Mercurial > hg > Members > kono > Proof > category
annotate monoidal.agda @ 730:e4ef69bae044
clean up
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 26 Nov 2017 21:57:41 +0900 |
| parents | 6bc9d68898ef |
| children | 117e5b392673 |
| rev | line source |
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696
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1 open import Level |
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2 open import Category |
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3 module monoidal where |
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4 |
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5 open import Data.Product renaming (_×_ to _*_) |
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6 open import Category.Constructions.Product |
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7 open import HomReasoning |
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8 open import cat-utility |
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9 open import Relation.Binary.Core |
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10 open import Relation.Binary |
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11 |
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12 open Functor |
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13 |
| 698 | 14 record Iso {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) |
| 15 (x y : Obj C ) | |
| 16 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where | |
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696
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17 field |
| 698 | 18 ≅→ : Hom C x y |
| 19 ≅← : Hom C y x | |
| 20 iso→ : C [ C [ ≅← o ≅→ ] ≈ id1 C x ] | |
| 21 iso← : C [ C [ ≅→ o ≅← ] ≈ id1 C y ] | |
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696
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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 | |
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700
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28 infixr 9 _□_ _■_ |
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29 _□_ : ( x y : Obj C ) → Obj C |
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30 _□_ x y = FObj BI ( x , y ) |
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31 _■_ : {a b c d : Obj C } ( f : Hom C a c ) ( g : Hom C b d ) → Hom C ( a □ b ) ( c □ d ) |
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32 _■_ f g = FMap BI ( f , g ) |
| 698 | 33 field |
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700
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34 mα-iso : {a b c : Obj C} → Iso C ( ( a □ b) □ c) ( a □ ( b □ c ) ) |
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35 mλ-iso : {a : Obj C} → Iso C ( I □ a) a |
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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' ) → |
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700
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38 C [ C [ ( f ■ id1 C ( b □ c )) o ≅→ (mα-iso {a} {b} {c}) ] ≈ |
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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' ) → |
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700
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41 C [ C [ ( id1 C a ■ ( f ■ id1 C c ) ) o ≅→ (mα-iso {a} {b} {c} ) ] ≈ |
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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' ) → |
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700
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44 C [ C [ ( id1 C a ■ ( id1 C b ■ f ) ) o ≅→ (mα-iso {a} {b} {c} ) ] ≈ |
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parents:
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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} ) ] ≈ | |
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700
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48 C [ ≅→ (mλ-iso ) o ( id1 C I ■ f ) ] ] |
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49 mρ→nat : {a a' : Obj C} → ( f : Hom C a a' ) → |
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50 C [ C [ f o ≅→ (mρ-iso {a} ) ] ≈ |
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51 C [ ≅→ (mρ-iso ) o ( f ■ id1 C I ) ] ] |
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705
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52 -- we should write naturalities for ≅← (maybe derived from above ) |
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700
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53 αABC□1D : {a b c d e : Obj C } → Hom C (((a □ b) □ c ) □ d) ((a □ (b □ c)) □ d) |
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701
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Monoidal Functor on going ...
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54 αABC□1D {a} {b} {c} {d} {e} = ( ≅→ mα-iso ■ id1 C d ) |
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700
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55 αAB□CD : {a b c d e : Obj C } → Hom C ((a □ (b □ c)) □ d) (a □ ((b □ c ) □ d)) |
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701
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Monoidal Functor on going ...
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56 αAB□CD {a} {b} {c} {d} {e} = ≅→ mα-iso |
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700
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57 1A□BCD : {a b c d e : Obj C } → Hom C (a □ ((b □ c ) □ d)) (a □ (b □ ( c □ d) )) |
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701
7a729bb62ce3
Monoidal Functor on going ...
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700
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58 1A□BCD {a} {b} {c} {d} {e} = ( id1 C a ■ ≅→ mα-iso ) |
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700
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59 αABC□D : {a b c d e : Obj C } → Hom C (a □ (b □ ( c □ d) )) ((a □ b ) □ (c □ d)) |
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701
7a729bb62ce3
Monoidal Functor on going ...
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parents:
700
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60 αABC□D {a} {b} {c} {d} {e} = ≅← mα-iso |
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700
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61 αA□BCD : {a b c d e : Obj C } → Hom C (((a □ b) □ c ) □ d) ((a □ b ) □ (c □ d)) |
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701
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Monoidal Functor on going ...
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parents:
700
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62 αA□BCD {a} {b} {c} {d} {e} = ≅→ mα-iso |
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700
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parents:
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63 αAIB : {a b : Obj C } → Hom C (( a □ I ) □ b ) (a □ ( I □ b )) |
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701
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Monoidal Functor on going ...
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700
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64 αAIB {a} {b} = ≅→ mα-iso |
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700
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65 1A□λB : {a b : Obj C } → Hom C (a □ ( I □ b )) ( a □ b ) |
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701
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Monoidal Functor on going ...
