<abstract xmlns="http://www.w3.org/1999/xhtml">This article is the final step of our attempts to formalize the negative solution of Hilbert’s tenth problem.In our approach, we work with the Pell’s Equation defined in [2]. We analyzed this equation in the general case to show its solvability as well as the cardinality and shape of all possible solutions. Then we focus on a special case of the equation, which has the form <italic>x</italic>2<italic>−</italic> (<italic>a</italic>2 − 1)<italic>y</italic>2 = 1 [8] and its solutions considered as two sequences <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forma-2019-0020_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mrow><mrow><mo>{</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup><mo>,</mo><msubsup><mrow><mrow><mrow><mo>{</mo><mrow><msub><mrow><mi>y</mi></mrow><mi>i</mi></msub><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></mrow></math><tex-math>$\left\{ {{x_i}(a)} \right\}_{i = 0}^\infty ,\left\{ {{y_i}(a)} \right\}_{i = 0}^\infty$</tex-math></alternatives></inline-formula>. We showed in [1] that the <italic>n</italic>-th element of these sequences can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities, or more precisely that the equation <italic>x</italic> = <italic>y</italic>i(<italic>a</italic>) is Diophantine. Following the post-Matiyasevich results we show that the equality determined by the value of the power function <italic>y</italic> = <italic>x</italic>z is Diophantine, and analogously property in cases of the binomial coe cient, factorial and several product [9].In this article, we combine analyzed so far Diophantine relation using conjunctions, alternatives as well as substitution to prove the bounded quantifier theorem. Based on this theorem we prove MDPR-theorem that <italic>every recursively enumerable set is Diophantine,</italic> where recursively enumerable sets have been defined by the Martin Davis normal form.The formalization by means of Mizar system [5], [7], [4] follows [10], Z. Adamowicz, P. Zbierski [3] as well as M. Davis [6].</abstract>

This article is the final step of our attempts to formalize the negative solution of Hilbert’s tenth problem.In our approach, we work with the Pell’s Equation defined in [2]. We analyzed this equation in the general case to show its solvability as well as the cardinality and shape of all possible solutions. Then we focus on a special case of the equation, which has the form x2− (a2 − 1)y2 = 1 [8] and its solutions considered as two sequences {xi(a)}i=0∞,{yi(a)}i=0∞$\left\{ {{x_i}(a)} \right\}_{i = 0}^\infty ,\left\{ {{y_i}(a)} \right\}_{i = 0}^\infty$. We showed in [1] that the n-th element of these sequences can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities, or more precisely that the equation x = yi(a) is Diophantine. Following the post-Matiyasevich results we show that the equality determined by the value of the power function y = xz is Diophantine, and analogously property in cases of the binomial coe cient, factorial and several product [9].In this article, we combine analyzed so far Diophantine relation using conjunctions, alternatives as well as substitution to prove the bounded quantifier theorem. Based on this theorem we prove MDPR-theorem that every recursively enumerable set is Diophantine, where recursively enumerable sets have been defined by the Martin Davis normal form.The formalization by means of Mizar system [5], [7], [4] follows [10], Z. Adamowicz, P. Zbierski [3] as well as M. Davis [6].

Formalization of the MRDP Theorem in the Mizar System

Formalized Mathematics

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{"article-title":"Formalization of the MRDP Theorem in the Mizar System"}

This article is the final step of our attempts to formalize the negative solution of Hilbert’s tenth problem.In our approach, we work with the Pell’s Equation...

Mizar系统中MRDP定理的形式化

在线发布:2019年7月20日

页面范围： 209 - 221

认可的:2019年5月27日

内政部： https://doi.org/10.2478/forma-2019-0020

关键词
希尔伯特的第十个问题,丢番图关系

©2019 Karol Pąk，由Sciendo出版

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