Mercurial > hg > Members > kono > Proof > category
annotate monoidal.agda @ 731:117e5b392673
Generalize Free Theorem
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Mon, 27 Nov 2017 14:42:49 +0900 |
parents | e4ef69bae044 |
children | 2439a142aba2 |
rev | line source |
---|---|
696
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
1 open import Level |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
2 open import Category |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
3 module monoidal where |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
4 |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
5 open import Data.Product renaming (_×_ to _*_) |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
6 open import Category.Constructions.Product |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
7 open import HomReasoning |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
8 open import cat-utility |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
9 open import Relation.Binary.Core |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
10 open import Relation.Binary |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
11 |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
12 open Functor |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
13 |
731
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
14 -- record Iso {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
15 -- (x y : Obj C ) |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
16 -- : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
17 -- field |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
18 -- ≅→ : Hom C x y |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
19 -- ≅← : Hom C y x |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
20 -- iso→ : C [ C [ ≅← o ≅→ ] ≈ id1 C x ] |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
21 -- iso← : C [ C [ ≅→ o ≅← ] ≈ id1 C y ] |
696
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
22 |
730 | 23 -- Monoidal Category |
698 | 24 |
25 record IsMonoidal {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) (I : Obj C) ( BI : Functor ( C × C ) C ) | |
26 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where | |
27 open Iso | |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
28 infixr 9 _□_ _■_ |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
29 _□_ : ( x y : Obj C ) → Obj C |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
30 _□_ x y = FObj BI ( x , y ) |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
31 _■_ : {a b c d : Obj C } ( f : Hom C a c ) ( g : Hom C b d ) → Hom C ( a □ b ) ( c □ d ) |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
32 _■_ f g = FMap BI ( f , g ) |
698 | 33 field |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
34 mα-iso : {a b c : Obj C} → Iso C ( ( a □ b) □ c) ( a □ ( b □ c ) ) |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
35 mλ-iso : {a : Obj C} → Iso C ( I □ a) a |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
36 mρ-iso : {a : Obj C} → Iso C ( a □ I) a |
698 | 37 mα→nat1 : {a a' b c : Obj C} → ( f : Hom C a a' ) → |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
38 C [ C [ ( f ■ id1 C ( b □ c )) o ≅→ (mα-iso {a} {b} {c}) ] ≈ |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
39 C [ ≅→ (mα-iso ) o ( (f ■ id1 C b ) ■ id1 C c ) ] ] |
698 | 40 mα→nat2 : {a b b' c : Obj C} → ( f : Hom C b b' ) → |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
41 C [ C [ ( id1 C a ■ ( f ■ id1 C c ) ) o ≅→ (mα-iso {a} {b} {c} ) ] ≈ |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
42 C [ ≅→ (mα-iso ) o ( (id1 C a ■ f ) ■ id1 C c ) ] ] |
698 | 43 mα→nat3 : {a b c c' : Obj C} → ( f : Hom C c c' ) → |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
44 C [ C [ ( id1 C a ■ ( id1 C b ■ f ) ) o ≅→ (mα-iso {a} {b} {c} ) ] ≈ |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
45 C [ ≅→ (mα-iso ) o ( id1 C ( a □ b ) ■ f ) ] ] |
698 | 46 mλ→nat : {a a' : Obj C} → ( f : Hom C a a' ) → |
47 C [ C [ f o ≅→ (mλ-iso {a} ) ] ≈ | |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
48 C [ ≅→ (mλ-iso ) o ( id1 C I ■ f ) ] ] |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
49 mρ→nat : {a a' : Obj C} → ( f : Hom C a a' ) → |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
50 C [ C [ f o ≅→ (mρ-iso {a} ) ] ≈ |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
51 C [ ≅→ (mρ-iso ) o ( f ■ id1 C I ) ] ] |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
52 -- we should write naturalities for ≅← (maybe derived from above ) |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
53 αABC□1D : {a b c d e : Obj C } → Hom C (((a □ b) □ c ) □ d) ((a □ (b □ c)) □ d) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
54 αABC□1D {a} {b} {c} {d} {e} = ( ≅→ mα-iso ■ id1 C d ) |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
55 αAB□CD : {a b c d e : Obj C } → Hom C ((a □ (b □ c)) □ d) (a □ ((b □ c ) □ d)) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
56 αAB□CD {a} {b} {c} {d} {e} = ≅→ mα-iso |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
57 1A□BCD : {a b c d e : Obj C } → Hom C (a □ ((b □ c ) □ d)) (a □ (b □ ( c □ d) )) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
58 1A□BCD {a} {b} {c} {d} {e} = ( id1 C a ■ ≅→ mα-iso ) |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
59 αABC□D : {a b c d e : Obj C } → Hom C (a □ (b □ ( c □ d) )) ((a □ b ) □ (c □ d)) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
60 αABC□D {a} {b} {c} {d} {e} = ≅← mα-iso |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
61 αA□BCD : {a b c d e : Obj C } → Hom C (((a □ b) □ c ) □ d) ((a □ b ) □ (c □ d)) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
62 αA□BCD {a} {b} {c} {d} {e} = ≅→ mα-iso |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
63 αAIB : {a b : Obj C } → Hom C (( a □ I ) □ b ) (a □ ( I □ b )) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
64 αAIB {a} {b} = ≅→ mα-iso |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
65 1A□λB : {a b : Obj C } → Hom C (a □ ( I □ b )) ( a □ b ) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
66 1A□λB {a} {b} = id1 C a ■ ≅→ mλ-iso |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
67 ρA□IB : {a b : Obj C } → Hom C (( a □ I ) □ b ) ( a □ b ) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
68 ρA□IB {a} {b} = ≅→ mρ-iso ■ id1 C b |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
69 |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
70 field |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
71 comm-penta : {a b c d e : Obj C} |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
72 → C [ C [ αABC□D {a} {b} {c} {d} {e} o C [ 1A□BCD {a} {b} {c} {d} {e} o C [ αAB□CD {a} {b} {c} {d} {e} o αABC□1D {a} {b} {c} {d} {e} ] ] ] |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
73 ≈ αA□BCD {a} {b} {c} {d} {e} ] |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
74 comm-unit : {a b : Obj C} |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
75 → C [ C [ 1A□λB {a} {b} o αAIB ] ≈ ρA□IB {a} {b} ] |
696
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
76 |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
77 record Monoidal {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
78 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
79 field |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
80 m-i : Obj C |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
81 m-bi : Functor ( C × C ) C |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
82 isMonoidal : IsMonoidal C m-i m-bi |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
83 |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
84 --------- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
85 -- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
86 -- Lax Monoidal Functor |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
87 -- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
88 -- N → M |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
89 -- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
90 --------- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
91 |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
92 --------- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
93 -- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
94 -- Two implementations of Functor ( C × C ) → D from F : Functor C → D (given) |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
95 -- dervied from F and two Monoidal Categories |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
96 -- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
97 -- F x ● F y |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
98 -- F ( x ⊗ y ) |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
99 -- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
100 -- and a given natural transformation for them |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
101 -- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
102 -- φ : F x ● F y → F ( x ⊗ y ) |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
103 -- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
104 -- TMap φ : ( x y : Obj C ) → Hom D ( F x ● F y ) ( F ( x ⊗ y )) |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
105 -- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
106 -- a given unit arrow |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
