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Klein-Beltrami model. III. (English) Zbl 1462.68222

Summary: Timothy Makarios (with Isabelle/HOL) and John Harrison (with HOL-Light) shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom”.
With the Mizar system we use some ideas taken from T. Makarios’ MSc thesis [A mechanical verification of the independence of Tarski’s Euclidean axiom. Wellington, NZ: Victoria University of Wellington (2012)] 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 [A. Grabowski and the author, “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 international conference on intelligent computer mathematics, CICM 2016. Aachen: RWTH Aachen. 4–9 (2016)]).
For Parts I and II, see the [the author, Formaliz. Math. 26, No. 1, 21–32 (2018; Zbl 1401.51001); ibid. 26, No. 1, 33–48 (2018; Zbl 1401.51002)].

MSC:

68V20 Formalization of mathematics in connection with theorem provers
51A05 General theory of linear incidence geometry and projective geometries
51M10 Hyperbolic and elliptic geometries (general) and generalizations
68V15 Theorem proving (automated and interactive theorem provers, deduction, resolution, etc.)

Software:

Mizar; Isabelle/HOL
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Full Text: DOI

References:

[1] 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:10.1007/978-3-319-20615-8_17. · Zbl 1417.68201
[2] Eugenio Beltrami. Saggio di interpetrazione della geometria non-euclidea. Giornale di Matematiche, 6:284-322, 1868.
[3] 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. · JFM 02.0334.03
[4] Karol Borsuk and Wanda Szmielew. Foundations of Geometry. North Holland, 1960. · Zbl 0093.33301
[5] Karol Borsuk and Wanda Szmielew. Podstawy geometrii. Państwowe Wydawnictwo Naukowe, Warszawa, 1955 (in Polish). · Zbl 0065.13203
[6] Roland Coghetto. Group of homography in real projective plane. Formalized Mathematics, 25(1):55-62, 2017. doi:10.1515/forma-2017-0005. · Zbl 1365.51014
[7] Roland Coghetto. Klein-Beltrami model. Part II. Formalized Mathematics, 26(1):33-48, 2018. doi:10.2478/forma-2018-0004. · Zbl 1401.51002
[8] 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. · Zbl 1352.51001
[9] Wojciech Leończuk and Krzysztof Prażmowski. Incidence projective spaces. Formalized Mathematics, 2(2):225-232, 1991. · Zbl 0741.51010
[10] Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean Axiom. Victoria University of Wellington, New Zealand, 2012. Master’s thesis.
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