10 research outputs found
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 BiaĆystok, PolandAdam Grabowski - Institute of Informatics, University of BiaĆystok, PolandGrzegorz Bancerek, CzesĆaw Bylinski, Adam Grabowski, Artur KorniĆowicz, 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 261â279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, CzesĆaw Bylinski, Adam Grabowski, Artur KorniĆowicz, 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):9â32, 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 GrĂ€tzer. Some examples of weakly associative lattices. Colloquium Mathematicum, 27:215â221, 1973. doi:10.4064/cm-27-2-215-221.Adam Grabowski. Mechanizing complemented lattices within Mizar system. Journal of Automated Reasoning, 55:211â221, 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 116â129. Springer, 2004. doi:10.1007/978-3-540-27818-4_9. 3rd International Conference on Mathematical Knowledge Management, Bialowieza, Poland, Sep. 19â21, 2004.Adam Grabowski and Damian Sawicki. On two alternative axiomatizations of lattices by McKenzie and Sholander. Formalized Mathematics, 26(2):193â198, 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 49â64. Springer, 2006. doi:https://doi.org/10.1007/11618027 4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15â17, 2005, Revised Selected Papers.Adam Grabowski, Artur KorniĆowicz, 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 45â54. IEEE, 2015. doi:10.15439/2015F229.George GrĂ€tzer. General Lattice Theory. Academic Press, New York, 1978.George GrĂ€tzer. Lattice Theory: Foundation. BirkhĂ€user, 2011.Dominik Kulesza and Adam Grabowski. Formalization of quasilattices. Formalized Mathematics, 28(2):217â225, 2020. doi:10.2478/forma-2020-0019.William McCune. Prover9 and Mace4. 2005â2010.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.StanisĆaw Zukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215â222, 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 BiaĆystok, PolandAdam Grabowski - Institute of Informatics, University of BiaĆystok, PolandGrzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, Karol PÄ
k, 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 261â279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, and Karol PÄ
k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9â32, 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, 55:211â221, 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):117â121, 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 116â129. Springer, 2004. doi:10.1007/978-3-540-27818-4_9. 3rd International Conference on Mathematical Knowledge Management, Bialowieza, Poland, Sep. 19â21, 2004.Adam Grabowski and Damian Sawicki. On two alternative axiomatizations of lattices by McKenzie and Sholander. Formalized Mathematics, 26(2):193â198, 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 49â64. Springer, 2006. doi:https://doi.org/10.1007/11618027_4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15â17, 2005, Revised Selected Papers.Adam Grabowski, Artur KorniĆowicz, 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 45â54. IEEE, 2015. doi:10.15439/2015F229.George GrĂ€tzer. General Lattice Theory. Academic Press, New York, 1978.George GrĂ€tzer. Lattice Theory: Foundation. BirkhĂ€user, 2011.William McCune. Prover9 and Mace4. 2005â2010.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.StanisĆaw Ć»ukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215â222, 1990.28221722
Elementary Number Theory Problems. Part XII â Primes in Arithmetic Progression
In this paper another twelve problems from W. SierpiĆskiâs book â250 Problems in Elementary Number Theoryâ are formalized, using the Mizar formalism, namely: 42, 43, 51, 51a, 57, 59, 72, 135, 136, and 153â155. Significant amount of the work is devoted to arithmetic progressions.Faculty of Computer Science, University of BiaĆystok, PolandLeonard Eugene Dickson. History of Theory of Numbers. New York, 1952.Adam Grabowski. Elementary number theory problems. Part VI. Formalized Mathematics, 30(3):235â244, 2022. doi:10.2478/forma-2022-0019.Adam Grabowski. Polygonal numbers. Formalized Mathematics, 21(2):103â113, 2013. doi:10.2478/forma-2013-0012.