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700
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66 1A□λB {a} {b} = id1 C a ■ ≅→ mλ-iso |
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700
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67 ρA□IB : {a b : Obj C } → Hom C (( a □ I ) □ b ) ( a □ b ) |
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701
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68 ρA□IB {a} {b} = ≅→ mρ-iso ■ id1 C b |
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700
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69 |
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70 field |
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71 comm-penta : {a b c d e : Obj C} |
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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} ] ] ] |
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73 ≈ αA□BCD {a} {b} {c} {d} {e} ] |
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74 comm-unit : {a b : Obj C} |
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75 → C [ C [ 1A□λB {a} {b} o αAIB ] ≈ ρA□IB {a} {b} ] |
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696
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76 |
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77 record Monoidal {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) |
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78 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where |
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79 field |
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80 m-i : Obj C |
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81 m-bi : Functor ( C × C ) C |
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82 isMonoidal : IsMonoidal C m-i m-bi |
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83 |
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84 --------- |
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85 -- |
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86 -- Lax Monoidal Functor |
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87 -- |
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88 -- N → M |
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89 -- |
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90 --------- |
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91 |
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92 --------- |
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93 -- |
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94 -- Two implementations of Functor ( C × C ) → D from F : Functor C → D (given) |
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95 -- dervied from F and two Monoidal Categories |
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96 -- |
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97 -- F x ● F y |
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98 -- F ( x ⊗ y ) |
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99 -- |
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100 -- and a given natural transformation for them |
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101 -- |
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102 -- φ : F x ● F y → F ( x ⊗ y ) |
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103 -- |
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104 -- TMap φ : ( x y : Obj C ) → Hom D ( F x ● F y ) ( F ( x ⊗ y )) |
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105 -- |
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106 -- a given unit arrow |
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107 -- |
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108 -- ψ : IN → IM |
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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) ) | |
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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 ∎ |
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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 ) | |
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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 | |
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198 |
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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 ) |
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201 ( ψ : Hom D (Monoidal.m-i N) (FObj MF (Monoidal.m-i M) ) ) |
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202 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where |
| 698 | 203 _⊗_ : (x y : Obj C ) → Obj C |
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204 _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y |
