All Liouville Numbers are Transcendental

In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and 0 < x − p q < 1 q n . It is easy to show that all Liouville numbers are irrational. The definition and basic notions are contained in [10], [1], and [12]. Liouvile constant, which is defined formally in [12], is the first explicit transcendental (not algebraic) number, another notable examples are e and π [5], [11], and [4]. Algebraic numbers were formalized with the help of the Mizar system [13] very recently, by Yasushige Watase in [23] and now we expand these techniques into the area of not only pure algebraic domains (as fields, rings and formal polynomials), but also for more settheoretic fields. Finally we show that all Liouville numbers are transcendental, based on Liouville’s theorem on Diophantine approximation.Korniłowicz Artur - Institute of Informatics, University of Białystok, Białystok, PolandNaumowicz Adam - Institute of Informatics, University of Białystok, Białystok, PolandGrabowski Adam - Institute of Informatics, University of Białystok, Białystok, PolandTom M. Apostol. Modular Functions and Dirichlet Series in Number Theory. Springer- Verlag, 2nd edition, 1997.Grzegorz 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.Sophie Bernard, Yves Bertot, Laurence Rideau, and Pierre-Yves Strub. Formal proofs of transcendence for e and _ as an application of multivariate and symmetric polynomials. In Jeremy Avigad and Adam Chlipala, editors, Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs, pages 76-87. ACM, 2016.Jesse Bingham. Formalizing a proof that e is transcendental. Journal of Formalized Reasoning, 4:71-84, 2011.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. 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.J.H. Conway and R.K. Guy. The Book of Numbers. Springer-Verlag, 1996.Manuel Eberl. Liouville numbers. Archive of Formal Proofs, December 2015. http://isa-afp.org/entries/Liouville_Numbers.shtml, Formal proof development.Adam Grabowski and Artur Korniłowicz. Introduction to Liouville numbers. Formalized Mathematics, 25(1):39-48, 2017.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.Joseph Liouville. Nouvelle démonstration d’un théorème sur les irrationnelles algébriques, inséré dans le Compte Rendu de la dernière séance. Compte Rendu Acad. Sci. Paris, Sér.A (18):910–911, 1844.Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265-269, 2001.Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391-395, 2001.Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001.Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97-104, 1991.Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1 (3):495-500, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Yasushige Watase. Algebraic numbers. Formalized Mathematics, 24(4):291-299, 2016.Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205-211, 1992

More on Divisibility Criteria for Selected Primes

This paper is a continuation of [19], where the divisibility criteria for initial prime numbers based on their representation in the decimal system were formalized. In the current paper we consider all primes up to 101 to demonstrate the method presented in [7].Naumowicz Adam - Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandPiliszek Radosław - 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. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281-290, 1990.Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711-714, 1990.Grzegorz Bancerek. Veblen hierarchy. Formalized Mathematics, 19(2):83-92, 2011. doi:10.2478/v10037-011-0014-5.C.C. Briggs. Simple divisibility rules for the 1st 1000 prime numbers. arXiv preprint arXiv:math/0001012, 2000.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. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321-328, 1990.Magdalena Jastrz¸ebska and Adam Grabowski. Some properties of Fibonacci numbers. Formalized Mathematics, 12(3):307-313, 2004.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.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 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.Adam Naumowicz. On the representation of natural numbers in positional numeral systems. Formalized Mathematics, 14(4):221-223, 2006. doi:10.2478/v10037-006-0025-9.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. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 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

A Note on the Seven Bridges of Königsberg Problem

In this paper we account for the formalization of the seven bridges of Königsberg puzzle. The problem originally posed and solved by Euler in 1735 is historically notable for having laid the foundations of graph theory, cf. [7]. Our formalization utilizes a simple set-theoretical graph representation with four distinct sets for the graph’s vertices and another seven sets that represent the edges (bridges). The work appends the article by Nakamura and Rudnicki [10] by introducing the classic example of a graph that does not contain an Eulerian path. This theorem is item #54 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandGrzegorz 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. 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 and Piotr Rudnicki. The correspondence between monotonic many sorted signatures and well-founded graphs. Part I. Formalized Mathematics, 5(4):577– 582, 1996.Gary Chartrand. Introductory Graph Theory. New York: Dover, 1985.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Krzysztof Hryniewiecki. Graphs. Formalized Mathematics, 2(3):365–370, 1991.Yatsuka Nakamura and Piotr Rudnicki. Euler circuits and paths. Formalized Mathematics, 6(3):417–425, 1997.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 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. Received June 16, 201

