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™: Keep your life in sync. Check it out.