{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T11:04:44Z","timestamp":1774868684131,"version":"3.50.1"},"reference-count":14,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2025,9,1]],"date-time":"2025-09-01T00:00:00Z","timestamp":1756684800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,9,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In this article, we develop the proof of the so-called Wallis formula using the Mizar formalism. The purpose of this formalization is to complete the proof of Stirling\u2019s formula using elementary techniques of calculus; however, the formal encoding of the Wallis-type integral is also included.<\/jats:p>","DOI":"10.2478\/forma-2025-0007","type":"journal-article","created":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T23:55:11Z","timestamp":1766620511000},"page":"85-94","source":"Crossref","is-referenced-by-count":1,"title":["Formalization of Wallis Infinite Product Formula for\n                    <i>\u03c0<\/i>\n                    and the Wallis Integral"],"prefix":"10.2478","volume":"33","author":[{"given":"Yasushige","family":"Watase","sequence":"first","affiliation":[{"name":"Faculty of Data Science , University of Rissho Magechi , Japan"}]}],"member":"374","published-online":{"date-parts":[[2025,12,23]]},"reference":[{"key":"2026033010152326198_j_forma-2025-0007_ref_001","doi-asserted-by":"crossref","unstructured":"Yves Bertot. A short presentation of Coq. In Otmane A\u00eft Mohamed, C\u00e9sar A. Mu\u00f1oz, and Sofi\u00e8ne Tahar, editors, Theorem Proving in Higher Order Logics (TPHOLs 2008), volume 5170 of LNCS, pages 12\u201316. Springer, 2008. doi:10.1007\/978-3-540-71067-73.","DOI":"10.1007\/978-3-540-71067-7_3"},{"key":"2026033010152326198_j_forma-2025-0007_ref_002","unstructured":"Richard Courant and Fritz John. Introduction to Calculus and Analysis I. John Wiley and Sons Inc., New York, 1st edition, 1965."},{"key":"2026033010152326198_j_forma-2025-0007_ref_003","unstructured":"Manuel Eberl. Stirling\u2019s formula. Archive of Formal Proofs, September 2016. https:\/\/isa-afp.org\/entries\/Stirling_Formula.html, Formal proof development."},{"key":"2026033010152326198_j_forma-2025-0007_ref_004","doi-asserted-by":"crossref","unstructured":"Adam Grabowski, Artur Korni\u0142owicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191\u2013198, 2015. doi:10.1007\/s10817-015-9345-1.","DOI":"10.1007\/s10817-015-9345-1"},{"key":"2026033010152326198_j_forma-2025-0007_ref_005","unstructured":"John Harrison. Stirling\u2019s approximation. 2010. Available online at https:\/\/github.com\/jrh13\/hol-light\/blob\/master\/100\/stirling.ml."},{"key":"2026033010152326198_j_forma-2025-0007_ref_006","unstructured":"Nobushige Kurokawa. Modern Trigonometric Function Theory. Iwanami Shoten, Publishers (Tokyo), 1st edition, 2013. In Japanese."},{"key":"2026033010152326198_j_forma-2025-0007_ref_007","unstructured":"Norman D. Megill and David A. Wheeler. Metamath: A Computer Language for Mathematical Proofs. Lulu Press, Morrisville, North Carolina, 2019. http:\/\/us.metamath.org\/downloads\/metamath.pdf."},{"key":"2026033010152326198_j_forma-2025-0007_ref_008","doi-asserted-by":"crossref","unstructured":"Leonardo de Moura and Sebastian Ullrich. The Lean 4 theorem prover and programming language. In Automated Deduction \u2013 CADE 28: 28th International Conference on Automated Deduction, Virtual Event, July 12\u201315, 2021, Proceedings, pages 625\u2013635, Berlin, Heidelberg, 2021. Springer-Verlag. doi:10.1007\/978-3-030-79876-537.","DOI":"10.1007\/978-3-030-79876-5_37"},{"key":"2026033010152326198_j_forma-2025-0007_ref_009","doi-asserted-by":"crossref","unstructured":"Karol P\u0105k. Leibniz series for \u03c0. Formalized Mathematics, 24(4):275\u2013280, 2016. doi:10.1515\/forma-2016-0023.","DOI":"10.1515\/forma-2016-0023"},{"key":"2026033010152326198_j_forma-2025-0007_ref_010","unstructured":"Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559\u2013564, 2001."},{"key":"2026033010152326198_j_forma-2025-0007_ref_011","doi-asserted-by":"crossref","unstructured":"Yasunari Shidama, Noboru Endou, and Katsumi Wasaki. Riemann indefinite integral of functions of real variable. Formalized Mathematics, 15(2):59\u201363, 2007. doi:10.2478\/v10037-007-0007-6.","DOI":"10.2478\/v10037-007-0007-6"},{"key":"2026033010152326198_j_forma-2025-0007_ref_012","doi-asserted-by":"crossref","unstructured":"Yasushige Watase. Formal proof of transcendence of the number e. Part II. Formalized Mathematics, 32(1):121\u2013131, 2024. doi:10.2478\/forma-2024-0009.","DOI":"10.2478\/forma-2024-0009"},{"key":"2026033010152326198_j_forma-2025-0007_ref_013","unstructured":"Freek Wiedijk. Formalizing 100 theorems. Available online at http:\/\/www.cs.ru.nl\/~freek\/100\/."},{"key":"2026033010152326198_j_forma-2025-0007_ref_014","unstructured":"Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255\u2013263, 1998."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.sciendo.com\/pdf\/10.2478\/forma-2025-0007","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T10:15:35Z","timestamp":1774865735000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.2478\/forma-2025-0007"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,1]]},"references-count":14,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,12,23]]},"published-print":{"date-parts":[[2025,9,1]]}},"alternative-id":["10.2478\/forma-2025-0007"],"URL":"https:\/\/doi.org\/10.2478\/forma-2025-0007","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,1]]}}}