6 research outputs found
Formalization of the MRDP Theorem in the Mizar System
This article is the final step of our attempts to formalize the negative solution of Hilbertâs tenth problem.In our approach, we work with the Pellâs 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 â (a2 â 1)y2 = 1 [8] and its solutions considered as two sequences {xi(a)}i=0â,{yi(a)}i=0â. 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].Institute of Informatics, University of BiaĆystok, PolandMarcin Acewicz and Karol PÄ
k. Basic Diophantine relations. Formalized Mathematics, 26(2):175â181, 2018. doi:10.2478/forma-2018-0015.Marcin Acewicz and Karol PÄ
k. Pellâs equation. Formalized Mathematics, 25(3):197â204, 2017. doi:10.1515/forma-2017-0019.Zofia Adamowicz and PaweĆ Zbierski. Logic of Mathematics: A Modern Course of Classical Logic. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. Wiley-Interscience, 1997.Grzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, Karol PÄ
k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261â279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, and Karol PÄ
k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9â32, 2018. doi:10.1007/s10817-017-9440-6.Martin Davis. Hilbertâs tenth problem is unsolvable. The American Mathematical Monthly, Mathematical Association of America, 80(3):233â269, 1973. doi:10.2307/2318447.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191â198, 2015. doi:10.1007/s10817-015-9345-1.Karol PÄ
k. The Matiyasevich theorem. Preliminaries. Formalized Mathematics, 25(4): 315â322, 2017. doi:10.1515/forma-2017-0029.Karol PÄ
k. Diophantine sets. Part II. Formalized Mathematics, 27(2):197â208, 2019. doi:10.2478/forma-2019-0019.Craig Alan Smorynski. Logical Number Theory I, An Introduction. Universitext. Springer-Verlag Berlin Heidelberg, 1991. ISBN 978-3-642-75462-3.27220922
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 productsz=âi=1x(1+iâ
y),ââââââââz=âi=1x(y+1-j),ââââââ(0.1)where y > x are Diophantine.The formalization follows [10], Z. Adamowicz, P. Zbierski [2] as well as M. Davis [5].Institute of Informatics, University of BiaĆystok, PolandMarcin Acewicz and Karol PÄ
k. Basic Diophantine relations. Formalized Mathematics, 26(2):175â181, 2018. doi:10.2478/forma-2018-0015.Zofia Adamowicz and PaweĆ Zbierski. Logic of Mathematics: A Modern Course of Classical Logic. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. Wiley-Interscience, 1997.Grzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, Karol PÄ
k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261â279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, and Karol PÄ
k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9â32, 2018. doi:10.1007/s10817-017-9440-6.Martin Davis. Hilbertâs tenth problem is unsolvable. The American Mathematical Monthly, Mathematical Association of America, 80(3):233â269, 1973. doi:10.2307/2318447.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191â198, 2015. doi:10.1007/s10817-015-9345-1.Artur KorniĆowicz and Karol PÄ
k. Basel problem â preliminaries. Formalized Mathematics, 25(2):141â147, 2017. doi:10.1515/forma-2017-0013.Xiquan Liang, Li Yan, and Junjie Zhao. Linear congruence relation and complete residue systems. Formalized Mathematics, 15(4):181â187, 2007. doi:10.2478/v10037-007-0022-7.Karol PÄ
k. Diophantine sets. Preliminaries. Formalized Mathematics, 26(1):81â90, 2018. doi:10.2478/forma-2018-0007.Craig Alan Smorynski. Logical Number Theory I, An Introduction. Universitext. Springer-Verlag Berlin Heidelberg, 1991. ISBN 978-3-642-75462-3.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825â829, 2001.RafaĆ Ziobro. On subnomials. Formalized Mathematics, 24(4):261â273, 2016. doi:10.1515/forma-2016-0022.27219720