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view applicative.agda @ 783:bded2347efa4
CCC by equation
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 17 Apr 2019 12:03:45 +0900 |
| parents | 06388660995b |
| children | dca4b29553cb |
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open import Level open import Category module applicative where open import Data.Product renaming (_×_ to _*_) open import Category.Constructions.Product open import HomReasoning open import cat-utility open import Relation.Binary.Core open import Relation.Binary open import monoidal open import Relation.Binary.PropositionalEquality hiding ( [_] ) ----- -- -- Applicative Functor -- -- is a monoidal functor on Sets and it can be constructoed from Haskell monoidal functor and vais versa -- ---- ----- -- -- To show Applicative functor is monoidal functor, uniquness of Functor is necessary, which is derived from the free theorem. -- -- they say it is not possible to prove FreeTheorem in Agda nor Coq -- https://stackoverflow.com/questions/24718567/is-it-possible-to-get-hold-of-free-theorems-as-propositional-equalities -- so we postulate this -- and we cannot indent a postulate ... open Functor postulate FreeTheorem : {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (C : Category c₁ c₂ ℓ) (D : Category c₁' c₂' ℓ') {a b c : Obj C } → (F : Functor C D ) → ( fmap : {a : Obj C } {b : Obj C } → Hom C a b → Hom D (FObj F a) ( FObj F b) ) → {h f : Hom C a b } → {g k : Hom C b c } → C [ C [ g o h ] ≈ C [ k o f ] ] → D [ D [ FMap F g o fmap h ] ≈ D [ fmap k o FMap F f ] ] UniquenessOfFunctor : {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (C : Category c₁ c₂ ℓ) (D : Category c₁' c₂' ℓ') (F : Functor C D) {a b : Obj C } { f : Hom C a b } → ( fmap : {a : Obj C } {b : Obj C } → Hom C a b → Hom D (FObj F a) ( FObj F b) ) → ( {b : Obj C } → D [ fmap (id1 C b) ≈ id1 D (FObj F b) ] ) → D [ fmap f ≈ FMap F f ] UniquenessOfFunctor C D F {a} {b} {f} fmap eq = begin fmap f ≈↑⟨ idL ⟩ id1 D (FObj F b) o fmap f ≈↑⟨ car ( IsFunctor.identity (isFunctor F )) ⟩ FMap F (id1 C b) o fmap f ≈⟨ FreeTheorem C D F fmap (IsEquivalence.refl (IsCategory.isEquivalence ( Category.isCategory C ))) ⟩ fmap (id1 C b) o FMap F f ≈⟨ car eq ⟩ id1 D (FObj F b) o FMap F f ≈⟨ idL ⟩ FMap F f ∎ where open ≈-Reasoning D open import Category.Sets import Relation.Binary.PropositionalEquality _・_ : {c₁ : Level} { a b c : Obj (Sets {c₁} ) } → (b → c) → (a → b) → a → c _・_ f g = λ x → f ( g x ) record IsApplicative {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) ( pure : {a : Obj Sets} → Hom Sets a ( FObj F a ) ) ( _<*>_ : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b ) : Set (suc (suc c₁)) where field identity : { a : Obj Sets } { u : FObj F a } → pure ( id1 Sets a ) <*> u ≡ u composition : { a b c : Obj Sets } { u : FObj F ( b → c ) } { v : FObj F (a → b ) } { w : FObj F a } → (( pure _・_ <*> u ) <*> v ) <*> w ≡ u <*> (v <*> w) homomorphism : { a b : Obj Sets } { f : Hom Sets a b } { x : a } → pure f <*> pure x ≡ pure (f x) interchange : { a b : Obj Sets } { u : FObj F ( a → b ) } { x : a } → u <*> pure x ≡ pure (λ f → f x) <*> u -- from http://www.staff.city.ac.uk/~ross/papers/Applicative.pdf record Applicative {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) : Set (suc (suc c₁)) where field pure : {a : Obj Sets} → Hom Sets a ( FObj F a ) <*> : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b isApplicative : IsApplicative F pure <*> ------ -- -- Appllicative Functor is a Monoidal Functor -- Applicative→Monoidal : {c : Level} ( F : Functor (Sets {c}) (Sets {c}) ) → (mf : Applicative F ) → IsApplicative F ( Applicative.pure mf ) ( Applicative.