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205 _□_ : {a b c d : Obj C } ( f : Hom C a c ) ( g : Hom C b d ) → Hom C ( a ⊗ b ) ( c ⊗ d ) |
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206 _□_ f g = FMap (Monoidal.m-bi M) ( f , g ) |
| 698 | 207 _●_ : (x y : Obj D ) → Obj D |
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208 _●_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal N) ) x y |
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209 _■_ : {a b c d : Obj D } ( f : Hom D a c ) ( g : Hom D b d ) → Hom D ( a ● b ) ( c ● d ) |
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210 _■_ f g = FMap (Monoidal.m-bi N) ( f , g ) |
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211 F● : Functor ( C × C ) D |
| 703 | 212 F● = Functor● C D N MF |
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213 F⊗ : Functor ( C × C ) D |
| 703 | 214 F⊗ = Functor⊗ C D M MF |
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215 field |
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216 φab : NTrans ( C × C ) D F● F⊗ |
| 698 | 217 open Iso |
| 218 open Monoidal | |
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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}) | |
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224 F : Obj C → Obj D |
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225 F x = FObj MF x |
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226 φ : ( x y : Obj C ) → Hom D ( FObj F● (x , y) ) ( FObj F⊗ ( x , y )) |
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227 φ x y = NTrans.TMap φab ( x , y ) |
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228 1●φBC : {a b c : Obj C} → Hom D ( F a ● ( F b ● F c ) ) ( F a ● ( F ( b ⊗ c ) )) |
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229 1●φBC {a} {b} {c} = id1 D (F a) ■ φ b c |
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230 φAB⊗C : {a b c : Obj C} → Hom D ( F a ● ( F ( b ⊗ c ) )) (F ( a ⊗ ( b ⊗ c ))) |
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231 φAB⊗C {a} {b} {c} = φ a (b ⊗ c ) |
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232 φAB●1 : {a b c : Obj C} → Hom D ( ( F a ● F b ) ● F c ) ( F ( a ⊗ b ) ● F c ) |
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233 φAB●1 {a} {b} {c} = φ a b ■ id1 D (F c) |
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234 φA⊗BC : {a b c : Obj C} → Hom D ( F ( a ⊗ b ) ● F c ) (F ( (a ⊗ b ) ⊗ c )) |
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235 φA⊗BC {a} {b} {c} = φ ( a ⊗ b ) c |
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236 FαC : {a b c : Obj C} → Hom D (F ( (a ⊗ b ) ⊗ c )) (F ( a ⊗ ( b ⊗ c ))) |
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237 FαC {a} {b} {c} = FMap MF ( ≅→ (mα-iso (isMonoidal M) {a} {b} {c}) ) |
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238 1●ψ : { a b : Obj C } → Hom D (F a ● Monoidal.m-i N ) ( F a ● F ( Monoidal.m-i M ) ) |
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239 1●ψ{a} {b} = id1 D (F a) ■ ψ |
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240 φAIC : { a b : Obj C } → Hom D ( F a ● F ( Monoidal.m-i M ) ) (F ( a ⊗ Monoidal.m-i M )) |
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241 φAIC {a} {b} = φ a ( Monoidal.m-i M ) |
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242 FρC : { a b : Obj C } → Hom D (F ( a ⊗ Monoidal.m-i M ))( F a ) |
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243 FρC {a} {b} = FMap MF ( ≅→ (mρ-iso (isMonoidal M) {a} ) ) |
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244 ρD : { a b : Obj C } → Hom D (F a ● Monoidal.m-i N ) ( F a ) |
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245 ρD {a} {b} = ≅→ (mρ-iso (isMonoidal N) {F a} ) |
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246 ψ●1 : { a b : Obj C } → Hom D (Monoidal.m-i N ● F b ) ( F ( Monoidal.m-i M ) ● F b ) |
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247 ψ●1 {a} {b} = ψ ■ id1 D (F b) |
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248 φICB : { a b : Obj C } → Hom D ( F ( Monoidal.m-i M ) ● F b ) ( F ( ( Monoidal.m-i M ) ⊗ b ) ) |
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249 φICB {a} {b} = φ ( Monoidal.m-i M ) b |
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250 FλD : { a b : Obj C } → Hom D ( F ( ( Monoidal.m-i M ) ⊗ b ) ) (F b ) |
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251 FλD {a} {b} = FMap MF ( ≅→ (mλ-iso (isMonoidal M) {b} ) ) |
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252 λD : { a b : Obj C } → Hom D (Monoidal.m-i N ● F b ) (F b ) |
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253 λD {a} {b} = ≅→ (mλ-iso (isMonoidal N) {F b} ) |
