Basic Diophantine Relations

The main purpose of formalization is to prove that two equations ya(z)= y, y = xz are Diophantine. These equations are explored in the proof of Matiyasevich’s negative solution of Hilbert’s tenth problem.

In our previous work [6], we showed that from the diophantine standpoint these equations can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities. In this formalization, we express these relations in terms of Diophantine set introduced in [7]. We prove that these relations are Diophantine and then we prove several second-order theorems that provide the ability to combine Diophantine relation using conjunctions and alternatives as well as to substitute the right-hand side of a given Diophantine equality as an argument in a given Diophantine relation. Finally, we investigate the possibilities of our approach to prove that the two equations, being the main purpose of this formalization, are Diophantine.

The formalization by means of Mizar system [3], [2] follows Z. Adamowicz, P. Zbierski [1] as well as M. Davis [4].