Tarski Geometry Axioms

This is the translation of the Mizar article containing readable Mizar proofs of some axiomatic geometry theorems formulated by the great Polish mathematician Alfred Tarski [8], and we hope to continue this work. The article is an extension and upgrading of the source code written by the first author with the help of miz3 tool; his primary goal was to use proof checkers to help teach rigorous axiomatic geometry in high school using Hilbert’s axioms. This is largely a Mizar port of Julien Narboux’s Coq pseudo-code [6]. We partially prove the theorem of [7] that Tarski’s (extremely weak!) plane geometry axioms imply Hilbert’s axioms. Specifically, we obtain Gupta’s amazing proof which implies Hilbert’s axiom I1 that two points determine a line. The primary Mizar coding was heavily influenced by [9] on axioms of incidence geometry. The original development was much improved using Mizar adjectives instead of predicates only, and to use this machinery in full extent, we have to construct some models of Tarski geometry. These are listed in the second section, together with appropriate registrations of clusters. Also models of Tarski’s geometry related to real planes were constructed.Richter William - Departament of Mathematics Nortwestern University Evanston, USAGrabowski Adam - Institute of Informatics University of Białystok Akademicka 2, 15-267 Białystok PolandAlama Jesse - Technical University of Vienna AustriaGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.Julien Narboux. Mechanical theorem proving in Tarski’s geometry. In F. Botana and T. Recio, editors, Automated Deduction in Geometry, volume 4869, pages 139–156, 2007.Wolfram Schwabhäuser, Wanda Szmielew, and Alfred Tarski. Metamathematische Methoden in der Geometrie. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.Alfred Tarski and Steven Givant. Tarski’s system of geometry. Bulletin of Symbolic Logic, 5(2):175–214, 1999.Wojciech A. Trybulec. Axioms of incidence. Formalized Mathematics, 1(1):205–213, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990. Received June 16, 201

Tarski Geometry Axioms – Part II

In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation),of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1),congruence equivalence relation (A2),congruence identity (A3),segment construction (A4),SAS (A5),betweenness identity (A6),Pasch (A7). Also a simple model, which satisfies these axioms, was previously constructed, and described in [6]. In this paper, we deal with four remaining axioms, namely: the lower dimension axiom (A8),the upper dimension axiom (A9),the Euclid axiom (A10),the continuity axiom (A11). They were introduced in the form of Mizar attributes. Additionally, the relation of congruence of triangles cong is introduced via congruence of sides (SSS).In order to show that the structure which satisfies all eleven Tarski’s axioms really exists, we provided a proof of the registration of a cluster that the Euclidean plane, or rather a natural [5] extension of ordinary metric structure Euclid 2 satisfies all these attributes.Although the tradition of the mechanization of Tarski’s geometry in Mizar is not as long as in Coq [11], first approaches to this topic were done in Mizar in 1990 [16] (even if this article started formal Hilbert axiomatization of geometry, and parallel development was rather unlikely at that time [8]). Connection with another proof assistant should be mentioned – we had some doubts about the proof of the Euclid’s axiom and inspection of the proof taken from Archive of Formal Proofs of Isabelle [10] clarified things a bit. Our development allows for the future faithful mechanization of [13] and opens the possibility of automatically generated Prover9 proofs which was useful in the case of lattice theory [7].Coghetto Roland - Rue de la Brasserie 5, 7100 La Louvière, BelgiumGrabowski Adam - Institute of Informatics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, PolandCzesław Byliński. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99–107, 2005.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Roland Coghetto. Circumcenter, circumcircle and centroid of a triangle. Formalized Mathematics, 24(1):17–26, 2016. doi:10.1515/forma-2016-0002.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.Adam Grabowski. Efficient rough set theory merging. Fundamenta Informaticae, 135(4): 371–385, 2014. doi:10.3233/FI-2014-1129.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.Adam Grabowski. Mechanizing complemented lattices within Mizar system. Journal of Automated Reasoning, 55:211–221, 2015. doi:10.1007/s10817-015-9333-5.Adam Grabowski and Christoph Schwarzweller. On duplication in mathematical repositories. In Serge Autexier, Jacques Calmet, David Delahaye, Patrick D. F. Ion, Laurence Rideau, Renaud Rioboo, and Alan P. Sexton, editors, Intelligent Computer Mathematics, 10th International Conference, AISC 2010, 17th Symposium, Calculemus 2010, and 9th International Conference, MKM 2010, Paris, France, July 5–10, 2010. Proceedings, volume 6167 of Lecture Notes in Computer Science, pages 300–314. Springer, 2010. doi:10.1007/978-3-642-14128-7_26.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.Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean Axiom. 2012. Master’s thesis.Julien Narboux. Mechanical theorem proving in Tarski’s geometry. In F. Botana and T. Recio, editors, Automated Deduction in Geometry, volume 4869 of Lecture Notes in Computer Science, pages 139–156. Springer, 2007.William Richter, Adam Grabowski, and Jesse Alama. Tarski geometry axioms. Formalized Mathematics, 22(2):167–176, 2014. doi:10.2478/forma-2014-0017.Wolfram Schwabhäuser, Wanda Szmielew, and Alfred Tarski. Metamathematische Methoden in der Geometrie. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.Alfred Tarski and Steven Givant. Tarski’s system of geometry. Bulletin of Symbolic Logic, 5(2):175–214, 1999.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.Wojciech A. Trybulec. Axioms of incidence. Formalized Mathematics, 1(1):205–213, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990