107 -- |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
108 -- ψ : IN → IM |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
109 |
703 | 110 Functor● : {c₁ c₂ ℓ : Level} (C D : Category c₁ c₂ ℓ) ( N : Monoidal D ) |
111 ( MF : Functor C D ) → Functor ( C × C ) D | |
112 Functor● C D N MF = record { | |
113 FObj = λ x → (FObj MF (proj₁ x) ) ● (FObj MF (proj₂ x) ) | |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
114 ; FMap = λ {x : Obj ( C × C ) } {y} f → ( FMap MF (proj₁ f ) ■ FMap MF (proj₂ f) ) |
703 | 115 ; isFunctor = record { |
116 ≈-cong = ≈-cong | |
117 ; identity = identity | |
118 ; distr = distr | |
119 } | |
120 } where | |
121 _●_ : (x y : Obj D ) → Obj D | |
122 _●_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal N) ) x y | |
704 | 123 _■_ : {a b c d : Obj D } ( f : Hom D a c ) ( g : Hom D b d ) → Hom D ( a ● b ) ( c ● d ) |
124 _■_ f g = FMap (Monoidal.m-bi N) ( f , g ) | |
125 F : { a b : Obj C } → ( f : Hom C a b ) → Hom D (FObj MF a) (FObj MF b ) | |
126 F f = FMap MF f | |
703 | 127 ≈-cong : {a b : Obj (C × C)} {f g : Hom (C × C) a b} → (C × C) [ f ≈ g ] → |
704 | 128 D [ (F (proj₁ f) ■ F (proj₂ f)) ≈ (F (proj₁ g) ■ F (proj₂ g)) ] |
703 | 129 ≈-cong {a} {b} {f} {g} f≈g = let open ≈-Reasoning D in begin |
704 | 130 F (proj₁ f) ■ F (proj₂ f) |
703 | 131 ≈⟨ fcong (Monoidal.m-bi N) ( fcong MF ( proj₁ f≈g ) , fcong MF ( proj₂ f≈g )) ⟩ |
704 | 132 F (proj₁ g) ■ F (proj₂ g) |
703 | 133 ∎ |
704 | 134 identity : {a : Obj (C × C)} → D [ (F (proj₁ (id1 (C × C) a)) ■ F (proj₂ (id1 (C × C) a))) |
703 | 135 ≈ id1 D (FObj MF (proj₁ a) ● FObj MF (proj₂ a)) ] |
136 identity {a} = let open ≈-Reasoning D in begin | |
704 | 137 F (proj₁ (id1 (C × C) a)) ■ F (proj₂ (id1 (C × C) a)) |
703 | 138 ≈⟨ fcong (Monoidal.m-bi N) ( IsFunctor.identity (isFunctor MF ) , IsFunctor.identity (isFunctor MF )) ⟩ |
704 | 139 id1 D (FObj MF (proj₁ a)) ■ id1 D (FObj MF (proj₂ a)) |
703 | 140 ≈⟨ IsFunctor.identity (isFunctor (Monoidal.m-bi N)) ⟩ |
141 id1 D (FObj MF (proj₁ a) ● FObj MF (proj₂ a)) | |
142 ∎ | |
143 distr : {a b c : Obj (C × C)} {f : Hom (C × C) a b} {g : Hom (C × C) b c} → | |
704 | 144 D [ (F (proj₁ ((C × C) [ g o f ])) ■ F (proj₂ ((C × C) [ g o f ]))) |
145 ≈ D [ (F (proj₁ g) ■ F (proj₂ g)) o (F (proj₁ f) ■ F (proj₂ f)) ] ] | |
703 | 146 distr {a} {b} {c} {f} {g} = let open ≈-Reasoning D in begin |
704 | 147 (F (proj₁ ((C × C) [ g o f ])) ■ F (proj₂ ((C × C) [ g o f ]))) |
703 | 148 ≈⟨ fcong (Monoidal.m-bi N) ( IsFunctor.distr ( isFunctor MF) , IsFunctor.distr ( isFunctor MF )) ⟩ |
704 | 149 ( F (proj₁ g) o F (proj₁ f) ) ■ ( F (proj₂ g) o F (proj₂ f) ) |
703 | 150 ≈⟨ IsFunctor.distr ( isFunctor (Monoidal.m-bi N)) ⟩ |
704 | 151 (F (proj₁ g) ■ F (proj₂ g)) o (F (proj₁ f) ■ F (proj₂ f)) |
703 | 152 ∎ |
696
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
153 |
703 | 154 Functor⊗ : {c₁ c₂ ℓ : Level} (C D : Category c₁ c₂ ℓ) ( M : Monoidal C ) |
155 ( MF : Functor C D ) → Functor ( C × C ) D | |
156 Functor⊗ C D M MF = record { | |
157 FObj = λ x → FObj MF ( proj₁ x ⊗ proj₂ x ) | |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
158 ; FMap = λ {a} {b} f → F ( proj₁ f □ proj₂ f ) |
703 | 159 ; isFunctor = record { |
160 ≈-cong = ≈-cong | |
161 ; identity = identity | |
162 ; distr = distr | |
163 } | |
164 } where | |
165 _⊗_ : (x y : Obj C ) → Obj C | |
166 _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y | |
704 | 167 _□_ : {a b c d : Obj C } ( f : Hom C a c ) ( g : Hom C b d ) → Hom C ( a ⊗ b ) ( c ⊗ d ) |
168 _□_ f g = FMap (Monoidal.m-bi M) ( f , g ) | |
169 F : { a b : Obj C } → ( f : Hom C a b ) → Hom D (FObj MF a) (FObj MF b ) | |
170 F f = FMap MF f | |
703 | 171 ≈-cong : {a b : Obj (C × C)} {f g : Hom (C × C) a b} → (C × C) [ f ≈ g ] → |
704 | 172 D [ F ( (proj₁ f □ proj₂ f)) ≈ F ( (proj₁ g □ proj₂ g)) ] |
703 | 173 ≈-cong {a} {b} {f} {g} f≈g = IsFunctor.≈-cong (isFunctor MF ) ( IsFunctor.≈-cong (isFunctor (Monoidal.m-bi M) ) f≈g ) |
704 | 174 identity : {a : Obj (C × C)} → D [ F ( (proj₁ (id1 (C × C) a) □ proj₂ (id1 (C × C) a))) |
703 | 175 ≈ id1 D (FObj MF (proj₁ a ⊗ proj₂ a)) ] |
176 identity {a} = let open ≈-Reasoning D in begin | |
704 | 177 F ( (proj₁ (id1 (C × C) a) □ proj₂ (id1 (C × C) a))) |
703 | 178 ≈⟨⟩ |
704 | 179 F (FMap (Monoidal.m-bi M) (id1 (C × C) a ) ) |
703 | 180 ≈⟨ fcong MF ( IsFunctor.identity (isFunctor (Monoidal.m-bi M) )) ⟩ |
704 | 181 F (id1 C (proj₁ a ⊗ proj₂ a)) |
703 | 182 ≈⟨ IsFunctor.identity (isFunctor MF) ⟩ |
183 id1 D (FObj MF (proj₁ a ⊗ proj₂ a)) | |
184 ∎ | |
185 distr : {a b c : Obj (C × C)} {f : Hom (C × C) a b} {g : Hom (C × C) b c} → D [ | |
704 | 186 F ( (proj₁ ((C × C) [ g o f ]) □ proj₂ ((C × C) [ g o f ]))) |
187 ≈ D [ F ( (proj₁ g □ proj₂ g)) o F ( (proj₁ f □ proj₂ f)) ] ] | |
703 | 188 distr {a} {b} {c} {f} {g} = let open ≈-Reasoning D in begin |
704 | 189 F ( (proj₁ ((C × C) [ g o f ]) □ proj₂ ((C × C) [ g o f ]))) |
703 | 190 ≈⟨⟩ |
704 | 191 F (FMap (Monoidal.m-bi M) ( (C × C) [ g o f ] )) |
703 | 192 ≈⟨ fcong MF ( IsFunctor.distr (isFunctor (Monoidal.m-bi M))) ⟩ |
704 | 193 F (C [ FMap (Monoidal.m-bi M) g o FMap (Monoidal.m-bi M) f ]) |
703 | 194 ≈⟨ IsFunctor.distr ( isFunctor MF ) ⟩ |
704 | 195 F ( proj₁ g □ proj₂ g) o F ( proj₁ f □ proj₂ f) |
703 | 196 ∎ |
197 | |
696
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
198 |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
199 record IsMonoidalFunctor {c₁ c₂ ℓ : Level} {C D : Category c₁ c₂ ℓ} ( M : Monoidal C ) ( N : Monoidal D ) |
698 | 200 ( MF : Functor C D ) |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
201 ( ψ : Hom D (Monoidal.m-i N) (FObj MF (Monoidal.m-i M) ) ) |
696
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
202 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where |
698 | 203 _⊗_ : (x y : Obj C ) → Obj C |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
204 _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
205 _□_ : {a b c d : Obj C } ( f : Hom C a c ) ( g : Hom C b d ) → Hom C ( a ⊗ b ) ( c ⊗ d ) |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
206 _□_ f g = FMap (Monoidal.m-bi M) ( f , g ) |
698 | 207 _●_ : (x y : Obj D ) → Obj D |
700
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
208 _●_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal N) ) x y |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
209 _■_ : {a b c d : Obj D } ( f : Hom D a c ) ( g : Hom D b d ) → Hom D ( a ● b ) ( c ● d ) |
cfd2d402c486
monodial cateogry and functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
699
diff
changeset
|
210 _■_ f g = FMap (Monoidal.m-bi N) ( f , g ) |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
211 F● : Functor ( C × C ) D |
703 | 212 F● = Functor● C D N MF |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
213 F⊗ : Functor ( C × C ) D |
703 | 214 F⊗ = Functor⊗ C D M MF |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
215 field |
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
216 φab : NTrans ( C × C ) D F● F⊗ |
698 | 217 open Iso |
218 open Monoidal | |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
219 open IsMonoidal hiding ( _■_ ; _□_ ) |
699 | 220 αC : {a b c : Obj C} → Hom C (( a ⊗ b ) ⊗ c ) ( a ⊗ ( b ⊗ c ) ) |
221 αC {a} {b} {c} = ≅→ (mα-iso (isMonoidal M) {a} {b} {c}) | |
222 αD : {a b c : Obj D} → Hom D (( a ● b ) ● c ) ( a ● ( b ● c ) ) | |
223 αD {a} {b} {c} = ≅→ (mα-iso (isMonoidal N) {a} {b} {c}) | |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
224 F : Obj C → Obj D |
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
225 F x = FObj MF x |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
226 φ : ( x y : Obj C ) → Hom D ( FObj F● (x , y) ) ( FObj F⊗ ( x , y )) |
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
227 φ x y = NTrans.TMap φab ( x , y ) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
228 1●φBC : {a b c : Obj C} → Hom D ( F a ● ( F b ● F c ) ) ( F a ● ( F ( b ⊗ c ) )) |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
229 1●φBC {a} {b} {c} = id1 D (F a) ■ φ b c |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
230 φAB⊗C : {a b c : Obj C} → Hom D ( F a ● ( F ( b ⊗ c ) )) (F ( a ⊗ ( b ⊗ c ))) |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
231 φAB⊗C {a} {b} {c} = φ a (b ⊗ c ) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
232 φAB●1 : {a b c : Obj C} → Hom D ( ( F a ● F b ) ● F c ) ( F ( a ⊗ b ) ● F c ) |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
233 φAB●1 {a} {b} {c} = φ a b ■ id1 D (F c) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
234 φA⊗BC : {a b c : Obj C} → Hom D ( F ( a ⊗ b ) ● F c ) (F ( (a ⊗ b ) ⊗ c )) |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
235 φA⊗BC {a} {b} {c} = φ ( a ⊗ b ) c |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
236 FαC : {a b c : Obj C} → Hom D (F ( (a ⊗ b ) ⊗ c )) (F ( a ⊗ ( b ⊗ c ))) |
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
237 FαC {a} {b} {c} = FMap MF ( ≅→ (mα-iso (isMonoidal M) {a} {b} {c}) ) |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
238 1●ψ : { a b : Obj C } → Hom D (F a ● Monoidal.m-i N ) ( F a ● F ( Monoidal.m-i M ) ) |
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
239 1●ψ{a} {b} = id1 D (F a) ■ ψ |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
240 φAIC : { a b : Obj C } → Hom D ( F a ● F ( Monoidal.m-i M ) ) (F ( a ⊗ Monoidal.m-i M )) |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
241 φAIC {a} {b} = φ a ( Monoidal.m-i M ) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
