On Subnomials

While discussing the sum of consecutive powers as a result of division of two binomials W.W. Sawyer [12] observes “It is a curious fact that most algebra textbooks give our ast result twice. It appears in two different chapters and usually there is no mention in either of these that it also occurs in the other. The first chapter, of course, is that on factors. The second is that on geometrical progressions. Geometrical progressions are involved in nearly all financial questions involving compound interest – mortgages, annuities, etc.” It’s worth noticing that the first issue involves a simple arithmetical division of 99...9 by 9. While the above notion seems not have changed over the last 50 years, it reflects only a special case of a broader class of problems involving two variables. It seems strange, that while binomial formula is discussed and studied widely [7], [8], little research is done on its counterpart with all coefficients equal to one, which we will call here the subnomial. The study focuses on its basic properties and applies it to some simple problems usually proven by induction [6].Department of Carbohydrate Technology, University of Agriculture, Krakow, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Czesław Byliński. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241–245, 1996.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Wenpai Chang, Hiroshi Yamazaki, and Yatsuka Nakamura. The inner product and conjugate of finite sequences of complex numbers. Formalized Mathematics, 13(3):367–373, 2005.Jacek Gancarzewicz. Arytmetyka. Wydawnictwo UJ, Kraków, 2000. In Polish.Cao Hui-Qin and Pan Hao. Factors of alternating binomial sums. Advances in Applied Mathematics, 45(1):96 – 107, 2010. doi:http://dx.doi.org/10.1016/j.aam.2009.09.004.Sudesh K. Khanduja, Ramneek Khassa, and Shanta Laishram. Some irreducibility results for truncated binomial expansions. Journal of Number Theory, 131(2):300 – 308, 2011. doi:http://dx.doi.org/10.1016/j.jnt.2010.08.004.Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4): 573–577, 1997.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829–832, 1990.W.W. Sawyer. The Search for Pattern. Penguin Books Ltd, Harmondsworth, Middlessex, England, 1970.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990

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=∏i=1x(1+i⋅y),        z=∏i=1x(y+1-j),      (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 Białystok, PolandMarcin Acewicz and Karol Pąk. Basic Diophantine relations. Formalized Mathematics, 26(2):175–181, 2018. doi:10.2478/forma-2018-0015.Zofia Adamowicz and Paweł 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, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, 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 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Martin Davis. Hilbert’s tenth problem is unsolvable. The American Mathematical Monthly, Mathematical Association of America, 80(3):233–269, 1973. doi:10.2307/2318447.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Artur Korniłowicz and Karol Pąk. Basel problem – preliminaries. Formalized Mathematics, 25(2):141–147, 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):181–187, 2007. doi:10.2478/v10037-007-0022-7.Karol Pąk. Diophantine sets. Preliminaries. Formalized Mathematics, 26(1):81–90, 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):825–829, 2001.Rafał Ziobro. On subnomials. Formalized Mathematics, 24(4):261–273, 2016. doi:10.1515/forma-2016-0022.27219720

Concatenation of Finite Sequences

The coexistence of “classical” finite sequences [1] and their zero-based equivalents finite 0-sequences [6] in Mizar has been regarded as a disadvantage. However the suggested replacement of the former type with the latter [5] has not yet been implemented, despite of several advantages of this form, such as the identity of length and domain operators [4]. On the other hand the number of theorems formalized using finite sequence notation is much larger then of those based on finite 0-sequences, so such translation would require quite an effort. The paper addresses this problem with another solution, using the Mizar system [3], [2]. Instead of removing one notation it is possible to introduce operators which would concatenate sequences of various types, and in this way allow utilization of the whole range of formalized theorems. While the operation could replace existing FS2XFS, XFS2FS commands (by using empty sequences as initial elements) its universal notation (independent on sequences that are concatenated to the initial object) allows to “forget” about the type of sequences that are concatenated on further positions, and thus simplify the proofs.Department of Carbohydrate Technology, University of Agriculture, Krakow, PolandGrzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6. Concatenation of finite sequences 13Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Artur Kornilowicz. How to define terms in Mizar effectively. Studies in Logic, Grammar and Rhetoric, 18:67–77, 2009.Piotr Rudnicki and Andrzej Trybulec. On the integrity of a repository of formalized mathematics. In Andrea Asperti, Bruno Buchberger, and James H. Davenport, editors, Mathematical Knowledge Management, volume 2594 of Lecture Notes in Computer Science, pages 162–174. Springer, Berlin, Heidelberg, 2003. doi:10.1007/3-540-36469-2_13.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825–829, 2001.Rafał Ziobro. Fermat’s Little Theorem via divisibility of Newton’s binomial. Formalized Mathematics, 23(3):215–229, 2015. doi:10.1515/forma-2015-0018.Rafał Ziobro. On subnomials. Formalized Mathematics, 24(4):261–273, 2016. doi:10.1515/forma-2016-0022.27111