{-# LANGUAGE AllowAmbiguousTypes #-} module Proarrow.Object.Exponential where import Data.Kind (Type) import qualified Prelude as P import Proarrow.Category.Instance.Product ((:**:)(..)) import Proarrow.Category.Opposite (OPPOSITE(..), Op (..)) import Proarrow.Core (PRO, CategoryOf (..), Profunctor(..), Promonad(..), UN, (//), tgt) import Proarrow.Object (Obj, obj, src) import Proarrow.Profunctor.Representable (Representable(..), dimapRep) import Proarrow.Category.Monoidal (leftUnitor, associator, Monoidal(..), MonoidalProfunctor (..)) import Proarrow.Category.Instance.Prof (Prof(..)) import Proarrow.Category.Instance.Unit qualified as U import Proarrow.Object.BinaryProduct (PROD(..), Prod (..), Cartesian, HasBinaryProducts (..)) import Proarrow.Profunctor.Exponential ((:~>:) (..)) import Proarrow.Profunctor.Product ((:*:)(..)) infixr 2 ~~> class Monoidal k => Closed k where type (a :: k) ~~> (b :: k) :: k curry' :: Obj (a :: k) -> Obj b -> a ** b ~> c -> a ~> b ~~> c uncurry' :: Obj (b :: k) -> Obj c -> a ~> b ~~> c -> a ** b ~> c (^^^) :: (b :: k) ~> y -> x ~> a -> a ~~> b ~> x ~~> y curry :: forall {k} (a :: k) b c. (Closed k, Ob a, Ob b) => a ** b ~> c -> a ~> b ~~> c curry :: forall {k} (a :: k) (b :: k) (c :: k). (Closed k, Ob a, Ob b) => ((a ** b) ~> c) -> a ~> (b ~~> c) curry = Obj a -> Obj b -> ((a ** b) ~> c) -> a ~> (b ~~> c) forall (a :: k) (b :: k) (c :: k). Obj a -> Obj b -> ((a ** b) ~> c) -> a ~> (b ~~> c) forall k (a :: k) (b :: k) (c :: k). Closed k => Obj a -> Obj b -> ((a ** b) ~> c) -> a ~> (b ~~> c) curry' (forall (a :: k). (CategoryOf k, Ob a) => Obj a forall {k} (a :: k). (CategoryOf k, Ob a) => Obj a obj @a) (forall (a :: k). (CategoryOf k, Ob a) => Obj a forall {k} (a :: k). (CategoryOf k, Ob a) => Obj a obj @b) uncurry :: forall {k} (a :: k) b c. (Closed k, Ob b, Ob c) => a ~> b ~~> c -> a ** b ~> c uncurry :: forall {k} (a :: k) (b :: k) (c :: k). (Closed k, Ob b, Ob c) => (a ~> (b ~~> c)) -> (a ** b) ~> c uncurry = Obj b -> Obj c -> (a ~> (b ~~> c)) -> (a ** b) ~> c forall (b :: k) (c :: k) (a :: k). Obj b -> Obj c -> (a ~> (b ~~> c)) -> (a ** b) ~> c forall k (b :: k) (c :: k) (a :: k). Closed k => Obj b -> Obj c -> (a ~> (b ~~> c)) -> (a ** b) ~> c uncurry' (forall (a :: k). (CategoryOf k, Ob a) => Obj a forall {k} (a :: k). (CategoryOf k, Ob a) => Obj a obj @b) (forall (a :: k). (CategoryOf k, Ob a) => Obj a forall {k} (a :: k). (CategoryOf k, Ob a) => Obj a obj @c) comp :: forall {k} (a :: k) b c. (Closed k, Ob