[Haskell-cafe] need help making sense of the type ...

[Brandon S. Allbery KF8NH]

On 2008 Sep 24, at 22:51, Daryoush Mehrtash wrote:
> I am having hard time making sense of the types in the following  
> example from the Applicative Programming paper: http://www.cs.nott.ac.uk/~ctm/IdiomLite.pdf
>
> ap :: Monad m ⇒ m (a → b ) → m a → m b
> ap mf mx = do
>     f ← mf
>     x ← mx
>     return (f x )
> Using this function we could rewrite sequence as:
>
> sequence :: [ IO a ] → IO [ a ]
> sequence [ ] = return [ ]
> sequence (c : cs ) = return (:) 'ap' c 'ap' sequence cs
>
>
> I am specifically confused over the type of "m" in:
>
>      return (:) 'ap' c
>
> "c" is obviously an  instance of IO a monad.   "return (:)"  on the  
> other hand (at least as I would expect it) is an instance of " ->"  
> monad.

Note that he first argument to ap is a function wrapped by a monad.   
"return (:)" wraps the function/operator (:) in an arbitrary monad  
(but then the type signature of sequence makes the monad IO).

     (:) :: a -> [a] -> [a]
     return (:) :: IO (a -> ([a] -> [a])) -- b is ([a] -> [a])
     ap (return (:)) :: IO a -> IO ([a] -> [a]) -- (IO a) is (m a) and  
(IO ([a] -> [a])) is (m b)
     return (:) `ap` c :: IO ([a] -> [a]) -- c is (IO a) per type  
signature
     return (:) `ap` c `ap` :: IO [a] -> IO [a] -- f in ap is the  
preceding IO ([a] -> [a])
     return (:) `ap` c `ap` sequence cs :: IO [a] -- (sequence cs) is  
(IO [a]) per type signature

-- 
brandon s. allbery [solaris,freebsd,perl,pugs,haskell] allbery at kf8nh.com
system administrator [openafs,heimdal,too many hats] allbery at ece.cmu.edu
electrical and computer engineering, carnegie mellon university    KF8NH


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