<*> mf ) → MonoidalFunctor {_} {c} {_} {Sets} {Sets} MonoidalSets MonoidalSets Applicative→Monoidal {l} F mf ismf = record { MF = F ; ψ = λ x → unit ; isMonodailFunctor = record { φab = record { TMap = λ x → φ ; isNTrans = record { commute = φab-comm } } ; associativity = λ {a b c} → associativity {a} {b} {c} ; unitarity-idr = λ {a b} → unitarity-idr {a} {b} ; unitarity-idl = λ {a b} → unitarity-idl {a} {b} } } where open Monoidal open IsMonoidal hiding ( _■_ ; _□_ ) M = MonoidalSets isM = Monoidal.isMonoidal MonoidalSets unit = Applicative.pure mf OneObj _⊗_ : (x y : Obj Sets ) → Obj Sets _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y _□_ : {a b c d : Obj Sets } ( f : Hom Sets a c ) ( g : Hom Sets b d ) → Hom Sets ( a ⊗ b ) ( c ⊗ d ) _□_ f g = FMap (m-bi M) ( f , g ) φ : {x : Obj (Sets × Sets) } → Hom Sets (FObj (Functor● Sets Sets MonoidalSets F) x) (FObj (Functor⊗ Sets Sets MonoidalSets F) x) φ x = Applicative.<*> mf (FMap F (λ j k → (j , k)) (proj₁ x )) (proj₂ x) _<*>_ : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b _<*>_ = Applicative.<*> mf left : {a b : Obj Sets} → {x y : FObj F ( a → b )} → {h : FObj F a } → ( x ≡ y ) → x <*> h ≡ y <*> h left {_} {_} {_} {_} {h} eq = ≡-cong ( λ k → k <*> h ) eq right : {a b : Obj Sets} → {h : FObj F ( a → b )} → {x y : FObj F a } → ( x ≡ y ) → h <*> x ≡ h <*> y right {_} {_} {h} {_} {_} eq = ≡-cong ( λ k → h <*> k ) eq id : { a : Obj Sets } → a → a id x = x pure : {a : Obj Sets } → Hom Sets a ( FObj F a ) pure a = Applicative.pure mf a -- special case F→pureid : {a b : Obj Sets } → (x : FObj F a ) → FMap F id x ≡ pure id <*> x F→pureid {a} {b} x = sym ( begin pure id <*> x ≡⟨ IsApplicative.identity ismf ⟩ x ≡⟨ ≡-cong ( λ k → k x ) (sym ( IsFunctor.identity (isFunctor F ) )) ⟩ FMap F id x ∎ ) where open Relation.Binary.PropositionalEquality open Relation.Binary.PropositionalEquality.≡-Reasoning F→pure : {a b : Obj Sets } → { f : a → b } → {x : FObj F a } → FMap F f x ≡ pure f <*> x F→pure {a} {b} {f} {x} = sym ( begin pure f <*> x ≡⟨ ≡-cong ( λ k → k x ) (UniquenessOfFunctor Sets Sets F ( λ f x → pure f <*> x ) ( extensionality Sets ( λ x → IsApplicative.identity ismf ))) ⟩ FMap F f x ∎ ) where open Relation.Binary.PropositionalEquality open Relation.Binary.PropositionalEquality.≡-Reasoning p*p : { a b : Obj Sets } { f : Hom Sets a b } { x : a } → pure f <*> pure x ≡ pure (f x) p*p = IsApplicative.homomorphism ismf comp = IsApplicative.composition ismf inter = IsApplicative.interchange ismf 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 pureAssoc f g h = trans ( trans (sym comp) (left (left p*p) )) ( left p*p ) where open Relation.Binary.PropositionalEquality φab-comm0 : {a b : Obj (Sets × Sets)} { f : Hom (Sets × Sets) a b} (x : ( FObj F (proj₁ a) * FObj F (proj₂ a)) ) → (Sets [ FMap (Functor⊗ Sets Sets MonoidalSets F) f o φ ]) x ≡ (Sets [ φ o FMap (Functor● Sets Sets MonoidalSets F) f ]) x φab-comm0 {a} {b} {(f , g)} (x , y) = begin ( FMap (Functor⊗ Sets Sets MonoidalSets F) (f , g) ) ( φ (x , y) ) ≡⟨⟩ FMap F (λ xy → f (proj₁ xy) , g (proj₂ xy)) ((FMap F (λ j k → j , k) x) <*> y) ≡⟨⟩ FMap F (map f g) ((FMap F (λ j k → j , k) x) <*> y) ≡⟨ F→pure ⟩ (pure (map f g) <*> (FMap F (λ j k → j , k) x <*> y)) ≡⟨ right ( left F→pure ) ⟩ (pure (map f g)) <*> ((pure (λ j k → j , k) <*> x) <*> y) ≡⟨ sym ( IsApplicative.composition ismf ) ⟩ (( pure _・_ <*> (pure (map f g))) <*> (pure (λ j k → j , k) <*> x)) <*> y ≡⟨ left ( sym ( IsApplicative.composition ismf )) ⟩ ((( pure _・_ <*> (( pure _・_ <*> (pure (map f g))))) <*> pure (λ j k → j , k)) <*> x) <*> y ≡⟨ trans ( trans (left ( left (left (right p*p )))) ( left ( left ( left p*p)))) (left (left p*p)) ⟩ (pure (( _・_ (( _・_ ((map f g))))) (λ j k → j , k)) <*> x) <*> y ≡⟨⟩ (pure (λ j k → f j , g k) <*> x) <*> y ≡⟨⟩ ( pure ((_・_ (( _・_ ( ( λ h → h g ))) ( _・_ ))) ((λ j k → f j , k))) <*> x ) <*> y ≡⟨ sym ( trans (left (left (left p*p))) (left ( left p*p)) ) ⟩ ((((pure _・_ <*> pure ((λ h → h g) ・ _・_)) <*> pure (λ j k → f j , k)) <*> x) <*> y) ≡⟨ sym (trans ( left ( left ( left (right (left p*p) )))) (left ( left (left (right p*p ))))) ⟩ (((pure _・_ <*> (( pure _・_ <*> ( pure ( λ h → h g ))) <*> ( pure _・_ ))) <*> (pure (λ j k → f j , k))) <*> x ) <*> y ≡⟨ left ( ( IsApplicative.composition ismf )) ⟩ ((( pure _・_ <*> ( pure ( λ h → h g ))) <*> ( pure _・_ )) <*> (pure (λ j k → f j , k) <*> x )) <*> y ≡⟨ left (IsApplicative.composition ismf ) ⟩ ( pure ( λ h → h g ) <*> ( pure _・_ <*> (pure (λ j k → f j , k) <*> x )) ) <*> y ≡⟨ left (sym (IsApplicative.interchange ismf )) ⟩ (( pure _・_ <*> (pure (λ j k → f j , k) <*> x )) <*> pure g) <*> y ≡⟨ IsApplicative.composition ismf ⟩ (pure (λ j k → f j , k) <*> x) <*> (pure g <*> y) ≡⟨ sym ( trans (left F→pure ) ( right F→pure ) ) ⟩ (FMap F (λ j k → f j , k) x) <*> (FMap F g y) ≡⟨ ≡-cong ( λ k → k x <*> (FMap F g y)) ( IsFunctor.distr (isFunctor F )) ⟩ (FMap F (λ j k → j , k) (FMap F f x)) <*> (FMap F g y) ≡⟨⟩ φ ( ( FMap (Functor● Sets Sets MonoidalSets F) (f , g) ) ( x , y ) ) ∎ where open Relation.Binary.PropositionalEquality open Relation.Binary.PropositionalEquality.≡-Reasoning φab-comm : {a b : Obj (Sets × Sets)} { f : Hom (Sets × Sets) a b} → Sets [ Sets [ FMap (Functor⊗ Sets Sets MonoidalSets F) f o φ ] ≈ Sets [ φ o FMap (Functor● Sets Sets MonoidalSets F) f ] ] φab-comm {a} {b} {f} = extensionality Sets ( λ (x : ( FObj F (proj₁ a) * FObj F (proj₂ a)) ) → φab-comm0 x ) associativity0 : {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 ≡ (Sets [ FMap F (Iso.≅→ (mα-iso isM)) o Sets [ φ o φ □ id1 Sets (FObj F c) ] ]) x associativity0 {x} {y} {f} ((a , b) , c ) = begin φ (( id □ φ ) ( ( Iso.≅→ (mα-iso isM) ) ((a , b) , c))) ≡⟨⟩ (FMap F (λ j k → j , k) a) <*> ( (FMap F (λ j k → j , k) b) <*> c) ≡⟨ trans (left F→pure) (right (left F→pure) ) ⟩ (pure (λ j k → j , k) <*> a) <*> ( (pure (λ j k → j , k) <*> b) <*> c) ≡⟨ sym comp ⟩ ( ( pure _・_ <*> (pure (λ j k → j , k) <*> a)) <*> (pure (λ j k → j , k) <*> b)) <*> c ≡⟨ sym ( left comp ) ⟩ (( ( pure _・_ <*> ( pure _・_ <*> (pure (λ j k → j , k) <*> a))) <*> (pure (λ j k → j , k))) <*> b) <*> c ≡⟨ sym ( left ( left ( left (right comp )))) ⟩ (( ( pure _・_ <*> (( (pure _・_ <*> pure _・_ ) <*> (pure (λ j k → j , k))) <*> a)) <*> (pure (λ j k → j , k))) <*> b) <*> c ≡⟨ trans (left ( left (left ( right (left ( left p*p )))))) (left ( left ( left (right (left p*p))))) ⟩ (( ( pure _・_ <*> ((pure ((_・_ ( _・_ )) ((λ j k → j , k)))) <*> a)) <*> (pure (λ j k → j , k))) <*> b) <*> c ≡⟨ sym (left ( left ( left comp ) )) ⟩ (((( ( pure _・_ <*> (pure _・_ )) <*> (pure ((_・_ ( _・_ )) ((λ j k → j , k))))) <*> a) <*> (pure (λ j k → j , k))) <*> b) <*> c ≡⟨ trans (left ( left ( left (left (left p*p))))) (left ( left ( left (left p*p )))) ⟩ ((((pure ( ( _・_ (_・_ )) (((_・_ ( _・_ )) ((λ j k → j , k)))))) <*> a) <*> (pure (λ j k → j , k))) <*> b) <*> c ≡⟨⟩ ((((pure (λ f g x y → f , g x y)) <*> a) <*> (pure (λ j k → j , k))) <*> b) <*> c ≡⟨ left ( left inter ) ⟩ (((pure (λ f → f (λ j k → j , k))) <*> ((pure (λ f g x y → f , g x y)) <*> a) ) <*> b) <*> c ≡⟨ sym ( left ( left comp )) ⟩ (((( pure _・_ <*> (pure (λ f → f (λ j k → j , k)))) <*> (pure (λ f g x y → f , g x y))) <*> a ) <*> b) <*> c ≡⟨ trans (left ( left (left (left p*p) ))) (left (left (left p*p ) )) ⟩ (((pure (λ f g h → f , g , h)) <*> a) <*> b) <*> c ≡⟨ sym (trans ( left ( left ( left (left (right (right p*p))) ) )) (trans (left (left( left (left (right p*p))))) (trans (left (left (left (left p*p)))) (trans ( left (left (left (right (left (right p*p )))))) (trans (left (left (left (right (left p*p))))) (trans (left (left (left (right p*p)))) (left (left (left p*p)))) ) ) ) ) ) ⟩ ((((pure _・_ <*> ((pure _・_ <*> ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))))))) <*> (( pure _・_ <*> ( pure _・_ <*> (pure (λ j k → j , k)))) <*> pure (λ j k → j , k))) <*> a) <*> b) <*> c ≡⟨ left (left comp ) ⟩ (((pure _・_ <*> ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))))) <*> ((( pure _・_ <*> ( pure _・_ <*> (pure (λ j k → j , k)))) <*> pure (λ j k → j , k)) <*> a)) <*> b) <*> c ≡⟨ left comp ⟩ ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))) <*> (((( pure _・_ <*> ( pure _・_ <*> (pure (λ j k → j , k)))) <*> pure (λ j k → j , k)) <*> a) <*> b)) <*> c ≡⟨ left ( right (left comp )) ⟩ ((pure _・_ <*> ( pure (λ abc → proj₁ (proj₁ abc) , proj₂ (proj₁ abc) , proj₂ abc))) <*> ((( pure _・_ <*> (pure (λ j k → j , k))) <*> (pure (λ j k → j , k) <*> a)) <*> b)) <*> c ≡⟨ left ( right comp ) ⟩ ((pure _・_ <*> ( 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 ≡⟨ comp ⟩ 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) ≡⟨ sym ( trans ( trans F→pure (right (left F→pure ))) ( right ( left (right (left F→pure ))))) ⟩ 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) ≡⟨⟩ ( FMap F (Iso.≅→ (mα-iso isM))) (φ (( φ □ id1 Sets (FObj F f) ) ((a , b) , c))) ∎ where open Relation.Binary.PropositionalEquality open Relation.Binary.PropositionalEquality.≡-Reasoning associativity : {a b c : Obj Sets} → Sets [ Sets [ φ o Sets [ (id1 Sets (FObj F a) □ φ ) o Iso.≅→ (mα-iso isM) ] ] ≈ Sets [ FMap F (Iso.≅→ (mα-iso isM)) o Sets [ φ o (φ □ id1 Sets (FObj F c)) ] ] ] associativity {a} {b} {c} = extensionality Sets ( λ x → associativity0 x ) unitarity-idr0 : {a b : Obj Sets} ( x : FObj F a * One ) → ( Sets [ FMap F (Iso.≅→ (mρ-iso isM)) o Sets [ φ o FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit )) ] ] ) x ≡ Iso.≅→ (mρ-iso isM) x unitarity-idr0 {a} {b} (x , OneObj ) = begin (FMap F (Iso.≅→ (mρ-iso isM))) ( φ (( FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit))) (x , OneObj) )) ≡⟨⟩ FMap F proj₁ ((FMap F (λ j k → j , k) x) <*> (pure OneObj)) ≡⟨ ≡-cong ( λ k → FMap F proj₁ k) ( IsApplicative.interchange ismf ) ⟩ FMap F proj₁ ((pure (λ f → f OneObj)) <*> (FMap F (λ j k → j , k) x)) ≡⟨ ( trans F→pure (right ( right F→pure )) ) ⟩ pure proj₁ <*> ((pure (λ f → f OneObj)) <*> (pure (λ j k → j , k) <*> x)) ≡⟨ sym ( right comp ) ⟩ pure proj₁ <*> (((pure _・_ <*> (pure (λ f → f OneObj))) <*> pure (λ j k → j , k)) <*> x) ≡⟨ sym comp ⟩ ( ( pure _・_ <*> (pure proj₁ ) ) <*> ((pure _・_ <*> (pure (λ f → f OneObj))) <*> pure (λ j k → j , k))) <*> x ≡⟨ trans ( trans ( trans ( left ( left p*p)) ( left ( right (left p*p) ))) (left (right p*p) ) ) (left p*p) ⟩ pure ( ( _・_ (proj₁ {l} {l})) ((_・_ ((λ f → f OneObj))) (λ j k → j , k))) <*> x ≡⟨⟩ pure id <*> x ≡⟨ IsApplicative.identity ismf ⟩ x ≡⟨⟩ Iso.