Some simple things to help me with my homework and to play around with Haskell.
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{-# 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