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254 field |
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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 ] ] ] |
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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} ] |
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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} ] |
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259 record MonoidalFunctor {c₁ c₂ ℓ : Level} {C D : Category c₁ c₂ ℓ} ( M : Monoidal C ) ( N : Monoidal D ) |
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260 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where |
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261 field |
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262 MF : Functor C D |
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263 ψ : Hom D (Monoidal.m-i N) (FObj MF (Monoidal.m-i M) ) |
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264 isMonodailFunctor : IsMonoidalFunctor M N MF ψ |
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265 |
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266 open import Category.Sets |
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267 |
| 706 | 268 import Relation.Binary.PropositionalEquality |
| 269 -- 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 ) | |
| 270 postulate extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Relation.Binary.PropositionalEquality.Extensionality c₂ c₂ | |
| 271 | |
| 730 | 272 ------ |
| 273 -- Data.Product as a Tensor Product for Monoidal Category | |
| 274 -- | |
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275 |
| 708 | 276 SetsTensorProduct : {c : Level} → Functor ( Sets {c} × Sets {c} ) (Sets {c}) |
| 277 SetsTensorProduct = record { | |
| 278 FObj = λ x → proj₁ x * proj₂ x | |
| 279 ; FMap = λ {x : Obj ( Sets × Sets ) } {y} f → map (proj₁ f) (proj₂ f) | |
| 280 ; isFunctor = record { | |
| 281 ≈-cong = ≈-cong | |
| 282 ; identity = refl | |
| 283 ; distr = refl | |
| 706 | 284 } |
| 708 | 285 } where |
| 286 ≈-cong : {a b : Obj (Sets × Sets)} {f g : Hom (Sets × Sets) a b} → | |
| 287 (Sets × Sets) [ f ≈ g ] → Sets [ map (proj₁ f) (proj₂ f) ≈ map (proj₁ g) (proj₂ g) ] | |
| 288 ≈-cong (refl , refl) = refl | |
| 289 | |
| 730 | 290 ----- |
| 291 -- | |
| 292 -- Sets as Monoidal Category | |
| 293 -- | |
| 294 -- almost all comutativities are refl | |
| 295 -- | |
| 296 -- | |
| 297 -- | |
| 708 | 298 |
| 730 | 299 data One {c : Level} : Set c where |
| 300 OneObj : One -- () in Haskell ( or any one object set ) | |
| 708 | 301 |
| 302 MonoidalSets : {c : Level} → Monoidal (Sets {c}) | |
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303 MonoidalSets {c} = record { |
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304 m-i = One {c} ; |
| 708 | 305 m-bi = SetsTensorProduct ; |
| 306 isMonoidal = record { | |
| 307 mα-iso = record { ≅→ = mα→ ; ≅← = mα← ; iso→ = refl ; iso← = refl } ; | |
| 308 mλ-iso = record { ≅→ = mλ→ ; ≅← = mλ← ; iso→ = extensionality Sets ( λ x → mλiso x ) ; iso← = refl } ; | |
| 309 mρ-iso = record { ≅→ = mρ→ ; ≅← = mρ← ; iso→ = extensionality Sets ( λ x → mρiso x ) ; iso← = refl } ; | |
| 310 mα→nat1 = λ f → refl ; | |
| 311 mα→nat2 = λ f → refl ; | |
| 312 mα→nat3 = λ f → refl ; | |
| 313 mλ→nat = λ f → refl ; | |
| 314 mρ→nat = λ f → refl ; | |
| 315 comm-penta = refl ; | |
| 316 comm-unit = refl | |
| 317 } | |
| 318 } where | |
| 319 _⊗_ : ( a b : Obj Sets ) → Obj Sets | |
| 320 _⊗_ a b = FObj SetsTensorProduct (a , b ) | |
| 730 | 321 -- association operations |
| 708 | 322 mα→ : {a b c : Obj Sets} → Hom Sets ( ( a ⊗ b ) ⊗ c ) ( a ⊗ ( b ⊗ c ) ) |
| 323 mα→ ((a , b) , c ) = (a , ( b , c ) ) | |
| 324 mα← : {a b c : Obj Sets} → Hom Sets ( a ⊗ ( b ⊗ c ) ) ( ( a ⊗ b ) ⊗ c ) | |
| 325 mα← (a , ( b , c ) ) = ((a , b) , c ) | |
| 730 | 326 -- (One , a) ⇔ a |
| 708 | 327 mλ→ : {a : Obj Sets} → Hom Sets ( One ⊗ a ) a |
| 328 mλ→ (_ , a) = a | |
| 329 mλ← : {a : Obj Sets} → Hom Sets a ( One ⊗ a ) | |
| 330 mλ← a = ( OneObj , a ) | |
| 331 mλiso : {a : Obj Sets} (x : One ⊗ a) → (Sets [ mλ← o mλ→ ]) x ≡ id1 Sets (One ⊗ a) x | |
| 332 mλiso (OneObj , _ ) = refl | |
| 730 | 333 -- (a , One) ⇔ a |
| 708 | 334 mρ→ : {a : Obj Sets} → Hom Sets ( a ⊗ One ) a |
| 335 mρ→ (a , _) = a | |
| 336 mρ← : {a : Obj Sets} → Hom Sets a ( a ⊗ One ) | |
| 337 mρ← a = ( a , OneObj ) | |
| 338 mρiso : {a : Obj Sets} (x : a ⊗ One ) → (Sets [ mρ← o mρ→ ]) x ≡ id1 Sets (a ⊗ One) x | |
| 339 mρiso (_ , OneObj ) = refl | |
| 340 | |
| 710 | 341 ≡-cong = Relation.Binary.PropositionalEquality.cong |
| 706 | 342 |
| 713 | 343 |
| 344 record IsHaskellMonoidalFunctor {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) | |