242 FρC : { a b : Obj C } → Hom D (F ( a ⊗ Monoidal.m-i M ))( F a ) |
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
243 FρC {a} {b} = FMap MF ( ≅→ (mρ-iso (isMonoidal M) {a} ) ) |
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
244 ρD : { a b : Obj C } → Hom D (F a ● Monoidal.m-i N ) ( F a ) |
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
245 ρD {a} {b} = ≅→ (mρ-iso (isMonoidal N) {F a} ) |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
246 ψ●1 : { a b : Obj C } → Hom D (Monoidal.m-i N ● F b ) ( F ( Monoidal.m-i M ) ● F b ) |
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
247 ψ●1 {a} {b} = ψ ■ id1 D (F b) |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
248 φICB : { a b : Obj C } → Hom D ( F ( Monoidal.m-i M ) ● F b ) ( F ( ( Monoidal.m-i M ) ⊗ b ) ) |
702
d16327b48b83
Monoidal Functor ( two functor remains )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
701
diff
changeset
|
249 φICB {a} {b} = φ ( Monoidal.m-i M ) b |
701
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
250 FλD : { a b : Obj C } → Hom D ( F ( ( Monoidal.m-i M ) ⊗ b ) ) (F b ) |
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
251 FλD {a} {b} = FMap MF ( ≅→ (mλ-iso (isMonoidal M) {b} ) ) |
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
252 λD : { a b : Obj C } → Hom D (Monoidal.m-i N ● F b ) (F b ) |
7a729bb62ce3
Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
diff
changeset
|
253 λD {a} {b} = ≅→ (mλ-iso (isMonoidal N) {F b} ) |
696
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
254 field |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
255 associativity : {a b c : Obj C } → D [ D [ φAB⊗C {a} {b} {c} o D [ 1●φBC o αD ] ] ≈ D [ FαC o D [ φA⊗BC o φAB●1 ] ] ] |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
256 unitarity-idr : {a b : Obj C } → D [ D [ FρC {a} {b} o D [ φAIC {a} {b} o 1●ψ{a} {b} ] ] ≈ ρD {a} {b} ] |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
257 unitarity-idl : {a b : Obj C } → D [ D [ FλD {a} {b} o D [ φICB {a} {b} o ψ●1 {a} {b} ] ] ≈ λD {a} {b} ] |
696
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
258 |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
259 record MonoidalFunctor {c₁ c₂ ℓ : Level} {C D : Category c₁ c₂ ℓ} ( M : Monoidal C ) ( N : Monoidal D ) |
10ccac3bc285
Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
260 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
261 field |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
262 MF : Functor C D |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
263 ψ : Hom D (Monoidal.m-i N) (FObj MF (Monoidal.m-i M) ) |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
264 isMonodailFunctor : IsMonoidalFunctor M N MF ψ |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
265 |
731
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
266 ----- |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
267 -- they say it is not possible to prove FreeTheorem in Agda nor Coq |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
268 -- https://stackoverflow.com/questions/24718567/is-it-possible-to-get-hold-of-free-theorems-as-propositional-equalities |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
269 -- so we postulate this |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
270 -- and we cannot indent postulate ... |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
271 postulate FreeTheorem : {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (C : Category c₁ c₂ ℓ) (D : Category c₁' c₂' ℓ') {a b c : Obj C } → (F : Functor C D ) → ( fmap : {a : Obj C } {b : Obj C } → Hom C a b → Hom D (FObj F a) ( FObj F b) ) → {h f : Hom C a b } → {g k : Hom C b c } → C [ C [ g o h ] ≈ C [ k o f ] ] → D [ D [ FMap F g o fmap h ] ≈ D [ fmap k o FMap F f ] ] |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
272 UniquenessOfFunctor : {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (C : Category c₁ c₂ ℓ) (D : Category c₁' c₂' ℓ') (F : Functor C D) |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
273 {a b : Obj C } { f : Hom C a b } → ( fmap : {a : Obj C } {b : Obj C } → Hom C a b → Hom D (FObj F a) ( FObj F b) ) |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
274 → ( {b : Obj C } → D [ fmap (id1 C b) ≈ id1 D (FObj F b) ] ) |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
275 → D [ fmap f ≈ FMap F f ] |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
276 UniquenessOfFunctor C D F {a} {b} {f} fmap eq = begin |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
277 fmap f |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
278 ≈↑⟨ idL ⟩ |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
279 id1 D (FObj F b) o fmap f |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
280 ≈↑⟨ car ( IsFunctor.identity (isFunctor F )) ⟩ |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
281 FMap F (id1 C b) o fmap f |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
282 ≈⟨ FreeTheorem C D F fmap (IsEquivalence.refl (IsCategory.isEquivalence ( Category.isCategory C ))) ⟩ |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
283 fmap (id1 C b) o FMap F f |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
284 ≈⟨ car eq ⟩ |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
285 id1 D (FObj F b) o FMap F f |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
286 ≈⟨ idL ⟩ |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
287 FMap F f |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
288 ∎ |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
289 where open ≈-Reasoning D |
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
290 |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
291 open import Category.Sets |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
292 |
706 | 293 import Relation.Binary.PropositionalEquality |
294 -- Extensionality a b = {A : Set a} {B : A → Set b} {f g : (x : A) → B x} → (∀ x → f x ≡ g x) → f ≡ g → ( λ x → f x ≡ λ x → g x ) | |
295 postulate extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Relation.Binary.PropositionalEquality.Extensionality c₂ c₂ | |
296 | |
730 | 297 ------ |
298 -- Data.Product as a Tensor Product for Monoidal Category | |
299 -- | |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
300 |
708 | 301 SetsTensorProduct : {c : Level} → Functor ( Sets {c} × Sets {c} ) (Sets {c}) |
302 SetsTensorProduct = record { | |
303 FObj = λ x → proj₁ x * proj₂ x | |
304 ; FMap = λ {x : Obj ( Sets × Sets ) } {y} f → map (proj₁ f) (proj₂ f) | |
305 ; isFunctor = record { | |
306 ≈-cong = ≈-cong | |
307 ; identity = refl | |
308 ; distr = refl | |
706 | 309 } |
708 | 310 } where |
311 ≈-cong : {a b : Obj (Sets × Sets)} {f g : Hom (Sets × Sets) a b} → | |
312 (Sets × Sets) [ f ≈ g ] → Sets [ map (proj₁ f) (proj₂ f) ≈ map (proj₁ g) (proj₂ g) ] | |
313 ≈-cong (refl , refl) = refl | |
314 | |
730 | 315 ----- |
316 -- | |
317 -- Sets as Monoidal Category | |
318 -- | |
319 -- almost all comutativities are refl | |
320 -- | |
321 -- | |
322 -- | |
708 | 323 |
730 | 324 data One {c : Level} : Set c where |
325 OneObj : One -- () in Haskell ( or any one object set ) | |
708 | 326 |
327 MonoidalSets : {c : Level} → Monoidal (Sets {c}) | |
727
ea84cc6c1797
monoidal functor and applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
726
diff
changeset
|
328 MonoidalSets {c} = record { |
ea84cc6c1797
monoidal functor and applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
726
diff
changeset
|
329 m-i = One {c} ; |
708 | 330 m-bi = SetsTensorProduct ; |
331 isMonoidal = record { | |
332 mα-iso = record { ≅→ = mα→ ; ≅← = mα← ; iso→ = refl ; iso← = refl } ; | |
333 mλ-iso = record { ≅→ = mλ→ ; ≅← = mλ← ; iso→ = extensionality Sets ( λ x → mλiso x ) ; iso← = refl } ; | |
334 mρ-iso = record { ≅→ = mρ→ ; ≅← = mρ← ; iso→ = extensionality Sets ( λ x → mρiso x ) ; iso← = refl } ; | |
335 mα→nat1 = λ f → refl ; | |
336 mα→nat2 = λ f → refl ; | |
337 mα→nat3 = λ f → refl ; | |
338 mλ→nat = λ f → refl ; | |
339 mρ→nat = λ f → refl ; | |
340 comm-penta = refl ; | |
341 comm-unit = refl | |
342 } | |
343 } where | |
344 _⊗_ : ( a b : Obj Sets ) → Obj Sets | |
345 _⊗_ a b = FObj SetsTensorProduct (a , b ) | |
730 | 346 -- association operations |
708 | 347 mα→ : {a b c : Obj Sets} → Hom Sets ( ( a ⊗ b ) ⊗ c ) ( a ⊗ ( b ⊗ c ) ) |
348 mα→ ((a , b) , c ) = (a , ( b , c ) ) | |
349 mα← : {a b c : Obj Sets} → Hom Sets ( a ⊗ ( b ⊗ c ) ) ( ( a ⊗ b ) ⊗ c ) | |
350 mα← (a , ( b , c ) ) = ((a , b) , c ) | |
730 | 351 -- (One , a) ⇔ a |
708 | 352 mλ→ : {a : Obj Sets} → Hom Sets ( One ⊗ a ) a |
353 mλ→ (_ , a) = a | |
354 mλ← : {a : Obj Sets} → Hom Sets a ( One ⊗ a ) | |
355 mλ← a = ( OneObj , a ) | |
356 mλiso : {a : Obj Sets} (x : One ⊗ a) → (Sets [ mλ← o mλ→ ]) x ≡ id1 Sets (One ⊗ a) x | |
357 mλiso (OneObj , _ ) = refl | |
730 | 358 -- (a , One) ⇔ a |
708 | 359 mρ→ : {a : Obj Sets} → Hom Sets ( a ⊗ One ) a |
360 mρ→ (a , _) = a | |
361 mρ← : {a : Obj Sets} → Hom Sets a ( a ⊗ One ) | |
362 mρ← a = ( a , OneObj ) | |
363 mρiso : {a : Obj Sets} (x : a ⊗ One ) → (Sets [ mρ← o mρ→ ]) x ≡ id1 Sets (a ⊗ One) x | |
364 mρiso (_ , OneObj ) = refl | |
365 | |
710 | 366 ≡-cong = Relation.Binary.PropositionalEquality.cong |
706 | 367 |
713 | 368 |
369 record IsHaskellMonoidalFunctor {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) | |
370 ( unit : FObj F One ) | |
371 ( φ : {a b : Obj Sets} → Hom Sets ((FObj F a) * (FObj F b )) ( FObj F ( a * b ) ) ) | |