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 49â64. Springer, 2006. doi:10.1007/11618027 4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15â17, 2005, Revised Selected Papers.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Mizar in a nutshell. Journal of Formalized Reasoning, 3(2):153â245, 2010.Artur KorniĆowicz. Flexary connectives in Mizar. Computer Languages, Systems & Structures, 44:238â250, December 2015. doi:10.1016/j.cl.2015.07.002.Artur KorniĆowicz and Adam Naumowicz. Elementary number theory problems. Part V. Formalized Mathematics, 30(3):229â234, 2022. doi:10.2478/forma-2022-0018.Adam Naumowicz. Dataset description: Formalization of elementary number theory in Mizar. In Christoph BenzmĂŒller and Bruce R. Miller, editors, Intelligent Computer Mathematics â 13th International Conference, CICM 2020, Bertinoro, Italy, July 26â31, 2020, Proceedings, volume 12236 of Lecture Notes in Computer Science, pages 303â308. Springer, 2020. doi:10.1007/978-3-030-53518-6_22.Adam Naumowicz. Extending numeric automation for number theory formalizations in Mizar. In Catherine Dubois and Manfred Kerber, editors, Intelligent Computer Mathematics â 16th International Conference, CICM 2023, Cambridge, UK, September 5â8, 2023, Proceedings, volume 14101 of Lecture Notes in Computer Science, pages 309â314. Springer, 2023. doi:10.1007/978-3-031-42753-4_23.Christoph Schwarzweller. Proth numbers. Formalized Mathematics, 22(2):111â118, 2014. doi:10.2478/forma-2014-0013.WacĆaw SierpiĆski. Elementary Theory of Numbers. PWN, Warsaw, 1964.WacĆaw SierpiĆski. 250 Problems in Elementary Number Theory. Elsevier, 1970.Nguyen Xuan Tho. On a remark of SierpiĆski. Rocky Mountain Journal of Mathematics, 52(2):717â726, 2022. doi:10.1216/rmj.2022.52.717.RafaĆ Ziobro. Fermatâs Little Theorem via divisibility of Newtonâs binomial. Formalized Mathematics, 23(3):215â229, 2015. doi:10.1515/forma-2015-0018.31127728
Multidimensional Measure Space and Integration
This paper introduces multidimensional measure spaces and the integration of functions on these spaces in Mizar. Integrals on the multidimensional Cartesian product measure space are defined and appropriate formal apparatus to deal with this notion is provided as well.Noboru Endou - National Institute of Technology, Gifu College, 2236-2 Kamimakuwa, Motosu, Gifu, JapanYasunari Shidama - Karuizawa Hotch 244-1, Nagano, JapanGrzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, Karol PÄ
k, 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 261â279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007.Sylvie Boldo, Catherine Lelay, and Guillaume Melquiond. Improving real analysis in Coq: A user-friendly approach to integrals and derivatives. In Chris Hawblitzel and Dale Miller, editors, Certified Programs and Proofs â Second International Conference, CPP 2012, Kyoto, Japan, December 13â15, 2012. Proceedings, volume 7679 of Lecture Notes in Computer Science, pages 289â304. Springer, 2012. doi:10.1007/978-3-642-35308-6_22.Sylvie Boldo, Catherine Lelay, and Guillaume Melquiond. Formalization of real analysis: A survey of proof assistants and libraries. Mathematical Structures in Computer Science, 26:1196â1233, 2015.Noboru Endou. Improper integral. Part II. Formalized Mathematics, 29(4):279â294, 2021. doi:10.2478/forma-2021-0024.Noboru Endou. Fubiniâs theorem on measure. Formalized Mathematics, 25(1):1â29, 2017. doi:10.1515/forma-2017-0001.Noboru Endou. Fubiniâs theorem. Formalized Mathematics, 27(1):67â74, 2019. doi:10.2478/forma-2019-0007.Noboru Endou. Absolutely integrable functions. Formalized Mathematics, 30(1):31â51, 2022. doi:10.2478/forma-2022-0004.Jacques D. Fleuriot. On the mechanization of real analysis in Isabelle/HOL. In Mark Aagaard and John Harrison, editors, Theorem Proving in Higher Order Logics, pages
145â161. Springer Berlin Heidelberg, 2000. ISBN 978-3-540-44659-0.Ruben Gamboa. Continuity and Differentiability, pages 301â315. Springer US, 2000. ISBN 978-1-4757-3188-0. doi:10.1007/978-1-4757-3188-0_18.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 49â64. Springer, 2006. doi:https://doi.org/10.1007/11618027 4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15â17, 2005, Revised Selected Papers.Johannes Hölzl and Armin Heller. Three chapters of measure theory in Isabelle/HOL. In Marko C. J. D. van Eekelen, Herman Geuvers, Julien Schmaltz, and Freek Wiedijk,