a, Ob b, Ob c) => (b ~~> c) ** (a ~~> b) ~> a ~~> c comp :: forall {k} (a :: k) (b :: k) (c :: k). (Closed k, Ob a, Ob b, Ob c) => ((b ~~> c) ** (a ~~> b)) ~> (a ~~> c) comp = let a :: Obj a a = forall (a :: k). (CategoryOf k, Ob a) => Obj a forall {k} (a :: k). (CategoryOf k, Ob a) => Obj a obj @a; b2c :: (b ~~> c) ~> (b ~~> c) b2c = forall (a :: k). (CategoryOf k, Ob a) => Obj a forall {k} (a :: k). (CategoryOf k, Ob a) => Obj a obj @c Obj c -> (b ~> b) -> (b ~~> c) ~> (b ~~> c) forall (b :: k) (y :: k) (x :: k) (a :: k). (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) forall k (b :: k) (y :: k) (x :: k) (a :: k). Closed k => (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) ^^^ forall (a :: k). (CategoryOf k, Ob a) => Obj a forall {k} (a :: k). (CategoryOf k, Ob a) => Obj a obj @b; a2b :: (a ~~> b) ~> (a ~~> b) a2b = forall (a :: k). (CategoryOf k, Ob a) => Obj a forall {k} (a :: k). (CategoryOf k, Ob a) => Obj a obj @b (b ~> b) -> Obj a -> (a ~~> b) ~> (a ~~> b) forall (b :: k) (y :: k) (x :: k) (a :: k). (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) forall k (b :: k) (y :: k) (x :: k) (a :: k). Closed k => (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) ^^^ Obj a a in forall (a :: k) (b :: k) (c :: k). (Closed k, Ob a, Ob b) => ((a ** b) ~> c) -> a ~> (b ~~> c) forall {k} (a :: k) (b :: k) (c :: k). (Closed k, Ob a, Ob b) => ((a ** b) ~> c) -> a ~> (b ~~> c) curry @_ @a @c (forall (a :: k) (b :: k). (Closed k, Ob a, Ob b) => ((a ~~> b) ** a) ~> b forall {k} (a :: k) (b :: k). (Closed k, Ob a, Ob b) => ((a ~~> b) ** a) ~> b eval @b @c (((b ~~> c) ** b) ~> c) -> ((((b ~~> c) ** (a ~~> b)) ** a) ~> ((b ~~> c) ** b)) -> (((b ~~> c) ** (a ~~> b)) ** a) ~> c forall (b :: k) (c :: k) (a :: k). (b ~> c) -> (a ~> b) -> a ~> c forall {k} (p :: PRO k k) (b :: k) (c :: k) (a :: k). Promonad p => p b c -> p a b -> p a c . ((b ~~> c) ~> (b ~~> c) b2c ((b ~~> c) ~> (b ~~> c)) -> (((a ~~> b) ** a) ~> b) -> ((b ~~> c) ** ((a ~~> b) ** a)) ~> ((b ~~> c) ** b) forall (a :: k) (b :: k) (c :: k) (d :: k). (a ~> b) -> (c ~> d) -> (a ** c) ~> (b ** d) forall k (a :: k) (b :: k) (c :: k) (d :: k). Monoidal k => (a ~> b) -> (c ~> d) -> (a ** c) ~> (b ** d) `par` forall (a :: k) (b :: k). (Closed k, Ob a, Ob b) => ((a ~~> b) ** a) ~> b forall {k} (a :: k) (b :: k). (Closed k, Ob a, Ob b) => ((a ~~> b) ** a) ~> b eval @a @b) (((b ~~> c) ** ((a ~~> b) ** a)) ~> ((b ~~> c) ** b)) -> ((((b ~~> c) ** (a ~~> b)) ** a) ~> ((b ~~> c) ** ((a ~~> b) ** a))) -> (((b ~~> c) ** (a ~~> b)) ** a) ~> ((b ~~> c) ** b) forall (b :: k) (c :: k) (a :: k). (b ~> c) -> (a ~> b) -> a ~> c forall {k} (p :: PRO k k) (b :: k) (c :: k) (a :: k). Promonad p => p b c -> p a b -> p a c . ((b ~~> c) ~> (b ~~> c)) -> ((a ~~> b) ~> (a ~~> b)) -> Obj a -> (((b ~~> c) ** (a ~~> b)) ** a) ~> ((b ~~> c) ** ((a ~~> b) ** a)) forall (a :: k) (b :: k) (c :: k). Obj a -> Obj b -> Obj c -> ((a ** b) ** c) ~> (a ** (b ** c)) forall k (a :: k) (b :: k) (c :: k). Monoidal k => Obj a -> Obj b -> Obj c -> ((a ** b) ** c) ~> (a ** (b ** c)) associator (b ~~> c) ~> (b ~~> c) b2c (a ~~> b) ~> (a ~~> b) a2b Obj a a) ((Ob (a ~~> b), Ob (a ~~> b)) => ((b ~~> c) ** (a ~~> b)) ~> (a ~~> c)) -> ((a ~~> b) ~> (a ~~> b)) -> ((b ~~> c) ** (a ~~> b)) ~> (a ~~> c) forall (a :: k) (b :: k) r. ((Ob a, Ob b) => r) -> (a ~> b) -> r forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) r. Profunctor p => ((Ob a, Ob b) => r) -> p a b -> r \\ (a ~~> b) ~> (a ~~> b) a2b ((Ob (b ~~> c), Ob (b ~~> c)) => ((b ~~> c) ** (a ~~> b)) ~> (a ~~> c)) -> ((b ~~> c) ~> (b ~~> c)) -> ((b ~~> c) ** (a ~~> b)) ~> (a ~~> c) forall (a :: k) (b :: k) r. ((Ob a, Ob b) => r) -> (a ~> b) -> r forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) r. Profunctor p => ((Ob a, Ob b) => r) -> p a b -> r \\ (b ~~> c) ~> (b ~~> c) b2c ((Ob ((b ~~> c) ** (a ~~> b)), Ob ((b ~~> c) ** (a ~~> b))) => ((b ~~> c) ** (a ~~> b)) ~> (a ~~> c)) -> (((b ~~> c) ** (a ~~> b)) ~> ((b ~~> c) ** (a ~~> b))) -> ((b ~~> c) ** (a ~~> b)) ~> (a ~~> c) forall (a :: k) (b :: k) r. ((Ob a, Ob b) => r) -> (a ~> b) -> r forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) r. Profunctor p => ((Ob a, Ob b) => r) -> p a b -> r \\ ((b ~~> c) ~> (b ~~> c) b2c ((b ~~> c) ~> (b ~~> c)) -> ((a ~~> b) ~> (a ~~> b)) -> ((b ~~> c) ** (a ~~> b)) ~> ((b ~~> c) ** (a ~~> b)) forall (a :: k) (b :: k) (c :: k) (d :: k). (a ~> b) -> (c ~> d) -> (a ** c) ~> (b ** d) forall k (a :: k) (b :: k) (c :: k) (d :: k). Monoidal k => (a ~> b) -> (c ~> d) -> (a ** c) ~> (b ** d) `par` (a ~~> b) ~> (a ~~> b) a2b) mkExponential :: forall {k} a b. Closed k => (a :: k) ~> b -> Unit ~> (a ~~> b) mkExponential :: forall {k} (a :: k) (b :: k). Closed k => (a ~> b) -> Unit ~> (a ~~> b) mkExponential a ~> b ab = forall (a :: k) (b :: k) (c :: k). (Closed k, Ob a, Ob b) => ((a ** b) ~> c) -> a ~> (b ~~> c) forall {k} (a :: k) (b :: k) (c :: k). (Closed k, Ob a, Ob b) => ((a ** b) ~> c) -> a ~> (b ~~> c) curry @_ @a (a ~> b