372 : Set (suc (suc c₁)) where | |
373 isM : IsMonoidal (Sets {c₁}) One SetsTensorProduct | |
374 isM = Monoidal.isMonoidal MonoidalSets | |
375 open IsMonoidal | |
376 field | |
715 | 377 natφ : { a b c d : Obj Sets } { x : FObj F a} { y : FObj F b} { f : a → c } { g : b → d } |
713 | 378 → FMap F (map f g) (φ (x , y)) ≡ φ (FMap F f x , FMap F g y) |
715 | 379 assocφ : { x y z : Obj Sets } { a : FObj F x } { b : FObj F y }{ c : FObj F z } |
713 | 380 → φ (a , φ (b , c)) ≡ FMap F (Iso.≅→ (mα-iso isM)) (φ (φ (a , b) , c)) |
715 | 381 idrφ : {a : Obj Sets } { x : FObj F a } → FMap F (Iso.≅→ (mρ-iso isM)) (φ (x , unit)) ≡ x |
382 idlφ : {a : Obj Sets } { x : FObj F a } → FMap F (Iso.≅→ (mλ-iso isM)) (φ (unit , x)) ≡ x | |
713 | 383 |
714 | 384 -- http://www.staff.city.ac.uk/~ross/papers/Applicative.pdf |
385 -- naturality of φ fmap(f × g)(φ u v) = φ ( fmap f u) ( fmap g v ) | |
386 -- left identity fmap snd (φ unit v) = v | |
387 -- right identity fmap fst (φ u unit) = u | |
388 -- associativity fmap assoc (φ u (φ v w)) = φ (φ u v) w | |
389 | |
390 | |
730 | 391 record HaskellMonoidalFunctor {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
392 : Set (suc (suc c₁)) where |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
393 field |
730 | 394 unit : FObj F One |
395 φ : {a b : Obj Sets} → Hom Sets ((FObj F a) * (FObj F b )) ( FObj F ( a * b ) ) | |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
396 |
730 | 397 HaskellMonoidalFunctor→MonoidalFunctor : {c : Level} ( F : Functor (Sets {c}) (Sets {c}) ) → (mf : HaskellMonoidalFunctor F ) |
713 | 398 → IsHaskellMonoidalFunctor F ( HaskellMonoidalFunctor.unit mf ) ( HaskellMonoidalFunctor.φ mf ) |
399 → MonoidalFunctor {_} {c} {_} {Sets} {Sets} MonoidalSets MonoidalSets | |
730 | 400 HaskellMonoidalFunctor→MonoidalFunctor {c} F mf ismf = record { |
709 | 401 MF = F |
402 ; ψ = λ _ → HaskellMonoidalFunctor.unit mf | |
403 ; isMonodailFunctor = record { | |
711 | 404 φab = record { TMap = λ x → φ ; isNTrans = record { commute = comm0 } } |
405 ; associativity = λ {a b c} → comm1 {a} {b} {c} | |
710 | 406 ; unitarity-idr = λ {a b} → comm2 {a} {b} |
407 ; unitarity-idl = λ {a b} → comm3 {a} {b} | |
709 | 408 } |
409 } where | |
410 open Monoidal | |
411 open IsMonoidal hiding ( _■_ ; _□_ ) | |
730 | 412 M : Monoidal (Sets {c}) |
709 | 413 M = MonoidalSets |
730 | 414 isM : IsMonoidal (Sets {c}) One SetsTensorProduct |
709 | 415 isM = Monoidal.isMonoidal MonoidalSets |
730 | 416 unit : FObj F One |
709 | 417 unit = HaskellMonoidalFunctor.unit mf |
418 _⊗_ : (x y : Obj Sets ) → Obj Sets | |
419 _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y | |
420 _□_ : {a b c d : Obj Sets } ( f : Hom Sets a c ) ( g : Hom Sets b d ) → Hom Sets ( a ⊗ b ) ( c ⊗ d ) | |
421 _□_ f g = FMap (m-bi M) ( f , g ) | |
711 | 422 φ : {x : Obj (Sets × Sets) } → Hom Sets (FObj (Functor● Sets Sets MonoidalSets F) x) (FObj (Functor⊗ Sets Sets MonoidalSets F) x) |
423 φ z = HaskellMonoidalFunctor.φ mf z | |
710 | 424 comm00 : {a b : Obj (Sets × Sets)} { f : Hom (Sets × Sets) a b} (x : ( FObj F (proj₁ a) * FObj F (proj₂ a)) ) → |
711 | 425 (Sets [ FMap (Functor⊗ Sets Sets MonoidalSets F) f o φ ]) x ≡ (Sets [ φ o FMap (Functor● Sets Sets MonoidalSets F) f ]) x |
710 | 426 comm00 {a} {b} {(f , g)} (x , y) = begin |
711 | 427 (FMap (Functor⊗ Sets Sets MonoidalSets F) (f , g) ) (φ (x , y)) |
710 | 428 ≡⟨⟩ |
711 | 429 (FMap F ( f □ g ) ) (φ (x , y)) |
710 | 430 ≡⟨⟩ |
711 | 431 FMap F ( map f g ) (φ (x , y)) |
715 | 432 ≡⟨ IsHaskellMonoidalFunctor.natφ ismf ⟩ |
711 | 433 φ ( FMap F f x , FMap F g y ) |
710 | 434 ≡⟨⟩ |
711 | 435 φ ( ( FMap F f □ FMap F g ) (x , y) ) |
710 | 436 ≡⟨⟩ |
711 | 437 φ ((FMap (Functor● Sets Sets MonoidalSets F) (f , g) ) (x , y) ) |
710 | 438 ∎ |
439 where | |
713 | 440 open Relation.Binary.PropositionalEquality.≡-Reasoning |
711 | 441 comm0 : {a b : Obj (Sets × Sets)} { f : Hom (Sets × Sets) a b} → Sets [ Sets [ FMap (Functor⊗ Sets Sets MonoidalSets F) f o φ ] |
442 ≈ Sets [ φ o FMap (Functor● Sets Sets MonoidalSets F) f ] ] | |
710 | 443 comm0 {a} {b} {f} = extensionality Sets ( λ (x : ( FObj F (proj₁ a) * FObj F (proj₂ a)) ) → comm00 x ) |
711 | 444 comm10 : {a b c : Obj Sets} → (x : ((FObj F a ⊗ FObj F b) ⊗ FObj F c) ) → (Sets [ φ o Sets [ id1 Sets (FObj F a) □ φ o Iso.≅→ (mα-iso isM) ] ]) x ≡ |
445 (Sets [ FMap F (Iso.≅→ (mα-iso isM)) o Sets [ φ o φ □ id1 Sets (FObj F c) ] ]) x | |
710 | 446 comm10 {x} {y} {f} ((a , b) , c ) = begin |
711 | 447 φ (( id1 Sets (FObj F x) □ φ ) ( ( Iso.≅→ (mα-iso isM) ) ((a , b) , c))) |
710 | 448 ≡⟨⟩ |
711 | 449 φ ( a , φ (b , c)) |
715 | 450 ≡⟨ IsHaskellMonoidalFunctor.assocφ ismf ⟩ |
711 | 451 ( FMap F (Iso.≅→ (mα-iso isM))) (φ (( φ (a , b)) , c )) |
710 | 452 ≡⟨⟩ |
711 | 453 ( FMap F (Iso.≅→ (mα-iso isM))) (φ (( φ □ id1 Sets (FObj F f) ) ((a , b) , c))) |
710 | 454 ∎ |
455 where | |
713 | 456 open Relation.Binary.PropositionalEquality.≡-Reasoning |
711 | 457 comm1 : {a b c : Obj Sets} → Sets [ Sets [ φ |
458 o Sets [ (id1 Sets (FObj F a) □ φ ) o Iso.≅→ (mα-iso isM) ] ] | |
459 ≈ Sets [ FMap F (Iso.≅→ (mα-iso isM)) o Sets [ φ o (φ □ id1 Sets (FObj F c)) ] ] ] | |
710 | 460 comm1 {a} {b} {c} = extensionality Sets ( λ x → comm10 x ) |
712 | 461 comm20 : {a b : Obj Sets} ( x : FObj F a * One ) → ( Sets [ |
462 FMap F (Iso.≅→ (mρ-iso isM)) o Sets [ φ o | |
463 FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit )) ] ] ) x ≡ Iso.≅→ (mρ-iso isM) x | |
464 comm20 {a} {b} (x , OneObj ) = begin | |
465 (FMap F (Iso.≅→ (mρ-iso isM))) ( φ ( x , unit ) ) | |
715 | 466 ≡⟨ IsHaskellMonoidalFunctor.idrφ ismf ⟩ |
712 | 467 x |
468 ≡⟨⟩ | |
469 Iso.≅→ (mρ-iso isM) ( x , OneObj ) | |
470 ∎ | |
471 where | |
713 | 472 open Relation.Binary.PropositionalEquality.≡-Reasoning |
709 | 473 comm2 : {a b : Obj Sets} → Sets [ Sets [ |
711 | 474 FMap F (Iso.≅→ (mρ-iso isM)) o Sets [ φ o |
709 | 475 FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit )) ] ] ≈ Iso.≅→ (mρ-iso isM) ] |
712 | 476 comm2 {a} {b} = extensionality Sets ( λ x → comm20 {a} {b} x ) |
477 comm30 : {a b : Obj Sets} ( x : One * FObj F b ) → ( Sets [ | |
478 FMap F (Iso.≅→ (mλ-iso isM)) o Sets [ φ o | |
479 FMap (m-bi MonoidalSets) ((λ _ → unit ) , id1 Sets (FObj F b) ) ] ] ) x ≡ Iso.≅→ (mλ-iso isM) x | |
480 comm30 {a} {b} ( OneObj , x) = begin | |
481 (FMap F (Iso.≅→ (mλ-iso isM))) ( φ ( unit , x ) ) | |
715 | 482 ≡⟨ IsHaskellMonoidalFunctor.idlφ ismf ⟩ |
712 | 483 x |
484 ≡⟨⟩ | |
485 Iso.≅→ (mλ-iso isM) ( OneObj , x ) | |
486 ∎ | |
711 | 487 where |
713 | 488 open Relation.Binary.PropositionalEquality.≡-Reasoning |
709 | 489 comm3 : {a b : Obj Sets} → Sets [ Sets [ FMap F (Iso.≅→ (mλ-iso isM)) o |
711 | 490 Sets [ φ o FMap (m-bi MonoidalSets) ((λ _ → unit ) , id1 Sets (FObj F b)) ] ] ≈ Iso.≅→ (mλ-iso isM) ] |
712 | 491 comm3 {a} {b} = extensionality Sets ( λ x → comm30 {a} {b} x ) |
709 | 492 |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
493 |
720 | 494 _・_ : {c₁ : Level} { a b c : Obj (Sets {c₁} ) } → (b → c) → (a → b) → a → c |
495 _・_ f g = λ x → f ( g x ) | |
713 | 496 |
497 record IsApplicative {c₁ : Level} ( f : Functor (Sets {c₁}) (Sets {c₁}) ) | |
498 ( pure : {a : Obj Sets} → Hom Sets a ( FObj f a ) ) | |
499 ( _<*>_ : {a b : Obj Sets} → FObj f ( a → b ) → FObj f a → FObj f b ) | |
500 : Set (suc (suc c₁)) where | |
501 field | |
502 identity : { a : Obj Sets } { u : FObj f a } → pure ( id1 Sets a ) <*> u ≡ u | |
503 composition : { a b c : Obj Sets } { u : FObj f ( b → c ) } { v : FObj f (a → b ) } { w : FObj f a } | |
504 → (( pure _・_ <*> u ) <*> v ) <*> w ≡ u <*> (v <*> w) | |
505 homomorphism : { a b : Obj Sets } { f : Hom Sets a b } { x : a } → pure f <*> pure x ≡ pure (f x) | |
506 interchange : { a b : Obj Sets } { u : FObj f ( a → b ) } { x : a } → u <*> pure x ≡ pure (λ f → f x) <*> u | |
730 | 507 -- from http://www.staff.city.ac.uk/~ross/papers/Applicative.pdf |
713 | 508 |
509 record Applicative {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) | |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
510 : Set (suc (suc c₁)) where |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
511 field |
713 | 512 pure : {a : Obj Sets} → Hom Sets a ( FObj F a ) |
513 <*> : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b | |
730 | 514 |
515 ------ | |
516 -- | |
517 -- Applicative ⇔ Monoidal | |
518 -- | |
519 -- | |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
520 |
713 | 521 lemma1 : {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) → Applicative F → HaskellMonoidalFunctor F |
522 lemma1 F app = record { unit = unit ; φ = φ } | |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
523 where |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
524 open Applicative |
713 | 525 unit : FObj F One |
706 | 526 unit = pure app OneObj |
713 | 527 φ : {a b : Obj Sets} → Hom Sets ((FObj F a) * (FObj F b )) ( FObj F ( a * b ) ) |
528 φ {a} {b} ( x , y ) = <*> app (FMap F (λ j k → (j , k)) x) y | |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
529 |
713 | 530 lemma2 : {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) → HaskellMonoidalFunctor F → Applicative F |
531 lemma2 F mono = record { pure = pure ; <*> = <*> } | |
705
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
532 where |
73a998711118
add Applicative and HaskellMonoidal Functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
704
diff
changeset
|
533 open HaskellMonoidalFunctor |
