{-# LANGUAGE MultiParamTypeClasses, FlexibleInstances #-}
module Coalgebra where

import Control.Monad

-- Definitions for (co)algebras
class Functor f => Algebra f m where
    phi :: f m -> m
 
class Functor f => Coalgebra f m where
    psi :: m -> f m

-- Fixpoint, ie f (Mu f) = Mu f
-- unfortunatly we need a data constructor
data Mu f = In (f (Mu f))

-- The fixpoint is both a algebra and coalgebra,
-- because there is an arrow id: X -> X = FX, if X is a fixpoint
instance Functor f => Algebra f (Mu f) where
    phi = In
 
instance Functor f => Coalgebra f (Mu f) where
    psi (In x) = x