First simple thing of coalgebra

Coalgebra.hs

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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

Streams.hs

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+{-# LANGUAGE MultiParamTypeClasses, FlexibleInstances, FlexibleContexts #-}
+
+import Control.Monad.Instances
+import Coalgebra
+
+-- F X = A x X, for a fixed A, The prelude already provides the functor instance :)
+type F a = (,) a
+
+-- This will give the fixpoint, ie a coalgebra.
+type Stream a = Mu (F a)
+
+(+:+) :: a -> Stream a -> Stream a
+(+:+) a s = In (a, s)
+
+-- For every other (F a)-coalgebra x, there is a arrow x -> Stream a
+-- and it is unique, so `Stream a` is the final (F a)-coalgebra!
+sem :: (Coalgebra (F a) x) => x -> Stream a
+sem x = x0 +:+ sem x' where (x0, x') = psi x
+
+-- auxilary functions
+toList :: Stream a -> [a]
+toList s = a0 : toList a' where (a0, a') = psi s
+
+
+-- example, with a very simple (F Int)-coalgebra, 1 -> 2, 2 -> 3, 3 -> 2
+data X = One | Two | Three
+instance Coalgebra (F Int) X where
+	psi One = (1, Two)
+	psi Two = (2, Three)
+	psi Three = (3, Two)
+
+main :: IO ()
+main = do
+	putStrLn $ show $ take 20 $ toList $ (sem One :: Stream Int)