Tarski Geometry Axioms. Part III

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Summary

In the article, we continue the formalization of the work devoted to Tarski’s geometry - the book “Metamathematische Methoden in der Geometrie” by W. Schwabhäuser, W. Szmielew, and A. Tarski. After we prepared some introductory formal framework in our two previous Mizar articles, we focus on the regular translation of underlying items faithfully following the abovementioned book (our encoding covers first seven chapters). Our development utilizes also other formalization efforts of the same topic, e.g. Isabelle/HOL by Makarios, Metamath or even proof objects obtained directly from Prover9. In addition, using the native Mizar constructions (cluster registrations) the propositions (“Satz”) are reformulated under weaker conditions, i.e. by using fewer axioms or by proposing an alternative version that uses just another axioms (ex. Satz 2.1 or Satz 2.2).

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  • [1] Michael Beeson and Larry Wos. OTTER proofs in Tarskian geometry. In International Joint Conference on Automated Reasoning volume 8562 of Lecture Notes in Computer Science pages 495-510. Springer 2014. doi: 10.1007/978-3-319-08587-6 38.

  • [2] Gabriel Braun and Julien Narboux. A synthetic proof of Pappus’ theorem in Tarski’s geometry. Journal of Automated Reasoning 58(2):23 2017. doi: 10.1007/s10817-016-9374-4.

  • [3] Roland Coghetto and Adam Grabowski. Tarski geometry axioms - Part II. Formalized Mathematics 24(2):157-166 2016. doi: 10.1515/forma-2016-0012.

  • [4] Sana Stojanovic Durdevic Julien Narboux and Predrag Janiˇcic. Automated generation of machine verifiable and readable proofs: a case study of Tarski’s geometry. Annals of Mathematics and Artificial Intelligence 74(3-4):249-269 2015.

  • [5] Adam Grabowski. Tarski’s geometry modelled in Mizar computerized proof assistant. In Maria Ganzha Leszek Maciaszek and Marcin Paprzycki editors Proceedings of the 2016 Federated Conference on Computer Science and Information Systems (FedCSIS) volume 8 of ACSIS - Annals of Computer Science and Information 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 simplification of Tarski’s axioms of geometry. Note di Matematica 33(2):123-132 2014.

  • [10] 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.

  • [11] William Richter Adam Grabowski and Jesse Alama. Tarski geometry axioms. Formalized Mathematics 22(2):167-176 2014. doi: 10.2478/forma-2014-0017.

  • [12] Wolfram Schwabhäuser Wanda Szmielew and Alfred Tarski. Metamathematische Methoden in der Geometrie. Springer-Verlag Berlin Heidelberg New York Tokyo 1983.

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