ab (a ~> b) -> ((Unit ** a) ~> a) -> (Unit ** a) ~> b forall (b :: k) (c :: k) (a :: k). (b ~> c) -> (a ~> b) -> a ~> c forall {k} (p :: PRO k k) (b :: k) (c :: k) (a :: k). Promonad p => p b c -> p a b -> p a c . Obj a -> (Unit ** a) ~> a forall (a :: k). Obj a -> (Unit ** a) ~> a forall k (a :: k). Monoidal k => Obj a -> (Unit ** a) ~> a leftUnitor ((a ~> b) -> Obj a forall {k1} {k2} (a :: k2) (b :: k1) (p :: PRO k2 k1). Profunctor p => p a b -> Obj a src a ~> b ab)) ((Ob a, Ob b) => Unit ~> (a ~~> b)) -> (a ~> b) -> Unit ~> (a ~~> b) forall (a :: k) (b :: k) r. ((Ob a, Ob b) => r) -> (a ~> b) -> r forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) r. Profunctor p => ((Ob a, Ob b) => r) -> p a b -> r \\ a ~> b ab eval' :: Closed k => Obj a -> Obj b -> ((a :: k) ~~> b) ** a ~> b eval' :: forall k (a :: k) (b :: k). Closed k => Obj a -> Obj b -> ((a ~~> b) ** a) ~> b eval' Obj a a Obj b b = Obj a -> Obj b -> ((a ~~> b) ~> (a ~~> b)) -> ((a ~~> b) ** a) ~> b forall (b :: k) (c :: k) (a :: k). Obj b -> Obj c -> (a ~> (b ~~> c)) -> (a ** b) ~> c forall k (b :: k) (c :: k) (a :: k). Closed k => Obj b -> Obj c -> (a ~> (b ~~> c)) -> (a ** b) ~> c uncurry' Obj a a Obj b b (Obj b b Obj b -> Obj a -> (a ~~> b) ~> (a ~~> b) forall (b :: k) (y :: k) (x :: k) (a :: k). (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) forall k (b :: k) (y :: k) (x :: k) (a :: k). Closed k => (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) ^^^ Obj a a) eval :: forall {k} a b. (Closed k, Ob a, Ob b) => ((a :: k) ~~> b) ** a ~> b eval :: forall {k} (a :: k) (b :: k). (Closed k, Ob a, Ob b) => ((a ~~> b) ** a) ~> b eval = Obj a -> Obj b -> ((a ~~> b) ** a) ~> b forall k (a :: k) (b :: k). Closed k => Obj a -> Obj b -> ((a ~~> b) ** a) ~> b eval' (forall (a :: k). (CategoryOf k, Ob a) => Obj a forall {k} (a :: k). (CategoryOf k, Ob a) => Obj a obj @a) (forall (a :: k). (CategoryOf k, Ob a) => Obj a forall {k} (a :: k). (CategoryOf k, Ob a) => Obj a obj @b) instance Closed Type where type a ~~> b = a -> b curry' :: forall a b c. Obj a -> Obj b -> ((a ** b) ~> c) -> a ~> (b ~~> c) curry' Obj a _ Obj b _ = ((a ** b) ~> c) -> a ~> (b ~~> c) ((a, b) -> c) -> a -> b -> c forall a b c. ((a, b) -> c) -> a -> b -> c P.curry uncurry' :: forall b c a. Obj b -> Obj c -> (a ~> (b ~~> c)) -> (a ** b) ~> c uncurry' Obj b _ Obj c _ = (a ~> (b ~~> c)) -> (a ** b) ~> c (a -> b -> c) -> (a, b) -> c forall a b c. (a -> b -> c) -> (a, b) -> c P.uncurry ^^^ :: forall b y x