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

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

Duality Notions in Real Projective Plane

This work has been supported by the Centre autonome de formation et de recherche en mathematiques et sciences avec assistants de preuve, ASBL (non-profit organization). Enterprise number: 0777.779.751. BelgiumIn this article, we check with the Mizar system [1], [2], the converse of Desargues’ theorem and the converse of Pappus’ theorem of the real projective plane. It is well known that in the projective plane, the notions of points and lines are dual [11], [9], [15], [8]. Some results (analytical, synthetic, combinatorial) of projective geometry are already present in some libraries Lean/Hott [5], Isabelle/Hol [3], Coq [13], [14], [4], Agda [6], . . . . Proofs of dual statements by proof assistants have already been proposed, using an axiomatic method (for example see in [13] - the section on duality: “[...] For every theorem we prove, we can easily derive its dual using our function swap [...]2”). In our formalisation, we use an analytical rather than a synthetic approach using the definitions of Leończuk and Prażmowski of the projective plane [12]. Our motivation is to show that it is possible by developing dual definitions to find proofs of dual theorems in a few lines of code. In the first part, rather technical, we introduce definitions that allow us to construct the duality between the points of the real projective plane and the lines associated to this projective plane. In the second part, we give a natural definition of line concurrency and prove that this definition is dual to the definition of alignment. Finally, we apply these results to find, in a few lines, the dual properties and theorems of those defined in the article [12] (transitive, Vebleian, at_least_3rank, Fanoian, Desarguesian, 2-dimensional). We hope that this methodology will allow us to continued more quickly the proof started in [7] that the Beltrami-Klein plane is a model satisfying the axioms of the hyperbolic plane (in the sense of Tarski geometry [10]).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:10.1007/978-3-319-20615-817.Grzegorz 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.Anthony Bordg. Projective geometry. Archive of Formal Proofs, jun 2018.David Braun. Approche combinatoire pour l’automatisation en Coq des preuves formelles en g´eom´etrie d’incidence projective. PhD thesis, Universit´e de Strasbourg, 2019.Ulrik Buchholtz and Egbert Rijke. The real projective spaces in homotopy type theory. In 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1–8. IEEE, 2017.Guillermo Calderón. Formalizing constructive projective geometry in Agda. Electronic Notes in Theoretical Computer Science, 338:61–77, 2018.Roland Coghetto. Klein-Beltrami model. Part I. Formalized Mathematics, 26(1):21–32, 2018. doi:10.2478/forma-2018-0003.Harold Scott Macdonald Coxeter. The real projective plane. Springer Science & Business Media, 1992.Nikolai Vladimirovich Efimov. G´eom´etrie sup´erieure. Mir, 1981.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.Robin Hartshorne. Foundations of projective geometry. Citeseer, 1967.Wojciech Leończuk and Krzysztof Prażmowski. Projective spaces – part I. Formalized Mathematics, 1(4):767–776, 1990.Nicolas Magaud, Julien Narboux, and Pascal Schreck. Formalizing projective plane geometry in Coq. In Automated Deduction in Geometry, pages 141–162. Springer, 2008.Nicolas Magaud, Julien Narboux, and Pascal Schreck. A case study in formalizing projective geometry in Coq: Desargues theorem. Computational Geometry, 45(8):406–424, 2012.Jurgen Richter-Gebert. Pappos’s Theorem: Nine Proofs and Three Variations, pages 3–31. Springer Berlin Heidelberg, 2011. ISBN 978-3-642-17286-1. doi:10.1007/978-3-642-17286-1-1.29416117

The Lean mathematical library

This paper describes mathlib, a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant. Among proof assistant libraries, it is distinguished by its dependently typed foundations, focus on classical mathematics, extensive hierarchy of structures, use of large- and small-scale automation, and distributed organization. We explain the architecture and design decisions of the library and the social organization that has led us here

A Collective Adaptive Approach to Decentralised k-Coverage in Multi-robot Systems

