{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T13:04:55Z","timestamp":1648991095464},"reference-count":15,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2017,10,1]],"date-time":"2017-10-01T00:00:00Z","timestamp":1506816000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,10,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>Some authors have formalized the integral in the Mizar Mathematical Library (MML). The first article in a series on the Darboux\/Riemann integral was written by Noboru Endou and Artur Korni\u0142owicz: [6]. The Lebesgue integral was formalized a little later [13] and recently the integral of Riemann-Stieltjes was introduced in the MML by Keiko Narita, Kazuhisa Nakasho and Yasunari Shidama [12].<\/jats:p>\n               <jats:p>A presentation of definitions of integrals in other proof assistants or proof checkers (ACL2, COQ, Isabelle\/HOL, HOL4, HOL Light, PVS, ProofPower) may be found in [10] and [4].<\/jats:p>\n               <jats:p>Using the Mizar system [1], we define the Gauge integral (Henstock-Kurzweil) of a real-valued function on a real interval [<jats:italic>a, b<\/jats:italic>] (see [2], [3], [15], [14], [11]). In the next section we formalize that the Henstock-Kurzweil integral is linear.<\/jats:p>\n               <jats:p>In the last section, we verified that a real-valued bounded integrable (in sense Darboux\/Riemann [6, 7, 8]) function over a interval <jats:italic>a, b<\/jats:italic> is Gauge integrable.<\/jats:p>\n               <jats:p>Note that, in accordance with the possibilities of the MML [9], we reuse a large part of demonstrations already present in another article. Instead of rewriting the proof already contained in [7] (MML Version: 5.42.1290), we slightly modified this article in order to use directly the expected results.<\/jats:p>","DOI":"10.1515\/forma-2017-0021","type":"journal-article","created":{"date-parts":[[2017,12,20]],"date-time":"2017-12-20T22:16:12Z","timestamp":1513808172000},"page":"217-225","source":"Crossref","is-referenced-by-count":0,"title":["Gauge Integral"],"prefix":"10.1515","volume":"25","author":[{"given":"Roland","family":"Coghetto","sequence":"first","affiliation":[{"name":"Rue de la Brasserie 5, 7100 La Louvi\u00e8re , Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,12,19]]},"reference":[{"key":"2021040814262538845_j_forma-2017-0021_ref_001_w2aab3b7b5b1b6b1ab1ab1Aa","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, 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\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.10.1007\/978-3-319-20615-8_17"},{"key":"2021040814262538845_j_forma-2017-0021_ref_002_w2aab3b7b5b1b6b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] Robert G. Bartle. Return to the Riemann integral. American Mathematical Monthly, pages 625\u2013632, 1996.","DOI":"10.1080\/00029890.1996.12004798"},{"key":"2021040814262538845_j_forma-2017-0021_ref_003_w2aab3b7b5b1b6b1ab1ab3Aa","doi-asserted-by":"crossref","unstructured":"[3] Robert G. Bartle. A modern theory of integration, volume 32. American Mathematical Society Providence, 2001.","DOI":"10.1090\/gsm\/032"},{"key":"2021040814262538845_j_forma-2017-0021_ref_004_w2aab3b7b5b1b6b1ab1ab4Aa","unstructured":"[4] Sylvie Boldo, Catherine Lelay, and Guillaume Melquiond. Formalization of real analysis: A survey of proof assistants and libraries. Mathematical Structures in Computer Science, 26(7):1196\u20131233, 2016."},{"key":"2021040814262538845_j_forma-2017-0021_ref_005_w2aab3b7b5b1b6b1ab1ab5Aa","unstructured":"[5] Roland Coghetto. Cousin\u2019s lemma. Formalized Mathematics, 24(2):107\u2013119, 2016. doi:10.1515\/forma-2016-0009.10.1515\/forma-2016-0009"},{"key":"2021040814262538845_j_forma-2017-0021_ref_006_w2aab3b7b5b1b6b1ab1ab6Aa","unstructured":"[6] Noboru Endou and Artur Korni\u0142owicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93\u2013102, 1999."},{"key":"2021040814262538845_j_forma-2017-0021_ref_007_w2aab3b7b5b1b6b1ab1ab7Aa","unstructured":"[7] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux\u2019s theorem. Formalized Mathematics, 9(1):197\u2013200, 2001."},{"key":"2021040814262538845_j_forma-2017-0021_ref_008_w2aab3b7b5b1b6b1ab1ab8Aa","unstructured":"[8] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Integrability of bounded total functions. Formalized Mathematics, 9(2):271\u2013274, 2001."},{"key":"2021040814262538845_j_forma-2017-0021_ref_009_w2aab3b7b5b1b6b1ab1ab9Aa","doi-asserted-by":"crossref","unstructured":"[9] Adam Grabowski and Christoph Schwarzweller. Revisions as an essential tool to maintain mathematical repositories. In M. Kauers, M. Kerber, R. Miner, and W. Windsteiger, editors, Towards Mechanized Mathematical Assistants. Lecture Notes in Computer Science, volume 4573, pages 235\u2013249. Springer: Berlin, Heidelberg, 2007.","DOI":"10.1007\/978-3-540-73086-6_20"},{"key":"2021040814262538845_j_forma-2017-0021_ref_010_w2aab3b7b5b1b6b1ab1ac10Aa","unstructured":"[10] John Harrison. Formalizing basic complex analysis. Studies in Logic, Grammar and Rhetoric, 23(10):151\u2013165, 2007."},{"key":"2021040814262538845_j_forma-2017-0021_ref_011_w2aab3b7b5b1b6b1ab1ac11Aa","unstructured":"[11] Jean Mawhin. L\u2019\u00e9ternel retour des sommes de Riemann-Stieltjes dans l\u2019\u00e9volution du calcul int\u00e9gral. Bulletin de la Soci\u00e9t\u00e9 Royale des Sciences de Li\u00e8ge, 70(4\u20136):345\u2013364, 2001."},{"key":"2021040814262538845_j_forma-2017-0021_ref_012_w2aab3b7b5b1b6b1ab1ac12Aa","unstructured":"[12] Keiko Narita, Kazuhisa Nakasho, and Yasunari Shidama. Riemann-Stieltjes integral. Formalized Mathematics, 24(3):199\u2013204, 2016. doi:10.1515\/forma-2016-0016.10.1515\/forma-2016-0016"},{"key":"2021040814262538845_j_forma-2017-0021_ref_013_w2aab3b7b5b1b6b1ab1ac13Aa","unstructured":"[13] Yasunari Shidama, Noburu Endou, and Pauline N. Kawamoto. On the formalization of Lebesgue integrals. Studies in Logic, Grammar and Rhetoric, 10(23):167\u2013177, 2007."},{"key":"2021040814262538845_j_forma-2017-0021_ref_014_w2aab3b7b5b1b6b1ab1ac14Aa","unstructured":"[14] Lee Peng Yee. The integral \u00e0 la Henstock. Scientiae Mathematicae Japonicae, 67(1): 13\u201321, 2008."},{"key":"2021040814262538845_j_forma-2017-0021_ref_015_w2aab3b7b5b1b6b1ab1ac15Aa","unstructured":"[15] Lee Peng Yee and Rudolf Vyborny. Integral: an easy approach after Kurzweil and Henstock, volume 14. 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