Mercurial > hg > Members > kono > Proof > category
comparison applicative.agda @ 768:9bcdbfbaaa39
clean up
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 12 Dec 2017 10:25:59 +0900 |
| parents | monoidal.agda@c30ca91f3a76 |
| children | 43138aead09b |
comparison
equal
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| 767:0f8e3b962c13 | 768:9bcdbfbaaa39 |
|---|---|
| 1 open import Level | |
| 2 open import Category | |
| 3 module applicative where | |
| 4 | |
| 5 open import Data.Product renaming (_×_ to _*_) | |
| 6 open import Category.Constructions.Product | |
| 7 open import HomReasoning | |
| 8 open import cat-utility | |
| 9 open import Relation.Binary.Core | |
| 10 open import Relation.Binary | |
| 11 open import monoidal | |
| 12 open import Category.Sets | |
| 13 import Relation.Binary.PropositionalEquality | |
| 14 | |
| 15 open Functor | |
| 16 | |
| 17 | |
| 18 _・_ : {c₁ : Level} { a b c : Obj (Sets {c₁} ) } → (b → c) → (a → b) → a → c | |
| 19 _・_ f g = λ x → f ( g x ) | |
| 20 | |
| 21 record IsApplicative {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) | |
| 22 ( pure : {a : Obj Sets} → Hom Sets a ( FObj F a ) ) | |
| 23 ( _<*>_ : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b ) | |
| 24 : Set (suc (suc c₁)) where | |
| 25 field | |
| 26 identity : { a : Obj Sets } { u : FObj F a } → pure ( id1 Sets a ) <*> u ≡ u | |
| 27 composition : { a b c : Obj Sets } { u : FObj F ( b → c ) } { v : FObj F (a → b ) } { w : FObj F a } | |
| 28 → (( pure _・_ <*> u ) <*> v ) <*> w ≡ u <*> (v <*> w) | |
| 29 homomorphism : { a b : Obj Sets } { f : Hom Sets a b } { x : a } → pure f <*> pure x ≡ pure (f x) | |
| 30 interchange : { a b : Obj Sets } { u : FObj F ( a → b ) } { x : a } → u <*> pure x ≡ pure (λ f → f x) <*> u | |
| 31 -- from http://www.staff.city.ac.uk/~ross/papers/Applicative.pdf | |
| 32 | |
| 33 record Applicative {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) | |
| 34 : Set (suc (suc c₁)) where | |
| 35 field | |
| 36 pure : {a : Obj Sets} → Hom Sets a ( FObj F a ) | |
| 37 <*> : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b | |
| 38 isApplicative : IsApplicative F pure <*> | |
| 39 | |
| 40 ------ | |
| 41 -- | |
| 42 -- Appllicative Functor is a Monoidal Functor | |
| 43 -- | |
| 44 | |
| 45 Applicative→Monoidal : {c : Level} ( F : Functor (Sets {c}) (Sets {c}) ) → (mf : Applicative F ) | |
| 46 → IsApplicative F ( Applicative.pure mf ) ( Applicative.<*> mf ) | |
| 47 → MonoidalFunctor {_} {c} {_} {Sets} {Sets} MonoidalSets MonoidalSets | |
| 48 Applicative→Monoidal {l} F mf ismf = record { | |
| 49 MF = F | |
| 50 ; ψ = λ x → unit | |
| 51 ; isMonodailFunctor = record { | |
| 52 φab = record { TMap = λ x → φ ; isNTrans = record { commute = comm0 } } | |
| 53 ; associativity = λ {a b c} → comm1 {a} {b} {c} | |
| 54 ; unitarity-idr = λ {a b} → comm2 {a} {b} | |
| 55 ; unitarity-idl = λ {a b} → comm3 {a} {b} | |
| 56 } | |
| 57 } where | |
| 58 open Monoidal | |
| 59 open IsMonoidal hiding ( _■_ ; _□_ ) | |
| 60 M = MonoidalSets | |
| 61 isM = Monoidal.isMonoidal MonoidalSets | |
| 62 unit = Applicative.pure mf OneObj | |
| 63 _⊗_ : (x y : Obj Sets ) → Obj Sets | |
| 64 _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y | |
| 65 _□_ : {a b c d : Obj Sets } ( f : Hom Sets a c ) ( g : Hom Sets b d ) → Hom Sets ( a ⊗ b ) ( c ⊗ d ) | |
| 66 _□_ f g = FMap (m-bi M) ( f , g ) | |
| 67 φ : {x : Obj (Sets × Sets) } → Hom Sets (FObj (Functor● Sets Sets MonoidalSets F) x) (FObj (Functor⊗ Sets Sets MonoidalSets F) x) | |
| 68 φ x = Applicative.<*> mf (FMap F (λ j k → (j , k)) (proj₁ x )) (proj₂ x) | |
| 69 _<*>_ : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b | |
| 70 _<*>_ = Applicative.<*> mf | |
| 71 left : {a b : Obj Sets} → {x y : FObj F ( a → b )} → {h : FObj F a } → ( x ≡ y ) → x <*> h ≡ y <*> h | |
| 72 left {_} {_} {_} {_} {h} eq = ≡-cong ( λ k → k <*> h ) eq | |
