Quantifier elimination for the theory of algebraically closed valued fields with analytic structure

The theory of algebraically closed non-Archimedean valued fields is proved to eliminate quantifiers in an analytic language similar to the one used by Cluckers, Lipshitz and Robinson. The proof makes use of a uniform parameterized normalization theorem which is also proved in this paper and which has far reaching consequences in the geometry of definable sets. This method of proving quantifier elimination in an analytic language does not require the algebraic quantifier elimination theorem of Weispfenning unlike the customary method of proof used in similar earlier analytic quantifier elimination theorems.