Klein-Beltrami model. Part III

Timothy Makarios (with Isabelle/HOL1) and John Harrison (with HOL-Light2) shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom” [2],[3],[4],[5]. With the Mizar system [1] we use some ideas taken from Tim Makarios’s MSc thesis [10] to formalize some definitions (like the absolute) and lemmas necessary for the verification of the independence of the parallel postulate. In this article we prove that our constructed model (we prefer “Beltrami-Klein” name over “Klein-Beltrami”, which can be seen in the naming convention for Mizar functors, and even MML identifiers) satisfies the congruence symmetry, the congruence equivalence relation, and the congruence identity axioms formulated by Tarski (and formalized in Mizar as described briefly in [8]).Rue de la Brasserie 5, 7100 La Louvière, BelgiumGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, 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.Eugenio Beltrami. Saggio di interpetrazione della geometria non-euclidea. Giornale di Matematiche, 6:284–322, 1868.Eugenio Beltrami. Essai d’interprétation de la géométrie non-euclidéenne. In Annales scientifiques de l’École Normale Supérieure. Trad. par J. Hoüel, volume 6, pages 251–288. Elsevier, 1869.Karol Borsuk and Wanda Szmielew. Foundations of Geometry. North Holland, 1960.Karol Borsuk and Wanda Szmielew. Podstawy geometrii. Panstwowe Wydawnictwo Naukowe, Warszawa, 1955 (in Polish).Roland Coghetto. Group of homography in real projective plane. Formalized Mathematics, 25(1):55–62, 2017. doi:10.1515/forma-2017-0005.Roland Coghetto. Klein-Beltrami model. Part II. Formalized Mathematics, 26(1):33–48, 2018. doi:10.2478/forma-2018-0004.Adam Grabowski and Roland Coghetto. Tarski’s geometry and the Euclidean plane in Mizar. In Joint Proceedings of the FM4M, MathUI, and ThEdu Workshops, Doctoral Program, and Work in Progress at the Conference on Intelligent Computer Mathematics 2016 co-located with the 9th Conference on Intelligent Computer Mathematics (CICM 2016), Białystok, Poland, July 25–29, 2016, volume 1785 of CEUR-WS, pages 4–9. CEURWS.org, 2016.Wojciech Leonczuk and Krzysztof Prazmowski. Incidence projective spaces. Formalized Mathematics, 2(2):225–232, 1991.Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean Axiom. Victoria University ofWellington, New Zealand, 2012. Master’s thesis.2811