Max Bolingbroke batterseapower at hotmail.com
Sun Jun 27 06:25:42 EDT 2010

By the way, you can use this stuff to solve the restricted monad
problem (e.g. make Set an instance of Monad). This is not that useful
until we find out what the mother of all MonadPlus is, though, because
we really need a MonadPlus Set instance.

Code below.

Cheers,
Max

\begin{code}
{-# LANGUAGE RankNTypes #-}
import Control.Applicative

import Data.Set (Set)
import qualified Data.Set as S

newtype CodensityOrd m a = CodensityOrd { runCodensityOrd :: forall b.
Ord b => (a -> m b) -> m b }

-- liftCodensityOrd :: Monad m => m a -> CodensityOrd m a
-- liftCodensityOrd m = CodensityOrd ((>>=) m)

-- lowerCodensityOrd :: (Ord a, Monad m) => CodensityOrd m a -> m a
-- lowerCodensityOrd m = runCodensityOrd m return

instance Functor (CodensityOrd f) where
fmap f m = CodensityOrd (\k -> runCodensityOrd m (k . f))

instance Applicative (CodensityOrd f) where
pure x = CodensityOrd (\k -> k x)
mf <*> mx = CodensityOrd (\k -> runCodensityOrd mf (\f ->
runCodensityOrd mx (\x -> k (f x))))

return = pure
m >>= k = CodensityOrd (\c -> runCodensityOrd m (\a ->
runCodensityOrd (k a) c))

liftSet :: Ord a => Set a -> CodensityOrd Set a
liftSet m = CodensityOrd (bind m)
where bind :: (Ord a, Ord b) => Set a -> (a -> Set b) -> Set b
mx bind fxmy = S.fold (\x my -> fxmy x S.union my) S.empty mx

lowerSet :: Ord a => CodensityOrd Set a -> Set a
lowerSet m = runCodensityOrd m S.singleton

main = print $lowerSet$ monadicPlus (liftSet $S.fromList [1, 2, 3]) (liftSet$ S.fromList [1, 2, 3])

monadicPlus :: Monad m => m Int -> m Int -> m Int