Mixed real-integer linear quantifier elimination

Consider t,hc elementary t.heory T of the real numl~ers in the language L having 0,l as constants, addition and subtract,ion and integer part, as operations; and cqualit~-: order and congruences module natural number constants as relations. We show that. T admits au effective quamifier elimination procedure and is decidable. Moreover this procedure provides sample answers for existentially quantified variables. The procedure comprises as special cases linear climinat:ion for the reals and for Prcsburger arithmet~ic. ~VC provide closely matching upper aud lower bounds for the complesity of the quantifier elimination and decision problem for T. .~pplicat.ions include a cliaracterixation of T-definable subs& of the real line, and the modeling of parametric mixed int,eger linear optimization, of continuous phenomena with periodici+; and the simulatmn and analysis of hybrid control systems. IVe also consitlcr the elementary theory of reals in varations of this langtmge in view of quantifier elimination and decidability and provide positive and ncgat,ive results for various variaIlt lauguages.