The Sum and Product of Finite Sequences of Complex Numbers

This article extends the [10]. We define the sum and the product of the sequence of complex numbers, and formalize these theorems. Our method refers to the [11].Miyajima Keiichi - Faculty of Engineering, Ibaraki University, Hitachi, JapanKato Takahiro - Faculty of Engineering, Graduate School of Ibaraki University, Hitachi, JapanKanchun and Yatsuka Nakamura. The inner product of finite sequences and of points of n-dimensional topological space. Formalized Mathematics, 11(2):179-183, 2003.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. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 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. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Keith E. Hirst. Numbers, Sequences and Series. Butterworth-Heinemann, 1984.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990

Basel Problem – Preliminaries

SummaryIn the article we formalize in the Mizar system [4] preliminary facts needed to prove the Basel problem [7, 1]. Facts that are independent from the notion of structure are included here.Korniłowicz Artur - Institute of Informatics, University of Białystok, PolandPąk Karol - Institute of Informatics, University of Białystok, PolandM. Aigner and G. M. Ziegler. Proofs from THE BOOK. Springer-Verlag, Berlin Heidelberg New York, 2004.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.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: 10.1007/978-3-319-20615-817.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.Augustin Louis Cauchy. Cours d’analyse de l’Ecole royale polytechnique. de l’Imprimerie royale, 1821.Wenpai Chang, Yatsuka Nakamura, and Piotr Rudnicki. Inner products and angles of complex numbers. Formalized Mathematics, 11(3):275–280, 2003.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.Noboru Endou. Double series and sums. Formalized Mathematics, 22(1):57–68, 2014. doi: 10.2478/forma-2014-0006.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.Adam Grabowski and Yatsuka Nakamura. Some properties of real maps. Formalized Mathematics, 6(4):455–459, 1997.Artur Korniłowicz and Yasunari Shidama. Inverse trigonometric functions arcsin and arccos. Formalized Mathematics, 13(1):73–79, 2005.Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 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.Robert Milewski. Trigonometric form of complex numbers. Formalized Mathematics, 9 (3):455–460, 2001.Keiichi Miyajima and Takahiro Kato. The sum and product of finite sequences of complex numbers. Formalized Mathematics, 18(2):107–111, 2010. doi: 10.2478/v10037-010-0014-x.Cuiying Peng, Fuguo Ge, and Xiquan Liang. Several integrability formulas of special functions. Formalized Mathematics, 15(4):189–198, 2007. doi: 10.2478/v10037-007-0023-6.Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125–130, 1991.Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787–791, 1990.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.Yasunari Shidama, Noboru Endou, and Katsumi Wasaki. Riemann indefinite integral of functions of real variable. Formalized Mathematics, 15(2):59–63, 2007. doi: 10.2478/v10037-007-0007-6.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.25214114