Elementary bounds for presburger arithmetic

We consider the first-order theory whose language has as nonlogical symbols the constant symbols 0 and 1, the binary relation symbols = and <, the unary function symbol − and the binary function symbol + This theory of integers under addition is commonly called the 'Presburger Arithmetic' and is known to be decidable for truth [Presburger (1929), Hilbert and Bernays (1968)]. We prove here that there exists a decision procedure for this theory, involving quantifier elimination, for which there is a superexponential upper bound on the size of formula produced when all variables have been eliminated.