≅→ (mρ-iso isM) (x , OneObj) ∎ where open Relation.Binary.PropositionalEquality open Relation.Binary.PropositionalEquality.≡-Reasoning unitarity-idr : {a b : Obj Sets} → Sets [ Sets [ FMap F (Iso.≅→ (mρ-iso isM)) o Sets [ φ o FMap (m-bi MonoidalSets) (id1 Sets (FObj F a) , (λ _ → unit )) ] ] ≈ Iso.≅→ (mρ-iso isM) ] unitarity-idr {a} {b} = extensionality Sets ( λ x → unitarity-idr0 {a} {b} x ) unitarity-idl0 : {a b : Obj Sets} ( x : One * FObj F b ) → ( Sets [ FMap F (Iso.≅→ (mλ-iso isM)) o Sets [ φ o FMap (m-bi MonoidalSets) ((λ _ → unit ) , id1 Sets (FObj F b) ) ] ] ) x ≡ Iso.≅→ (mλ-iso isM) x unitarity-idl0 {a} {b} ( OneObj , x) = begin (FMap F (Iso.≅→ (mλ-iso isM))) ( φ ( unit , x ) ) ≡⟨⟩ FMap F proj₂ ((FMap F (λ j k → j , k) (pure OneObj)) <*> x) ≡⟨ ( trans F→pure (right ( left F→pure )) ) ⟩ pure proj₂ <*> ((pure (λ j k → j , k) <*> (pure OneObj)) <*> x) ≡⟨ sym comp ⟩ ((pure _・_ <*> (pure proj₂)) <*> (pure (λ j k → j , k) <*> (pure OneObj))) <*> x ≡⟨ trans (trans (left (left p*p )) (left ( right p*p)) ) (left p*p) ⟩ pure ((_・_ (proj₂ {l}) )((λ (j : One {l}) (k : b ) → j , k) OneObj)) <*> x ≡⟨⟩ pure id <*> x ≡⟨ IsApplicative.identity ismf ⟩ x ≡⟨⟩ Iso.≅→ (mλ-iso isM) ( OneObj , x ) ∎ where open Relation.Binary.PropositionalEquality open Relation.Binary.PropositionalEquality.≡-Reasoning unitarity-idl : {a b : Obj Sets} → Sets [ Sets [ FMap F (Iso.≅→ (mλ-iso isM)) o Sets [ φ o FMap (m-bi MonoidalSets) ((λ _ → unit ) , id1 Sets (FObj F b)) ] ] ≈ Iso.≅→ (mλ-iso isM) ] unitarity-idl {a} {b} = extensionality Sets ( λ x → unitarity-idl0 {a} {b} x ) ---- -- -- Monoidal laws implies Applicative laws -- HaskellMonoidal→Applicative : {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) ( Mono : HaskellMonoidalFunctor F ) → Applicative F HaskellMonoidal→Applicative {c₁} F Mono = record { pure = pure ; <*> = _<*>_ ; isApplicative = record { identity = identity ; composition = composition ; homomorphism = homomorphism ; interchange = interchange } } where unit : FObj F One unit = HaskellMonoidalFunctor.unit Mono φ : {a b : Obj Sets} → Hom Sets ((FObj F a) * (FObj F b )) ( FObj F ( a * b ) ) φ = HaskellMonoidalFunctor.φ Mono mono : IsHaskellMonoidalFunctor F unit φ mono = HaskellMonoidalFunctor.isHaskellMonoidalFunctor Mono id : { a : Obj Sets } → a → a id x = x isM : IsMonoidal (Sets {c₁}) One SetsTensorProduct isM = Monoidal.isMonoidal MonoidalSets pure : {a : Obj Sets} → Hom Sets a ( FObj F a ) pure {a} x = FMap F ( λ y → x ) (unit ) _<*>_ : {a b : Obj Sets} → FObj F ( a → b ) → FObj F a → FObj F b _<*>_ {a} {b} x y = FMap F ( λ r → ( proj₁ r ) ( proj₂ r ) ) (φ ( x , y )) -- right does not work right it makes yellows. why? -- right : {n : Level} { a b : Set n} → { x y : a } { h : a → b } → ( x ≡ y ) → h x ≡ h y -- right {_} {_} {_} {_} {_} {h} eq = ≡-cong ( λ k → h k ) eq left : {n : Level} { a b : Set n} → { x y : a → b } { h : a } → ( x ≡ y ) → x h ≡ y h left {_} {_} {_} {_} {_} {h} eq = ≡-cong ( λ k → k h ) eq open Relation.Binary.PropositionalEquality FφF→F : { a b c d e : Obj Sets } { g : Hom Sets a c } { h : Hom Sets b d } { f : Hom Sets (c * d) e } { x : FObj F a } { y : FObj F b } → FMap F f ( φ ( FMap F g x , FMap F h y ) ) ≡ FMap F ( f o map g h ) ( φ ( x , y ) ) FφF→F {a} {b} {c} {d} {e} {g} {h} {f} {x} {y} = sym ( begin FMap F ( f o map g h ) ( φ ( x , y ) ) ≡⟨ ≡-cong ( λ k → k ( φ ( x , y ))) ( IsFunctor.distr (isFunctor F) ) ⟩ FMap F f (( FMap F ( map g h ) ) ( φ ( x , y ))) ≡⟨ ≡-cong ( λ k → FMap F f k ) ( IsHaskellMonoidalFunctor.natφ mono ) ⟩ FMap F f ( φ ( FMap F g x , FMap F h y ) ) ∎ ) where open Relation.Binary.PropositionalEquality.