a. (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) (^^^) = ((x -> a) -> (b -> y) -> (a -> b) -> x -> y) -> (b -> y) -> (x -> a) -> (a -> b) -> x -> y forall a b c. (a -> b -> c) -> b -> a -> c P.flip (x ~> a) -> (b ~> y) -> (a -> b) -> x -> y (x -> a) -> (b -> y) -> (a -> b) -> x -> y forall c a b d. (c ~> a) -> (b ~> d) -> (a -> b) -> c -> d forall {j} {k} (p :: PRO j k) (c :: j) (a :: j) (b :: k) (d :: k). Profunctor p => (c ~> a) -> (b ~> d) -> p a b -> p c d dimap instance Closed U.UNIT where type U.U ~~> U.U = U.U curry' :: forall (a :: UNIT) (b :: UNIT) (c :: UNIT). Obj a -> Obj b -> ((a ** b) ~> c) -> a ~> (b ~~> c) curry' Obj a Unit a a U.Unit Obj b Unit b b U.Unit (a ** b) ~> c Unit U c U.Unit = a ~> (b ~~> c) Unit U U U.Unit uncurry' :: forall (b :: UNIT) (c :: UNIT) (a :: UNIT). Obj b -> Obj c -> (a ~> (b ~~> c)) -> (a ** b) ~> c uncurry' Obj b Unit b b U.Unit Obj c Unit c c U.Unit a ~> (b ~~> c) Unit a U U.Unit = (a ** b) ~> c Unit U U U.Unit b ~> y Unit b y U.Unit ^^^ :: forall (b :: UNIT) (y :: UNIT) (x :: UNIT) (a :: UNIT). (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) ^^^ x ~> a Unit x a U.Unit = (a ~~> b) ~> (x ~~> y) Unit U U U.Unit instance (CategoryOf j, CategoryOf k) => Closed (PROD (PRO j k)) where type p ~~> q = PR (UN PR p :~>: UN PR q) curry' :: forall (a :: PROD (PRO j k)) (b :: PROD (PRO j k)) (c :: PROD (PRO j k)). Obj a -> Obj b -> ((a ** b) ~> c) -> a ~> (b ~~> c) curry' (Prod Prof{}) (Prod Prof{}) (Prod (Prof (UN 'PR a :*: UN 'PR b) :~> UN 'PR c n)) = (UN 'PR ('PR (UN 'PR a)) ~> UN 'PR ('PR (UN 'PR b :~>: UN 'PR c))) -> Prod ('PR (UN 'PR a)) ('PR (UN 'PR b :~>: UN 'PR c)) forall {k} (a :: PROD k) (b :: PROD k). (Ob a, Ob b) => (UN 'PR a ~> UN 'PR b) -> Prod a b Prod ((UN 'PR a :~> (UN 'PR b :~>: UN 'PR c)) -> Prof (UN 'PR a) (UN 'PR b :~>: UN 'PR c) forall {j} {k} (p :: PRO j k) (q :: PRO j k). (Profunctor p, Profunctor q) => (p :~> q) -> Prof p q Prof \UN 'PR a a b p -> UN 'PR a a b p UN 'PR a a b -> ((Ob a, Ob b) => (:~>:) (UN 'PR b) (UN 'PR c) a b) -> (:~>:) (UN 'PR b) (UN 'PR c) a b forall {k1} {k2} (p :: PRO k1 k2) (a :: k1) (b :: k2) r. Profunctor p => p a b -> ((Ob a, Ob b) => r) -> r // (forall (c :: j) (d :: k). (c ~> a) -> (b ~> d) -> UN 'PR b c d -> UN 'PR c c d) -> (:~>:) (UN 'PR b) (UN 'PR c) a b forall {k} {k1} (a :: k) (b :: k1) (p :: k -> k1 -> Type) (q :: k -> k1 -> Type). (Ob a, Ob b) => (forall (c :: k) (d :: k1). (c ~> a) -> (b ~> d) -> p c d -> q c d) -> (:~>:) p q a b Exp \c ~> a