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))))
instance Monad (CodensityOrd f) where
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
monadicPlus mx my = do
x <- mx
y <- my
return (x + y)
\end{code}
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