editors, Interactive Theorem Proving (ITP 2011), volume 6898 of LNCS, pages 135â151, 2011.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51â59, 2011. doi:10.2478/v10037-011-0009-2.M.M. Rao. Measure Theory and Integration. Marcel Dekker, 2nd edition, 2004.31118119
Simple Extensions
In this article we continue the formalization of field theory in Mizar. We introduce simple extensions: an extension E of F is simple if E is generated over F by a single element of E, that is E = F(a) for some a â E. First, we prove that a finite extension E of F is simple if and only if there are only finitely many intermediate fields between E and F [7]. Second, we show that finite extensions of a field F with characteristic 0 are always simple [1]. For this we had to prove, that irreducible polynomials over F have single roots only, which required extending results on divisibility and gcds of polynomials [14], [13] and formal derivation of polynomials [15].Christoph Schwarzweller - Institute of Informatics, University of GdaĆsk, PolandAgnieszka RowiĆska-Schwarzweller - Institute of Informatics, University of GdaĆsk, PolandAndreas Gathmann. Einfšuhrung in die Algebra. Lecture Notes, University of Kaiserslautern, Germany, 2011.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 49â64. Springer, 2006. doi:10.1007/11618027 4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15â17, 2005, Revised Selected Papers.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Mizar in a nutshell. Journal of Formalized Reasoning, 3(2):153â245, 2010.Adam Grabowski, Artur KorniĆowicz, 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 363â371, 2016. doi:10.15439/2016F520.Artur KorniĆowicz. Flexary connectives in Mizar. Computer Languages, Systems & Structures, 44:238â250, December 2015. doi:10.1016/j.cl.2015.07.002.Serge Lang. Algebra. PWN, Warszawa, 1984.Serge Lang. Algebra. Springer Verlag, 2002 (Revised Third Edition).Heinz Lšuneburg. Gruppen, Ringe, Kšorper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1999.Christoph Schwarzweller. Normal extensions. Formalized Mathematics, 31(1):121â130, 2023. doi:10.2478/forma-2023-0011.Christoph Schwarzweller. Renamings and a condition-free formalization of Kroneckerâs construction. Formalized Mathematics, 28(2):129â135, 2020. doi:10.2478/forma-2020-0012.Christoph Schwarzweller. Ring and field adjunctions, algebraic elements and minimal poynomials. Formalized Mathematics, 28(3):251â261, 2020. doi:10.2478/forma-2020-0022.Christoph Schwarzweller. Splitting fields. Formalized Mathematics, 29(3):129â139, 2021. doi:10.2478/forma-2021-0013.Christoph Schwarzweller. On roots of polynomials and algebraically closed fields. Formalized Mathematics, 25(3):185â195, 2017. doi:10.1515/forma-2017-0018.Christoph Schwarzweller, Artur KorniĆowicz, and Agnieszka RowiĆska-Schwarzweller. Some algebraic properties of polynomial rings. Formalized Mathematics, 24(3):227â237, 2016. doi:10.1515/forma-2016-0019.Yasushige Watase. Derivation of commutative rings and the Leibniz formula for power of derivation. Formalized Mathematics, 29(1):1â8, 2021. doi:10.2478/forma-2021-0001.31128729
Symmetrical Piecewise Linear Functions Composed by Absolute Value Function
We continue the formal development of the application of piecewise linear functions and centroids in the area of fuzzy set theory. The corresponding piecewise linear functions are symmetrical and composed by absolute function. In this paper we prove that the membership functions of isosceles triangle type and isosceles trapezoid type can be constructed by functions of this type.Faculty of Business and Informatics, Nagano University, JapanDidier Dubois and Henri Prade. Operations on fuzzy numbers. International Journal of Systems Science, 9(6):613â626, 1978. doi:10.1080/00207727808941724.Didier Dubois and Henri Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980.Ronald E. Giachetti and Robert E. Young. A parametric representation of fuzzy numbers and their arithmetic operators. Fuzzy Sets and Systems, 91(2):185â202, 1997.
