Improving legibility of natural deduction proofs is not trivial

In formal proof checking environments such as Mizar it is not merely the validity of mathematical formulas that is evaluated in the process of adoption to the body of accepted formalizations, but also the readability of the proofs that witness validity. As in case of computer programs, such proof scripts may sometimes be more and sometimes be less readable. To better understand the notion of readability of formal proofs, and to assess and improve their readability, we propose in this paper a method of improving proof readability based on Behaghel's First Law of sentence structure. Our method maximizes the number of local references to the directly preceding statement in a proof linearisation. It is shown that our optimization method is NP-complete.Comment: 33 page

Continuity of Barycentric Coordinates in Euclidean Topological Spaces

In this paper we present selected properties of barycentric coordinates in the Euclidean topological space. We prove the topological correspondence between a subset of an affine closed space of εn and the set of vectors created from barycentric coordinates of points of this subset.Institute of Informatics, University of Białystok, PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Czesław Byliński. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99-107, 2005.Jing-Chao Chen. The Steinitz theorem and the dimension of a real linear space. Formalized Mathematics, 6(3):411-415, 1997.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53-58, 2003.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23-28, 2003.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Artur Korniłowicz. The correspondence between n-dimensional Euclidean space and the product of n real lines. Formalized Mathematics, 18(1):81-85, 2010, doi: 10.2478/v10037-010-0011-0.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Anna Lango and Grzegorz Bancerek. Product of families of groups and vector spaces. Formalized Mathematics, 3(2):235-240, 1992.Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339-345, 1996.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Karol Pąk. Affine independence in vector spaces. Formalized Mathematics, 18(1):87-93, 2010, doi: 10.2478/v10037-010-0012-z.Karol Pąk. Linear transformations of Euclidean topological spaces. Formalized Mathematics, 19(2):103-108, 2011, doi: 10.2478/v10037-011-0016-3.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847-850, 1990.Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883-885, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992.Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205-211, 1992

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 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∞. 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 Białystok, PolandMarcin Acewicz and Karol Pąk. Basic Diophantine relations. Formalized Mathematics, 26(2):175–181, 2018. doi:10.2478/forma-2018-0015.Marcin Acewicz and Karol Pąk. Pell’s equation. Formalized Mathematics, 25(3):197–204, 2017. doi:10.1515/forma-2017-0019.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.Karol Pąk. The Matiyasevich theorem. Preliminaries. Formalized Mathematics, 25(4): 315–322, 2017. doi:10.1515/forma-2017-0029.Karol Pąk. Diophantine sets. Part II. Formalized Mathematics, 27(2):197–208, 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

Bertrand’s Ballot Theorem

In this article we formalize the Bertrand’s Ballot Theorem based on [17]. Suppose that in an election we have two candidates: A that receives n votes and B that receives k votes, and additionally n ≥ k. Then this theorem states that the probability of the situation where A maintains more votes than B throughout the counting of the ballots is equal to (n − k)/(n + k). This theorem is item #30 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.The paper has been financed by the resources of the Polish National Science Centre granted by decision no DEC-2012/07/N/ST6/02147.Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181–187, 2005.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.Karol Pąk. Cardinal numbers and finite sets. Formalized Mathematics, 13(3):399–406, 2005.Karol Pąk. The Catalan numbers. Part II. Formalized Mathematics, 14(4):153–159, 2006. doi:10.2478/v10037-006-0019-7.Jan Popiołek. Introduction to probability. Formalized Mathematics, 1(4):755–760, 1990.M. Renault. Four proofs of the ballot theorem. Mathematics Magazine, 80(5):345–352, December 2007.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.Andrzej Trybulec. On the decomposition of finite sequences. Formalized Mathematics, 5 (3):317–322, 1996.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825–829, 2001.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990

Basic Properties of Metrizable Topological Spaces

We continue Mizar formalization of general topology according to the book [11] by Engelking. In the article, we present the final theorem of Section 4.1. Namely, the paper includes the formalization of theorems on the correspondence between the cardinalities of the basis and of some open subcover, and a discreet (closed) subspaces, and the weight of that metrizable topological space. We also define Lindel¨of spaces and state the above theorem in this special case. We also introduce the concept of separation among two subsets (see [12]).Institute of Computer Science, University of Białystok, PolandGrzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Józef Białas and Yatsuka Nakamura. The theorem of Weierstrass. Formalized Mathematics, 5(3):353-359, 1996.Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481-485, 1991.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Ryszard Engelking. General Topology, volume 60 of Monografie Matematyczne. PWN-Polish Scientific Publishers, Warsaw, 1977.Ryszard Engelking. Teoria wymiaru. PWN, 1981.Adam Grabowski. Properties of the product of compact topological spaces. Formalized Mathematics, 8(1):55-59, 1999.Adam Grabowski. On the Borel families of subsets of topological spaces. Formalized Mathematics, 13(4):453-461, 2005.Adam Grabowski. On the boundary and derivative of a set. Formalized Mathematics, 13(1):139-146, 2005.Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607-610, 1990.Zbigniew Karno. Maximal discrete subspaces of almost discrete topological spaces. Formalized Mathematics, 4(1):125-135, 1993.Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285-294, 1998.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics, 5(2):233-236, 1996.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535-545, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Flexary Operations

