A Mechanical Verification of the Independence of Tarski's Euclidean Axiom

This thesis describes the mechanization of Tarski's axioms of plane geometry in the proof verification program Isabelle. The real Cartesian plane is mechanically verified to be a model of Tarski's axioms, thus verifying the consistency of the axiom system. The Klein–Beltrami model of the hyperbolic plane is also defined in Isabelle; in order to achieve this, the projective plane is defined and several theorems about it are proven. The Klein–Beltrami model is then shown in Isabelle to be a model of all of Tarski's axioms except his Euclidean axiom, thus mechanically verifying the independence of the Euclidean axiom — the primary goal of this project. For some of Tarski's axioms, only an insufficient or an inconvenient published proof was found for the theorem that states that the Klein–Beltrami model satisfies the axiom; in these cases, alternative proofs were devised and mechanically verified. These proofs are described in this thesis — most notably, the proof that the model satisfies the axiom of segment construction, and the proof that it satisfies the five-segments axiom. The proof that the model satisfies the upper 2-dimensional axiom also uses some of the lemmas that were used to prove that the model satisfies the five-segments axiom

Group of Homography in Real Projective Plane

Using the Mizar system [2], we formalized that homographies of the projective real plane (as defined in [5]), form a group. Then, we prove that, using the notations of Borsuk and Szmielew in [3] “Consider in space ℝℙ2 points P1, P2, P3, P4 of which three points are not collinear and points Q1,Q2,Q3,Q4 each three points of which are also not collinear. There exists one homography h of space ℝℙ2 such that h(Pi) = Qi for i = 1, 2, 3, 4. ”(Existence Statement 52 and Existence Statement 53) [3]. Or, using notations of Richter [11] “Let [a], [b], [c], [d] in ℝℙ2 be four points of which no three are collinear and let [a′],[b′],[c′],[d′] in ℝℙ2 be another four points of which no three are collinear, then there exists a 3 × 3 matrix M such that [Ma] = [a′], [Mb] = [b′], [Mc] = [c′], and [Md] = [d′] ”Makarios has formalized the same results in Isabelle/Isar (the collineations form a group, lemma statement52-existence and lemma statement 53-existence) and published it in Archive of Formal Proofs [10], [9].Rue de la Brasserie 5, 7100, La Louvière, BelgiumGrzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz 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.Karol Borsuk and Wanda Szmielew. Foundations of Geometry. North Holland, 1960.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Roland Coghetto. Homography in RP2. Formalized Mathematics, 24(4):239-251, 2016.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381-383, 2003.Wojciech Leończuk and Krzysztof Prazmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761-766, 1990.Timothy James McKenzie Makarios. The independence of Tarski’s Euclidean Axiom. Archive of Formal Proofs, October 2012.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.Jürgen Richter-Gebert. Perspectives on projective geometry: a guided tour through real and complex geometry. Springer Science & Business Media, 2011

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

A Mechanical Verification of the Independence of Tarski's Euclidean Axiom

This thesis describes the mechanization of Tarski's axioms of plane geometry in the proof verification program Isabelle. The real Cartesian plane is mechanically verified to be a model of Tarski's axioms, thus verifying the consistency of the axiom system. The Klein–Beltrami model of the hyperbolic plane is also defined in Isabelle; in order to achieve this, the projective plane is defined and several theorems about it are proven. The Klein–Beltrami model is then shown in Isabelle to be a model of all of Tarski's axioms except his Euclidean axiom, thus mechanically verifying the independence of the Euclidean axiom — the primary goal of this project. For some of Tarski's axioms, only an insufficient or an inconvenient published proof was found for the theorem that states that the Klein–Beltrami model satisfies the axiom; in these cases, alternative proofs were devised and mechanically verified. These proofs are described in this thesis — most notably, the proof that the model satisfies the axiom of segment construction, and the proof that it satisfies the five-segments axiom. The proof that the model satisfies the upper 2-dimensional axiom also uses some of the lemmas that were used to prove that the model satisfies the five-segments axiom

A Mechanical Verification of the Independence of Tarski's Euclidean Axiom

This thesis describes the mechanization of Tarski's axioms of plane geometry in the proof verification program Isabelle. The real Cartesian plane is mechanically verified to be a model of Tarski's axioms, thus verifying the consistency of the axiom system. The Klein–Beltrami model of the hyperbolic plane is also defined in Isabelle; in order to achieve this, the projective plane is defined and several theorems about it are proven. The Klein–Beltrami model is then shown in Isabelle to be a model of all of Tarski's axioms except his Euclidean axiom, thus mechanically verifying the independence of the Euclidean axiom — the primary goal of this project. For some of Tarski's axioms, only an insufficient or an inconvenient published proof was found for the theorem that states that the Klein–Beltrami model satisfies the axiom; in these cases, alternative proofs were devised and mechanically verified. These proofs are described in this thesis — most notably, the proof that the model satisfies the axiom of segment construction, and the proof that it satisfies the five-segments axiom. The proof that the model satisfies the upper 2-dimensional axiom also uses some of the lemmas that were used to prove that the model satisfies the five-segments axiom.</p