[Haskell-cafe] Confusion on the third monad law when using lambda abstractions

[wren ng thornton]

Hans van Thiel wrote:
> On Wed, 2009-06-17 at 21:26 -0500, Jake McArthur wrote:
>> Jon Strait wrote:
>>> I'm reading the third (bind associativity) law for monads in this form:
>>>
>>> m >>= (\x -> k x >>= h)  =  (m >>= k) >>= h
>> Arguably, that law would be better stated as:
>>
>>      (h <=< k) <=< m  =  h <=< (k <=< m)
>>
>> This wouldn't be so unintuitive.
> Hi, 
> The only place I've ever seen Kleisli composition, or its flip, used is
> in demonstrating the monad laws. Yet it is so elegant and, even having
> its own name, it must have some practical use. Do you, or anybody else,
> have some pointers?

     import Prelude hiding   (mapM)
     import Data.Traversable (mapM)
     import Control.Monad    ((<=<))
     newtype Fix f = Fix { unFix :: f (Fix f) }


     cata  phi  = phi   .   fmap (cata  phi)  . unFix

     cataM phiM = phiM <=< (mapM (cataM phiM) . unFix)


     ana   psi  =        Fix . fmap (ana  psi)    .  psi

     anaM  psiM = (liftM Fix . mapM (anaM psiM)) <=< psiM


etc. It's great for anyone who enjoys point-free style but wants to work 
with monads.

-- 
Live well,
~wren