Extended Natural Numbers and Counters

This article introduces extended natural numbers, i.e. the set ℕ ∪ {+∞}, in Mizar [4], [3] and formalizes a way to list a cardinal numbers of cardinals. Both concepts have applications in graph [email protected] Gutenberg University, Mainz, GermanyGrzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.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.John Adrian Bondy and U. S. R. Murty. Graph Theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. ISBN 978-1-84628-969-9.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.Pavol Hell and Jaroslav Nesetril. Graphs and homomorphisms. Oxford Lecture Series in Mathematics and Its Applications; 28. Oxford University Press, Oxford, 2004. ISBN 0-19-852817-5.Ulrich Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41 of De Gruyter Studies in Mathematics. Walter de Gruyter, 2011.Library Committee of the Association of Mizar Users. Number-valued functions. Mizar Mathematical Library, 2007.Library Committee of the Association of Mizar Users. Introduction to arithmetic of extended real numbers. Mizar Mathematical Library, 2006.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341–347, 2003.Andrzej Trybulec. Subsets of complex numbers. Mizar Mathematical Library, 2003.Andrzej Trybulec. Basic operations on extended real numbers. Mizar Mathematical Library, 2008.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825–829, 2001.28323924

Binary Representation of Natural Numbers

This study was supported in part by JSPS KAKENHI Grant Numbers JP17K00182. The author would also like to express gratitude to Prof. Yasunari Shidama for his support and encouragement.Binary representation of integers [5], [3] and arithmetic operations on them have already been introduced in Mizar Mathematical Library [8, 7, 6, 4]. However, these articles formalize the notion of integers as mapped into a certain length tuple of boolean values.In this article we formalize, by means of Mizar system [2], [1], the binary representation of natural numbers which maps ℕ into bitstreams.Shinshu University, Nagano, JapanGrzegorz 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.Donald E. Knuth. The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third Edition. Addison-Wesley, 1997.Hisayoshi Kunimune and Yatsuka Nakamura. A representation of integers by binary arithmetics and addition of integers. Formalized Mathematics, 11(2):175–178, 2003.Gottfried Wilhelm Leibniz. Explication de l’Arithmétique Binaire, volume 7. C. Gerhardt, Die Mathematische Schriften edition, 223 pages, 1879.Robert Milewski. Binary arithmetics. Binary sequences. Formalized Mathematics, 7(1): 23–26, 1998.Yasuho Mizuhara and Takaya Nishiyama. Binary arithmetics, addition and subtraction of integers. Formalized Mathematics, 5(1):27–29, 1996.Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4 (1):83–86, 1993.26322322

Arithmetic Operations on Short Finite Sequences

In contrast to other proving systems Mizar Mathematical Library, considered as one of the largest formal mathematical libraries [4], is maintained as a single base of theorems, which allows the users to benefit from earlier formalized items [3], [2]. This eventually leads to a development of certain branches of articles using common notation and ideas. Such formalism for finite sequences has been developed since 1989 [1] and further developed despite of the controversy over indexing which excludes zero [6], also for some advanced and new mathematics [5].The article aims to add some new machinery for dealing with finite sequences, especially those of short length.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, 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.Alexander Elizarov, Alexander Kirillovich, Evgeny Lipachev, and Olga Nevzorova. Digital ecosystem OntoMath: Mathematical knowledge analytics and management. In Leonid Kalinichenko, Sergei O. Kuznetsov, and Yannis Manolopoulos, editors, Data Analytics and Management in Data Intensive Domains, pages 33–46. Springer International Publishing, 2017.Adam Naumowicz and Artur Korniłowicz. A brief overview of Mizar. In International Conference on Theorem Proving in Higher Order Logics, pages 67–72. Springer, 2009. doi:10.1007/978-3-642-03359-9_5.Piotr Rudnicki and Andrzej Trybulec. On the integrity of a repository of formalized mathematics. Mathematical Knowledge Management, 2003. doi:10.1007/3-540-36469-2_13.26319920

