  We define a class of pre-ordered abelian groups that we call
finite-by-Presburger groups, and prove that their theory is model-complete. We
show that certain quotients of the multiplicative group of a local field of
characteristic zero are finite-by-Presburger and interpret the higher residue
rings of the local field. We apply these results to give a new proof of the
model completeness in the ring language of a local field of characteristic zero
(a result that follows also from work of Prestel-Roquette).