| 345 ( unit : FObj F One ) | |
| 346 ( φ : {a b : Obj Sets} → Hom Sets ((FObj F a) * (FObj F b )) ( FObj F ( a * b ) ) ) | |
| 347 : Set (suc (suc c₁)) where | |
| 348 isM : IsMonoidal (Sets {c₁}) One SetsTensorProduct | |
| 349 isM = Monoidal.isMonoidal MonoidalSets | |
| 350 open IsMonoidal | |
| 351 field | |
| 715 | 352 natφ : { a b c d : Obj Sets } { x : FObj F a} { y : FObj F b} { f : a → c } { g : b → d } |
| 713 | 353 → FMap F (map f g) (φ (x , y)) ≡ φ (FMap F f x , FMap F g y) |
| 715 | 354 assocφ : { x y z : Obj Sets } { a : FObj F x } { b : FObj F y }{ c : FObj F z } |
| 713 | 355 → φ (a , φ (b , c)) ≡ FMap F (Iso.≅→ (mα-iso isM)) (φ (φ (a , b) , c)) |
| 715 | 356 idrφ : {a : Obj Sets } { x : FObj F a } → FMap F (Iso.≅→ (mρ-iso isM)) (φ (x , unit)) ≡ x |
| 357 idlφ : {a : Obj Sets } { x : FObj F a } → FMap F (Iso.≅→ (mλ-iso isM)) (φ (unit , x)) ≡ x | |
| 713 | 358 |
| 714 | 359 -- http://www.staff.city.ac.uk/~ross/papers/Applicative.pdf |
| 360 -- naturality of φ fmap(f × g)(φ u v) = φ ( fmap f u) ( fmap g v ) | |
| 361 -- left identity fmap snd (φ unit v) = v | |
| 362 -- right identity fmap fst (φ u unit) = u | |
| 363 -- associativity fmap assoc (φ u (φ v w)) = φ (φ u v) w | |
| 364 | |
| 365 | |
| 730 | 366 record HaskellMonoidalFunctor {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) |
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367 : Set (suc (suc c₁)) where |
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368 field |
| 730 | 369 unit : FObj F One |
| 370 φ : {a b : Obj Sets} → Hom Sets ((FObj F a) * (FObj F b )) ( FObj F ( a * b ) ) | |
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371 |
| 730 | 372 HaskellMonoidalFunctor→MonoidalFunctor : {c : Level} ( F : Functor (Sets {c}) (Sets {c}) ) → (mf : HaskellMonoidalFunctor F ) |
| 713 | 373 → IsHaskellMonoidalFunctor F ( HaskellMonoidalFunctor.unit mf ) ( HaskellMonoidalFunctor.φ mf ) |
| 374 → MonoidalFunctor {_} {c} {_} {Sets} {Sets} MonoidalSets MonoidalSets | |
| 730 | 375 HaskellMonoidalFunctor→MonoidalFunctor {c} F mf ismf = record { |
| 709 | 376 MF = F |
| 377 ; ψ = λ _ → HaskellMonoidalFunctor.unit mf | |
| 378 ; isMonodailFunctor = record { | |
| 711 | 379 φab = record { TMap = λ x → φ ; isNTrans = record { commute = comm0 } } |
| 380 ; associativity = λ {a b c} → comm1 {a} {b} {c} | |
| 710 | 381 ; unitarity-idr = λ {a b} → comm2 {a} {b} |
| 382 ; unitarity-idl = λ {a b} → comm3 {a} {b} | |
| 709 | 383 } |
| 384 } where | |
| 385 open Monoidal | |
| 386 open IsMonoidal hiding ( _■_ ; _□_ ) | |
| 730 | 387 M : Monoidal (Sets {c}) |
| 709 | 388 M = MonoidalSets |
| 730 | 389 isM : IsMonoidal (Sets {c}) One SetsTensorProduct |
| 709 | 390 isM = Monoidal.isMonoidal MonoidalSets |
| 730 | 391 unit : FObj F One |
| 709 | 392 unit = HaskellMonoidalFunctor.unit mf |
| 393 _⊗_ : (x y : Obj Sets ) → Obj Sets | |
| 394 _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y | |
| 395 _□_ : {a b c d : Obj Sets } ( f : Hom Sets a c ) ( g : Hom Sets b d ) → Hom Sets ( a ⊗ b ) ( c ⊗ d ) | |
| 396 _□_ f g = FMap (m-bi M) ( f , g ) | |
| 711 | 397 φ : {x : Obj (Sets × Sets) } → Hom Sets (FObj (Functor● Sets Sets MonoidalSets F) x) (FObj (Functor⊗ Sets Sets MonoidalSets F) x) |
| 398 φ z = HaskellMonoidalFunctor.φ mf z | |
| 710 | 399 comm00 : {a b : Obj (Sets × Sets)} { f : Hom (Sets × Sets) a b} (x : ( FObj F (proj₁ a) * FObj F (proj₂ a)) ) → |
| 711 | 400 (Sets [ FMap (Functor⊗ Sets Sets MonoidalSets F) f o φ ]) x ≡ (Sets [ φ o FMap (Functor● Sets Sets MonoidalSets F) f ]) x |
| 710 | 401 comm00 {a} {b} {(f , g)} (x , y) = begin |
| 711 | 402 (FMap (Functor⊗ Sets Sets MonoidalSets F) (f , g) ) (φ (x , y)) |
| 710 | 403 ≡⟨⟩ |
| 711 | 404 (FMap F ( f □ g ) ) (φ (x , y)) |
| 710 | 405 ≡⟨⟩ |
| 711 | 406 FMap F ( map f g ) (φ (x , y)) |
| 715 | 407 ≡⟨ IsHaskellMonoidalFunctor.natφ ismf ⟩ |
| 711 | 408 φ ( FMap F f x , FMap F g y ) |
| 710 | 409 ≡⟨⟩ |
| 711 | 410 φ ( ( FMap F f □ FMap F g ) (x , y) ) |
| 710 | 411 ≡⟨⟩ |
| 711 | 412 φ ((FMap (Functor● Sets Sets MonoidalSets F) (f , g) ) (x , y) ) |
| 710 | 413 ∎ |
| 414 where | |
| 713 | 415 open Relation.Binary.PropositionalEquality.≡-Reasoning |
| 711 | 416 comm0 : {a b : Obj (Sets × Sets)} { f : Hom (Sets × Sets) a b} → Sets [ Sets [ FMap (Functor⊗ Sets Sets MonoidalSets F) f o φ ] |
| 417 ≈ Sets [ φ o FMap (Functor● Sets Sets MonoidalSets F) f ] ] | |
| 710 | 418 comm0 {a} {b} {f} = extensionality Sets ( λ (x : ( FObj F (proj₁ a) * FObj F (proj₂ a)) ) → comm00 x ) |
| 711 | 419 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 ≡ |
| 420 (Sets [ FMap F (Iso.≅→ (mα-iso isM)) o Sets [ φ o φ □ id1 Sets (FObj F c) ] ]) x | |
| 710 | 421 comm10 {x} {y} {f} ((a , b) , c ) = begin |