713 | 534 pure : {a : Obj Sets} → Hom Sets a ( FObj F a ) |
535 pure {a} x = FMap F ( λ y → x ) (unit mono) | |
730 | 536 <*> : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b |
715 | 537 <*> {a} {b} x y = FMap F ( λ r → ( proj₁ r ) ( proj₂ r ) ) (φ mono ( x , y )) |
713 | 538 |
730 | 539 ------ |
540 -- | |
541 -- Appllicative Functor is a Monoidal Functor | |
542 -- | |
543 | |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
544 Applicative→Monoidal : {c : Level} ( F : Functor (Sets {c}) (Sets {c}) ) → (mf : Applicative F ) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
545 → IsApplicative F ( Applicative.pure mf ) ( Applicative.<*> mf ) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
546 → MonoidalFunctor {_} {c} {_} {Sets} {Sets} MonoidalSets MonoidalSets |
727
ea84cc6c1797
monoidal functor and applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
726
diff
changeset
|
547 Applicative→Monoidal {l} F mf ismf = record { |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
548 MF = F |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
549 ; ψ = λ x → unit |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
550 ; isMonodailFunctor = record { |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
551 φab = record { TMap = λ x → φ ; isNTrans = record { commute = comm0 } } |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
552 ; associativity = λ {a b c} → comm1 {a} {b} {c} |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
553 ; unitarity-idr = λ {a b} → comm2 {a} {b} |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
554 ; unitarity-idl = λ {a b} → comm3 {a} {b} |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
555 } |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
556 } where |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
557 open Monoidal |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
558 open IsMonoidal hiding ( _■_ ; _□_ ) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
559 M = MonoidalSets |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
560 isM = Monoidal.isMonoidal MonoidalSets |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
561 unit = Applicative.pure mf OneObj |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
562 _⊗_ : (x y : Obj Sets ) → Obj Sets |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
563 _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
564 _□_ : {a b c d : Obj Sets } ( f : Hom Sets a c ) ( g : Hom Sets b d ) → Hom Sets ( a ⊗ b ) ( c ⊗ d ) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
565 _□_ f g = FMap (m-bi M) ( f , g ) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
566 φ : {x : Obj (Sets × Sets) } → Hom Sets (FObj (Functor● Sets Sets MonoidalSets F) x) (FObj (Functor⊗ Sets Sets MonoidalSets F) x) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
567 φ x = Applicative.<*> mf (FMap F (λ j k → (j , k)) (proj₁ x )) (proj₂ x) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
568 _<*>_ : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
569 _<*>_ = Applicative.<*> mf |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
570 left : {a b : Obj Sets} → {x y : FObj F ( a → b )} → {h : FObj F a } → ( x ≡ y ) → x <*> h ≡ y <*> h |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
571 left {_} {_} {_} {_} {h} eq = ≡-cong ( λ k → k <*> h ) eq |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
572 right : {a b : Obj Sets} → {h : FObj F ( a → b )} → {x y : FObj F a } → ( x ≡ y ) → h <*> x ≡ h <*> y |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
573 right {_} {_} {h} {_} {_} eq = ≡-cong ( λ k → h <*> k ) eq |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
574 id : { a : Obj Sets } → a → a |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
575 id x = x |
720 | 576 pure : {a : Obj Sets } → Hom Sets a ( FObj F a ) |
577 pure a = Applicative.pure mf a | |
725 | 578 -- special case |
579 F→pureid : {a b : Obj Sets } → (x : FObj F a ) → FMap F id x ≡ pure id <*> x | |
580 F→pureid {a} {b} x = sym ( begin | |
722
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
581 pure id <*> x |
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
582 ≡⟨ IsApplicative.identity ismf ⟩ |
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
583 x |
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
584 ≡⟨ ≡-cong ( λ k → k x ) (sym ( IsFunctor.identity (isFunctor F ) )) ⟩ FMap F id x |
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
585 ∎ ) |
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
586 where |
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
587 open Relation.Binary.PropositionalEquality |
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
588 open Relation.Binary.PropositionalEquality.≡-Reasoning |
725 | 589 F→pure : {a b : Obj Sets } → { f : a → b } → {x : FObj F a } → FMap F f x ≡ pure f <*> x |
590 F→pure {a} {b} {f} {x} = sym ( begin | |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
591 pure f <*> x |
731
117e5b392673
Generalize Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
730
diff
changeset
|
592 ≡⟨ ≡-cong ( λ k → k x ) (UniquenessOfFunctor Sets Sets F ( λ f x → pure f <*> x ) ( extensionality Sets ( λ x → IsApplicative.identity ismf ))) ⟩ |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
593 FMap F f x |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
594 ∎ ) |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
595 where |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
596 open Relation.Binary.PropositionalEquality |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
597 open Relation.Binary.PropositionalEquality.≡-Reasoning |
725 | 598 p*p : { a b : Obj Sets } { f : Hom Sets a b } { x : a } → pure f <*> pure x ≡ pure (f x) |
599 p*p = IsApplicative.homomorphism ismf | |
600 comp = IsApplicative.composition ismf | |
601 inter = IsApplicative.interchange ismf | |
729 | 602 pureAssoc : {a b c : Obj Sets } ( f : b → c ) ( g : a → b ) ( h : FObj F a ) → pure f <*> ( pure g <*> h ) ≡ pure ( f ・ g ) <*> h |
603 pureAssoc f g h = trans ( trans (sym comp) (left (left p*p) )) ( left p*p ) | |
604 where | |
605 open Relation.Binary.PropositionalEquality | |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
606 comm00 : {a b : Obj (Sets × Sets)} { f : Hom (Sets × Sets) a b} (x : ( FObj F (proj₁ a) * FObj F (proj₂ a)) ) → |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
607 (Sets [ FMap (Functor⊗ Sets Sets MonoidalSets F) f o φ ]) x ≡ (Sets [ φ o FMap (Functor● Sets Sets MonoidalSets F) f ]) x |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
608 comm00 {a} {b} {(f , g)} (x , y) = begin |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
609 ( FMap (Functor⊗ Sets Sets MonoidalSets F) (f , g) ) ( φ (x , y) ) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
610 ≡⟨⟩ |
725 | 611 FMap F (λ xy → f (proj₁ xy) , g (proj₂ xy)) ((FMap F (λ j k → j , k) x) <*> y) |
612 ≡⟨⟩ | |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
613 FMap F (map f g) ((FMap F (λ j k → j , k) x) <*> y) |
725 | 614 ≡⟨ F→pure ⟩ |
615 (pure (map f g) <*> (FMap F (λ j k → j , k) x <*> y)) | |
616 ≡⟨ right ( left F→pure ) ⟩ | |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
617 (pure (map f g)) <*> ((pure (λ j k → j , k) <*> x) <*> y) |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
618 ≡⟨ sym ( IsApplicative.composition ismf ) ⟩ |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
619 (( pure _・_ <*> (pure (map f g))) <*> (pure (λ j k → j , k) <*> x)) <*> y |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
620 ≡⟨ left ( sym ( IsApplicative.composition ismf )) ⟩ |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
621 ((( pure _・_ <*> (( pure _・_ <*> (pure (map f g))))) <*> pure (λ j k → j , k)) <*> x) <*> y |
725 | 622 ≡⟨ trans ( trans (left ( left (left (right p*p )))) ( left ( left ( left p*p)))) (left (left p*p)) ⟩ |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
623 (pure (( _・_ (( _・_ ((map f g))))) (λ j k → j , k)) <*> x) <*> y |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
624 ≡⟨⟩ |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
625 (pure (λ j k → f j , g k) <*> x) <*> y |
725 | 626 ≡⟨⟩ |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
627 ( pure ((_・_ (( _・_ ( ( λ h → h g ))) ( _・_ ))) ((λ j k → f j , k))) <*> x ) <*> y |
725 | 628 ≡⟨ sym ( trans (left (left (left p*p))) (left ( left p*p)) ) ⟩ |
629 ((((pure _・_ <*> pure ((λ h → h g) ・ _・_)) <*> pure (λ j k → f j , k)) <*> x) <*> y) | |
630 ≡⟨ sym (trans ( left ( left ( left (right (left p*p) )))) (left ( left (left (right p*p ))))) ⟩ | |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
631 (((pure _・_ <*> (( pure _・_ <*> ( pure ( λ h → h g ))) <*> ( pure _・_ ))) <*> (pure (λ j k → f j , k))) <*> x ) <*> y |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
632 ≡⟨ left ( ( IsApplicative.composition ismf )) ⟩ |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
633 ((( pure _・_ <*> ( pure ( λ h → h g ))) <*> ( pure _・_ )) <*> (pure (λ j k → f j , k) <*> x )) <*> y |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
634 ≡⟨ left (IsApplicative.composition ismf ) ⟩ |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
635 ( pure ( λ h → h g ) <*> ( pure _・_ <*> (pure (λ j k → f j , k) <*> x )) ) <*> y |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
636 ≡⟨ left (sym (IsApplicative.interchange ismf )) ⟩ |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
