Discovering the Impact of Knowledge in Recommender Systems: A Comparative Study

Recommender systems engage user profiles and appropriate filtering techniques to assist users in finding more relevant information over the large volume of information. User profiles play an important role in the success of recommendation process since they model and represent the actual user needs. However, a comprehensive literature review of recommender systems has demonstrated no concrete study on the role and impact of knowledge in user profiling and filtering approache. In this paper, we review the most prominent recommender systems in the literature and examine the impression of knowledge extracted from different sources. We then come up with this finding that semantic information from the user context has substantial impact on the performance of knowledge based recommender systems. Finally, some new clues for improvement the knowledge-based profiles have been proposed.Comment: 14 pages, 3 tables; International Journal of Computer Science & Engineering Survey (IJCSES) Vol.2, No.3, August 201

Aspects of Assembly and Cascaded Aspects of Assembly: Logical and Temporal Properties

Highly dynamic computing environments, like ubiquitous and pervasive computing environments, require frequent adaptation of applications. This has to be done in a timely fashion, and the adaptation process must be as fast as possible and mastered. Moreover the adaptation process has to ensure a consistent result when finished whereas adaptations to be implemented cannot be anticipated at design time. In this paper we present our mechanism for self-adaptation based on the aspect oriented programming paradigm called Aspect of Assembly (AAs). Using AAs: (1) the adaptations process is fast and its duration is mastered; (2) adaptations' entities are independent of each other thanks to the weaver logical merging mechanism; and (3) the high variability of the software infrastructure can be managed using a mono or multi-cycle weaving approach.Comment: 14 pages, published in International Journal of Computer Science, Volume 8, issue 4, Jul 2011, ISSN 1694-081

Combinatorics on Words 10th International Conference

This volume contains the Local Proceedings of the Tenth International Conference on WORDS, that took place at the Kiel University, Germany, from the 14th to the 17th September 2015. WORDS is the main conference series devoted to the mathematical theory of words, and it takes place every two years. The first conference in the series was organised in 1997 in Rouen, France, with the following editions taking place in Rouen, Palermo,Turku, Montreal, Marseille, Salerno, Prague, and Turku. The main object in the scope of the conference, words, are finite or infinite sequences of symbols over a finite alphabet. They appear as natural and basic mathematical model in many areas, theoretical or applicative. Accordingly, the WORDS conference is open to both theoretical contributions related to combinatorial, algebraic, and algorithmic aspects of words, as well as to contributions presenting application of the theory of words, for instance, in other fields of computer science, inguistics, biology and bioinformatics, or physics. For the second time in the history of WORDS, after the 2013 edition, a refereed proceedings volume was published in Springer’s Lecture Notes in Computer Science series. In addition, this local proceedings volume was published in the Kiel Computer Science Series of the Kiel University. Being a conference at the border between theoretical computer science and mathematics, WORDS tries to capture in its two proceedings volumes the characteristics of the conferences from both these worlds. While the Lecture Notes in Computer Science volume was dedicated to formal contributions, this local proceedings volume allows, in the spirit of mathematics conferences, the publication of several contributions informing on current research and work in progress in areas closely connected to the core topics of WORDS. All the papers, the ones published in the Lecture Notes in Computer Science proceedings volume or the ones from this volume, were refereed to high standards by the members of the Program Committee. Following the conference, a special issue of the Theoretical Computer Science journal will be edited, containing extended versions of papers from both proceedings volumes. In total, the conference hosted 18 contributed talks. The papers on which 14 of these talks were based, were published in th LNCS volume; the other 4 are published in this volume. In addition to the contributed talks, the conference program included six invited talks given by leading experts in the areas covered by the WORDS conference: Jörg Endrullis (Amsterdam), Markus Lohrey (Siegen), Jean Néraud (Rouen), Dominique Perrin (Paris), Michaël Rao (Lyon), Thomas Stoll (Nancy). WORDS 2015 was the tenth conference in the series, so we were extremely happy to welcome, as invited speaker at this anniversary edition, Jean Néraud, one of the initiators of the series and the main organiser of the first two editions of this conference. We thank all the invited speakers and all the authors of submitted papers for their contributions to the the success of the conference. We are grateful to the members of the Program Committee for their work that lead to the selection of the contributed talks, and, implicitly, of the papers published in this volume. They were assisted in their task by a series of external referees, gratefully acknowledged below. The submission and reviewing process used the Easychair system; we thank Andrej Voronkov for this system which facilitated the work of the Programme Committee and the editors considerably. We grateful thank Gheorghe Iosif for designing the logo, poster, and banner of WORDS 2015; the logo of the conference can be seen on the front cover of this book. We also thank the editors of the Kiel Computer Science Series, especially Lasse Kliemann, for their support in editing this volume. Finally, we thank the Organising Committee of WORDS 2015 for ensuring the smooth run of the conference

