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dc.contributor.authorCoghetto, Roland-
dc.contributor.authorGrabowski, Adam-
dc.date.accessioned2019-05-21T07:18:03Z-
dc.date.available2019-05-21T07:18:03Z-
dc.date.issued2019-
dc.identifier.citationFormalized Mathematics, Volume 27, Issue 1, Pages 75-85-
dc.identifier.issn1426-2630-
dc.identifier.urihttp://hdl.handle.net/11320/7842-
dc.description.abstractIn the article, we continue [7] the formalization of the work devoted to Tarski’s geometry – the book “Metamathematische Methoden in der Geometrie” (SST for short) by W. Schwabhäuser, W. Szmielew, and A. Tarski [14], [9], [10]. We use the Mizar system to systematically formalize Chapter 8 of the SST book.We define the notion of right angle and prove some of its basic properties, a theory of intersecting lines (including orthogonality). Using the notion of perpendicular foot, we prove the existence of the midpoint (Satz 8.22), which will be used in the form of the Mizar functor (as the uniqueness can be easily shown) in Chapter 10. In the last section we give some lemmas proven by means of Otter during Tarski Formalization Project by M. Beeson (the so-called Section 8A of SST).-
dc.language.isoen-
dc.publisherDeGruyter Open-
dc.subjectTarski geometry-
dc.subjectfoundations of geometry-
dc.subjectright angle-
dc.titleTarski Geometry Axioms. Part IV – Right Angle-
dc.typeArticle-
dc.identifier.doi10.2478/forma-2019-0008-
dc.description.AffiliationRoland Coghetto - Rue de la Brasserie 5, 7100 La Louvi`ere, Belgium-
dc.description.AffiliationAdam Grabowski - Institute of Informatics, University of Białystok, Poland-
dc.description.referencesGrzegorz 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.-
dc.description.referencesGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.-
dc.description.referencesMichael 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.-
dc.description.referencesMichael Beeson, Julien Narboux, and Freek Wiedijk. Proof-checking Euclid. Annals of Mathematics and Artificial Intelligence, Jan 2019. doi:10.1007/s10472-018-9606-x.-
dc.description.referencesPierre Boutry, Gabriel Braun, and Julien Narboux. Formalization of the Arithmetization of Euclidean Plane Geometry and Applications. Journal of Symbolic Computation, 90: 149–168, 2019. doi:10.1016/j.jsc.2018.04.007.-
dc.description.referencesPierre Boutry, Charly Gries, Julien Narboux, and Pascal Schreck. Parallel postulates and continuity axioms: a mechanized study in intuitionistic logic using Coq. Journal of Automated Reasoning, 62(1):1–68, 2019.-
dc.description.referencesRoland Coghetto and Adam Grabowski. Tarski geometry axioms. Part III. Formalized Mathematics, 25(4):289–313, 2017. doi:10.1515/forma-2017-0028.-
dc.description.referencesSana Stojanovic Durdevic, Julien Narboux, and Predrag Janičić. 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.-
dc.description.referencesAdam 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.-
dc.description.referencesAdam 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, 2016.-
dc.description.referencesHaragauri Narayan Gupta. Contributions to the Axiomatic Foundations of Geometry. PhD thesis, University of California-Berkeley, 1965.-
dc.description.referencesJulien 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.-
dc.description.referencesWilliam Richter, Adam Grabowski, and Jesse Alama. Tarski geometry axioms. Formalized Mathematics, 22(2):167–176, 2014. doi:10.2478/forma-2014-0017.-
dc.description.referencesWolfram Schwabhcuser, Wanda Szmielew, and Alfred Tarski. Metamathematische Methoden in der Geometrie. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.-
dc.identifier.eissn1898-9934-
dc.description.volume27-
dc.description.issue1-
dc.description.firstpage75-
dc.description.lastpage85-
dc.identifier.citation2Formalized Mathematics-
dc.identifier.orcid0000-0002-4901-0766-
dc.identifier.orcid0000-0001-5026-3990-
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Formalized Mathematics, 2019, Volume 27, Issue 1

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