637 (( pure _・_ <*> (pure (λ j k → f j , k) <*> x )) <*> pure g) <*> y |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
638 ≡⟨ IsApplicative.composition ismf ⟩ |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
639 (pure (λ j k → f j , k) <*> x) <*> (pure g <*> y) |
725 | 640 ≡⟨ sym ( trans (left F→pure ) ( right F→pure ) ) ⟩ |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
641 (FMap F (λ j k → f j , k) x) <*> (FMap F g y) |
720 | 642 ≡⟨ ≡-cong ( λ k → k x <*> (FMap F g y)) ( IsFunctor.distr (isFunctor F )) ⟩ |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
643 (FMap F (λ j k → j , k) (FMap F f x)) <*> (FMap F g y) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
644 ≡⟨⟩ |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
645 φ ( ( FMap (Functor● Sets Sets MonoidalSets F) (f , g) ) ( x , y ) ) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
646 ∎ |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
647 where |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
648 open Relation.Binary.PropositionalEquality |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
649 open Relation.Binary.PropositionalEquality.≡-Reasoning |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
650 comm0 : {a b : Obj (Sets × Sets)} { f : Hom (Sets × Sets) a b} → Sets [ Sets [ FMap (Functor⊗ Sets Sets MonoidalSets F) f o φ ] |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
651 ≈ Sets [ φ o FMap (Functor● Sets Sets MonoidalSets F) f ] ] |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
652 comm0 {a} {b} {f} = extensionality Sets ( λ (x : ( FObj F (proj₁ a) * FObj F (proj₂ a)) ) → comm00 x ) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
653 comm10 : {a b c : Obj Sets} → (x : ((FObj F a ⊗ FObj F b) ⊗ FObj F c) ) → (Sets [ φ o Sets [ id1 Sets (FObj F a) □ φ o Iso.≅→ (mα-iso isM) ] ]) x ≡ |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
654 (Sets [ FMap F (Iso.≅→ (mα-iso isM)) o Sets [ φ o φ □ id1 Sets (FObj F c) ] ]) x |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
655 comm10 {x} {y} {f} ((a , b) , c ) = begin |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
656 φ (( id □ φ ) ( ( Iso.≅→ (mα-iso isM) ) ((a , b) , c))) |
720 | 657 ≡⟨⟩ |
658 (FMap F (λ j k → j , k) a) <*> ( (FMap F (λ j k → j , k) b) <*> c) | |
726 | 659 ≡⟨ trans (left F→pure) (right (left F→pure) ) ⟩ |
725 | 660 (pure (λ j k → j , k) <*> a) <*> ( (pure (λ j k → j , k) <*> b) <*> c) |
726 | 661 ≡⟨ sym comp ⟩ |
725 | 662 ( ( pure _・_ <*> (pure (λ j k → j , k) <*> a)) <*> (pure (λ j k → j , k) <*> b)) <*> c |
726 | 663 ≡⟨ sym ( left comp ) ⟩ |
725 | 664 (( ( pure _・_ <*> ( pure _・_ <*> (pure (λ j k → j , k) <*> a))) <*> (pure (λ j k → j , k))) <*> b) <*> c |
726 | 665 ≡⟨ sym ( left ( left ( left (right comp )))) ⟩ |
725 | 666 (( ( pure _・_ <*> (( (pure _・_ <*> pure _・_ ) <*> (pure (λ j k → j , k))) <*> a)) <*> (pure (λ j k → j , k))) <*> b) <*> c |
726 | 667 ≡⟨ trans (left ( left (left ( right (left ( left p*p )))))) (left ( left ( left (right (left p*p))))) ⟩ |
725 | 668 (( ( pure _・_ <*> ((pure ((_・_ ( _・_ )) ((λ j k → j , k)))) <*> a)) <*> (pure (λ j k → j , k))) <*> b) <*> c |
726 | 669 ≡⟨ sym (left ( left ( left comp ) )) ⟩ |
725 | 670 (((( ( pure _・_ <*> (pure _・_ )) <*> (pure ((_・_ ( _・_ )) ((λ j k → j , k))))) <*> a) <*> (pure (λ j k → j , k))) <*> b) <*> c |
726 | 671 ≡⟨ trans (left ( left ( left (left (left p*p))))) (left ( left ( left (left p*p )))) ⟩ |
725 | 672 ((((pure ( ( _・_ (_・_ )) (((_・_ ( _・_ )) ((λ j k → j , k)))))) <*> a) <*> (pure (λ j k → j , k))) <*> b) <*> c |
673 ≡⟨⟩ | |
674 ((((pure (λ f g x y → f , g x y)) <*> a) <*> (pure (λ j k → j , k))) <*> b) <*> c | |
726 | 675 ≡⟨ left ( left inter ) ⟩ |
725 | 676 (((pure (λ f → f (λ j k → j , k))) <*> ((pure (λ f g x y → f , g x y)) <*> a) ) <*> b) <*> c |
726 | 677 ≡⟨ sym ( left ( left comp )) ⟩ |
725 | 678 (((( pure _・_ <*> (pure (λ f → f (λ j k → j , k)))) <*> (pure (λ f g x y → f , g x y))) <*> a ) <*> b) <*> c |
726 | 679 ≡⟨ trans (left ( left (left (left p*p) ))) (left (left (left p*p ) )) ⟩ |
725 | 680 ((pure (( _・_ (λ f → f (λ j k → j , k))) (λ f g x y → f , g x y)) <*> a ) <*> b) <*> c |
681 ≡⟨⟩ | |
682 (((pure (λ f g h → f , g , h)) <*> a) <*> b) <*> c | |
683 ≡⟨⟩ | |
684 ((pure ((_・_ ((_・_ ((_・_ ( (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))))))) | |
685 (( _・_ ( _・_ ((λ j k → j , k)))) (λ j k → j , k))) <*> a) <*> b) <*> c | |
726 | 686 ≡⟨ sym (trans ( left ( left ( left (left (right (right p*p))) ) )) (trans (left (left( left (left (right p*p))))) |
687 (trans (left (left (left (left p*p)))) (trans ( left (left (left (right (left (right p*p )))))) | |
688 (trans (left (left (left (right (left p*p))))) (trans (left (left (left (right p*p)))) (left (left (left p*p)))) ) ) ) | |
689 ) ) ⟩ | |
725 | 690 ((((pure _・_ <*> ((pure _・_ <*> ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))))))) <*> |
691 (( pure _・_ <*> ( pure _・_ <*> (pure (λ j k → j , k)))) <*> pure (λ j k → j , k))) <*> a) <*> b) <*> c | |
726 | 692 ≡⟨ left (left comp ) ⟩ |
725 | 693 (((pure _・_ <*> ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))))) <*> |
694 ((( pure _・_ <*> ( pure _・_ <*> (pure (λ j k → j , k)))) <*> pure (λ j k → j , k)) <*> a)) <*> b) <*> c | |
726 | 695 ≡⟨ left comp ⟩ |
725 | 696 ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))) <*> |
697 (((( pure _・_ <*> ( pure _・_ <*> (pure (λ j k → j , k)))) <*> pure (λ j k → j , k)) <*> a) <*> b)) <*> c | |
726 | 698 ≡⟨ left ( right (left comp )) ⟩ |
725 | 699 ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))) <*> |
700 ((( pure _・_ <*> (pure (λ j k → j , k))) <*> (pure (λ j k → j , k) <*> a)) <*> b)) <*> c | |
726 | 701 ≡⟨ left ( right comp ) ⟩ |
725 | 702 ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))) <*> |
703 (pure (λ j k → j , k) <*> ( (pure (λ j k → j , k) <*> a) <*> b))) <*> c | |
726 | 704 ≡⟨ comp ⟩ |
725 | 705 pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc) <*> ( (pure (λ j k → j , k) <*> ( (pure (λ j k → j , k) <*> a) <*> b)) <*> c) |
726 | 706 ≡⟨ sym ( trans ( trans F→pure (right (left F→pure ))) ( right ( left (right (left F→pure ))))) ⟩ |
720 | 707 FMap F (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc) ( (FMap F (λ j k → j , k) ( (FMap F (λ j k → j , k) a) <*> b)) <*> c) |
708 ≡⟨⟩ | |
709 ( FMap F (Iso.≅→ (mα-iso isM))) (φ (( φ □ id1 Sets (FObj F f) ) ((a , b) , c))) | |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
710 ∎ |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
711 where |
720 | 712 open Relation.Binary.PropositionalEquality |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
713 open Relation.Binary.PropositionalEquality.≡-Reasoning |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
714 comm1 : {a b c : Obj Sets} → Sets [ Sets [ φ |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
715 o Sets [ (id1 Sets (FObj F a) □ φ ) o Iso.≅→ (mα-iso isM) ] ] |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
716 ≈ Sets [ FMap F (Iso.≅→ (mα-iso isM)) o Sets [ φ o (φ □ id1 Sets (FObj F c)) ] ] ] |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
717 comm1 {a} {b} {c} = extensionality Sets ( λ x → comm10 x ) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
718 comm20 : {a b : Obj Sets} ( x : FObj F a * One ) → ( Sets [ |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
719 FMap F (Iso.≅→ (mρ-iso isM)) o Sets [ φ o |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
720 FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit )) ] ] ) x ≡ Iso.≅→ (mρ-iso isM) x |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
721 comm20 {a} {b} (x , OneObj ) = begin |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
722 (FMap F (Iso.≅→ (mρ-iso isM))) ( φ (( FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit))) (x , OneObj) )) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
723 ≡⟨⟩ |
720 | 724 FMap F proj₁ ((FMap F (λ j k → j , k) x) <*> (pure OneObj)) |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
725 ≡⟨ ≡-cong ( λ k → FMap F proj₁ k) ( IsApplicative.interchange ismf ) ⟩ |
720 | 726 FMap F proj₁ ((pure (λ f → f OneObj)) <*> (FMap F (λ j k → j , k) x)) |
725 | 727 ≡⟨ ( trans F→pure (right ( right F→pure )) ) ⟩ |
728 pure proj₁ <*> ((pure (λ f → f OneObj)) <*> (pure (λ j k → j , k) <*> x)) | |
729 ≡⟨ sym ( right comp ) ⟩ | |
730 pure proj₁ <*> (((pure _・_ <*> (pure (λ f → f OneObj))) <*> pure (λ j k → j , k)) <*> x) | |
731 ≡⟨ sym comp ⟩ | |
732 ( ( pure _・_ <*> (pure proj₁ ) ) <*> ((pure _・_ <*> (pure (λ f → f OneObj))) <*> pure (λ j k → j , k))) <*> x | |
733 ≡⟨ trans ( trans ( trans ( left ( left p*p)) ( left ( right (left p*p) ))) (left (right p*p) ) ) (left p*p) ⟩ | |
727
ea84cc6c1797
monoidal functor and applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
726
diff
changeset
|
734 pure ( ( _・_ (proj₁ {l} {l})) ((_・_ ((λ f → f OneObj))) (λ j k → j , k))) <*> x |
725 | 735 ≡⟨⟩ |
736 pure id <*> x | |
737 ≡⟨ IsApplicative.identity ismf ⟩ | |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
738 x |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
739 ≡⟨⟩ |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
740 Iso.≅→ (mρ-iso isM) (x , OneObj) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
741 ∎ |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
742 where |
725 | 743 open Relation.Binary.PropositionalEquality |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
