A Proof of Bel’tyukov–Lipshitz Theorem by Quasi-Quantifier Elimination. I. Definitions and GCD-Lemma

Abstract

This paper is the first part of a new proof of decidability of the existential theory of the structure 〈 $$\mathbb{Z}$$ ; 0, 1, +, –, ≤, |〉, where | corresponds to the binary divisibility relation. The decidability was proved independently in 1976 by A.P. Bel’tyukov and L. Lipshitz. In 1977, V.I. Mart’yanov proved an equivalent result by considering the ternary GCD predicate instead of divisibility (the predicates are interchangeable with respect to existential definability). Generalizing in some sense the notion of quantifier elimination (QE) algorithm, we construct a quasi-QE algorithm to prove decidability of the positive existential theory of the structure 〈 $${{\mathbb{Z}}_{{ > 0}}}$$ ; 1, $${{\{ a{\kern 1pt} \cdot \} }_{{a \in {{\mathbb{Z}}_{{ > 0}}}}}}$$ , GCD〉. We reduce to the decision problem for this theory the decision problem for the existential theory of the structure 〈 $$\mathbb{Z}$$ ; 0, 1, +, –, ≤, GCD〉. A quasi-QE algorithm, which performs this reduction, will be constructed in the second part of the proof. Transformations of formulas are based on a generalization of the Chinese remainder theorem to systems of the form GCD(ai, bi + x) = di for every $$i \in [1..m]$$ , where ai, bi, di are some integers such that $${{a}_{i}} \ne 0$$ , $${{d}_{i}} > 0$$ .