| 73 right : {a b : Obj Sets} → {h : FObj F ( a → b )} → {x y : FObj F a } → ( x ≡ y ) → h <*> x ≡ h <*> y | |
| 74 right {_} {_} {h} {_} {_} eq = ≡-cong ( λ k → h <*> k ) eq | |
| 75 id : { a : Obj Sets } → a → a | |
| 76 id x = x | |
| 77 pure : {a : Obj Sets } → Hom Sets a ( FObj F a ) | |
| 78 pure a = Applicative.pure mf a | |
| 79 -- special case | |
| 80 F→pureid : {a b : Obj Sets } → (x : FObj F a ) → FMap F id x ≡ pure id <*> x | |
| 81 F→pureid {a} {b} x = sym ( begin | |
| 82 pure id <*> x | |
| 83 ≡⟨ IsApplicative.identity ismf ⟩ | |
| 84 x | |
| 85 ≡⟨ ≡-cong ( λ k → k x ) (sym ( IsFunctor.identity (isFunctor F ) )) ⟩ FMap F id x | |
| 86 ∎ ) | |
| 87 where | |
| 88 open Relation.Binary.PropositionalEquality | |
| 89 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 90 F→pure : {a b : Obj Sets } → { f : a → b } → {x : FObj F a } → FMap F f x ≡ pure f <*> x | |
| 91 F→pure {a} {b} {f} {x} = sym ( begin | |
| 92 pure f <*> x | |
| 93 ≡⟨ ≡-cong ( λ k → k x ) (UniquenessOfFunctor Sets Sets F ( λ f x → pure f <*> x ) ( extensionality Sets ( λ x → IsApplicative.identity ismf ))) ⟩ | |
| 94 FMap F f x | |
| 95 ∎ ) | |
| 96 where | |
| 97 open Relation.Binary.PropositionalEquality | |
| 98 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 99 p*p : { a b : Obj Sets } { f : Hom Sets a b } { x : a } → pure f <*> pure x ≡ pure (f x) | |
| 100 p*p = IsApplicative.homomorphism ismf | |
| 101 comp = IsApplicative.composition ismf | |
| 102 inter = IsApplicative.interchange ismf | |
| 103 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 | |
| 104 pureAssoc f g h = trans ( trans (sym comp) (left (left p*p) )) ( left p*p ) | |
| 105 where | |
| 106 open Relation.Binary.PropositionalEquality | |
| 107 comm00 : {a b : Obj (Sets × Sets)} { f : Hom (Sets × Sets) a b} (x : ( FObj F (proj₁ a) * FObj F (proj₂ a)) ) → | |
| 108 (Sets [ FMap (Functor⊗ Sets Sets MonoidalSets F) f o φ ]) x ≡ (Sets [ φ o FMap (Functor● Sets Sets MonoidalSets F) f ]) x | |
| 109 comm00 {a} {b} {(f , g)} (x , y) = begin | |
| 110 ( FMap (Functor⊗ Sets Sets MonoidalSets F) (f , g) ) ( φ (x , y) ) | |
| 111 ≡⟨⟩ | |
| 112 FMap F (λ xy → f (proj₁ xy) , g (proj₂ xy)) ((FMap F (λ j k → j , k) x) <*> y) | |
| 113 ≡⟨⟩ | |
| 114 FMap F (map f g) ((FMap F (λ j k → j , k) x) <*> y) | |
| 115 ≡⟨ F→pure ⟩ | |
| 116 (pure (map f g) <*> (FMap F (λ j k → j , k) x <*> y)) | |
| 117 ≡⟨ right ( left F→pure ) ⟩ | |
| 118 (pure (map f g)) <*> ((pure (λ j k → j , k) <*> x) <*> y) | |
| 119 ≡⟨ sym ( IsApplicative.composition ismf ) ⟩ | |
| 120 (( pure _・_ <*> (pure (map f g))) <*> (pure (λ j k → j , k) <*> x)) <*> y | |
| 121 ≡⟨ left ( sym ( IsApplicative.composition ismf )) ⟩ | |
| 122 ((( pure _・_ <*> (( pure _・_ <*> (pure (map f g))))) <*> pure (λ j k → j , k)) <*> x) <*> y | |
| 123 ≡⟨ trans ( trans (left ( left (left (right p*p )))) ( left ( left ( left p*p)))) (left (left p*p)) ⟩ | |
| 124 (pure (( _・_ (( _・_ ((map f g))))) (λ j k → j , k)) <*> x) <*> y | |
| 125 ≡⟨⟩ | |
| 126 (pure (λ j k → f j , g k) <*> x) <*> y | |
| 127 ≡⟨⟩ | |
| 128 ( pure ((_・_ (( _・_ ( ( λ h → h g ))) ( _・_ ))) ((λ j k → f j , k))) <*> x ) <*> y | |
| 129 ≡⟨ sym ( trans (left (left (left p*p))) (left ( left p*p)) ) ⟩ | |
| 130 ((((pure _・_ <*> pure ((λ h → h g) ・ _・_)) <*> pure (λ j k → f j , k)) <*> x) <*> y) | |
| 131 ≡⟨ sym (trans ( left ( left ( left (right (left p*p) )))) (left ( left (left (right p*p ))))) ⟩ | |
| 132 (((pure _・_ <*> (( pure _・_ <*> ( pure ( λ h → h g ))) <*> ( pure _・_ ))) <*> (pure (λ j k → f j , k))) <*> x ) <*> y | |
| 133 ≡⟨ left ( ( IsApplicative.composition ismf )) ⟩ | |
| 134 ((( pure _・_ <*> ( pure ( λ h → h g ))) <*> ( pure _・_ )) <*> (pure (λ j k → f j , k) <*> x )) <*> y | |
| 135 ≡⟨ left (IsApplicative.composition ismf ) ⟩ | |
| 136 ( pure ( λ h → h g ) <*> ( pure _・_ <*> (pure (λ j k → f j , k) <*> x )) ) <*> y | |
| 137 ≡⟨ left (sym (IsApplicative.interchange ismf )) ⟩ | |