| 711 | 422 φ (( id1 Sets (FObj F x) □ φ ) ( ( Iso.≅→ (mα-iso isM) ) ((a , b) , c))) |
| 710 | 423 ≡⟨⟩ |
| 711 | 424 φ ( a , φ (b , c)) |
| 715 | 425 ≡⟨ IsHaskellMonoidalFunctor.assocφ ismf ⟩ |
| 711 | 426 ( FMap F (Iso.≅→ (mα-iso isM))) (φ (( φ (a , b)) , c )) |
| 710 | 427 ≡⟨⟩ |
| 711 | 428 ( FMap F (Iso.≅→ (mα-iso isM))) (φ (( φ □ id1 Sets (FObj F f) ) ((a , b) , c))) |
| 710 | 429 ∎ |
| 430 where | |
| 713 | 431 open Relation.Binary.PropositionalEquality.≡-Reasoning |
| 711 | 432 comm1 : {a b c : Obj Sets} → Sets [ Sets [ φ |
| 433 o Sets [ (id1 Sets (FObj F a) □ φ ) o Iso.≅→ (mα-iso isM) ] ] | |
| 434 ≈ Sets [ FMap F (Iso.≅→ (mα-iso isM)) o Sets [ φ o (φ □ id1 Sets (FObj F c)) ] ] ] | |
| 710 | 435 comm1 {a} {b} {c} = extensionality Sets ( λ x → comm10 x ) |
| 712 | 436 comm20 : {a b : Obj Sets} ( x : FObj F a * One ) → ( Sets [ |
| 437 FMap F (Iso.≅→ (mρ-iso isM)) o Sets [ φ o | |
| 438 FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit )) ] ] ) x ≡ Iso.≅→ (mρ-iso isM) x | |
| 439 comm20 {a} {b} (x , OneObj ) = begin | |
| 440 (FMap F (Iso.≅→ (mρ-iso isM))) ( φ ( x , unit ) ) | |
| 715 | 441 ≡⟨ IsHaskellMonoidalFunctor.idrφ ismf ⟩ |
| 712 | 442 x |
| 443 ≡⟨⟩ | |
| 444 Iso.≅→ (mρ-iso isM) ( x , OneObj ) | |
| 445 ∎ | |
| 446 where | |
| 713 | 447 open Relation.Binary.PropositionalEquality.≡-Reasoning |
| 709 | 448 comm2 : {a b : Obj Sets} → Sets [ Sets [ |
| 711 | 449 FMap F (Iso.≅→ (mρ-iso isM)) o Sets [ φ o |
| 709 | 450 FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit )) ] ] ≈ Iso.≅→ (mρ-iso isM) ] |
| 712 | 451 comm2 {a} {b} = extensionality Sets ( λ x → comm20 {a} {b} x ) |
| 452 comm30 : {a b : Obj Sets} ( x : One * FObj F b ) → ( Sets [ | |
| 453 FMap F (Iso.≅→ (mλ-iso isM)) o Sets [ φ o | |
| 454 FMap (m-bi MonoidalSets) ((λ _ → unit ) , id1 Sets (FObj F b) ) ] ] ) x ≡ Iso.≅→ (mλ-iso isM) x | |
| 455 comm30 {a} {b} ( OneObj , x) = begin | |
| 456 (FMap F (Iso.≅→ (mλ-iso isM))) ( φ ( unit , x ) ) | |
| 715 | 457 ≡⟨ IsHaskellMonoidalFunctor.idlφ ismf ⟩ |
| 712 | 458 x |
| 459 ≡⟨⟩ | |
| 460 Iso.≅→ (mλ-iso isM) ( OneObj , x ) | |
| 461 ∎ | |
| 711 | 462 where |
| 713 | 463 open Relation.Binary.PropositionalEquality.≡-Reasoning |
| 709 | 464 comm3 : {a b : Obj Sets} → Sets [ Sets [ FMap F (Iso.≅→ (mλ-iso isM)) o |
| 711 | 465 Sets [ φ o FMap (m-bi MonoidalSets) ((λ _ → unit ) , id1 Sets (FObj F b)) ] ] ≈ Iso.≅→ (mλ-iso isM) ] |
| 712 | 466 comm3 {a} {b} = extensionality Sets ( λ x → comm30 {a} {b} x ) |
| 709 | 467 |
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468 |
| 720 | 469 _・_ : {c₁ : Level} { a b c : Obj (Sets {c₁} ) } → (b → c) → (a → b) → a → c |
| 470 _・_ f g = λ x → f ( g x ) | |
| 713 | 471 |
| 472 record IsApplicative {c₁ : Level} ( f : Functor (Sets {c₁}) (Sets {c₁}) ) | |
| 473 ( pure : {a : Obj Sets} → Hom Sets a ( FObj f a ) ) | |
| 474 ( _<*>_ : {a b : Obj Sets} → FObj f ( a → b ) → FObj f a → FObj f b ) | |
| 475 : Set (suc (suc c₁)) where | |
| 476 field | |
| 477 identity : { a : Obj Sets } { u : FObj f a } → pure ( id1 Sets a ) <*> u ≡ u | |
| 478 composition : { a b c : Obj Sets } { u : FObj f ( b → c ) } { v : FObj f (a → b ) } { w : FObj f a } | |
| 479 → (( pure _・_ <*> u ) <*> v ) <*> w ≡ u <*> (v <*> w) | |
| 480 homomorphism : { a b : Obj Sets } { f : Hom Sets a b } { x : a } → pure f <*> pure x ≡ pure (f x) | |
| 481 interchange : { a b : Obj Sets } { u : FObj f ( a → b ) } { x : a } → u <*> pure x ≡ pure (λ f → f x) <*> u | |
| 730 | 482 -- from http://www.staff.city.ac.uk/~ross/papers/Applicative.pdf |
| 713 | 483 |
| 484 record Applicative {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) | |
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485 : Set (suc (suc c₁)) where |
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486 field |
| 713 | 487 pure : {a : Obj Sets} → Hom Sets a ( FObj F a ) |
| 488 <*> : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b | |
| 730 | 489 |
| 490 ------ | |
| 491 -- | |
| 492 -- Applicative ⇔ Monoidal | |
| 493 -- | |
| 494 -- | |
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495 |
| 713 | 496 lemma1 : {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) → Applicative F → HaskellMonoidalFunctor F |
| 497 lemma1 F app = record { unit = unit ; φ = φ } | |
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498 where |
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499 open Applicative |
| 713 | 500 unit : FObj F One |
| 706 | 501 unit = pure app OneObj |
| 713 | 502 φ : {a b : Obj Sets} → Hom Sets ((FObj F a) * (FObj F b )) ( FObj F ( a * b ) ) |
| 503 φ {a} {b} ( x , y ) = <*> app (FMap F (λ j k → (j , k)) x) y | |
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504 |
