[Haskell-cafe] Naturality condition for inits

[R J]

Here's another Bird problem that's stymied me:

The function "inits" computes the list of initial segments of a list; its type is inits :: [a] -> [[a]].  What is the appropriate naturality condition for "inits"?

The only discussion in the text concerning naturality conditions concerns map, where the naturality conditions are stated in what seem to be quasi-commutativity laws over the composition operator, as follows:

   f . head            =  head . map f, where f is strict (i.e., f _|_ = _|_)
   map f . tail        =  tail . map f
   map f (xs ++ ys)    =  map f xs ++ map f ys
   map f . reverse     =  reverse . map f
   map f . concat      =  concat . map (map f)

I believe that none of the following naturality conditions, extrapolated from those above, hold:

   a. head . inits     =  inits [head]
   b. tail . inits     =  inits . tail    
   c. reverse . inits  =  inits . reverse
   d. concat . inits   =  inits . concat

In case the definition of "inits" is relevant, my definition is:



inits                  :: [a] -> [[a]]

inits xs               =  [take n xs | n <- seglengths]

                          where

                              seglengths = [0..length xs]

Thanks.

_________________________________________________________________
Windows Live™: Life without walls.
http://windowslive.com/explore?ocid=TXT_TAGLM_WL_allup_1a_explore_032009
-------------- next part --------------
An HTML attachment was scrubbed...
URL: http://www.haskell.org/pipermail/haskell-cafe/attachments/20090307/76e7d3c5/attachment.htm