Diophantine Sets. Part II

The article is the next in a series aiming to formalize the MDPR-theorem using the Mizar proof assistant [3], [6], [4]. We analyze four equations from the Diophantine standpoint that are crucial in the bounded quantifier theorem, that is used in one of the approaches to solve the problem.

Based on our previous work [1], we prove that the value of a given binomial coefficient and factorial can be determined by its arguments in a Diophantine way. Then we prove that two products

z=∏i=1x(1+i⋅y), z=∏i=1x(y+1-j), (0.1)$$z = \prod\limits_{i = 1}^x {\left( {1 + i \cdot y} \right),\,\,\,\,\,\,\,\,} z = \prod\limits_{i = 1}^x {\left( {y + 1 - j} \right),\,\,\,\,\,\,(0.1)} $$

where y > x are Diophantine.

The formalization follows [10], Z. Adamowicz, P. Zbierski [2] as well as M. Davis [5].