| 713 | 505 lemma2 : {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) → HaskellMonoidalFunctor F → Applicative F |
| 506 lemma2 F mono = record { pure = pure ; <*> = <*> } | |
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507 where |
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508 open HaskellMonoidalFunctor |
| 713 | 509 pure : {a : Obj Sets} → Hom Sets a ( FObj F a ) |
| 510 pure {a} x = FMap F ( λ y → x ) (unit mono) | |
| 730 | 511 <*> : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b |
| 715 | 512 <*> {a} {b} x y = FMap F ( λ r → ( proj₁ r ) ( proj₂ r ) ) (φ mono ( x , y )) |
| 713 | 513 |
| 730 | 514 |
| 515 ------ | |
| 516 -- | |
| 517 -- Appllicative Functor is a Monoidal Functor | |
| 518 -- | |
| 519 | |
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520 Applicative→Monoidal : {c : Level} ( F : Functor (Sets {c}) (Sets {c}) ) → (mf : Applicative F ) |
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521 → IsApplicative F ( Applicative.pure mf ) ( Applicative.<*> mf ) |
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522 → MonoidalFunctor {_} {c} {_} {Sets} {Sets} MonoidalSets MonoidalSets |
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|
523 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
|
524 MF = F |
|
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
525 ; ψ = λ x → unit |
|
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
526 ; isMonodailFunctor = record { |
|
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
527 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
|
528 ; 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
|
529 ; 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
|
530 ; 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
|
531 } |
|
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
532 } where |
|
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
533 open Monoidal |
|
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
534 open IsMonoidal hiding ( _■_ ; _□_ ) |
|
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
535 M = MonoidalSets |
|
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
536 isM = Monoidal.isMonoidal MonoidalSets |
|
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
537 unit = Applicative.pure mf OneObj |
|
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
538 _⊗_ : (x y : Obj Sets ) → Obj Sets |
|
a017ed40dd77
Applicative law → Monoidal law begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
717
diff
changeset
|
539 _⊗_ 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
|
540 _□_ : {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
|
541 _□_ 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
|
542 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
|
543 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
|
544 _<*>_ : {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
|
545 _<*>_ = Applicative.<*> mf |
|
721
a8b595fb4905
use FMap F f x ≡ pure f <*> x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
720
diff
changeset
|
546 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
|
547 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
|
548 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
|
549 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
|
550 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
|
551 id x = x |
| 720 | 552 pure : {a : Obj Sets } → Hom Sets a ( FObj F a ) |
| 553 pure a = Applicative.pure mf a | |
| 725 | 554 -- special case |
| 555 F→pureid : {a b : Obj Sets } → (x : FObj F a ) → FMap F id x ≡ pure id <*> x | |
| 556 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
|
557 pure id <*> x |
|
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
558 ≡⟨ IsApplicative.identity ismf ⟩ |
|
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
559 x |
|
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
560 ≡⟨ ≡-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
|
561 ∎ ) |
|
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
562 where |
|
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
563 open Relation.Binary.PropositionalEquality |
|
69f01b82dfc9
uniquness of functor fmap
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
721
diff
changeset
|
564 open Relation.Binary.PropositionalEquality.≡-Reasoning |
|
723
40122e9c7fc9
postulate Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
722
diff
changeset
|
565 ----- |
| 730 | 566 -- they say it is not possible to prove FreeTheorem in Agda nor Coq |
|
723
40122e9c7fc9
postulate Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
722
diff
changeset
|
567 -- https://stackoverflow.com/questions/24718567/is-it-possible-to-get-hold-of-free-theorems-as-propositional-equalities |
