We studied formulas of elementary number theory resulting from formulas of Presburger arithmetic PrA (additive elementary theory of integers with order) by substituting for some variables, polynomials and integer values of rational functions in a single new variable y, and quantifying over y. We show that the extension ALA (almost linear arithmetic) of PrA obtained in this way, has essentially the same upper and lower complexity bounds as the original theory. The same applies to the fragments of ALA obtained by restricting the number or type of quantifiers in formulas. We also show new upper complexity bounds for quantifier elimination in PrA and its fragments. The results form a common extension of known complexity bounds for PrA and for the existential almost linear problems studied by Gurari & Ibarra. The method is applicable also to other extensions of PrA without order, related to the famous bounds for binary forms of A. Baker.
