Recursive.hs

data CoalgT f a b = CoalgT (forall z. f b z -> a -> z)
--   a -> ExprF b
-- ~ a -> (forall z. (ExprF b -> z) -> z)
-- ~ a -> (forall z. ExprA b z -> z)
-- ~ CoalgT ExprA a b

data PatternF f a = PatternF (forall z. f a z -> z)
--   ExprF a
-- ~ forall z. (ExprF a -> z) -> z
-- ~ forall z. ExprA a z -> z

instance Profunctor f => Functor (PatternF f) where
  fmap f (PatternF run) = PatternF (\alg -> run (dimap f id alg))

instance Profunctor f => Profunctor (CoalgT f) where
  dimap l r (CoalgT run) = CoalgT $ \alg -> \a -> run (dimap r id alg) (l a)

class Profunctor f => Recursive t f | t -> f where
  embed   :: f t t
  cata    :: forall z. f z z -> t -> z
  project :: forall z. f t z -> t -> z
  
  embedF :: PatternF f t -> t
  embedF (PatternF run) = run embed
  
  patternF :: forall a b. f a b -> (PatternF f a -> b)
  patternF alg = \(PatternF run) -> run alg
  
  unPatternF :: forall a b. (PatternF f a -> b) -> f a b
  unPatternF alg = undefined