744 open Relation.Binary.PropositionalEquality.≡-Reasoning |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
745 comm2 : {a b : Obj Sets} → Sets [ Sets [ |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
746 FMap F (Iso.≅→ (mρ-iso isM)) o Sets [ φ o |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
747 FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit )) ] ] ≈ Iso.≅→ (mρ-iso isM) ] |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
748 comm2 {a} {b} = extensionality Sets ( λ x → comm20 {a} {b} x ) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
749 comm30 : {a b : Obj Sets} ( x : One * FObj F b ) → ( Sets [ |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
750 FMap F (Iso.≅→ (mλ-iso isM)) o Sets [ φ o |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
751 FMap (m-bi MonoidalSets) ((λ _ → unit ) , id1 Sets (FObj F b) ) ] ] ) x ≡ Iso.≅→ (mλ-iso isM) x |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
752 comm30 {a} {b} ( OneObj , x) = begin |
720 | 753 (FMap F (Iso.≅→ (mλ-iso isM))) ( φ ( unit , x ) ) |
754 ≡⟨⟩ | |
755 FMap F proj₂ ((FMap F (λ j k → j , k) (pure OneObj)) <*> x) | |
725 | 756 ≡⟨ ( trans F→pure (right ( left F→pure )) ) ⟩ |
757 pure proj₂ <*> ((pure (λ j k → j , k) <*> (pure OneObj)) <*> x) | |
758 ≡⟨ sym comp ⟩ | |
759 ((pure _・_ <*> (pure proj₂)) <*> (pure (λ j k → j , k) <*> (pure OneObj))) <*> x | |
760 ≡⟨ trans (trans (left (left p*p )) (left ( right p*p)) ) (left p*p) ⟩ | |
727
ea84cc6c1797
monoidal functor and applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
726
diff
changeset
|
761 pure ((_・_ (proj₂ {l}) )((λ (j : One {l}) (k : b ) → j , k) OneObj)) <*> x |
725 | 762 ≡⟨⟩ |
763 pure id <*> x | |
764 ≡⟨ IsApplicative.identity ismf ⟩ | |
720 | 765 x |
766 ≡⟨⟩ | |
767 Iso.≅→ (mλ-iso isM) ( OneObj , x ) | |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
768 ∎ |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
769 where |
725 | 770 open Relation.Binary.PropositionalEquality |
719
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
771 open Relation.Binary.PropositionalEquality.≡-Reasoning |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
772 comm3 : {a b : Obj Sets} → Sets [ Sets [ FMap F (Iso.≅→ (mλ-iso isM)) o |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
773 Sets [ φ o FMap (m-bi MonoidalSets) ((λ _ → unit ) , id1 Sets (FObj F b)) ] ] ≈ Iso.≅→ (mλ-iso isM) ] |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
774 comm3 {a} {b} = extensionality Sets ( λ x → comm30 {a} {b} x ) |
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
775 |
730 | 776 ---- |
777 -- | |
778 -- Monoidal laws imples Applicative laws | |
779 -- | |
713 | 780 |
781 HaskellMonoidal→Applicative : {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) | |
782 ( unit : FObj F One ) | |
783 ( φ : {a b : Obj Sets} → Hom Sets ((FObj F a) * (FObj F b )) ( FObj F ( a * b ) ) ) | |
784 ( mono : IsHaskellMonoidalFunctor F unit φ ) | |
715 | 785 → IsApplicative F (λ x → FMap F ( λ y → x ) unit) (λ x y → FMap F ( λ r → ( proj₁ r ) ( proj₂ r ) ) (φ ( x , y ))) |
713 | 786 HaskellMonoidal→Applicative {c₁} F unit φ mono = record { |
787 identity = identity | |
788 ; composition = composition | |
789 ; homomorphism = homomorphism | |
790 ; interchange = interchange | |
791 } | |
792 where | |
714 | 793 id : { a : Obj Sets } → a → a |
794 id x = x | |
713 | 795 isM : IsMonoidal (Sets {c₁}) One SetsTensorProduct |
796 isM = Monoidal.isMonoidal MonoidalSets | |
797 pure : {a : Obj Sets} → Hom Sets a ( FObj F a ) | |
798 pure {a} x = FMap F ( λ y → x ) (unit ) | |
799 _<*>_ : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b | |
715 | 800 _<*>_ {a} {b} x y = FMap F ( λ r → ( proj₁ r ) ( proj₂ r ) ) (φ ( x , y )) |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
801 -- right does not work right it makes yellows. why? |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
802 -- right : {n : Level} { a b : Set n} → { x y : a } { h : a → b } → ( x ≡ y ) → h x ≡ h y |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
803 -- right {_} {_} {_} {_} {_} {h} eq = ≡-cong ( λ k → h k ) eq |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
804 left : {n : Level} { a b : Set n} → { x y : a → b } { h : a } → ( x ≡ y ) → x h ≡ y h |
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
805 left {_} {_} {_} {_} {_} {h} eq = ≡-cong ( λ k → k h ) eq |
715 | 806 open Relation.Binary.PropositionalEquality |
717
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
807 FφF→F : { a b c d e : Obj Sets } { g : Hom Sets a c } { h : Hom Sets b d } |
715 | 808 { f : Hom Sets (c * d) e } |
809 { x : FObj F a } { y : FObj F b } | |
810 → FMap F f ( φ ( FMap F g x , FMap F h y ) ) ≡ FMap F ( f o map g h ) ( φ ( x , y ) ) | |
717
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
811 FφF→F {a} {b} {c} {d} {e} {g} {h} {f} {x} {y} = sym ( begin |
715 | 812 FMap F ( f o map g h ) ( φ ( x , y ) ) |
813 ≡⟨ ≡-cong ( λ k → k ( φ ( x , y ))) ( IsFunctor.distr (isFunctor F) ) ⟩ | |
814 FMap F f (( FMap F ( map g h ) ) ( φ ( x , y ))) | |
815 ≡⟨ ≡-cong ( λ k → FMap F f k ) ( IsHaskellMonoidalFunctor.natφ mono ) ⟩ | |
816 FMap F f ( φ ( FMap F g x , FMap F h y ) ) | |
817 ∎ ) | |
818 where | |
819 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
716 | 820 u→F : {a : Obj Sets } {u : FObj F a} → u ≡ FMap F id u |
821 u→F {a} {u} = sym ( ≡-cong ( λ k → k u ) ( IsFunctor.identity ( isFunctor F ) ) ) | |
822 φunitr : {a : Obj Sets } {u : FObj F a} → φ ( unit , u) ≡ FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) u | |
823 φunitr {a} {u} = sym ( begin | |
824 FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) u | |
825 ≡⟨ ≡-cong ( λ k → FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) k ) (sym (IsHaskellMonoidalFunctor.idlφ mono)) ⟩ | |
826 FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) ( FMap F (Iso.≅→ (IsMonoidal.mλ-iso isM)) ( φ ( unit , u) ) ) | |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
827 ≡⟨ left ( sym ( IsFunctor.distr ( isFunctor F ) )) ⟩ |
716 | 828 (FMap F ( (Iso.≅← (IsMonoidal.mλ-iso isM)) o (Iso.≅→ (IsMonoidal.mλ-iso isM)))) ( φ ( unit , u) ) |
829 ≡⟨ ≡-cong ( λ k → FMap F k ( φ ( unit , u) )) (Iso.iso→ ( (IsMonoidal.mλ-iso isM) )) ⟩ | |
830 FMap F id ( φ ( unit , u) ) | |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
831 ≡⟨ left ( IsFunctor.identity ( isFunctor F ) ) ⟩ |
716 | 832 id ( φ ( unit , u) ) |
833 ≡⟨⟩ | |
834 φ ( unit , u) | |
835 ∎ ) | |
836 where | |
837 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
838 φunitl : {a : Obj Sets } {u : FObj F a} → φ ( u , unit ) ≡ FMap F (Iso.≅← (IsMonoidal.mρ-iso isM)) u | |
839 φunitl {a} {u} = sym ( begin | |
840 FMap F (Iso.≅← (IsMonoidal.mρ-iso isM)) u | |
841 ≡⟨ ≡-cong ( λ k → FMap F (Iso.≅← (IsMonoidal.mρ-iso isM)) k ) (sym (IsHaskellMonoidalFunctor.idrφ mono)) ⟩ | |
842 FMap F (Iso.≅← (IsMonoidal.mρ-iso isM)) ( FMap F (Iso.≅→ (IsMonoidal.mρ-iso isM)) ( φ ( u , unit ) ) ) | |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
843 ≡⟨ left ( sym ( IsFunctor.distr ( isFunctor F ) )) ⟩ |
716 | 844 (FMap F ( (Iso.≅← (IsMonoidal.mρ-iso isM)) o (Iso.≅→ (IsMonoidal.mρ-iso isM)))) ( φ ( u , unit ) ) |
845 ≡⟨ ≡-cong ( λ k → FMap F k ( φ ( u , unit ) )) (Iso.iso→ ( (IsMonoidal.mρ-iso isM) )) ⟩ | |
846 FMap F id ( φ ( u , unit ) ) | |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
847 ≡⟨ left ( IsFunctor.identity ( isFunctor F ) ) ⟩ |
716 | 848 id ( φ ( u , unit ) ) |
849 ≡⟨⟩ | |
850 φ ( u , unit ) | |
851 ∎ ) | |
852 where | |
853 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
715 | 854 open IsMonoidal |
713 | 855 identity : { a : Obj Sets } { u : FObj F a } → pure ( id1 Sets a ) <*> u ≡ u |
856 identity {a} {u} = begin | |
714 | 857 pure id <*> u |
713 | 858 ≡⟨⟩ |
715 | 859 ( FMap F ( λ r → ( proj₁ r ) ( proj₂ r )) ) ( φ ( FMap F ( λ y → id ) unit , u ) ) |
716 | 860 ≡⟨ ≡-cong ( λ k → ( FMap F ( λ r → ( proj₁ r ) ( proj₂ r )) ) ( φ ( FMap F ( λ y → id ) unit , k ))) u→F ⟩ |
715 | 861 ( FMap F ( λ r → ( proj₁ r ) ( proj₂ r )) ) ( φ ( FMap F ( λ y → id ) unit , FMap F id u ) ) |
717
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
862 ≡⟨ FφF→F ⟩ |
713 | 863 FMap F (λ x → proj₂ x ) (φ (unit , u ) ) |
864 ≡⟨⟩ | |
865 FMap F (Iso.≅→ (mλ-iso isM)) (φ (unit , u )) | |
715 | 866 ≡⟨ IsHaskellMonoidalFunctor.idlφ mono ⟩ |
713 | 867 u |
868 ∎ | |
869 where | |
870 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
871 composition : { a b c : Obj Sets } { u : FObj F ( b → c ) } { v : FObj F (a → b ) } { w : FObj F a } | |
872 → (( pure _・_ <*> u ) <*> v ) <*> w ≡ u <*> (v <*> w) | |
873 composition {a} {b} {c} {u} {v} {w} = begin | |
715 | 874 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ |
875 (FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f g x → f (g x)) unit , u)) , v)) , w)) | |
716 | 876 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f g x → f (g x)) unit , k)) , v)) , w)) ) u→F ⟩ |
715 | 877 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ |
716 | 878 (FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f g x → f (g x)) unit , FMap F id u )) , v)) , w)) |
717
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
879 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ ( k , v)) , w)) ) FφF→F ⟩ |
715 | 880 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ |
716 | 881 (FMap F ( λ x → (λ r → proj₁ r (proj₂ r)) ((map (λ y f g x → f (g x)) id ) x)) (φ ( unit , u)) , v)) , w)) |
882 ≡⟨ ≡-cong ( λ k → ( FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ | |
