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-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. Foundations of Geometry . North Holland, 1960. [5] Karol Borsuk and Wanda Szmielew. Podstawy geometrii . Państwowe Wydawnictwo Naukowe, Warszawa, 1955 (in Polish). [6] Roland Coghetto. Group of homography in real projective plane. Formalized Mathematics , 25( 1 ):55–62, 2017. doi:10.1515/forma

-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. Foundations of Geometry . North Holland, 1960. [5] Karol Borsuk and Wanda Szmielew. Podstawy geometrii . Państwowe Wydawnictwo Naukowe, Warszawa, 1955 (in Polish). [6] Roland Coghetto. Homography in 𝕉𝕇 2 . Formalized Mathematics , 24( 4 ):239–251, 2016. doi:10.1515/forma-2016-0020. [7] Roland

Axiomatic Foundations of Geometry . PhD thesis, University of California-Berkeley, 1965. [12] Julien Narboux. Mechanical theorem proving in Tarski’s geometry. In Francisco Botana and Tomas Recio, editors, Automated Deduction in Geometry , pages 139–156, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg. ISBN 978-3-540-77356-6. [13] William Richter, Adam Grabowski, and Jesse Alama. Tarski geometry axioms. Formalized Mathematics , 22( 2 ):167–176, 2014. doi:10.2478/forma-2014-0017. [14] Wolfram Schwabhcuser, Wanda Szmielew, and Alfred Tarski. Metamathematische

Systems, pages 373-381, 2016. doi: 10.15439/2016F290. [6] Haragauri Narayan Gupta. Contributions to the Axiomatic Foundations of Geometry. PhD thesis, University of California-Berkeley, 1965. [7] Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean Axiom. Victoria University ofWellington, New Zealand, 2012. Master’s thesis. [8] Timothy James McKenzie Makarios. The independence of Tarski’s Euclidean Axiom. Archive of Formal Proofs, October 2012. Formal proof development. [9] Timothy James McKenzie Makarios. A further

Summary

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.

Summary

In our earlier article , the first part of axioms of geometry proposed by Alfred Tarski was formally introduced by means of Mizar proof assistant . 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 . 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 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 , first approaches to this topic were done in Mizar in 1990 (even if this article started formal Hilbert axiomatization of geometry, and parallel development was rather unlikely at that time ). 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 clarified things a bit. Our development allows for the future faithful mechanization of and opens the possibility of automatically generated Prover9 proofs which was useful in the case of lattice theory .

Summary

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” [], [], [], [].

With the Mizar system [], [] we use some ideas are taken from Tim Makarios’ MSc thesis [] 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 []. Note that the model presented here, may also be called “Beltrami-Klein Model”, “Klein disk model”, and the “Cayley-Klein model” [].

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].

References 1. Matiss, I. (2014). Multi-element sensor for non-destructive measurement of the dielectric permittivity and thickness of dielectric plates and shells. NDT and E International, 66, 99-105. 2. Armitage D. H., & Gardiner, S. J. (2001). Classical Potential Theory. Springer, pp. 89-118. 3. Matiss, I. (2011). Electrical Measurement Techniques -a New Challenge for Non- Destructive Testing. Publishing House of Riga Technical University, Riga (in Latvian), pp.44-56. 4. Venema, G.A. (2005). The Foundations of Geometry. Prentice Hall, New Jersey, pp. 111-113.

References [1] D. Barbilian - Pagini inedite, volume 2, Editors G. Barbilian and V. G. Vodă, Editura Albatros, 1984. [2] W. Boskoff, S. Barcanescu - On the Tzitzeica-Johnson configuration, Jour- nal of Geometry, 96(2009), 57-61. [3] W. Boskoff and P. Horja - The characterization of some special Barbilian spaces using the Tzitzeica construction, Stud. Cere. Mat. 46 (1994), No. 5, 503-514. [4] A. Emch - Remarks on the foregoing circle theorem, American Mathemat- ical Monthly, 23(1916), 162-164. [5] M. J. Greenberg - Anstotle’s axiom in the foundations of geometry J