1. bookVolume 25 (2017): Issue 2 (July 2017)
Journal Details
License
Format
Journal
eISSN
1898-9934
ISSN
1426-2630
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English
access type Open Access

Basel Problem – Preliminaries

Published Online: 23 Sep 2017
Volume & Issue: Volume 25 (2017) - Issue 2 (July 2017)
Page range: 141 - 147
Received: 27 Jun 2017
Journal Details
License
Format
Journal
eISSN
1898-9934
ISSN
1426-2630
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English
Summary

In 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.

Keywords

MSC 2010

[1] M. Aigner and G. M. Ziegler. Proofs from THE BOOK. Springer-Verlag, Berlin Heidelberg New York, 2004.10.1007/978-3-662-05412-3Search in Google Scholar

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Search in Google Scholar

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Search in Google Scholar

[4] 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: 10.1007/978-3-319-20615-817.10.1007/978-3-319-20615-817Open DOISearch in Google Scholar

[5] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Search in Google Scholar

[6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Search in Google Scholar

[7] Augustin Louis Cauchy. Cours d’analyse de l’Ecole royale polytechnique. de l’Imprimerie royale, 1821.Search in Google Scholar

[8] Wenpai Chang, Yatsuka Nakamura, and Piotr Rudnicki. Inner products and angles of complex numbers. Formalized Mathematics, 11(3):275–280, 2003.Search in Google Scholar

[9] 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.Search in Google Scholar

[10] Noboru Endou. Double series and sums. Formalized Mathematics, 22(1):57–68, 2014. doi: 10.2478/forma-2014-0006.10.2478/forma-2014-0006Search in Google Scholar

[11] 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.Search in Google Scholar

[12] Adam Grabowski and Yatsuka Nakamura. Some properties of real maps. Formalized Mathematics, 6(4):455–459, 1997.Search in Google Scholar

[13] Artur Korniłowicz and Yasunari Shidama. Inverse trigonometric functions arcsin and arccos. Formalized Mathematics, 13(1):73–79, 2005.Search in Google Scholar

[14] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.Search in Google Scholar

[15] Xiquan Liang and Bing Xie. Inverse trigonometric functions arctan and arccot. Formalized Mathematics, 16(2):147–158, 2008. doi: 10.2478/v10037-008-0021-3.10.2478/v10037-008-0021-3Open DOISearch in Google Scholar

[16] Robert Milewski. Trigonometric form of complex numbers. Formalized Mathematics, 9 (3):455–460, 2001.Search in Google Scholar

[17] 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.10.2478/v10037-010-0014-xOpen DOISearch in Google Scholar

[18] Cuiying Peng, Fuguo Ge, and Xiquan Liang. Several integrability formulas of special functions. Formalized Mathematics, 15(4):189–198, 2007. doi: 10.2478/v10037-007-0023-6.10.2478/v10037-007-0023-6Open DOISearch in Google Scholar

[19] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125–130, 1991.Search in Google Scholar

[20] Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787–791, 1990.Search in Google Scholar

[21] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.Search in Google Scholar

[22] 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/v10037-007-0007-6.10.2478/v10037-007-0007-6Open DOISearch in Google Scholar

[23] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.Search in Google Scholar

[24] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Search in Google Scholar

[25] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Search in Google Scholar

[26] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.Search in Google Scholar

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