Partial Correctness of a Power Algorithm

This work continues a formal verification of algorithms written in terms of simple-named complex-valued nominative data [6],[8],[15],[11],[12],[13]. In this paper we present a formalization in the Mizar system [3],[1] of the partial correctness of the algorithm: i := val.1 j := val.2 b := val.3 n := val.4 s := val.5 while (i n) i := i + j s := s * b return s computing the natural n power of given complex number b, where variables i, b, n, s are located as values of a V-valued Function, loc, as: loc/.1 = i, loc/.3 = b, loc/.4 = n and loc/.5 = s, and the constant 1 is located in the location loc/.2 = j (set V represents simple names of considered nominative data [17]).The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2],[4] with partial pre- and post-conditions [14],[16],[7],[5].Institute of Informatics, University of Białystok, PolandGrzegorz 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.R.W. Floyd. Assigning meanings to programs. Mathematical aspects of computer science, 19(19–32), 1967.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.C.A.R. Hoare. An axiomatic basis for computer programming. Commun. ACM, 12(10): 576–580, 1969.Ievgen Ivanov and Mykola Nikitchenko. On the sequence rule for the Floyd-Hoare logic with partial pre- and post-conditions. In Proceedings of the 14th International Conference on ICT in Education, Research and Industrial Applications. Integration, Harmonization and Knowledge Transfer. Volume II: Workshops, Kyiv, Ukraine, May 14–17, 2018, volume 2104 of CEUR Workshop Proceedings, pages 716–724, 2018.Ievgen Ivanov, Mykola Nikitchenko, Andrii Kryvolap, and Artur Korniłowicz. Simple-named complex-valued nominative data – definition and basic operations. Formalized Mathematics, 25(3):205–216, 2017. doi:10.1515/forma-2017-0020.Ievgen Ivanov, Artur Korniłowicz, and Mykola Nikitchenko. Implementation of the composition-nominative approach to program formalization in Mizar. The Computer Science Journal of Moldova, 26(1):59–76, 2018.Ievgen Ivanov, Artur Korniłowicz, and Mykola Nikitchenko. On an algorithmic algebra over simple-named complex-valued nominative data. Formalized Mathematics, 26(2):149–158, 2018. doi:10.2478/forma-2018-0012.Ievgen Ivanov, Artur Korniłowicz, and Mykola Nikitchenko. An inference system of an extension of Floyd-Hoare logic for partial predicates. Formalized Mathematics, 26(2): 159–164, 2018. doi:10.2478/forma-2018-0013.Ievgen Ivanov, Artur Korniłowicz, and Mykola Nikitchenko. Partial correctness of GCD algorithm. Formalized Mathematics, 26(2):165–173, 2018. doi:10.2478/forma-2018-0014.Ievgen Ivanov, Artur Korniłowicz, and Mykola Nikitchenko. On algebras of algorithms and specifications over uninterpreted data. Formalized Mathematics, 26(2):141–147, 2018. doi:10.2478/forma-2018-0011.Artur Kornilowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. Formalization of the algebra of nominative data in Mizar. In Maria Ganzha, Leszek A. Maciaszek, and Marcin Paprzycki, editors, Proceedings of the 2017 Federated Conference on Computer Science and Information Systems, FedCSIS 2017, Prague, Czech Republic, September 3–6, 2017., pages 237–244, 2017. ISBN 978-83-946253-7-5. doi:10.15439/2017F301.Artur Kornilowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. Formalization of the nominative algorithmic algebra in Mizar. In Leszek Borzemski, Jerzy Świątek, and Zofia Wilimowska, editors, Information Systems Architecture and Technology: Proceedings of 38th International Conference on Information Systems Architecture and Technology – ISAT 2017 – Part II, Szklarska Poręba, Poland, September 17–19, 2017, volume 656 of Advances in Intelligent Systems and Computing, pages 176–186. Springer, 2017. ISBN 978-3-319-67228-1. doi:10.1007/978-3-319-67229-8_16.Artur Korniłowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. An approach to formalization of an extension of Floyd-Hoare logic. In Vadim Ermolayev, Nick Bassiliades, Hans-Georg Fill, Vitaliy Yakovyna, Heinrich C. Mayr, Vyacheslav Kharchenko, Vladimir Peschanenko, Mariya Shyshkina, Mykola Nikitchenko, and Aleksander Spivakovsky, editors, Proceedings of the 13th International Conference on ICT in Education, Research and Industrial Applications. Integration, Harmonization and Knowledge Transfer, Kyiv, Ukraine, May 15–18, 2017, volume 1844 of CEUR Workshop Proceedings, pages 504–523. CEUR-WS.org, 2017.Artur Korniłowicz, Ievgen Ivanov, and Mykola Nikitchenko. Kleene algebra of partial predicates. Formalized Mathematics, 26(1):11–20, 2018. doi:10.2478/forma-2018-0002.Andrii Kryvolap, Mykola Nikitchenko, and Wolfgang Schreiner. Extending Floyd-Hoare logic for partial pre- and postconditions. In Vadim Ermolayev, Heinrich C. Mayr, Mykola Nikitchenko, Aleksander Spivakovsky, and Grygoriy Zholtkevych, editors, Information and Communication Technologies in Education, Research, and Industrial Applications: 9th International Conference, ICTERI 2013, Kherson, Ukraine, June 19–22, 2013, Revised Selected Papers, pages 355–378. Springer International Publishing, 2013. ISBN 978-3-319-03998-5. doi:10.1007/978-3-319-03998-5_18.Volodymyr G. Skobelev, Mykola Nikitchenko, and Ievgen Ivanov. On algebraic properties of nominative data and functions. In Vadim Ermolayev, Heinrich C. Mayr, Mykola Nikitchenko, Aleksander Spivakovsky, and Grygoriy Zholtkevych, editors, Information and Communication Technologies in Education, Research, and Industrial Applications – 10th International Conference, ICTERI 2014, Kherson, Ukraine, June 9–12, 2014, Revised Selected Papers, volume 469 of Communications in Computer and Information Science, pages 117–138. Springer, 2014. ISBN 978-3-319-13205-1. doi:10.1007/978-3-319-13206-8_6.27218919