Elementary Number Theory Problems. Part I

In this paper we demonstrate the feasibility of formalizing recreational mathematics in Mizar ([1], [2]) drawing examples from W. Sierpinski’s book “250 Problems in Elementary Number Theory” [4]. The current work contains proofs of initial ten problems from the chapter devoted to the divisibility of numbers. Included are problems on several levels of difficulty.Institute of Informatics, University of Białystok, PolandGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, 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 Bylinski, 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.Marco Riccardi. The perfect number theorem and Wilson’s theorem. Formalized Mathematics, 17(2):123–128, 2009. doi:10.2478/v10037-009-0013-y.Wacław Sierpinski. 250 Problems in Elementary Number Theory. Elsevier, 1970.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825–829, 2001.Li Yan, Xiquan Liang, and Junjie Zhao. Gauss lemma and law of quadratic reciprocity. Formalized Mathematics, 16(1):23–28, 2008. doi:10.2478/v10037-008-0004-4.11512

Niven’s Theorem

This article formalizes the proof of Niven’s theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9].Korniłowicz Artur - Institute of Informatics, University of Białystok, PolandNaumowicz Adam - Institute of Informatics, University of Białystok, 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.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996.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.Yuzhong Ding and Xiquan Liang. Formulas and identities of trigonometric functions. Formalized Mathematics, 12(3):243–246, 2004.Magdalena Jastrzębska and Adam Grabowski. Some properties of Fibonacci numbers. Formalized Mathematics, 12(3):307–313, 2004.J.D. King. Integer roots of polynomials. The Mathematical Gazette, 90(519):455–456, 2006. doi:http://dx.doi.org/10.1017/S0025557200180295.Serge Lang. Algebra. Addison-Wesley, 1980.Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391–395, 2001.Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461–470, 2001.Ivan Niven. Irrational numbers. The Carus Mathematical Monographs, No. 11. The Mathematical Association of America. Distributed by John Wiley and Sons, Inc., New York, N.Y., 1956.Piotr Rudnicki. Little Bezout theorem (factor theorem). Formalized Mathematics, 12(1): 49–58, 2004.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. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990

Syntactic-Semantic Form of Mizar Articles

Mizar Mathematical Library is most appreciated for the wealth of mathematical knowledge it contains. However, accessing this publicly available huge corpus of formalized data is not straightforward due to the complexity of the underlying Mizar language, which has been designed to resemble informal mathematical papers. For this reason, most systems exploring the library are based on an internal XML representation format used by semantic modules of Mizar. This representation is easily accessible, but it lacks certain syntactic information available only in the original human-readable Mizar source files. In this paper we propose a new XML-based format which combines both syntactic and semantic data. It is intended to facilitate various applications of the Mizar library requiring fullest possible information to be retrieved from the formalization files

Prime Factorization of Sums and Differences of Two Like Powers

Representation of a non zero integer as a signed product of primes is unique similarly to its representations in various types of positional notations [4], [3]. The study focuses on counting the prime factors of integers in the form of sums or differences of two equal powers (thus being represented by 1 and a series of zeroes in respective digital bases).Although the introduced theorems are not particularly important, they provide a couple of shortcuts useful for integer factorization, which could serve in further development of Mizar projects [2]. This could be regarded as one of the important benefits of proof formalization [9].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, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, 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.Paul Erdős and János Surányi. Topics in the Theory of Numbers, chapter Divisibility, the Fundamental Theorem of Number Theory, pages 1-37. Springer New York, 2003.Jacek Gancarzewicz. Arytmetyka, 2000. In Polish.Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4): 573-577, 1997.Artur Korniłowicz and Piotr Rudnicki. Fundamental Theorem of Arithmetic. Formalized Mathematics, 12(2):179-186, 2004.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.Adam Naumowicz. An example of formalizing recent mathematical results in Mizar. Journal of Applied Logic, 4(4):396-413, 2006.Akira Nishino and Yasunari Shidama. The Maclaurin expansions. Formalized Mathematics, 13(3):421-425, 2005.Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.Marco Riccardi. Pocklington’s theorem and Bertrand’s postulate. Formalized Mathematics, 14(2):47-52, 2006.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Rafał Ziobro. Fermat’s Little Theorem via divisibility of Newton’s binomial. Formalized Mathematics, 23(3):215-229, 2015