module Proarrow.Profunctor.Coyoneda where

import Data.Kind (Type)

import Proarrow.Core (PRO, CategoryOf(..), Promonad(..), Profunctor(..))

type Coyoneda :: (j -> k -> Type) -> PRO j k
data Coyoneda p a b where
  Coyoneda :: (a ~> c) -> (d ~> b) -> p c d -> Coyoneda p a b

instance (CategoryOf j, CategoryOf k) => Profunctor (Coyoneda (p :: j -> k -> Type)) where
  dimap :: forall (c :: j) (a :: j) (b :: k) (d :: k).
(c ~> a) -> (b ~> d) -> Coyoneda p a b -> Coyoneda p c d
dimap c ~> a
l b ~> d
r (Coyoneda a ~> c
f d ~> b
g p c d
p) = (c ~> c) -> (d ~> d) -> p c d -> Coyoneda p c d
forall {k} {k} (a :: k) (c :: k) (d :: k) (b :: k)
       (p :: k -> k -> Type).
(a ~> c) -> (d ~> b) -> p c d -> Coyoneda p a b
Coyoneda (a ~> c
f (a ~> c) -> (c ~> a) -> c ~> c
forall (b :: j) (c :: j) (a :: j). (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
. c ~> a
l) (b ~> d
r (b ~> d) -> (d ~> b) -> d ~> d
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
. d ~> b
g) p c d
p
  (Ob a, Ob b) => r
r \\ :: forall (a :: j) (b :: k) r.
((Ob a, Ob b) => r) -> Coyoneda p a b -> r
\\ Coyoneda a ~> c
f d ~> b
g p c d
_ = r
(Ob a, Ob c) => r
(Ob a, Ob b) => r
r ((Ob a, Ob c) => r) -> (a ~> c) -> r
forall (a :: j) (b :: j) 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
f ((Ob d, Ob b) => r) -> (d ~> b) -> 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
\\ d ~> b
g

coyoneda :: (CategoryOf j, CategoryOf k, Ob a, Ob b) => p a b -> Coyoneda (p :: j -> k -> Type) a b
coyoneda :: forall j k (a :: j) (b :: k) (p :: j -> k -> Type).
(CategoryOf j, CategoryOf k, Ob a, Ob b) =>
p a b -> Coyoneda p a b
coyoneda = (a ~> a) -> (b ~> b) -> p a b -> Coyoneda p a b
forall {k} {k} (a :: k) (c :: k) (d :: k) (b :: k)
       (p :: k -> k -> Type).
(a ~> c) -> (d ~> b) -> p c d -> Coyoneda p a b
Coyoneda 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