On Weakly Associative Lattices and Near Lattices

The main aim of this article is to introduce formally two generalizations of lattices, namely weakly associative lattices and near lattices, which can be obtained from the former by certain weakening of the usual well-known axioms. We show selected propositions devoted to weakly associative lattices and near lattices from Chapter 6 of [15], dealing also with alternative versions of classical axiomatizations. Some of the results were proven in the Mizar [1], [2] system with the help of Prover9 [14] proof assistant.Damian Sawicki - Institute of Informatics, University of Białystok, PolandAdam Grabowski - Institute of Informatics, University of Białystok, PolandGrzegorz 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. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. 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.Garrett Birkhoff. Lattice Theory. Providence, Rhode Island, New York, 1967.B.A. Davey and H.A. Priestley. Introduction to Lattices and Order. Cambridge University Press, 2002.Ervin Fried and George Grätzer. Some examples of weakly associative lattices. Colloquium Mathematicum, 27:215–221, 1973. doi:10.4064/cm-27-2-215-221.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 Markus Moschner. Managing heterogeneous theories within a mathematical knowledge repository. In Andrea Asperti, Grzegorz Bancerek, and Andrzej Trybulec, editors, Mathematical Knowledge Management Proceedings, volume 3119 of Lecture Notes in Computer Science, pages 116–129. Springer, 2004. doi:10.1007/978-3-540-27818-4_9. 3rd International Conference on Mathematical Knowledge Management, Bialowieza, Poland, Sep. 19–21, 2004.Adam Grabowski and Damian Sawicki. On two alternative axiomatizations of lattices by McKenzie and Sholander. Formalized Mathematics, 26(2):193–198, 2018. doi:10.2478/forma-2018-0017.Adam Grabowski and Christoph Schwarzweller. Translating mathematical vernacular into knowledge repositories. In Michael Kohlhase, editor, Mathematical Knowledge Management, volume 3863 of Lecture Notes in Computer Science, pages 49–64. Springer, 2006. doi:https://doi.org/10.1007/11618027 4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15–17, 2005, Revised Selected Papers.Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. Equality in computer proof-assistants. In Ganzha, Maria and Maciaszek, Leszek and Paprzycki, Marcin, editor, Proceedings of the 2015 Federated Conference on Computer Science and Information Systems, volume 5 of ACSIS-Annals of Computer Science and Information Systems, pages 45–54. IEEE, 2015. doi:10.15439/2015F229.George Grätzer. General Lattice Theory. Academic Press, New York, 1978.George Grätzer. Lattice Theory: Foundation. Birkhäuser, 2011.Dominik Kulesza and Adam Grabowski. Formalization of quasilattices. Formalized Mathematics, 28(2):217–225, 2020. doi:10.2478/forma-2020-0019.William McCune. Prover9 and Mace4. 2005–2010.William McCune and Ranganathan Padmanabhan. Automated Deduction in Equational Logic and Cubic Curves. Springer-Verlag, Berlin, 1996.Ranganathan Padmanabhan and Sergiu Rudeanu. Axioms for Lattices and Boolean Algebras. World Scientific Publishers, 2008.Piotr Rudnicki and Josef Urban. Escape to ATP for Mizar. In First International Workshop on Proof eXchange for Theorem Proving-PxTP 2011, 2011.Stanisław Zukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215–222, 1990.292778

