-preserve-id (identity x) = refl
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
60
10
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
61 identity-covariant : {l : Level} {A B C : Set l}
7
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
62 (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
63 identity-covariant f g (identity x) = refl
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
64
10
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
65 identity-is-functor : {l : Level} {A : Set l} -> Functor {l} {A} Identity
7
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
66 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
67 preserve-id = identity-preserve-id ;
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
68 covariant = identity-covariant }
6
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
69
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
70
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
71
90abb3f53c03 Add functor example
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
72
10
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
73 record NaturalTransformation {l : Level} (F G : {l' : Level} -> Set l' -> Set l')
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
74 {fmapF : {A B : Set l} -> (A -> B) -> (F A) -> (F B)}
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
75 {fmapG : {A B : Set l} -> (A -> B) -> (G A) -> (G B)}
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
76 (natural-transformation : {A : Set l} -> F A -> G A)
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
77 : Set (suc l) where
7
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
78 field
10
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
79 commute : {A B : Set l} -> (f : A -> B) -> (x : F A) ->
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
80 natural-transformation (fmapF f x) ≡ fmapG f (natural-transformation x)
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
81 open NaturalTransformation
7
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
82
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
83 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
84 tail nil = nil
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
85 tail (cons _ xs) = xs
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
86
10
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
87 tail-commute : {l : Level} {A B : Set l} -> (f : A -> B) -> (xs : List A) ->
7
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
88 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
89 tail-commute f nil = refl
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
90 tail-commute f (cons x xs) = refl
c11c259916b7 Example for natural transformation
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
91
10
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
92
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
93 tail-is-natural-transformation : {l : Level} -> NaturalTransformation List List {list-fmap} {list-fmap {l}} tail
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
94 tail-is-natural-transformation = record { commute = tail-commute}
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
95
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
96
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
97 append : {l : Level} {A : Set l} -> List A -> List A -> List A
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
98 append nil ys = ys
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
99 append (cons x xs) ys = cons x (append xs ys)
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
100
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
101 append-with-fmap : {l : Level} {A B : Set l} -> (f : A -> B) -> (xs : List A) -> (ys : List A) ->
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
102 append (list-fmap f xs) (list-fmap f ys) ≡ list-fmap f (append xs ys)
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
103 append-with-fmap f nil ys = refl
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
104 append-with-fmap f (cons x xs) ys = begin
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
105 append (list-fmap f (cons x xs)) (list-fmap f ys) ≡⟨ refl ⟩
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
106 append (cons (f x) (list-fmap f xs)) (list-fmap f ys) ≡⟨ refl ⟩
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
107 cons (f x) (append (list-fmap f xs) (list-fmap f ys)) ≡⟨ cong (\li -> cons (f x) li) (append-with-fmap f xs ys) ⟩
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
108 list-fmap f (append (cons x xs) ys) ∎
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
109
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
110
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
111
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
112 concat : {l : Level} {A : Set l} -> List (List A) -> List A
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
113 concat nil = nil
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
114 concat (cons x xs) = append x (concat xs)
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
115
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
116 concat-commute : {l : Level} {A B : Set l} -> (f : A -> B) -> (xs : List (List A)) ->
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
117 concat ((list-fmap ∙ list-fmap) f xs) ≡ list-fmap f (concat xs)
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
118 concat-commute f nil = refl
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
119 concat-commute f (cons x xs) = begin
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
120 concat ((list-fmap ∙ list-fmap) f (cons x xs)) ≡⟨ refl ⟩
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
121 concat (cons (list-fmap f x) ((list-fmap ∙ list-fmap) f xs)) ≡⟨ refl ⟩
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
122 append (list-fmap f x) (concat ((list-fmap ∙ list-fmap) f xs)) ≡⟨ cong (\li -> append (list-fmap f x) li) (concat-commute f xs) ⟩
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
123 append (list-fmap f x) (list-fmap f (concat xs)) ≡⟨ append-with-fmap f x (concat xs) ⟩
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
124 list-fmap f (append x (concat xs)) ≡⟨ refl ⟩
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
125 list-fmap f (concat (cons x xs))
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
126
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
127
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
128 concat-is-natural-transformation : {l : Level} -> NaturalTransformation (\A -> List (List A)) List
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
129 {list-fmap ∙ list-fmap} {list-fmap {l}} concat
7c7659d4521d Improve NaturalTransformation definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
130 concat-is-natural-transformation = record {commute = concat-commute}