| 138 (( pure _・_ <*> (pure (λ j k → f j , k) <*> x )) <*> pure g) <*> y | |
| 139 ≡⟨ IsApplicative.composition ismf ⟩ | |
| 140 (pure (λ j k → f j , k) <*> x) <*> (pure g <*> y) | |
| 141 ≡⟨ sym ( trans (left F→pure ) ( right F→pure ) ) ⟩ | |
| 142 (FMap F (λ j k → f j , k) x) <*> (FMap F g y) | |
| 143 ≡⟨ ≡-cong ( λ k → k x <*> (FMap F g y)) ( IsFunctor.distr (isFunctor F )) ⟩ | |
| 144 (FMap F (λ j k → j , k) (FMap F f x)) <*> (FMap F g y) | |
| 145 ≡⟨⟩ | |
| 146 φ ( ( FMap (Functor● Sets Sets MonoidalSets F) (f , g) ) ( x , y ) ) | |
| 147 ∎ | |
| 148 where | |
| 149 open Relation.Binary.PropositionalEquality | |
| 150 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 151 comm0 : {a b : Obj (Sets × Sets)} { f : Hom (Sets × Sets) a b} → Sets [ Sets [ FMap (Functor⊗ Sets Sets MonoidalSets F) f o φ ] | |
| 152 ≈ Sets [ φ o FMap (Functor● Sets Sets MonoidalSets F) f ] ] | |
| 153 comm0 {a} {b} {f} = extensionality Sets ( λ (x : ( FObj F (proj₁ a) * FObj F (proj₂ a)) ) → comm00 x ) | |
| 154 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 ≡ | |
| 155 (Sets [ FMap F (Iso.≅→ (mα-iso isM)) o Sets [ φ o φ □ id1 Sets (FObj F c) ] ]) x | |
| 156 comm10 {x} {y} {f} ((a , b) , c ) = begin | |
| 157 φ (( id □ φ ) ( ( Iso.≅→ (mα-iso isM) ) ((a , b) , c))) | |
| 158 ≡⟨⟩ | |
| 159 (FMap F (λ j k → j , k) a) <*> ( (FMap F (λ j k → j , k) b) <*> c) | |
| 160 ≡⟨ trans (left F→pure) (right (left F→pure) ) ⟩ | |
| 161 (pure (λ j k → j , k) <*> a) <*> ( (pure (λ j k → j , k) <*> b) <*> c) | |
| 162 ≡⟨ sym comp ⟩ | |
| 163 ( ( pure _・_ <*> (pure (λ j k → j , k) <*> a)) <*> (pure (λ j k → j , k) <*> b)) <*> c | |
| 164 ≡⟨ sym ( left comp ) ⟩ | |
| 165 (( ( pure _・_ <*> ( pure _・_ <*> (pure (λ j k → j , k) <*> a))) <*> (pure (λ j k → j , k))) <*> b) <*> c | |
| 166 ≡⟨ sym ( left ( left ( left (right comp )))) ⟩ | |
| 167 (( ( pure _・_ <*> (( (pure _・_ <*> pure _・_ ) <*> (pure (λ j k → j , k))) <*> a)) <*> (pure (λ j k → j , k))) <*> b) <*> c | |
| 168 ≡⟨ trans (left ( left (left ( right (left ( left p*p )))))) (left ( left ( left (right (left p*p))))) ⟩ | |
| 169 (( ( pure _・_ <*> ((pure ((_・_ ( _・_ )) ((λ j k → j , k)))) <*> a)) <*> (pure (λ j k → j , k))) <*> b) <*> c | |
| 170 ≡⟨ sym (left ( left ( left comp ) )) ⟩ | |
| 171 (((( ( pure _・_ <*> (pure _・_ )) <*> (pure ((_・_ ( _・_ )) ((λ j k → j , k))))) <*> a) <*> (pure (λ j k → j , k))) <*> b) <*> c | |
| 172 ≡⟨ trans (left ( left ( left (left (left p*p))))) (left ( left ( left (left p*p )))) ⟩ | |
| 173 ((((pure ( ( _・_ (_・_ )) (((_・_ ( _・_ )) ((λ j k → j , k)))))) <*> a) <*> (pure (λ j k → j , k))) <*> b) <*> c | |
| 174 ≡⟨⟩ | |
| 175 ((((pure (λ f g x y → f , g x y)) <*> a) <*> (pure (λ j k → j , k))) <*> b) <*> c | |
| 176 ≡⟨ left ( left inter ) ⟩ | |
| 177 (((pure (λ f → f (λ j k → j , k))) <*> ((pure (λ f g x y → f , g x y)) <*> a) ) <*> b) <*> c | |
| 178 ≡⟨ sym ( left ( left comp )) ⟩ | |
| 179 (((( pure _・_ <*> (pure (λ f → f (λ j k → j , k)))) <*> (pure (λ f g x y → f , g x y))) <*> a ) <*> b) <*> c | |
| 180 ≡⟨ trans (left ( left (left (left p*p) ))) (left (left (left p*p ) )) ⟩ | |
| 181 ((pure (( _・_ (λ f → f (λ j k → j , k))) (λ f g x y → f , g x y)) <*> a ) <*> b) <*> c | |
| 182 ≡⟨⟩ | |
| 183 (((pure (λ f g h → f , g , h)) <*> a) <*> b) <*> c | |
| 184 ≡⟨⟩ | |
| 185 ((pure ((_・_ ((_・_ ((_・_ ( (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))))))) | |
| 186 (( _・_ ( _・_ ((λ j k → j , k)))) (λ j k → j , k))) <*> a) <*> b) <*> c | |
| 187 ≡⟨ sym (trans ( left ( left ( left (left (right (right p*p))) ) )) (trans (left (left( left (left (right p*p))))) | |
| 188 (trans (left (left (left (left p*p)))) (trans ( left (left (left (right (left (right p*p )))))) | |
| 189 (trans (left (left (left (right (left p*p))))) (trans (left (left (left (right p*p)))) (left (left (left p*p)))) ) ) ) | |
| 190 ) ) ⟩ | |
| 191 ((((pure _・_ <*> ((pure _・_ <*> ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))))))) <*> | |
