{-# 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)