{-# LANGUAGE MultiParamTypeClasses, FlexibleInstances, FlexibleContexts #-}

import Control.Monad.Instances
import Coalgebra

-- F X = 1 + X
type F = Maybe

-- This will give the fixpoint, ie a coalgebra, because F is a functor
type Natinfi = Nu F

-- The semantics from the following coalgebra to Natinfi is "the selection function" (I hope)
-- In some sense this behaviour searches through all natural numbers
instance Coalgebra F (Natinfi -> Bool) where
	psi p
		| p (phi Nothing) == False = Nothing
		| otherwise = Just (\x -> p $ phi $ Just x)

-- On "numbers" bigger that one, return True
test :: Natinfi -> Bool
test p = case q of
		Nothing -> True
		Just y -> False
	where q = psi p

-- Of course this will not always terminate!
toInt :: Natinfi -> Int
toInt s = case q of
		Nothing -> 0
		Just y -> 1 + toInt y
	where q = psi s

main :: IO ()
main = do
	putStrLn $ show $ toInt $ (semantics test :: Natinfi)