| 192 (( pure _・_ <*> ( pure _・_ <*> (pure (λ j k → j , k)))) <*> pure (λ j k → j , k))) <*> a) <*> b) <*> c | |
| 193 ≡⟨ left (left comp ) ⟩ | |
| 194 (((pure _・_ <*> ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))))) <*> | |
| 195 ((( pure _・_ <*> ( pure _・_ <*> (pure (λ j k → j , k)))) <*> pure (λ j k → j , k)) <*> a)) <*> b) <*> c | |
| 196 ≡⟨ left comp ⟩ | |
| 197 ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))) <*> | |
| 198 (((( pure _・_ <*> ( pure _・_ <*> (pure (λ j k → j , k)))) <*> pure (λ j k → j , k)) <*> a) <*> b)) <*> c | |
| 199 ≡⟨ left ( right (left comp )) ⟩ | |
| 200 ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))) <*> | |
| 201 ((( pure _・_ <*> (pure (λ j k → j , k))) <*> (pure (λ j k → j , k) <*> a)) <*> b)) <*> c | |
| 202 ≡⟨ left ( right comp ) ⟩ | |
| 203 ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))) <*> | |
| 204 (pure (λ j k → j , k) <*> ( (pure (λ j k → j , k) <*> a) <*> b))) <*> c | |
| 205 ≡⟨ comp ⟩ | |
| 206 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) | |
| 207 ≡⟨ sym ( trans ( trans F→pure (right (left F→pure ))) ( right ( left (right (left F→pure ))))) ⟩ | |
| 208 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) | |
| 209 ≡⟨⟩ | |
| 210 ( FMap F (Iso.≅→ (mα-iso isM))) (φ (( φ □ id1 Sets (FObj F f) ) ((a , b) , c))) | |
| 211 ∎ | |
| 212 where | |
| 213 open Relation.Binary.PropositionalEquality | |
| 214 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 215 comm1 : {a b c : Obj Sets} → Sets [ Sets [ φ | |
| 216 o Sets [ (id1 Sets (FObj F a) □ φ ) o Iso.≅→ (mα-iso isM) ] ] | |
| 217 ≈ Sets [ FMap F (Iso.≅→ (mα-iso isM)) o Sets [ φ o (φ □ id1 Sets (FObj F c)) ] ] ] | |
| 218 comm1 {a} {b} {c} = extensionality Sets ( λ x → comm10 x ) | |
| 219 comm20 : {a b : Obj Sets} ( x : FObj F a * One ) → ( Sets [ | |
| 220 FMap F (Iso.≅→ (mρ-iso isM)) o Sets [ φ o | |
| 221 FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit )) ] ] ) x ≡ Iso.≅→ (mρ-iso isM) x | |
| 222 comm20 {a} {b} (x , OneObj ) = begin | |
| 223 (FMap F (Iso.≅→ (mρ-iso isM))) ( φ (( FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit))) (x , OneObj) )) | |
| 224 ≡⟨⟩ | |
| 225 FMap F proj₁ ((FMap F (λ j k → j , k) x) <*> (pure OneObj)) | |
| 226 ≡⟨ ≡-cong ( λ k → FMap F proj₁ k) ( IsApplicative.interchange ismf ) ⟩ | |
| 227 FMap F proj₁ ((pure (λ f → f OneObj)) <*> (FMap F (λ j k → j , k) x)) | |
| 228 ≡⟨ ( trans F→pure (right ( right F→pure )) ) ⟩ | |
| 229 pure proj₁ <*> ((pure (λ f → f OneObj)) <*> (pure (λ j k → j , k) <*> x)) | |
| 230 ≡⟨ sym ( right comp ) ⟩ | |
| 231 pure proj₁ <*> (((pure _・_ <*> (pure (λ f → f OneObj))) <*> pure (λ j k → j , k)) <*> x) | |
| 232 ≡⟨ sym comp ⟩ | |
| 233 ( ( pure _・_ <*> (pure proj₁ ) ) <*> ((pure _・_ <*> (pure (λ f → f OneObj))) <*> pure (λ j k → j , k))) <*> x | |
| 234 ≡⟨ trans ( trans ( trans ( left ( left p*p)) ( left ( right (left p*p) ))) (left (right p*p) ) ) (left p*p) ⟩ | |
| 235 pure ( ( _・_ (proj₁ {l} {l})) ((_・_ ((λ f → f OneObj))) (λ j k → j , k))) <*> x | |
| 236 ≡⟨⟩ | |
| 237 pure id <*> x | |
| 238 ≡⟨ IsApplicative.identity ismf ⟩ | |
| 239 x | |
| 240 ≡⟨⟩ | |
| 241 Iso.≅→ (mρ-iso isM) (x , OneObj) | |
| 242 ∎ | |
| 243 where | |
| 244 open Relation.Binary.PropositionalEquality | |
| 245 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 246 comm2 : {a b : Obj Sets} → Sets [ Sets [ | |
| 247 FMap F (Iso.≅→ (mρ-iso isM)) o Sets [ φ o | |
| 248 FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit )) ] ] ≈ Iso.≅→ (mρ-iso isM) ] | |
| 249 comm2 {a} {b} = extensionality Sets ( λ x → comm20 {a} {b} x ) | |
| 250 comm30 : {a b : Obj Sets} ( x : One * FObj F b ) → ( Sets [ | |
| 251 FMap F (Iso.≅→ (mλ-iso isM)) o Sets [ φ o | |
| 252 FMap (m-bi MonoidalSets) ((λ _ → unit ) , id1 Sets (FObj F b) ) ] ] ) x ≡ Iso.≅→ (mλ-iso isM) x | |
| 253 comm30 {a} {b} ( OneObj , x) = begin | |
| 254 (FMap F (Iso.≅→ (mλ-iso isM))) ( φ ( unit , x ) ) | |