≡-Reasoning u→F : {a : Obj Sets } {u : FObj F a} → u ≡ FMap F id u u→F {a} {u} = sym ( ≡-cong ( λ k → k u ) ( IsFunctor.identity ( isFunctor F ) ) ) φunitr : {a : Obj Sets } {u : FObj F a} → φ ( unit , u) ≡ FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) u φunitr {a} {u} = sym ( begin FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) u ≡⟨ ≡-cong ( λ k → FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) k ) (sym (IsHaskellMonoidalFunctor.idlφ mono)) ⟩ FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) ( FMap F (Iso.≅→ (IsMonoidal.mλ-iso isM)) ( φ ( unit , u) ) ) ≡⟨ left ( sym ( IsFunctor.distr ( isFunctor F ) )) ⟩ (FMap F ( (Iso.≅← (IsMonoidal.mλ-iso isM)) o (Iso.≅→ (IsMonoidal.mλ-iso isM)))) ( φ ( unit , u) ) ≡⟨ ≡-cong ( λ k → FMap F k ( φ ( unit , u) )) (Iso.iso→ ( (IsMonoidal.mλ-iso isM) )) ⟩ FMap F id ( φ ( unit , u) ) ≡⟨ left ( IsFunctor.identity ( isFunctor F ) ) ⟩ id ( φ ( unit , u) ) ≡⟨⟩ φ ( unit , u) ∎ ) where open Relation.Binary.PropositionalEquality.≡-Reasoning φunitl : {a : Obj Sets } {u : FObj F a} → φ ( u , unit ) ≡ FMap F (Iso.≅← (IsMonoidal.mρ-iso isM)) u φunitl {a} {u} = sym ( begin FMap F (Iso.≅← (IsMonoidal.mρ-iso isM)) u ≡⟨ ≡-cong ( λ k → FMap F (Iso.≅← (IsMonoidal.mρ-iso isM)) k ) (sym (IsHaskellMonoidalFunctor.idrφ mono)) ⟩ FMap F (Iso.≅← (IsMonoidal.mρ-iso isM)) ( FMap F (Iso.≅→ (IsMonoidal.mρ-iso isM)) ( φ ( u , unit ) ) ) ≡⟨ left ( sym ( IsFunctor.distr ( isFunctor F ) )) ⟩ (FMap F ( (Iso.≅← (IsMonoidal.mρ-iso isM)) o (Iso.≅→ (IsMonoidal.mρ-iso isM)))) ( φ ( u , unit ) ) ≡⟨ ≡-cong ( λ k → FMap F k ( φ ( u , unit ) )) (Iso.iso→ ( (IsMonoidal.mρ-iso isM) )) ⟩ FMap F id ( φ ( u , unit ) ) ≡⟨ left ( IsFunctor.identity ( isFunctor F ) ) ⟩ id ( φ ( u , unit ) ) ≡⟨⟩ φ ( u , unit ) ∎ ) where open Relation.Binary.PropositionalEquality.≡-Reasoning open IsMonoidal identity : { a : Obj Sets } { u : FObj F a } → pure ( id1 Sets a ) <*> u ≡ u identity {a} {u} = begin pure id <*> u ≡⟨⟩ ( FMap F ( λ r → ( proj₁ r ) ( proj₂ r )) ) ( φ ( FMap F ( λ y → id ) unit , u ) ) ≡⟨ ≡-cong ( λ k → ( FMap F ( λ r → ( proj₁ r ) ( proj₂ r )) ) ( φ ( FMap F ( λ y → id ) unit , k ))) u→F ⟩ ( FMap F ( λ r → ( proj₁ r ) ( proj₂ r )) ) ( φ ( FMap F ( λ y → id ) unit , FMap F id u ) ) ≡⟨ FφF→F ⟩ FMap F (λ x → proj₂ x ) (φ (unit , u ) ) ≡⟨⟩ FMap F (Iso.≅→ (mλ-iso isM)) (φ (unit , u )) ≡⟨ IsHaskellMonoidalFunctor.idlφ mono ⟩ u ∎ where open Relation.Binary.PropositionalEquality.≡-Reasoning composition : { a b c : Obj Sets } { u : FObj F ( b → c ) } { v : FObj F (a → b ) } { w : FObj F a } → (( pure _・_ <*> u ) <*> v ) <*> w ≡ u <*> (v <*> w) composition {a} {b} {c} {u} {v} {w} = begin 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 , u)) , v)) , w)) ≡⟨ ≡-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 ⟩ 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 , FMap F id u )) , v)) , w)) ≡⟨ ≡-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 ⟩ FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F ( λ x → (λ (r : ((b → c) → _ ) * (b → c) ) → proj₁ r (proj₂ r)) ((map (λ y f g x → f (g x)) id ) x)) (φ ( unit , u)) , v)) , w)) ≡⟨ ≡-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 → (λ (r : ((b → c) → _ ) * (b → c) ) → proj₁ r (proj₂ r)) ((map (λ y f g x → f (g x)) id ) x)) k , v)) , w)) ) ) φunitr ⟩ FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ ( (FMap F ( λ x → (λ (r : ((b → c) → _ ) * (b → c) ) → proj₁ r (proj₂ r)) ((map (λ y f g x → f (g x)) id ) x)) (FMap F (Iso.