Klein-Beltrami Model. Part II

Roland Coghetto 1
  • 1 , 7100, La Louvière, Belgium

Summary

Tim Makarios (with Isabelle/HOL1) and John Harrison (with HOL-Light2) have 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 are taken from Tim Makarios’ MSc thesis [] for formalized some definitions (like the tangent) and lemmas necessary for the verification of the independence of the parallel postulate. This work can be also treated as a further development of Tarski’s geometry in the formal setting [9].

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  • [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.

  • [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.

  • [4] Karol Borsuk and Wanda Szmielew. Podstawy geometrii. Państwowe Wydawnictwo Naukowe, Warszawa, 1955 (in Polish).

  • [5] Roland Coghetto. Homography in ℝℙ2. Formalized Mathematics, 24(4):239–251, 2016. doi:10.1515/forma-2016-0020.

  • [6] Roland Coghetto. Group of homography in real projective plane. Formalized Mathematics, 25(1):55–62, 2017. doi:10.1515/forma-2017-0005.

  • [7] Roland Coghetto. Klein-Beltrami model. Part I. Formalized Mathematics, 26(1):21–32, 2018. doi:10.2478/forma-2018-0003.

  • [8] Roland Coghetto. Pascal’s theorem in real projective plane. Formalized Mathematics, 25 (2):107–119, 2017. doi:10.1515/forma-2017-0011.

  • [9] Adam Grabowski. Tarski’s geometry modelled in Mizar computerized proof assistant. In Proceedings of the 2016 Federated Conference on Computer Science and Information Systems, FedCSIS 2016, Gdańsk, Poland, September 11–14, 2016, pages 373–381, 2016. doi:10.15439/2016F290.

  • [10] Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.

  • [11] Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381–383, 2003.

  • [12] 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.

  • [13] Yatsuka Nakamura. Graph theoretical properties of arcs in the plane and Fashoda Meet Theorem. Formalized Mathematics, 7(2):193–201, 1998.

  • [14] Karol Pąk and Andrzej Trybulec. Laplace expansion. Formalized Mathematics, 15(3): 143–150, 2007. doi:10.2478/v10037-007-0016-5.

  • [15] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4): 535–545, 1991.

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