| 255 ≡⟨⟩ | |
| 256 FMap F proj₂ ((FMap F (λ j k → j , k) (pure OneObj)) <*> x) | |
| 257 ≡⟨ ( trans F→pure (right ( left F→pure )) ) ⟩ | |
| 258 pure proj₂ <*> ((pure (λ j k → j , k) <*> (pure OneObj)) <*> x) | |
| 259 ≡⟨ sym comp ⟩ | |
| 260 ((pure _・_ <*> (pure proj₂)) <*> (pure (λ j k → j , k) <*> (pure OneObj))) <*> x | |
| 261 ≡⟨ trans (trans (left (left p*p )) (left ( right p*p)) ) (left p*p) ⟩ | |
| 262 pure ((_・_ (proj₂ {l}) )((λ (j : One {l}) (k : b ) → j , k) OneObj)) <*> x | |
| 263 ≡⟨⟩ | |
| 264 pure id <*> x | |
| 265 ≡⟨ IsApplicative.identity ismf ⟩ | |
| 266 x | |
| 267 ≡⟨⟩ | |
| 268 Iso.≅→ (mλ-iso isM) ( OneObj , x ) | |
| 269 ∎ | |
| 270 where | |
| 271 open Relation.Binary.PropositionalEquality | |
| 272 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 273 comm3 : {a b : Obj Sets} → Sets [ Sets [ FMap F (Iso.≅→ (mλ-iso isM)) o | |
| 274 Sets [ φ o FMap (m-bi MonoidalSets) ((λ _ → unit ) , id1 Sets (FObj F b)) ] ] ≈ Iso.≅→ (mλ-iso isM) ] | |
| 275 comm3 {a} {b} = extensionality Sets ( λ x → comm30 {a} {b} x ) | |
| 276 | |
| 277 ---- | |
| 278 -- | |
| 279 -- Monoidal laws imples Applicative laws | |
| 280 -- | |
| 281 | |
| 282 HaskellMonoidal→Applicative : {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) | |
| 283 ( Mono : HaskellMonoidalFunctor F ) | |
| 284 → Applicative F | |
| 285 HaskellMonoidal→Applicative {c₁} F Mono = record { | |
| 286 pure = pure ; | |
| 287 <*> = _<*>_ ; | |
| 288 isApplicative = record { | |
| 289 identity = identity | |
| 290 ; composition = composition | |
| 291 ; homomorphism = homomorphism | |
| 292 ; interchange = interchange | |
| 293 } | |
| 294 } | |
| 295 where | |
| 296 unit : FObj F One | |
| 297 unit = HaskellMonoidalFunctor.unit Mono | |
| 298 φ : {a b : Obj Sets} → Hom Sets ((FObj F a) * (FObj F b )) ( FObj F ( a * b ) ) | |
| 299 φ = HaskellMonoidalFunctor.φ Mono | |
| 300 mono : IsHaskellMonoidalFunctor F unit φ | |
| 301 mono = HaskellMonoidalFunctor.isHaskellMonoidalFunctor Mono | |
| 302 id : { a : Obj Sets } → a → a | |
| 303 id x = x | |
| 304 isM : IsMonoidal (Sets {c₁}) One SetsTensorProduct | |
| 305 isM = Monoidal.isMonoidal MonoidalSets | |
| 306 pure : {a : Obj Sets} → Hom Sets a ( FObj F a ) | |
| 307 pure {a} x = FMap F ( λ y → x ) (unit ) | |
| 308 _<*>_ : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b | |
| 309 _<*>_ {a} {b} x y = FMap F ( λ r → ( proj₁ r ) ( proj₂ r ) ) (φ ( x , y )) | |
| 310 -- right does not work right it makes yellows. why? | |
| 311 -- right : {n : Level} { a b : Set n} → { x y : a } { h : a → b } → ( x ≡ y ) → h x ≡ h y | |
| 312 -- right {_} {_} {_} {_} {_} {h} eq = ≡-cong ( λ k → h k ) eq | |
| 313 left : {n : Level} { a b : Set n} → { x y : a → b } { h : a } → ( x ≡ y ) → x h ≡ y h | |
| 314 left {_} {_} {_} {_} {_} {h} eq = ≡-cong ( λ k → k h ) eq | |
| 315 open Relation.Binary.PropositionalEquality | |
| 316 FφF→F : { a b c d e : Obj Sets } { g : Hom Sets a c } { h : Hom Sets b d } | |
| 317 { f : Hom Sets (c * d) e } | |
| 318 { x : FObj F a } { y : FObj F b } | |
| 319 → FMap F f ( φ ( FMap F g x , FMap F h y ) ) ≡ FMap F ( f o map g h ) ( φ ( x , y ) ) | |
| 320 FφF→F {a} {b} {c} {d} {e} {g} {h} {f} {x} {y} = sym ( begin | |
| 321 FMap F ( f o map g h ) ( φ ( x , y ) ) | |
| 322 ≡⟨ ≡-cong ( λ k → k ( φ ( x , y ))) ( IsFunctor.distr (isFunctor F) ) ⟩ | |
| 323 FMap F f (( FMap F ( map g h ) ) ( φ ( x , y ))) | |
| 324 ≡⟨ ≡-cong ( λ k → FMap F f k ) ( IsHaskellMonoidalFunctor.natφ mono ) ⟩ | |
| 325 FMap F f ( φ ( FMap F g x , FMap F h y ) ) | |
| 326 ∎ ) | |
| 327 where | |
| 328 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 329 u→F : {a : Obj Sets } {u : FObj F a} → u ≡ FMap F id u | |
| 330 u→F {a} {u} = sym ( ≡-cong ( λ k → k u ) ( IsFunctor.identity ( isFunctor F ) ) ) | |
| 331 φunitr : {a : Obj Sets } {u : FObj F a} → φ ( unit , u) ≡ FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) u | |