ca b ~> d bd UN 'PR b c d q -> (:*:) (UN 'PR a) (UN 'PR b) c d -> UN 'PR c c d (UN 'PR a :*: UN 'PR b) :~> UN 'PR c n ((c ~> a) -> (b ~> d) -> UN 'PR a a b -> UN 'PR a c d forall (c :: j) (a :: j) (b :: k) (d :: k). (c ~> a) -> (b ~> d) -> UN 'PR a a b -> UN 'PR a c d forall {j} {k} (p :: PRO j k) (c :: j) (a :: j) (b :: k) (d :: k). Profunctor p => (c ~> a) -> (b ~> d) -> p a b -> p c d dimap c ~> a ca b ~> d bd UN 'PR a a b p UN 'PR a c d -> UN 'PR b c d -> (:*:) (UN 'PR a) (UN 'PR b) c d forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) (q :: PRO j k). p a b -> q a b -> (:*:) p q a b :*: UN 'PR b c d q)) uncurry' :: forall (b :: PROD (PRO j k)) (c :: PROD (PRO j k)) (a :: PROD (PRO j k)). Obj b -> Obj c -> (a ~> (b ~~> c)) -> (a ** b) ~> c uncurry' (Prod Prof{}) (Prod Prof{}) (Prod (Prof UN 'PR a :~> (UN 'PR b :~>: UN 'PR c) n)) = (UN 'PR ('PR (UN 'PR a :*: UN 'PR b)) ~> UN 'PR ('PR (UN 'PR c))) -> Prod ('PR (UN 'PR a :*: UN 'PR b)) ('PR (UN 'PR c)) forall {k} (a :: PROD k) (b :: PROD k). (Ob a, Ob b) => (UN 'PR a ~> UN 'PR b) -> Prod a b Prod (((UN 'PR a :*: UN 'PR b) :~> UN 'PR c) -> Prof (UN 'PR a :*: UN 'PR b) (UN 'PR c) forall {j} {k} (p :: PRO j k) (q :: PRO j k). (Profunctor p, Profunctor q) => (p :~> q) -> Prof p q Prof \(UN 'PR a a b p :*: UN 'PR b a b q) -> case UN 'PR a a b -> (:~>:) (UN 'PR b) (UN 'PR c) a b UN 'PR a :~> (UN 'PR b :~>: UN 'PR c) n UN 'PR a a b p of Exp forall (c :: j) (d :: k). (c ~> a) -> (b ~> d) -> UN 'PR b c d -> UN 'PR c c d f -> (a ~> a) -> (b ~> b) -> UN 'PR b a b -> UN 'PR c a b forall (c :: j) (d :: k). (c ~> a) -> (b ~> d) -> UN 'PR b c d -> UN 'PR c c d f a ~> a forall (a :: j). Ob a => a ~> a forall {k} (p :: PRO k k) (a :: k). (Promonad p, Ob a) => p a a id b ~> b forall (a :: k). Ob a => a ~> a forall {k} (p :: PRO k k) (a :: k). (Promonad p, Ob a) => p a a id UN 'PR b a b q ((Ob a, Ob b) => UN 'PR c a b) -> UN 'PR b a b -> UN 'PR c a b forall (a :: j) (b :: k) r. ((Ob a, Ob b) => r) -> UN 'PR b a b -> r forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) r. Profunctor p => ((Ob a, Ob b) => r) -> p a b -> r \\ UN 'PR b a b q) Prod (Prof UN 'PR b :~> UN 'PR y m) ^^^ :: forall (b :: PROD (PRO j k)) (y :: PROD (PRO j k)) (x :: PROD (PRO j k)) (a :: PROD (PRO j k)). (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) ^^^ Prod (Prof UN 'PR x :~> UN 'PR a n) = (UN 'PR ('PR (UN 'PR a :~>: UN 'PR b)) ~> UN 'PR ('PR (UN 'PR x :~>: UN 'PR y))) -> Prod ('PR (UN 'PR a :~>: UN 'PR b)) ('PR (UN 