Formalization of Quasilattices

The main aim of this article is to introduce formally one of the generalizations of lattices, namely quasilattices, which can be obtained from the axiomatization of the former class by certain weakening of ordinary absorption laws. We show propositions QLT-1 to QLT-7 from [15], presenting also some short variants of corresponding axiom systems. Some of the results were proven in the Mizar [1], [2] system with the help of Prover9 [14] proof assistant.Dominik Kulesza - Institute of Informatics, University of Białystok, PolandAdam Grabowski - Institute of Informatics, University of Białystok, PolandGrzegorz 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.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.Garrett Birkhoff. Lattice Theory. Providence, Rhode Island, New York, 1967.B.A. Davey and H.A. Priestley. Introduction to Lattices and Order. Cambridge University Press, 2002.G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott. A Compendium of Continuous Lattices. Springer-Verlag, Berlin, Heidelberg, New York, 1980.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 Robert Milewski. Boolean posets, posets under inclusion and products of relational structures. Formalized Mathematics, 6(1):117–121, 1997.Adam Grabowski and Markus Moschner. Managing heterogeneous theories within a mathematical knowledge repository. In Andrea Asperti, Grzegorz Bancerek, and Andrzej Trybulec, editors, Mathematical Knowledge Management Proceedings, volume 3119 of Lecture Notes in Computer Science, pages 116–129. Springer, 2004. doi:10.1007/978-3-540-27818-4_9. 3rd International Conference on Mathematical Knowledge Management, Bialowieza, Poland, Sep. 19–21, 2004.Adam Grabowski and Damian Sawicki. On two alternative axiomatizations of lattices by McKenzie and Sholander. Formalized Mathematics, 26(2):193–198, 2018. doi:10.2478/forma-2018-0017.Adam Grabowski and Christoph Schwarzweller. Translating mathematical vernacular into knowledge repositories. In Michael Kohlhase, editor, Mathematical Knowledge Management, volume 3863 of Lecture Notes in Computer Science, pages 49–64. Springer, 2006. doi:https://doi.org/10.1007/11618027_4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15–17, 2005, Revised Selected Papers.Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. Equality in computer proof-assistants. In Ganzha, Maria and Maciaszek, Leszek and Paprzycki, Marcin, editor, Proceedings of the 2015 Federated Conference on Computer Science and Information Systems, volume 5 of ACSIS-Annals of Computer Science and Information Systems, pages 45–54. IEEE, 2015. doi:10.15439/2015F229.George Grätzer. General Lattice Theory. Academic Press, New York, 1978.George Grätzer. Lattice Theory: Foundation. Birkhäuser, 2011.William McCune. Prover9 and Mace4. 2005–2010.William McCune and Ranganathan Padmanabhan. Automated Deduction in Equational Logic and Cubic Curves. Springer-Verlag, Berlin, 1996.Ranganathan Padmanabhan and Sergiu Rudeanu. Axioms for Lattices and Boolean Algebras. World Scientific Publishers, 2008.Piotr Rudnicki and Josef Urban. Escape to ATP for Mizar. In First International Workshop on Proof eXchange for Theorem Proving-PxTP 2011, 2011.Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215–222, 1990.28221722

On Roots of Polynomials and Algebraically Closed Fields

SummaryIn this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].Institute of Informatics, University of Gdańsk, PolandGrzegorz 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.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.Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. On algebraic hierarchies in mathematical repository of Mizar. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2016 Federated Conference on Computer Science and Information Systems (FedCSIS), volume 8 of Annals of Computer Science and Information Systems, pages 363–371, 2016. doi:10.15439/2016F520.H. Heuser. Lehrbuch der Analysis. B.G. Teubner Stuttgart, 1990.Nathan Jacobson. Basic Algebra I. 2nd edition. Dover Publications Inc., 2009.Heinz Lüneburg. Gruppen, Ringe, Körper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1990.Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991.Christoph Schwarzweller and Agnieszka Rowińska-Schwarzweller. Schur’s theorem on the stability of networks. Formalized Mathematics, 14(4):135–142, 2006. doi:10.2478/v10037-006-0017-9.Christoph Schwarzweller, Artur Korniłowicz, and Agnieszka Rowińska-Schwarzweller. Some algebraic properties of polynomial rings. Formalized Mathematics, 24(3):227–237, 2016. doi:10.1515/forma-2016-0019.25318519

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