| 332 φunitr {a} {u} = sym ( begin | |
| 333 FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) u | |
| 334 ≡⟨ ≡-cong ( λ k → FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) k ) (sym (IsHaskellMonoidalFunctor.idlφ mono)) ⟩ | |
| 335 FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) ( FMap F (Iso.≅→ (IsMonoidal.mλ-iso isM)) ( φ ( unit , u) ) ) | |
| 336 ≡⟨ left ( sym ( IsFunctor.distr ( isFunctor F ) )) ⟩ | |
| 337 (FMap F ( (Iso.≅← (IsMonoidal.mλ-iso isM)) o (Iso.≅→ (IsMonoidal.mλ-iso isM)))) ( φ ( unit , u) ) | |
| 338 ≡⟨ ≡-cong ( λ k → FMap F k ( φ ( unit , u) )) (Iso.iso→ ( (IsMonoidal.mλ-iso isM) )) ⟩ | |
| 339 FMap F id ( φ ( unit , u) ) | |
| 340 ≡⟨ left ( IsFunctor.identity ( isFunctor F ) ) ⟩ | |
| 341 id ( φ ( unit , u) ) | |
| 342 ≡⟨⟩ | |
| 343 φ ( unit , u) | |
| 344 ∎ ) | |
| 345 where | |
| 346 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 347 φunitl : {a : Obj Sets } {u : FObj F a} → φ ( u , unit ) ≡ FMap F (Iso.≅← (IsMonoidal.mρ-iso isM)) u | |
| 348 φunitl {a} {u} = sym ( begin | |
| 349 FMap F (Iso.≅← (IsMonoidal.mρ-iso isM)) u | |
| 350 ≡⟨ ≡-cong ( λ k → FMap F (Iso.≅← (IsMonoidal.mρ-iso isM)) k ) (sym (IsHaskellMonoidalFunctor.idrφ mono)) ⟩ | |
| 351 FMap F (Iso.≅← (IsMonoidal.mρ-iso isM)) ( FMap F (Iso.≅→ (IsMonoidal.mρ-iso isM)) ( φ ( u , unit ) ) ) | |
| 352 ≡⟨ left ( sym ( IsFunctor.distr ( isFunctor F ) )) ⟩ | |
| 353 (FMap F ( (Iso.≅← (IsMonoidal.mρ-iso isM)) o (Iso.≅→ (IsMonoidal.mρ-iso isM)))) ( φ ( u , unit ) ) | |
| 354 ≡⟨ ≡-cong ( λ k → FMap F k ( φ ( u , unit ) )) (Iso.iso→ ( (IsMonoidal.mρ-iso isM) )) ⟩ | |
| 355 FMap F id ( φ ( u , unit ) ) | |
| 356 ≡⟨ left ( IsFunctor.identity ( isFunctor F ) ) ⟩ | |
| 357 id ( φ ( u , unit ) ) | |
| 358 ≡⟨⟩ | |
| 359 φ ( u , unit ) | |
| 360 ∎ ) | |
| 361 where | |
| 362 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 363 open IsMonoidal | |
| 364 identity : { a : Obj Sets } { u : FObj F a } → pure ( id1 Sets a ) <*> u ≡ u | |
| 365 identity {a} {u} = begin | |
| 366 pure id <*> u | |
| 367 ≡⟨⟩ | |
| 368 ( FMap F ( λ r → ( proj₁ r ) ( proj₂ r )) ) ( φ ( FMap F ( λ y → id ) unit , u ) ) | |
| 369 ≡⟨ ≡-cong ( λ k → ( FMap F ( λ r → ( proj₁ r ) ( proj₂ r )) ) ( φ ( FMap F ( λ y → id ) unit , k ))) u→F ⟩ | |
| 370 ( FMap F ( λ r → ( proj₁ r ) ( proj₂ r )) ) ( φ ( FMap F ( λ y → id ) unit , FMap F id u ) ) | |
| 371 ≡⟨ FφF→F ⟩ | |
| 372 FMap F (λ x → proj₂ x ) (φ (unit , u ) ) | |
| 373 ≡⟨⟩ | |
| 374 FMap F (Iso.≅→ (mλ-iso isM)) (φ (unit , u )) | |
| 375 ≡⟨ IsHaskellMonoidalFunctor.idlφ mono ⟩ | |
| 376 u | |
| 377 ∎ | |
| 378 where | |
| 379 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 380 composition : { a b c : Obj Sets } { u : FObj F ( b → c ) } { v : FObj F (a → b ) } { w : FObj F a } | |
| 381 → (( pure _・_ <*> u ) <*> v ) <*> w ≡ u <*> (v <*> w) | |
| 382 composition {a} {b} {c} {u} {v} {w} = begin | |
| 383 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ | |
| 384 (FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f g x → f (g x)) unit , u)) , v)) , w)) | |
| 385 ≡⟨ ≡-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 ⟩ | |
| 386 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ | |
| 387 (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)) | |
| 388 ≡⟨ ≡-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 ⟩ | |
| 389 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ | |
| 390 (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)) | |
| 391 ≡⟨ ≡-cong ( λ k → ( FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ | |
| 392 (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 ⟩ | |
| 393 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ | |
| 394 ( (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)) | |
| 395 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ | |
| 396 (k u , v)) , w)) ) (sym ( IsFunctor.distr (isFunctor F ))) ⟩ | |
