  We generalize Cooper's method of quantifier elimination for classical
Presburger arithmetic to give a new proof that all parametric Presburger
families (as defined by Kevin Woods) are definable by formulas with
polynomially bounded quantifiers in an expanded language with predicates for
divisibility by f(t) for every polynomial f over the integers. In fact, this
quantifier bounding method works more generally in expansions of Presburger
arithmetic with multiplication by scalars from any ring of functions into Z.
