A Proof of Bel’tyukov–Lipshitz Theorem by Quasi-Quantifier Elimination. II. The Main Reduction

This paper is the second part of a new proof Bel’tyukov–Lipshitz theorem, which states that existential theory structure $$\left\langle {\mathbb{Z};0,1, + , - \leqslant ,|{\kern 1pt} } \right\rangle $$ decidable. We construct quasi-quantifier elimination algorithm (the notion was introduced in first proof) to reduce decision problem for ,{\text{GCD}}} positive {{{\mathbb{Z}}_{{ > 0}}};1,{{{\{ a{\kern \cdot {\kern \} }}_{{a \in {{\mathbb{Z}}_{{ 0}}}}}},{\text{GCD}}} . Since latter proved decidable part, this reduction completes theorem. Analogues two lemmas Lipshitz’s are used step variable isolation quasi-elimination. In quasi-elimination we apply GCD-lemma, part.