| 397 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ | |
| 398 ( FMap F (λ x → ((λ y f g x₁ → f (g x₁)) unit x) ) u , v)) , w)) | |
| 399 ≡⟨⟩ | |
| 400 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ | |
| 401 ( FMap F (λ x g h → x (g h) ) u , v)) , w)) | |
| 402 ≡⟨ ≡-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 ⟩ | |
| 403 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)) | |
| 404 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (k , w)) ) FφF→F ⟩ | |
| 405 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)) | |
| 406 ≡⟨⟩ | |
| 407 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ x h → proj₁ x (proj₂ x h)) (φ (u , v)) , w)) | |
| 408 ≡⟨ ≡-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 ⟩ | |
| 409 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)) | |
| 410 ≡⟨ FφF→F ⟩ | |
| 411 FMap F ((λ r → proj₁ r (proj₂ r)) o map (λ x h → proj₁ x (proj₂ x h)) id) (φ (φ (u , v) , w)) | |
| 412 ≡⟨⟩ | |
| 413 FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (φ (φ (u , v) , w)) | |
| 414 ≡⟨ ≡-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 ))) ⟩ | |
| 415 FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (FMap F id (φ (φ (u , v) , w)) ) | |
| 416 ≡⟨ ≡-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))) ⟩ | |
| 417 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)) ) | |
| 418 ≡⟨ ≡-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 )) ⟩ | |
| 419 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)) )) | |
| 420 ≡⟨ ≡-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 ) ) ⟩ | |
| 421 FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (FMap F (Iso.≅← (mα-iso isM)) (φ (u , φ (v , w)))) | |
| 422 ≡⟨⟩ | |
| 423 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)))) | |
| 424 ≡⟨ left (sym ( IsFunctor.distr (isFunctor F ))) ⟩ | |
| 425 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))) | |
| 426 ≡⟨⟩ | |
| 427 FMap F (λ y → proj₁ y (proj₁ (proj₂ y) (proj₂ (proj₂ y)))) (φ (u , φ (v , w))) | |
| 428 ≡⟨⟩ | |
| 429 FMap F ( λ x → (proj₁ x) ((λ r → proj₁ r (proj₂ r)) ( proj₂ x))) ( φ ( u , (φ (v , w)))) | |
| 430 ≡⟨ sym FφF→F ⟩ | |
| 431 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F id u , FMap F (λ r → proj₁ r (proj₂ r)) (φ (v , w)))) | |
| 432 ≡⟨ ≡-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 ) ⟩ | |
| 433 FMap F (λ r → proj₁ r (proj₂ r)) (φ (u , FMap F (λ r → proj₁ r (proj₂ r)) (φ (v , w)))) | |
| 434 ∎ | |
| 435 where | |
| 436 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 437 homomorphism : { a b : Obj Sets } { f : Hom Sets a b } { x : a } → pure f <*> pure x ≡ pure (f x) | |
| 438 homomorphism {a} {b} {f} {x} = begin | |
| 439 pure f <*> pure x | |
| 440 ≡⟨⟩ | |
| 441 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y → f) unit , FMap F (λ y → x) unit)) | |
| 442 ≡⟨ FφF→F ⟩ | |
| 443 FMap F ((λ r → proj₁ r (proj₂ r)) o map (λ y → f) (λ y → x)) (φ (unit , unit)) | |
| 444 ≡⟨⟩ | |
| 445 FMap F (λ y → f x) (φ (unit , unit)) | |
| 446 ≡⟨ ≡-cong ( λ k → FMap F (λ y → f x) k ) φunitl ⟩ | |
| 447 FMap F (λ y → f x) (FMap F (Iso.≅← (mρ-iso isM)) unit) | |
| 448 ≡⟨⟩ | |
| 449 FMap F (λ y → f x) (FMap F (λ y → (y , OneObj)) unit) | |
| 450 ≡⟨ left ( sym ( IsFunctor.distr (isFunctor F ))) ⟩ | |
| 451 FMap F (λ y → f x) unit | |
| 452 ≡⟨⟩ | |
| 453 pure (f x) | |
| 454 ∎ | |
| 455 where | |
| 456 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 457 interchange : { a b : Obj Sets } { u : FObj F ( a → b ) } { x : a } → u <*> pure x ≡ pure (λ f → f x) <*> u | |
| 458 interchange {a} {b} {u} {x} = begin | |
| 459 u <*> pure x | |
| 460 ≡⟨⟩ | |
| 461 FMap F (λ r → proj₁ r (proj₂ r)) (φ (u , FMap F (λ y → x) unit)) | |
| 462 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (k , FMap F (λ y → x) unit)) ) u→F ⟩ | |
| 463 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F id u , FMap F (λ y → x) unit)) | |
| 464 ≡⟨ FφF→F ⟩ | |
| 465 FMap F ((λ r → proj₁ r (proj₂ r)) o map id (λ y → x)) (φ (u , unit)) | |
