biappmon.hs

class (Bifunctor f) => Biapplicative f where
  bipure :: a -> b -> f a b
  biap :: f (a -> b) (c -> d) -> f a c -> f b d

class (Bifunctor l, Bifunctor r) => Bimonad l r where
  bireturnl :: a -> l a b
  bireturnr :: b -> r a b
  bijoinl :: l (l a b) (r a b) -> l a b
  bijoinr :: r (l a b) (r a b) -> r a b

-- m >>= f = join (fmap f m)

bibindl :: l a b -> (a -> l c d) -> (b -> r c d) -> l c d
bibindl lab l r = bijoinl (bimap l r lab)
bibindr :: r a b -> (a -> l c d) -> (b -> r c d) -> r c d
bibindr rab l r = bijoinr (bimap l r rab)

-- the above definition of Bimonad necessitates that the Bimonad is an endofunctor (B : Hask -> Hask)
-- relative monads do not have this requirement.
class RelativeBimonad j m where
  bireturn :: j a b -> m a b
  bibind :: m a b -> (j a b -> m c d) -> m c d

--laws:
--bireturn jab `bibind` k = k jab
--m `bibind` bireturn = m
--m `bibind` (\jab -> k jab `bibind` h) = (m `bibind` k) `bibind` h