'PR x :~>: UN 'PR y)) forall {k} (a :: PROD k) (b :: PROD k). (Ob a, Ob b) => (UN 'PR a ~> UN 'PR b) -> Prod a b Prod (((UN 'PR a :~>: UN 'PR b) :~> (UN 'PR x :~>: UN 'PR y)) -> Prof (UN 'PR a :~>: UN 'PR b) (UN 'PR x :~>: UN 'PR y) forall {j} {k} (p :: PRO j k) (q :: PRO j k). (Profunctor p, Profunctor q) => (p :~> q) -> Prof p q Prof \(Exp forall (c :: j) (d :: k). (c ~> a) -> (b ~> d) -> UN 'PR a c d -> UN 'PR b c d f) -> (forall (c :: j) (d :: k). (c ~> a) -> (b ~> d) -> UN 'PR x c d -> UN 'PR y c d) -> (:~>:) (UN 'PR x) (UN 'PR y) a b forall {k} {k1} (a :: k) (b :: k1) (p :: k -> k1 -> Type) (q :: k -> k1 -> Type). (Ob a, Ob b) => (forall (c :: k) (d :: k1). (c ~> a) -> (b ~> d) -> p c d -> q c d) -> (:~>:) p q a b Exp \c ~> a ca b ~> d bd UN 'PR x c d p -> UN 'PR b c d -> UN 'PR y c d UN 'PR b :~> UN 'PR y m ((c ~> a) -> (b ~> d) -> UN 'PR a c d -> UN 'PR b c d forall (c :: j) (d :: k). (c ~> a) -> (b ~> d) -> UN 'PR a c d -> UN 'PR b c d f c ~> a ca b ~> d bd (UN 'PR x c d -> UN 'PR a c d UN 'PR x :~> UN 'PR a n UN 'PR x c d p))) type ExponentialFunctor :: PRO k (OPPOSITE k, k) data ExponentialFunctor a b where ExponentialFunctor :: (Ob c, Ob d) => a ~> (c ~~> d) -> ExponentialFunctor a '(OP c, d) instance Closed k => Profunctor (ExponentialFunctor :: PRO k (OPPOSITE k, k)) where dimap :: forall (c :: k) (a :: k) (b :: (OPPOSITE k, k)) (d :: (OPPOSITE k, k)). (c ~> a) -> (b ~> d) -> ExponentialFunctor a b -> ExponentialFunctor c d dimap = (c ~> a) -> (b ~> d) -> ExponentialFunctor a b -> ExponentialFunctor c d forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) (c :: j) (d :: k). Representable p => (c ~> a) -> (b ~> d) -> p a b -> p c d dimapRep (Ob a, Ob b) => r r \\ :: forall (a :: k) (b :: (OPPOSITE k, k)) r. ((Ob a, Ob b) => r) -> ExponentialFunctor a b -> r \\ ExponentialFunctor a ~> (c ~~> d) f = r (Ob a, Ob (c ~~> d)) => r (Ob a, Ob b) => r r ((Ob a, Ob (c ~~> d)) => r) -> (a ~> (c ~~> d)) -> r forall (a :: k) (b :: k) r. ((Ob a, Ob b) => r) -> (a ~> b) -> r forall {j} {k} (p :: PRO j k) (a :: j) (b :: k) r. Profunctor p => ((Ob a, Ob b) => r) -> p a b -> r \\ a ~> (c ~~> d) f instance Closed k => Representable (ExponentialFunctor :: PRO k (OPPOSITE k, k)) where type ExponentialFunctor % '(OP a, b) = a ~~> b index :: forall (a :: k) (b :: (OPPOSITE k, k)). ExponentialFunctor a b -> a ~> (ExponentialFunctor % b) index (ExponentialFunctor a ~> (c ~~> d) f) = a ~> (ExponentialFunctor % b) a ~> (c ~~> d) f tabulate :: forall (b :: (OPPOSITE k, k)) (a :: k). Ob b => (a ~> (ExponentialFunctor % b)) -> ExponentialFunctor a b tabulate = (a ~> (ExponentialFunctor % b)) -> ExponentialFunctor a b (a ~> (UN 'OP (Fst b) ~~> Snd b)) -> ExponentialFunctor a '( 'OP (UN 'OP (Fst b)), Snd b) forall {k} (c :: k) (d :: k) (a :: k). (Ob c, Ob d) => (a ~> (c ~~> d)) -> ExponentialFunctor a '( 'OP c, d) ExponentialFunctor repMap :: forall (a :: (OPPOSITE k, k)) (b :: (OPPOSITE k, k)). (a ~> b) -> (ExponentialFunctor % a) ~> (ExponentialFunctor % b) repMap (Op b1 ~> a1 f :**: a2 ~> b2 g) = a2 ~> b2 g (a2 ~> b2) -> (b1 ~> a1) -> (a1 ~~> a2) ~> (b1 ~~> b2) forall (b :: k) (y :: k) (x :: k) (a :: k). (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) forall k (b :: k) (y :: k) (x :: k) (a :: k). Closed k => (b ~> y) -> (x ~> a) -> (a ~~> b) ~> (x ~~> y) ^^^ b1 ~> a1 f ap :: forall {j} {k} y a x p . (Cartesian j, Closed k, MonoidalProfunctor (p :: PRO j k), Ob y) => p a (x ~~> y) -> p a x -> p a y ap :: forall {j} {k} (y :: k) (a :: j) (x :: k) (p :: PRO j k). (Cartesian j, Closed k, MonoidalProfunctor p, Ob y) => p a (x ~~> y) -> p a x -> p a y ap p a (x ~~> y) pf p a x px = let a :: Obj a a = p a (x ~~> y) -> Obj a forall {k1} {k2} (a :: k2) (b :: k1) (p :: PRO k2 k1). Profunctor p => p a b -> Obj a src p a (x ~~> y) pf in (a ~> (a && a)) -> (((x ~~> y) ** x) ~> y) -> p (a && a) ((x ~~> y) ** x) -> p a y forall (c :: j) (a :: j) (b :: k) (d :: k). (c ~> a) -> (b ~> d) -> p a b -> p c d forall {j} {k} (p :: PRO j k) (c :: j) (a :: j) (b :: k) (d :: k). Profunctor p => (c ~> a) -> (b ~> d) -> p a b -> p c d dimap (Obj a a Obj a -> Obj a -> a ~> (a && a) forall (a :: j) (x :: j) (y :: j). (a ~> x) -> (a ~> y) -> a ~> (x && y) forall k (a :: k) (x :: k) (y :: k). HasBinaryProducts k => (a ~> x) -> (a ~> y) -> a ~> (x && y) &&& Obj a a) (Obj x -> Obj y -> ((x ~~> y) ** x) ~> y forall k (a :: k) (b :: k). Closed k => Obj a -> Obj b -> ((a ~~> b) ** a) ~> b eval' (p a x -> Obj x forall {k1} {k2} (a :: k2) (b :: k1) (p :: PRO k2 k1). Profunctor p => p a b -> Obj b tgt p a x px) (forall (a :: k). (CategoryOf k, Ob a) => Obj a forall {k} (a :: k). (CategoryOf k, Ob a) => Obj a obj @y)) (p a (x ~~> y) -> p a x -> p (a ** a) ((x ~~> y) ** x) forall (x1 :: j) (x2 :: k) (y1 :: j) (y2 :: k). p x1 x2 -> p y1 y2 -> p (x1 ** y1) (x2 ** y2) forall j k (p :: PRO j k) (x1 :: j) (x2 :: k) (y1 :: j) (y2 :: k). MonoidalProfunctor p => p x1 x2 -> p y1 y2 -> p (x1 ** y1) (x2 ** y2) lift2 p a (x ~~> y) pf p a x px)