AbstractIn this article we introduce necessary notation and definitions to prove the Euler’s Partition Theorem according to H.S. Wilf’s lecture notes [31]. Our aim is to create an environment which allows to formalize the theorem in a way that is as similar as possible to the original informal proof. Euler’s Partition Theorem is listed as item #45 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/ [30].Institute of Informatics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213-225, 1992.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Marco B. Caminati. Preliminaries to classical first order model theory. Formalized Mathematics, 19(3):155-167, 2011. doi:10.2478/v10037-011-0025-2. [Crossref]Marco B. Caminati. First order languages: Further syntax and semantics. Formalized Mathematics, 19(3):179-192, 2011. doi:10.2478/v10037-011-0027-0. [Crossref]Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Public-key cryptography and Pepin’s test for the primality of Fermat numbers. Formalized Mathematics, 7(2):317-321, 1998.Artur Korniłowicz. Arithmetic operations on functions from sets into functional sets. Formalized Mathematics, 17(1):43-60, 2009. doi:10.2478/v10037-009-0005-y. [Crossref]Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Yatsuka Nakamura and Hisashi Ito. Basic properties and concept of selected subsequence of zero based finite sequences. Formalized Mathematics, 16(3):283-288, 2008. doi:10.2478/v10037-008-0034-y. [Crossref]Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Wojciech A. Trybulec. Binary operations on finite sequences. Formalized Mathematics, 1 (5):979-981, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Freek Wiedijk. Formalizing 100 theorems.Herbert S. Wilf. Lectures on integer partitions.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Euler’s Partition Theorem

AbstractIn this article we prove the Euler’s Partition Theorem which states that the number of integer partitions with odd parts equals the number of partitions with distinct parts. The formalization follows H.S. Wilf’s lecture notes [28] (see also [1]). Euler’s Partition Theorem is listed as item #45 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/ [27].Institute of Informatics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, PolandGeorge E. Andrews and Kimmo Eriksson. Integer Partitions. ISBN 9780521600903.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. Countable sets and Hessenberg’s theorem. Formalized Mathematics, 2(1):65-69, 1991.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Marco B. Caminati. Preliminaries to classical first order model theory. Formalized Mathematics, 19(3):155-167, 2011. doi:10.2478/v10037-011-0025-2. [Crossref]Marco B. Caminati. First order languages: Further syntax and semantics. Formalized Mathematics, 19(3):179-192, 2011. doi:10.2478/v10037-011-0027-0. [Crossref]Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Magdalena Jastrzȩbska and Adam Grabowski. Some properties of Fibonacci numbers. Formalized Mathematics, 12(3):307-313, 2004.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Karol Pak. Flexary operations. Formalized Mathematics, 23(2):81-92, 2015. doi:10.1515 /forma-2015-0008. [Crossref]Karol Pak. The Nagata-Smirnov theorem. Part II. Formalized Mathematics, 12(3):385-389, 2004.Karol Pak. Stirling numbers of the second kind. Formalized Mathematics, 13(2):337-345, 2005.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Freek Wiedijk. Formalizing 100 theorems.Herbert S. Wilf. Lectures on integer partitions.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Bo Zhang and Yatsuka Nakamura. The definition of finite sequences and matrices of probability, and addition of matrices of real elements. Formalized Mathematics, 14(3): 101-108, 2006. doi:10.2478/v10037-006-0012-1. [Crossref

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

Leibniz Series for π

In this article we prove the Leibniz series for π which states that π4=∑n=0∞(−1)n2⋅n+1. The formalization follows K. Knopp [8], [1] and [6]. Leibniz’s Series for Pi is item #26 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.Pąk Karol - Institute of Informatics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, PolandGeorge E. Andrews, Richard Askey, and Ranjan Roy. Special Functions. Cambridge University Press, 1999.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Lokenath Debnath. The Legacy of Leonhard Euler: A Tricentennial Tribute. World Scientific, 2010.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from ℝ to ℝ and integrability for continuous functions. Formalized Mathematics, 9(2):281–284, 2001.Konrad Knopp. Infinite Sequences and Series. Dover Publications, 1956. ISBN 978-0-486-60153-3.Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1 (3):471–475, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.Xiquan Liang and Bing Xie. Inverse trigonometric functions arctan and arccot. Formalized Mathematics, 16(2):147–158, 2008. doi:10.2478/v10037-008-0021-3.Akira Nishino and Yasunari Shidama. The Maclaurin expansions. Formalized Mathematics, 13(3):421–425, 2005.Chanapat Pacharapokin, Kanchun, and Hiroshi Yamazaki. Formulas and identities of trigonometric functions. Formalized Mathematics, 12(2):139–141, 2004.Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125–130, 1991.Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998