content: {-# LANGUAGE RankNTypes #-}
  {-# LANGUAGE TypeOperators #-}
  
  module Data.Profunctor.Strong where
  
  import Control.Arrow
  import Control.Category
  import Data.Tuple
  import Prelude hiding (id,(.))
  
  infixr 0 :->
  type p :-> q = forall a b. p a b -> q a b
  
  class Profunctor p where
    dimap :: (a -> b) -> (c -> d) -> p b c -> p a d
  
  class ProfunctorFunctor t where
    promap    :: Profunctor p => (p :-> q) -> t p :-> t q
  
  class ProfunctorFunctor t => ProfunctorMonad t where
    proreturn :: Profunctor p => p       :-> t p
    projoin   :: Profunctor p => t (t p) :-> t p
  
  class ProfunctorFunctor t => ProfunctorComonad t where
    proextract   :: Profunctor p => t p :-> p
    produplicate :: Profunctor p => t p :-> t (t p)
  
  class Profunctor p => Strong p where
    first' :: p a b  -> p (a, c) (b, c)
    first' = dimap swap swap . second'
  
    second' :: p a b -> p (c, a) (c, b)
    second' = dimap swap swap . first'
  
  ----------------------------------------------------------------------------
  
  newtype Tambara p a b = Tambara { runTambara :: forall c. p (a, c) (b, c) }
  
  instance Profunctor p => Profunctor (Tambara p) where
    dimap f g (Tambara p) = Tambara $ dimap (first f) (first g) p
  
  instance ProfunctorFunctor Tambara where
    promap f (Tambara p) = Tambara (f p)
  
  instance ProfunctorComonad Tambara where
    proextract (Tambara p) = dimap (\a -> (a,())) fst p
  
    produplicate (Tambara p) = Tambara (Tambara $ dimap hither yon p)
      where
        hither :: ((a, b), c) -> (a, (b, c))
        hither ~(~(x,y),z) = (x,(y,z))
  
        yon    :: (a, (b, c)) -> ((a, b), c)
        yon    ~(x,~(y,z)) = ((x,y),z)
  
  instance Profunctor p => Strong (Tambara p) where
    first' = (\x -> runTambara x) . produplicate
    -- Simple subsumption (#17775) requires eta expansion here
  