≅← (mλ-iso isM)) u) ) , v)) , w)) ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ (k u , v)) , w)) ) (sym ( IsFunctor.distr (isFunctor F ))) ⟩ FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ r → proj₁ r (proj₂ r)) (φ ( FMap F (λ x → ((λ y f g x₁ → f (g x₁)) unit x) ) u , v)) , w)) ≡⟨⟩ 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 , v)) , w)) ≡⟨ ≡-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 ⟩ 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)) ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (k , w)) ) FφF→F ⟩ 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)) ≡⟨⟩ FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ x h → proj₁ x (proj₂ x h)) (φ (u , v)) , w)) ≡⟨ ≡-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 ⟩ 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)) ≡⟨ FφF→F ⟩ FMap F ((λ r → proj₁ r (proj₂ r)) o map (λ x h → proj₁ x (proj₂ x h)) id) (φ (φ (u , v) , w)) ≡⟨⟩ FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (φ (φ (u , v) , w)) ≡⟨ ≡-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 ))) ⟩ FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (FMap F id (φ (φ (u , v) , w)) ) ≡⟨ ≡-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))) ⟩ 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)) ) ≡⟨ ≡-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 )) ⟩ 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)) )) ≡⟨ ≡-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 ) ) ⟩ FMap F (λ x → proj₁ (proj₁ x) (proj₂ (proj₁ x) (proj₂ x))) (FMap F (Iso.≅← (mα-iso isM)) (φ (u , φ (v , w)))) ≡⟨⟩ 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)))) ≡⟨ left (sym ( IsFunctor.distr (isFunctor F ))) ⟩ 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))) ≡⟨⟩ FMap F (λ y → proj₁ y (proj₁ (proj₂ y) (proj₂ (proj₂ y)))) (φ (u , φ (v , w))) ≡⟨⟩ FMap F ( λ x → (proj₁ x) ((λ r → proj₁ r (proj₂ r)) ( proj₂ x))) ( φ ( u , (φ (v , w)))) ≡⟨ sym FφF→F ⟩ FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F id u , FMap F (λ r → proj₁ r (proj₂ r)) (φ (v , w)))) ≡⟨ ≡-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 ) ⟩ FMap F (λ r → proj₁ r (proj₂ r)) (φ (u , FMap F (λ r → proj₁ r (proj₂ r)) (φ (v , w)))) ∎ where open Relation.Binary.PropositionalEquality.≡-Reasoning homomorphism : { a b : Obj Sets } { f : Hom Sets a b } { x : a } → pure f <*> pure x ≡ pure (f x) homomorphism {a} {b} {f} {x} = begin pure f <*> pure x ≡⟨⟩ FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y → f) unit , FMap F (λ y → x) unit)) ≡⟨ FφF→F ⟩ FMap F ((λ r → proj₁ r (proj₂ r)) o map (λ y → f) (λ y → x)) (φ (unit , unit)) ≡⟨⟩ FMap F (λ y → f x) (φ (unit , unit)) ≡⟨ ≡-cong ( λ k → FMap F (λ y → f x) k ) φunitl ⟩ FMap F (λ y → f x) (FMap F (Iso.≅← (mρ-iso isM)) unit) ≡⟨⟩ FMap F (λ y → f x) (FMap F (λ y → (y , OneObj)) unit) ≡⟨ left ( sym ( IsFunctor.distr (isFunctor F ))) ⟩ FMap F (λ y → f x) unit ≡⟨⟩ pure (f x) ∎ where open Relation.Binary.PropositionalEquality.