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

Klein-Beltrami Model. Part I

Tim 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” [3], [4], [14], [5]. With the Mizar system [2], [7] we use some ideas are taken from Tim Makarios’ MSc thesis [13] for the formalization of some definitions (like the absolute) and lemmas necessary for the verification of the independence of the parallel postulate. This work can be also treated as further development of Tarski’s geometry in the formal setting [6]. Note that the model presented here, may also be called “Beltrami-Klein Model”, “Klein disk model”, and the “Cayley-Klein model” [1].Rue de la Brasserie 5, 7100 La Louvière, BelgiumNorbert A’Campo and Athanase Papadopoulos. On Klein’s so-called non-Euclidean geometry. arXiv preprint arXiv:1406.7309, 2014.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.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. Podstawy geometrii. Państwowe Wydawnictwo Naukowe, Warszawa, 1955 (in Polish).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.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.Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381–383, 2003.Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in Tn . Formalized Mathematics, 12(3):301–306, 2004.Artur Korniłowicz and Yasunari Shidama. Some properties of circles on the plane. Formalized Mathematics, 13(1):117–124, 2005.Akihiro Kubo. Lines in n -dimensional Euclidean spaces. Formalized Mathematics, 11(4): 371–376, 2003.Xiquan Liang, Piqing Zhao, and Ou Bai. Vector functions and their differentiation formulas in 3-dimensional Euclidean spaces. Formalized Mathematics, 18(1):1–10, 2010. doi:10.2478/v10037-010-0001-2.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.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4): 535–545, 1991.Xiaopeng Yue, Xiquan Liang, and Zhongpin Sun. Some properties of some special matrices. Formalized Mathematics, 13(4):541–547, 2005.261213

Klein-Beltrami model. Part IV

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 and lemmas necessary for the verification of the independence of the parallel postulate. In this article, which is the continuation of [8], we prove that our constructed model satisfies the axioms of segment construction, the axiom of betweenness identity, and the axiom of Pasch due to Tarski, as formalized in [11] and related Mizar articles.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. Homography in RP2. Formalized Mathematics, 24(4):239–251, 2016.doi:10.1515/forma-2016-0020.Roland Coghetto. Klein-Beltrami model. Part I. Formalized Mathematics, 26(1):21–32, 2018. doi:10.2478/forma-2018-0003.Roland Coghetto. Klein-Beltrami model. Part III. Formalized Mathematics, 28(1):1–7, 2020. doi:10.2478/forma-2020-0001.Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381–383, 2003.Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean Axiom. Victoria University ofWellington, New Zealand, 2012. Master’s thesis.William Richter, Adam Grabowski, and Jesse Alama. Tarski geometry axioms. Formalized Mathematics, 22(2):167–176, 2014. doi:10.2478/forma-2014-0017.92