Model Completeness of O-Minimal Fields with Convex Valuations

We let R be an o-minimal expansion of a field, V a convex subring, and (R0, V0) an elementary substructure of (R, V ). We let L be the language consisting of a language for R, in which R has elimination of quantifiers, and a predicate for V , and we let LR0 be the language L expanded by constants for all elements of R0. Our main result is that (R, V ) considered as an LR0-structure is model complete provided that kR, the corresponding residue field with structure induced from R, is o-minimal. Along the way we show that o-minimality of kR implies that the sets definable in kR are the same as the sets definable in k with structure induced from (R, V ). We also give a criterion for a superstructure of (R, V ) being an elementary extension of (R, V ).