Invertible Operators on Banach Spaces

In this article, using the Mizar system [2], [1], we discuss invertible operators on Banach spaces. In the first chapter, we formalized the theorem that denotes any operators that are close enough to an invertible operator are also invertible by using the property of Neumann series.In the second chapter, we formalized the continuity of an isomorphism that maps an invertible operator on Banach spaces to its inverse. These results are used in the proof of the implicit function theorem. We referred to [3], [10], [6], [7] in this formalization.Yamaguchi University, Yamaguchi, JapanGrzegorz 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.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.Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972.Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Implicit function theorem. Part I. Formalized Mathematics, 25(4):269–281, 2017. doi:10.1515/forma-2017-0026.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul différentiel, tome 2. Analyse. Hermann, 1997.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2004.Yasunari Shidama. The Banach algebra of bounded linear operators. Formalized Mathematics, 12(2):103–108, 2004.Kosaku Yoshida. Functional Analysis. Springer, 1980.27210711

Grothendieck Universes

The foundation of the Mizar Mathematical Library [2], is first-order Tarski-Grothendieck set theory. However, the foundation explicitly refers only to Tarski’s Axiom A, which states that for every set X there is a Tarski universe U such that X ∈ U. In this article, we prove, using the Mizar [3] formalism, that the Grothendieck name is justified. We show the relationship between Tarski and Grothendieck universe.This work has been supported by the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473.Institute of Informatics, University of Białystok, PolandGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.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.Chad E. Brown and Karol Pąk. A tale of two set theories. In Cezary Kaliszyk, Edwin Brady, Andrea Kohlhase, and Claudio Sacerdoti Coen, editors, Intelligent Computer Mathematics – 12th International Conference, CICM 2019, CIIRC, Prague, Czech Republic, July 8-12, 2019, Proceedings, volume 11617 of Lecture Notes in Computer Science, pages 44–60. Springer, 2019. doi:10.1007/978-3-030-23250-4_4.N. H. Williams. On Grothendieck universes. Compositio Mathematica, 21(1):1–3, 1969.28221121

The 3-Fold Product Space of Real Normed Spaces and its Properties

In this article, we formalize in Mizar [1], [2] the 3-fold product space of real normed spaces for usefulness in application fields such as engineering, although the formalization of the 2-fold product space of real normed spaces has been stored in the Mizar Mathematical Library [3]. First, we prove some theorems about the 3-variable function and 3-fold Cartesian product for preparation. Then we formalize the definition of 3-fold product space of real linear spaces. Finally, we formulate the definition of 3-fold product space of real normed spaces. We referred to [7] and [6] in the formalization.Hiroyuki Okazaki - Shinshu University, Nagano, JapanKazuhisa Nakasho, Yamaguchi University, Yamaguchi, JapanGrzegorz 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:10.1007/978-3-319-20615-817.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.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. The product space of real normed spaces and its properties. Formalized Mathematics, 15(3):81–85, 2007. doi:10.2478/v10037-007-0010-y.Artur Korniłowicz. Compactness of the bounded closed subsets of Ε²т. Formalized Mathematics, 8(1):61–68, 1999.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.Michael Read and Barry Simon. Functional Analysis (Methods of Modern Mathematical Physics). Academic Press, 1980.Kosaku Yosida. Functional Analysis. Springer, 1980.29424124

Isomorphisms from the Space of Multilinear Operators

In this article, using the Mizar system [5], [2], the isomorphisms from the space of multilinear operators are discussed. In the first chapter, two isomorphisms are formalized. The former isomorphism shows the correspondence between the space of multilinear operators and the space of bilinear operators.The latter shows the correspondence between the space of multilinear operators and the space of the composition of linear operators. In the last chapter, the above isomorphisms are extended to isometric mappings between the normed spaces. We referred to [6], [11], [9], [3], [10] in this formalization.Yamaguchi University, Yamaguchi, JapanGrzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996.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.Nelson Dunford and Jacob T. Schwartz. Linear operators I. Interscience Publ., 1958.Yuichi Futa, Noboru Endou, and Yasunari Shidama. Isometric differentiable functions on real normed space. Formalized Mathematics, 21(4):249–260, 2013. doi:10.2478/forma-2013-0027.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.Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972.Kazuhisa Nakasho. Bilinear operators on normed linear spaces. Formalized Mathematics, 27(1):15–23, 2019. doi:10.2478/forma-2019-0002.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul différentiel, tome 2. Analyse. Hermann, 1997.Kosaku Yoshida. Functional Analysis. Springer, 1980.27210110