-- | A 'Cocone' is a list of arrows sharing a single target (the coapex), as a profunctor from lists of
-- objects to objects; a 'Sink' is a cocone with the coapex hidden existentially.
module Proarrow.Profunctor.Instance.Cocone where

import Proarrow.Category.Monoidal (MonoidalProfunctor (..))
import Proarrow.Colimit.BinaryCoproduct (COPROD (..), Coprod (..), HasBinaryCoproducts (..), HasCoproducts)
import Proarrow.Core (CategoryOf (..), Profunctor (..), Promonad (..), UN, rmap, type (+->))
import Proarrow.Profunctor.Instance.List (LIST (..), List (..))

-- | A cocone is a bunch of arrows with a shared target.
data Cocone (bs :: LIST k) (a :: COPROD k) where
  Coapex :: (Ob a) => Cocone (L '[]) (COPR a)
  Coleg :: b ~> a -> Cocone (L bs) (COPR a) -> Cocone (L (b : bs)) (COPR a)

instance (CategoryOf k) => Profunctor (Cocone :: COPROD k +-> LIST k) where
  dimap Nil r Coapex = Coapex \\ r
  dimap (Cons l ls) r@(Coprod r') (Coleg f fs) = Coleg (r' . f . l) (dimap ls r fs)
  r \\ Coapex = r
  r \\ Coleg l Coapex = r \\ l
  r \\ Coleg l c@(Coleg _ c1) = r \\ l \\ c \\ c1

instance (HasCoproducts k) => MonoidalProfunctor (Cocone :: COPROD k +-> LIST k) where
  one = Coapex
  Coapex @l ** rs = rmap (Coprod (rgt @_ @l)) rs \\ rs
  Coleg l ls ** (rs :: Cocone rs r) = Coleg (lft @_ @_ @(UN COPR r) . l) (ls ** rs) \\ l \\ rs

-- | A sink is a cocone, but with the apex type hidden by an existential.
data Sink (as :: [k]) where
  Cocone :: Cocone (L as) (COPR a) -> Sink as