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