Date: Tue, 27 Mar 2012 11:03:54 +1300 From: "Richard O'Keefe" Subject: Re: [Haskell-cafe] adding the elements of two lists To: Cc: haskell-cafe@haskell.org

And *that* is why I stopped trying to define instance Num t => Num [t]. If "I KNEW that the length of the lists is ... fixed ..." then the type wasn't *really* [t], but some abstract type that happened to be implemented as [t], and that abstract type deserved a newtype name of its own.

Naming the type - makes the author's intent clearer to readers - lets the compiler check it's used consistently - lets you have instances that don't match instances for other abstract types that happen to have the same implementation - provides a natural place to document the purpose of the type - gives you a way to enforce the intended restrictions all for zero run-time overhead.

Quite taken by this manifesto for high style and morality, I resolved to do right by some old code, of which I had been quite proud: www.cs.dartmouth.edu/~doug/powser.html. Sadly, the exercise took some bloom off the rose. What the code gained in honesty and safety, it lost in beauty and readability. Here's the contrast, seen in overloading arithmetic to handle addition and multiplication of power series. --------- without newtype toSeries f = f : repeat 0 -- coerce scalar to series instance Num a => Num [a] where (f:fs) + (g:gs) = f+g : fs+gs (f:fs') * gs@(g:gs') = f*g : fs'*gs + (toSeries f)*gs' --------- with newtype newtype Num a => PS a = PS [a] deriving (Show, Eq) fromPS (PS fs) = fs -- extract list toPS f = PS (f : repeat 0) -- coerce scalar to series instance Num a => Num (PS a) where (PS (f:fs)) + (PS (g:gs)) = PS (f+g : fs+gs) (PS (f:fs)) * gPS@(PS (g:gs)) = PS $ f*g : fromPS ((PS fs)*gPS + (toPS f)*(PS gs)) The code suffers a pox of PS. When one goes on to introduce efficiencies and generalize to handle finite series (polynomials), it only gets worse. One unpleasant technical problem: the deriving clause is almost useless--one can't print or detect equality of infinite objects. For output one must apply something like (take 10 . fromPS). Yet this modest package would likely pose problems if it were combined with others in a grander suite for automated mathematics. What to do? I think I might write the package without newtype, and then wrap it in PS for export in hopes of exploiting the advantages of both styles.