| 466 ≡⟨⟩ | |
| 467 FMap F (λ r → proj₁ r x) (φ (u , unit)) | |
| 468 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r x) k ) φunitl ⟩ | |
| 469 FMap F (λ r → proj₁ r x) (( FMap F (Iso.≅← (mρ-iso isM))) u ) | |
| 470 ≡⟨ left ( sym ( IsFunctor.distr (isFunctor F )) ) ⟩ | |
| 471 FMap F (( λ r → proj₁ r x) o ((Iso.≅← (mρ-iso isM) ))) u | |
| 472 ≡⟨⟩ | |
| 473 FMap F (( λ r → proj₂ r x) o ((Iso.≅← (mλ-iso isM) ))) u | |
| 474 ≡⟨ left ( IsFunctor.distr (isFunctor F )) ⟩ | |
| 475 FMap F (λ r → proj₂ r x) (FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) u) | |
| 476 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₂ r x) k ) (sym φunitr ) ⟩ | |
| 477 FMap F (λ r → proj₂ r x) (φ (unit , u)) | |
| 478 ≡⟨⟩ | |
| 479 FMap F ((λ r → proj₁ r (proj₂ r)) o map (λ y f → f x) id) (φ (unit , u)) | |
| 480 ≡⟨ sym FφF→F ⟩ | |
| 481 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f → f x) unit , FMap F id u)) | |
| 482 ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f → f x) unit , k)) ) (sym u→F) ⟩ | |
| 483 FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f → f x) unit , u)) | |
| 484 ≡⟨⟩ | |
| 485 pure (λ f → f x) <*> u | |
| 486 ∎ | |
| 487 where | |
| 488 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 489 | |
| 490 ---- | |
| 491 -- | |
| 492 -- Applicative laws imples Monoidal laws | |
| 493 -- | |
| 494 | |
| 495 Applicative→HaskellMonoidal : {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) | |
| 496 ( App : Applicative F ) | |
| 497 → HaskellMonoidalFunctor F | |
| 498 Applicative→HaskellMonoidal {l} F App = record { | |
| 499 unit = unit ; | |
| 500 φ = φ ; | |
| 501 isHaskellMonoidalFunctor = record { | |
| 502 natφ = natφ | |
| 503 ; assocφ = assocφ | |
| 504 ; idrφ = idrφ | |
| 505 ; idlφ = idlφ | |
| 506 } | |
| 507 } where | |
| 508 pure = Applicative.pure App | |
| 509 <*> = Applicative.<*> App | |
| 510 app = Applicative.isApplicative App | |
| 511 unit : FObj F One | |
| 512 unit = pure OneObj | |
| 513 φ : {a b : Obj Sets} → Hom Sets ((FObj F a) * (FObj F b )) ( FObj F ( a * b ) ) | |
| 514 φ = λ x → <*> (FMap F (λ j k → (j , k)) ( proj₁ x)) ( proj₂ x) | |
| 515 isM : IsMonoidal (Sets {l}) One SetsTensorProduct | |
| 516 isM = Monoidal.isMonoidal MonoidalSets | |
| 517 MF : MonoidalFunctor {_} {l} {_} {Sets} {Sets} MonoidalSets MonoidalSets | |
| 518 MF = Applicative→Monoidal F App app | |
| 519 isMF = MonoidalFunctor.isMonodailFunctor MF | |
| 520 natφ : { a b c d : Obj Sets } { x : FObj F a} { y : FObj F b} { f : a → c } { g : b → d } | |
| 521 → FMap F (map f g) (φ (x , y)) ≡ φ (FMap F f x , FMap F g y) | |
| 522 natφ {a} {b} {c} {d} {x} {y} {f} {g} = begin | |
| 523 FMap F (map f g) (φ (x , y)) | |
| 524 ≡⟨⟩ | |
| 525 FMap F (λ xy → f (proj₁ xy) , g (proj₂ xy)) (<*> (FMap F (λ j k → j , k) x) y) | |
| 526 ≡⟨ ≡-cong ( λ h → h (x , y)) ( IsNTrans.commute ( NTrans.isNTrans ( IsMonoidalFunctor.φab isMF ))) ⟩ | |
| 527 <*> (FMap F (λ j k → j , k) (FMap F f x)) (FMap F g y) | |
| 528 ≡⟨⟩ | |
| 529 φ (FMap F f x , FMap F g y) | |
| 530 ∎ | |
| 531 where | |
| 532 open Relation.Binary.PropositionalEquality.≡-Reasoning | |
| 533 assocφ : { x y z : Obj Sets } { a : FObj F x } { b : FObj F y }{ c : FObj F z } | |
| 534 → φ (a , φ (b , c)) ≡ FMap F (Iso.≅→ (IsMonoidal.mα-iso isM)) (φ (φ (a , b) , c)) | |
| 535 assocφ {x} {y} {z} {a} {b} {c} = ≡-cong ( λ h → h ((a , b) , c ) ) ( IsMonoidalFunctor.associativity isMF ) | |
| 536 idrφ : {a : Obj Sets } { x : FObj F a } → FMap F (Iso.≅→ (IsMonoidal.mρ-iso isM)) (φ (x , unit)) ≡ x | |
| 537 idrφ {a} {x} = ≡-cong ( λ h → h (x , OneObj ) ) ( IsMonoidalFunctor.unitarity-idr isMF {a} {One} ) | |
| 538 idlφ : {a : Obj Sets } { x : FObj F a } → FMap F (Iso.≅→ (IsMonoidal.mλ-iso isM)) (φ (unit , x)) ≡ x | |
| 539 idlφ {a} {x} = ≡-cong ( λ h → h (OneObj , x ) ) ( IsMonoidalFunctor.unitarity-idl isMF {One} {a} ) | |
| 540 | |
| 541 |