doi:10.1016/S0165-0114(97)00140-1.Eikou Gonda, Hitoshi Miyata, and Masaaki Ohkita. Self-turning of fuzzy rules with different types of MSFs (in Japanese). Journal of Japan Society for Fuzzy Theory and
Intelligent Informatics, 16(6):540â550, 2004. doi:10.3156/jsoft.16.540.Adam Grabowski. The formal construction of fuzzy numbers. Formalized Mathematics, 22(4):321â327, 2014. doi:10.2478/forma-2014-0032.Adam Grabowski. Fuzzy implications in the Mizar system. In 30th IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2021, Luxembourg, July 11â14, 2021, pages
1â6. IEEE, 2021. doi:10.1109/FUZZ45933.2021.9494593.Adam Grabowski. On the computer certification of fuzzy numbers. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, 2013 Federated Conference on Computer Science and Information Systems (FedCSIS), Federated Conference on Computer Science and Information Systems, pages 51â54, 2013.Adam Grabowski and Takashi Mitsuishi. Initial comparison of formal approaches to fuzzy and rough sets. In Leszek Rutkowski, Marcin Korytkowski, Rafal Scherer, Ryszard
Tadeusiewicz, Lotfi A. Zadeh, and Jacek M. Zurada, editors, Artificial Intelligence and Soft Computing â 14th International Conference, ICAISC 2015, Zakopane, Poland, June 14-18, 2015, Proceedings, Part I, volume 9119 of Lecture Notes in Computer Science, pages 160â171. Springer, 2015. doi:10.1007/978-3-319-19324-3_15.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 5â10, 2010. Proceedings, volume 6167 of Lecture Notes in Computer Science, pages 300â314. Springer, 2010. doi:10.1007/978-3-642-14128-7_26.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 49â64. Springer, 2006. doi:10.1007/11618027 4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15â17, 2005, Revised Selected Papers.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Mizar in a nutshell. Journal of Formalized Reasoning, 3(2):153â245, 2010.Tetsuro Katafuchi, Kiyoji Asai, and Hiroshi Fujita. Investigation of deffuzification in fuzzy inference: Proposal of a new defuzzification method (in Japanese). Medical Imaging and Information Sciences, 18(1):19â30, 2001. doi:10.11318/mii1984.18.19.Ebrahim H. Mamdani. Application of fuzzy algorithms for control of simple dynamic plant. IEE Proceedings, 121:1585â1588, 1974.Takashi Mitsuishi. Some properties of membership functions composed of triangle functions and piecewise linear functions. Formalized Mathematics, 29(2):103â115, 2021. doi:10.2478/forma-2021-0011.Takashi Mitsuishi. Definition of centroid method as defuzzification. Formalized Mathematics, 30(2):125â134, 2022. doi:10.2478/forma-2022-0010.Takashi Mitsuishi. Isosceles triangular and isosceles trapezoidal membership functions using centroid method. Formalized Mathematics, 31(1):59â66, 2023. doi:10.2478/forma-2023-0006.Takashi Mitsuishi, Takanori Terashima, Nami Shimada, Toshimichi Homma, and Yasunari Shidama. Approximate reasoning using LR fuzzy number as input for sensorless fuzzy control. In 2016 IEEE Symposium on Sensorless Control for Electrical Drives (SLED), pages 1â5, 2016. doi:10.1109/SLED.2016.7518804.Masaharu Mizumoto. Improvement of fuzzy control (IV)-case by product-sum-gravity method. In Proc. 6th Fuzzy System Symposium, 1990, pages 9â13, 1990.Timothy J. Ross. Fuzzy Logic with Engineering Applications. John Wiley and Sons Ltd, 2010.Luciano Stefanini and Laerte Sorini. Fuzzy arithmetic with parametric LR fuzzy numbers. In Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, pages 600â605, 2009.Werner Van Leekwijck and Etienne E. Kerre. Defuzzification: Criteria and classification. Fuzzy Sets and Systems, 108(2):159â178, 1999.Lotfi Zadeh. Fuzzy sets. Information and Control, 8(3):338â353, 1965. doi:10.1016/S0019-9958(65)90241-X.31129930