883 (FMap F ( λ x → (λ r → proj₁ r (proj₂ r)) ((map (λ y f g x → f (g x)) id ) x)) k , v)) , w)) ) ) φunitr ⟩ | |
715 | 884 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ |
716 | 885 ( (FMap F ( λ x → (λ r → proj₁ r (proj₂ r)) ((map (λ y f g x → f (g x)) id ) x)) (FMap F (Iso.≅← (mλ-iso isM)) u) ) , v)) , w)) |
886 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ | |
887 (k u , v)) , w)) ) (sym ( IsFunctor.distr (isFunctor F ))) ⟩ | |
888 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ | |
889 ( FMap F (λ x → ((λ y f g x₁ → f (g x₁)) unit x) ) u , v)) , w)) | |
714 | 890 ≡⟨⟩ |
715 | 891 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ |
716 | 892 ( FMap F (λ x g h → x (g h) ) u , v)) , w)) |
893 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ ( FMap F (λ x g h → x (g h) ) u , k)) , w)) ) u→F ⟩ | |
894 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ x g h → x (g h)) u , FMap F id v)) , w)) | |
717
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
895 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (k , w)) ) FφF→F ⟩ |
716 | 896 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F ((λ r → proj₁ r (proj₂ r)) o map (λ x g h → x (g h)) id) (φ (u , v)) , w)) |
715 | 897 ≡⟨⟩ |
716 | 898 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ x h → proj₁ x (proj₂ x h)) (φ (u , v)) , w)) |
899 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ x h → proj₁ x (proj₂ x h)) (φ (u , v)) , k)) ) u→F ⟩ | |
900 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ x h → proj₁ x (proj₂ x h)) (φ (u , v)) , FMap F id w)) | |
717
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
901 ≡⟨ FφF→F ⟩ |
716 | 902 FMap F ((λ r → proj₁ r (proj₂ r)) o map (λ x h → proj₁ x (proj₂ x h)) id) (φ (φ (u , v) , w)) |
714 | 903 ≡⟨⟩ |
716 | 904 FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (φ (φ (u , v) , w)) |
905 ≡⟨ ≡-cong ( λ k → FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (k (φ (φ (u , v) , w)) )) (sym (IsFunctor.identity (isFunctor F ))) ⟩ | |
906 FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (FMap F id (φ (φ (u , v) , w)) ) | |
907 ≡⟨ ≡-cong ( λ k → FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (FMap F k (φ (φ (u , v) , w)) ) ) (sym (Iso.iso→ (mα-iso isM))) ⟩ | |
908 FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (FMap F ( (Iso.≅← (mα-iso isM)) o (Iso.≅→ (mα-iso isM))) (φ (φ (u , v) , w)) ) | |
909 ≡⟨ ≡-cong ( λ k → FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (k (φ (φ (u , v) , w)))) ( IsFunctor.distr (isFunctor F )) ⟩ | |
910 FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (FMap F (Iso.≅← (mα-iso isM)) ( FMap F (Iso.≅→ (mα-iso isM)) (φ (φ (u , v) , w)) )) | |
911 ≡⟨ ≡-cong ( λ k → FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (FMap F (Iso.≅← (mα-iso isM)) k) ) (sym ( IsHaskellMonoidalFunctor.assocφ mono ) ) ⟩ | |
912 FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (FMap F (Iso.≅← (mα-iso isM)) (φ (u , φ (v , w)))) | |
715 | 913 ≡⟨⟩ |
716 | 914 FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (FMap F (λ r → (proj₁ r , proj₁ (proj₂ r)) , proj₂ (proj₂ r)) (φ (u , φ (v , w)))) |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
915 ≡⟨ left (sym ( IsFunctor.distr (isFunctor F ))) ⟩ |
716 | 916 FMap F (λ y → (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) ((λ r → (proj₁ r , proj₁ (proj₂ r)) , proj₂ (proj₂ r)) y )) (φ (u , φ (v , w))) |
715 | 917 ≡⟨⟩ |
716 | 918 FMap F (λ y → proj₁ y (proj₁ (proj₂ y) (proj₂ (proj₂ y)))) (φ (u , φ (v , w))) |
715 | 919 ≡⟨⟩ |
920 FMap F ( λ x → (proj₁ x) ((λ r → proj₁ r (proj₂ r)) ( proj₂ x))) ( φ ( u , (φ (v , w)))) | |
717
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
921 ≡⟨ sym FφF→F ⟩ |
715 | 922 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F id u , FMap F (λ r → proj₁ r (proj₂ r)) (φ (v , w)))) |
716 | 923 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (k , FMap F (λ r → proj₁ r (proj₂ r)) (φ (v , w)))) ) (sym u→F ) ⟩ |
715 | 924 FMap F (λ r → proj₁ r (proj₂ r)) (φ (u , FMap F (λ r → proj₁ r (proj₂ r)) (φ (v , w)))) |
713 | 925 ∎ |
926 where | |
927 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
928 homomorphism : { a b : Obj Sets } { f : Hom Sets a b } { x : a } → pure f <*> pure x ≡ pure (f x) | |
717
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
929 homomorphism {a} {b} {f} {x} = begin |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
930 pure f <*> pure x |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
931 ≡⟨⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
932 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y → f) unit , FMap F (λ y → x) unit)) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
933 ≡⟨ FφF→F ⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
934 FMap F ((λ r → proj₁ r (proj₂ r)) o map (λ y → f) (λ y → x)) (φ (unit , unit)) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
935 ≡⟨⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
936 FMap F (λ y → f x) (φ (unit , unit)) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
937 ≡⟨ ≡-cong ( λ k → FMap F (λ y → f x) k ) φunitl ⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
938 FMap F (λ y → f x) (FMap F (Iso.≅← (mρ-iso isM)) unit) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
939 ≡⟨⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
940 FMap F (λ y → f x) (FMap F (λ y → (y , OneObj)) unit) |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
941 ≡⟨ left ( sym ( IsFunctor.distr (isFunctor F ))) ⟩ |
717
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
942 FMap F (λ y → f x) unit |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
943 ≡⟨⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
944 pure (f x) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
945 ∎ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
946 where |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
947 open Relation.Binary.PropositionalEquality.≡-Reasoning |
713 | 948 interchange : { a b : Obj Sets } { u : FObj F ( a → b ) } { x : a } → u <*> pure x ≡ pure (λ f → f x) <*> u |
717
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
949 interchange {a} {b} {u} {x} = begin |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
950 u <*> pure x |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
951 ≡⟨⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
952 FMap F (λ r → proj₁ r (proj₂ r)) (φ (u , FMap F (λ y → x) unit)) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
953 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (k , FMap F (λ y → x) unit)) ) u→F ⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
954 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F id u , FMap F (λ y → x) unit)) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
955 ≡⟨ FφF→F ⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
956 FMap F ((λ r → proj₁ r (proj₂ r)) o map id (λ y → x)) (φ (u , unit)) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
957 ≡⟨⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
958 FMap F (λ r → proj₁ r x) (φ (u , unit)) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
959 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r x) k ) φunitl ⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
960 FMap F (λ r → proj₁ r x) (( FMap F (Iso.≅← (mρ-iso isM))) u ) |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
961 ≡⟨ left ( sym ( IsFunctor.distr (isFunctor F )) ) ⟩ |
717
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
962 FMap F (( λ r → proj₁ r x) o ((Iso.≅← (mρ-iso isM) ))) u |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
963 ≡⟨⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
964 FMap F (( λ r → proj₂ r x) o ((Iso.≅← (mλ-iso isM) ))) u |
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
965 ≡⟨ left ( IsFunctor.distr (isFunctor F )) ⟩ |
717
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
966 FMap F (λ r → proj₂ r x) (FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) u) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
967 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₂ r x) k ) (sym φunitr ) ⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
968 FMap F (λ r → proj₂ r x) (φ (unit , u)) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
969 ≡⟨⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
970 FMap F ((λ r → proj₁ r (proj₂ r)) o map (λ y f → f x) id) (φ (unit , u)) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
971 ≡⟨ sym FφF→F ⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
972 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f → f x) unit , FMap F id u)) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
973 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f → f x) unit , k)) ) (sym u→F) ⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
974 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f → f x) unit , u)) |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
975 ≡⟨⟩ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
976 pure (λ f → f x) <*> u |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
977 ∎ |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
978 where |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
979 open Relation.Binary.PropositionalEquality.≡-Reasoning |
a41b2b9b0407
Haskell Monoidal Funtor to Applicative done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
716
diff
changeset
|
980 |
730 | 981 -- |