| 730 | 568 -- so we postulate this |
|
723
40122e9c7fc9
postulate Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
722
diff
changeset
|
569 -- and we cannot indent postulate ... |
|
40122e9c7fc9
postulate Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
722
diff
changeset
|
570 postulate FreeTheorem : {l : Level } {a b c : Obj (Sets {l}) } → (F : Functor (Sets {l}) (Sets {l} ) ) → ( fmap : {a : Obj Sets } {b : Obj Sets } → Hom Sets a b → Hom Sets (FObj F a) ( FObj F b) ) → {h f : Hom Sets a b } → {g k : Hom Sets b c } → Sets [ g o h ≈ k o f ] → Sets [ FMap F g o fmap h ≈ fmap k o FMap F f ] |
| 729 | 571 UniquenessOfFunctor : (F : Functor Sets Sets) |
|
723
40122e9c7fc9
postulate Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
722
diff
changeset
|
572 {a b : Obj Sets } { f : Hom Sets a b } → ( fmap : {a : Obj Sets } {b : Obj Sets } → Hom Sets a b → Hom Sets (FObj F a) ( FObj F b) ) |
| 724 | 573 → ( {b : Obj Sets } → Sets [ fmap (id1 Sets b) ≈ id1 Sets (FObj F b) ] ) |
|
723
40122e9c7fc9
postulate Free Theorem
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
722
diff
changeset
|
574 → Sets [ fmap f ≈ FMap F f ] |
| 729 | 575 UniquenessOfFunctor F {a} {b} {f} fmap eq = begin |
| 724 | 576 fmap f |
| 577 ≈↑⟨ idL ⟩ | |
| 578 id1 Sets (FObj F b) o fmap f | |
| 579 ≈↑⟨ car ( IsFunctor.identity (isFunctor F )) ⟩ | |
| 580 FMap F (id1 Sets b) o fmap f | |
| 581 ≈⟨ FreeTheorem F fmap refl-hom ⟩ | |
| 582 fmap (id1 Sets b) o FMap F f | |
| 583 ≈⟨ car eq ⟩ | |
| 584 id1 Sets (FObj F b) o FMap F f | |
| 585 ≈⟨ idL ⟩ | |
| 586 FMap F f | |
| 587 ∎ | |
| 588 where open ≈-Reasoning Sets hiding ( _o_ ) | |
| 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 |
| 729 | 592 ≡⟨ ≡-cong ( λ k → k x ) (UniquenessOfFunctor 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) |
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diff
changeset
|
644 ≡⟨⟩ |
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|
645 φ ( ( FMap (Functor● Sets Sets MonoidalSets F) (f , g) ) ( x , y ) ) |
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diff
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|
646 ∎ |
|
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Applicative law → Monoidal law begin
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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
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|
649 open Relation.Binary.PropositionalEquality.≡-Reasoning |
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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 ] |
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Applicative law → Monoidal law begin
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diff
changeset
|
651 ≈ Sets [ φ o FMap (Functor● Sets Sets MonoidalSets F) f ] ] |
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|
652 comm0 {a} {b} {f} = extensionality Sets ( λ (x : ( FObj F (proj₁ a) * FObj F (proj₂ a)) ) → comm00 x ) |
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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 ≡ |
|
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Applicative law → Monoidal law begin
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|
654 (Sets [ FMap F (Iso.≅→ (mα-iso isM)) o Sets [ φ o φ □ id1 Sets (FObj F c) ] ]) x |
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|
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
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diff
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|
710 ∎ |
|
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parents:
717
diff
changeset
|
711 where |
| 720 | 712 open Relation.Binary.PropositionalEquality |
|
719
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diff
changeset
|
713 open Relation.Binary.PropositionalEquality.≡-Reasoning |
|
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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
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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>
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717
diff
changeset
|
718 comm20 : {a b : Obj Sets} ( x : FObj F a * One ) → ( Sets [ |
|
a017ed40dd77
Applicative law → Monoidal law begin
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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
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parents:
717
diff
changeset
|
723 ≡⟨⟩ |
| 720 | 724 FMap F proj₁ ((FMap F (λ j k → j , k) x) <*> (pure OneObj)) |
|
719
a017ed40dd77
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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
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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 -- |