We focus on the online multi-object k-coverage problem (OMOkC), where mobile robots are required to sense a mobile target from k diverse points of view, coordinating themselves in a scalable and possibly decentralised way. There is active research on OMOkC, particularly in the design of decentralised algorithms for solving it. We propose a new take on the issue: Rather than classically developing new algorithms, we apply a macro-level paradigm, called aggregate computing, specifically designed to directly program the global behaviour of a whole ensemble of devices at once. To understand the potential of the application of aggregate computing to OMOkC, we extend the Alchemist simulator (supporting aggregate computing natively) with a novel toolchain component supporting the simulation of mobile robots. This way, we build a software engineering toolchain comprising language and simulation tooling for addressing OMOkC. Finally, we exercise our approach and related toolchain by introducing new algorithms for OMOkC; we show that they can be expressed concisely, reuse existing software components and perform better than the current state-of-the-art in terms of coverage over time and number of objects covered overall

Formalizing the Ring of Witt Vectors

The ring of Witt vectors $\mathbb{W} R$ over a base ring $R$ is an important tool in algebraic number theory and lies at the foundations of modern $p$-adic Hodge theory. $\mathbb{W} R$ has the interesting property that it constructs a ring of characteristic $0$ out of a ring of characteristic $p > 1$, and it can be used more specifically to construct from a finite field containing $\mathbb{Z}/p\mathbb{Z}$ the corresponding unramified field extension of the $p$-adic numbers $\mathbb{Q}_p$ (which is unique up to isomorphism). We formalize the notion of a Witt vector in the Lean proof assistant, along with the corresponding ring operations and other algebraic structure. We prove in Lean that, for prime $p$, the ring of Witt vectors over $\mathbb{Z}/p\mathbb{Z}$ is isomorphic to the ring of $p$-adic integers $\mathbb{Z}_p$. In the process we develop idioms to cleanly handle calculations of identities between operations on the ring of Witt vectors. These calculations are intractable with a naive approach, and require a proof technique that is usually skimmed over in the informal literature. Our proofs resemble the informal arguments while being fully rigorous

Proceedings of the 4th Workshop of the MPM4CPS COST Action

Proceedings of the 4th Workshop of the MPM4CPS COST Action with the presentations delivered during the workshop and papers with extended versions of some of them

Upowszechnianie wyników badań naukowych w międzynarodowych bazach danych : analiza biometryczna na przykładzie nauk technicznych, ze szczególnym uwzględnieniem elektrotechniki

The issues of bibliometrics, scientometrics, informetrics and webometrics have an important place among research subject undertaken by Polish and foreign scholars. Initially, these notions were used only by researchers in the fields of library science, scientometrics and information science. However, at the turn of the 20th and 21st centuries, quantitative methods became a fundamental tool for evaluation of, among others, sources of academic communication, academic research, research and academic centers. One of the elements of the evaluation is a quantitative analysis of academic publications in databases with international access. It is of particular importance in the case of technical sciences. This work is an attempt at a quantitative analysis of publications by Polish authors (affiliated to Polish technical universities) and Polish journals on electrotechnics in international databases. The contents are organized into four chapters with an introduction, conclusions, bibliography, name index and a list of figures and illustrations. Chapters one and two are devoted to theoretical issues, whereas chapters three and four – to practical issues. In the first chapter, selected issues concerning quantitative methods were presented, including an analysis of literature and a discussion over terminology carried out in book publications and journals. Moreover, selected examples of research conducted with the use of quantitative methods (including rankings and scientific reports) were discussed in this chapter). In chapter two, sources of information on academic publications, their origins and development (from bibliographic bases to citation indexes) were presented. A separate subchapter was devoted to databases of academic publications created by libraries of technical universities, and to indicators in the assessment of academic publications. Chapter three deals with electrotechnics as a field of science. An analysis was conducted with regard to the place of electrotechnics in science classifications based on selected examples, and the development of the teaching of electrotechnics at university level was shown. In this chapter, early and contemporary Polish journals on electrotechnics were presented, including journals published by technical universities themselves. Chapter four contains the results of an analysis of international databases (Scopus, WoS, CC), focusing on the representation of Polish journals, including their citations, and publications of authors with affiliation to Polish technical universities. Final conclusions of research and analyses have brought an answer to questions raised with regard to the assessment of representation in international databases of publications by Polish authors affiliated to Polish technical universities (in its various aspects, e.g., a publication type, language of publication, publication dynamics taking into account years of publications, cooperation with representatives of other European and non-European countries), and of Polish journals

Journal of Telecommunications and Information Technology, 2019, nr 2

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