≡-Reasoning interchange : { a b : Obj Sets } { u : FObj F ( a → b ) } { x : a } → u <*> pure x ≡ pure (λ f → f x) <*> u interchange {a} {b} {u} {x} = begin u <*> pure x ≡⟨⟩ FMap F (λ r → proj₁ r (proj₂ r)) (φ (u , FMap F (λ y → x) unit)) ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (k , FMap F (λ y → x) unit)) ) u→F ⟩ FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F id u , FMap F (λ y → x) unit)) ≡⟨ FφF→F ⟩ FMap F ((λ r → proj₁ r (proj₂ r)) o map id (λ y → x)) (φ (u , unit)) ≡⟨⟩ FMap F (λ r → proj₁ r x) (φ (u , unit)) ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r x) k ) φunitl ⟩ FMap F (λ r → proj₁ r x) (( FMap F (Iso.≅← (mρ-iso isM))) u ) ≡⟨ left ( sym ( IsFunctor.distr (isFunctor F )) ) ⟩ FMap F (( λ r → proj₁ r x) o ((Iso.≅← (mρ-iso isM) ))) u ≡⟨⟩ FMap F (( λ r → proj₂ r x) o ((Iso.≅← (mλ-iso isM) ))) u ≡⟨ left ( IsFunctor.distr (isFunctor F )) ⟩ FMap F (λ r → proj₂ r x) (FMap F (Iso.≅← (IsMonoidal.mλ-iso isM)) u) ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₂ r x) k ) (sym φunitr ) ⟩ FMap F (λ r → proj₂ r x) (φ (unit , u)) ≡⟨⟩ FMap F ((λ r → proj₁ r (proj₂ r)) o map (λ y f → f x) id) (φ (unit , u)) ≡⟨ sym FφF→F ⟩ FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f → f x) unit , FMap F id u)) ≡⟨ ≡-cong ( λ k → FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f → f x) unit , k)) ) (sym u→F) ⟩ FMap F (λ r → proj₁ r (proj₂ r)) (φ (FMap F (λ y f → f x) unit , u)) ≡⟨⟩ pure (λ f → f x) <*> u ∎ where open Relation.Binary.PropositionalEquality.≡-Reasoning ---- -- -- Applicative functor implements Haskell Monoidal functor -- Haskell Monoidal functor is directly represents monoidal functor, it is easy to make it from a monoidal functor -- Applicative→HaskellMonoidal : {c₁ : Level} ( F : Functor (Sets {c₁}) (Sets {c₁}) ) ( App : Applicative F ) → HaskellMonoidalFunctor F Applicative→HaskellMonoidal {l} F App = record { unit = unit ; φ = φ ; isHaskellMonoidalFunctor = record { natφ = natφ ; assocφ = assocφ ; idrφ = idrφ ; idlφ = idlφ } } where pure = Applicative.pure App <*> = Applicative.<*> App app = Applicative.isApplicative App unit : FObj F One unit = pure OneObj φ : {a b : Obj Sets} → Hom Sets ((FObj F a) * (FObj F b )) ( FObj F ( a * b ) ) φ = λ x → <*> (FMap F (λ j k → (j , k)) ( proj₁ x)) ( proj₂ x) isM : IsMonoidal (Sets {l}) One SetsTensorProduct isM = Monoidal.isMonoidal MonoidalSets MF : MonoidalFunctor {_} {l} {_} {Sets} {Sets} MonoidalSets MonoidalSets MF = Applicative→Monoidal F App app isMF = MonoidalFunctor.isMonodailFunctor MF natφ : { a b c d : Obj Sets } { x : FObj F a} { y : FObj F b} { f : a → c } { g : b → d } → FMap F (map f g) (φ (x , y)) ≡ φ (FMap F f x , FMap F g y) natφ {a} {b} {c} {d} {x} {y} {f} {g} = begin FMap F (map f g) (φ (x , y)) ≡⟨⟩ FMap F (λ xy → f (proj₁ xy) , g (proj₂ xy)) (<*> (FMap F (λ j k → j , k) x) y) ≡⟨ ≡-cong ( λ h → h (x , y)) ( IsNTrans.commute ( NTrans.isNTrans ( IsMonoidalFunctor.φab isMF ))) ⟩ <*> (FMap F (λ j k → j , k) (FMap F f x)) (FMap F g y) ≡⟨⟩ φ (FMap F f x , FMap F g y) ∎ where open Relation.Binary.PropositionalEquality.≡-Reasoning assocφ : { x y z : Obj Sets } { a : FObj F x } { b : FObj F y }{ c : FObj F z } → φ (a , φ (b , c)) ≡ FMap F (Iso.≅→ (IsMonoidal.mα-iso isM)) (φ (φ (a , b) , c)) assocφ {x} {y} {z} {a} {b} {c} = ≡-cong ( λ h → h ((a , b) , c ) ) ( IsMonoidalFunctor.associativity isMF ) idrφ : {a : Obj Sets } { x : FObj F a } → FMap F (Iso.≅→ (IsMonoidal.mρ-iso isM)) (φ (x , unit)) ≡ x idrφ {a} {x} = ≡-cong ( λ h → h (x , OneObj ) ) ( IsMonoidalFunctor.unitarity-idr isMF {a} {One} ) idlφ : {a : Obj Sets } { x : FObj F a } → FMap F (Iso.≅→ (IsMonoidal.mλ-iso isM)) (φ (unit , x)) ≡ x idlφ {a} {x} = ≡-cong ( λ h → h (OneObj , x ) ) ( IsMonoidalFunctor.unitarity-idl isMF {One} {a} ) -- end