Nicolas Frisby wrote:

My question is: Given products and a fixed point combinator, can any pure expression be transformed into a corresponding expression that has just a single use of fix?

Albert Y. C. Lai pointed out model-theoretical and CPU-practical answers. There is also a Haskell-automatic answer. That is, polyvariadic fixpoint, a combinator to obtain a mutually least fixpoint of several terms, is expressible in Haskell itself, via the regular fix: http://okmij.org/ftp/Computation/fixed-point-combinators.html#Poly-variadic I like its inferred type fix':: [[a->b]->a->b] -> [a->b] which is truly the type of the (applicative, or eta-expanded) regular fix with some round parentheses replaced with square ones.