352,248 research outputs found
Ordered Rings and Fields
We introduce ordered rings and fields following Artin-Schreierâs approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial rings can be ordered in (at least) two different ways [8, 5, 4, 9]. This is the continuation of the development of algebraic hierarchy in Mizar [2, 3].Schwarzweller Christoph - Institute of Informatics, University of Gdansk, Gdansk, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015.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 363-371, 2016.Nathan Jacobson. Lecture Notes in Abstract Algebra, III. Theory of Fields and Galois Theory. Springer-Verlag, 1964.Manfred Knebusch and Claus Scheiderer. Einfšuhrung in die reelle Algebra. Vieweg-Verlag, 1989.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Eugeniusz Kusak, Wojciech Leonczuk, and MichaĆ Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Alexander Prestel. Lectures on Formally Real Fields. Springer-Verlag, 1984.Knut Radbruch. Geordnete Kšorper. Lecture Notes, University of Kaiserslautern, Germany, 1991.MichaĆ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990
Lattice of â€-module
In this article, we formalize the definition of lattice of â€-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers â. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of â€-module is necessary for lattice problems, LLL (Lenstra, Lenstra and LovĂĄsz) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].Futa Yuichi - Japan Advanced Institute of Science and Technology Ishikawa, JapanShidama Yasunari - Shinshu University Nagano, JapanGrzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3): 537-541, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz 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 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.CzesĆaw ByliĆski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Wolfgang Ebeling. Lattices and Codes. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 2013.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. â€-modules. Formalized Mathematics, 20(1):47-59, 2012. doi:10.2478/v10037-012-0007-z.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of â€-module. Formalized Mathematics, 20(3):205-214, 2012. doi:10.2478/v10037-012-0024-y.Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, and Yasunari Shidama. Torsion â€-module and torsion-free â€-module. Formalized Mathematics, 22(4):277-289, 2014. doi:10.2478/forma-2014-0028.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Matrix of â€-module. Formalized Mathematics, 23(1):29-49, 2015. doi:10.2478/forma-2015-0003.A. K. Lenstra, H. W. Lenstra Jr., and L. LovĂĄsz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4), 1982.Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: A cryptographic perspective. The International Series in Engineering and Computer Science, 2002.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990
Formally Real Fields
Summary We extend the algebraic theory of ordered fields [7, 6] in Mizar [1, 2, 3]: we show that every preordering can be extended into an ordering, i.e. that formally real and ordered fields coincide.We further prove some characterizations of formally real fields, in particular the one by Artin and Schreier using sums of squares [4]. In the second part of the article we define absolute values and the square root function [5].Institute of Informatics, Faculty of Mathematics, Physics and Informatics, University of Gdansk Wita Stwosza 57, 80-308 Gdansk, 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 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: 10.1007/978-3-319-20615-8 17.Adam Grabowski, Artur KorniĆowicz, 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 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 Infor mation Systems (FedCSIS), volume 8 of Annals of Computer Science and Information Systems, pages 363-371, 2016. doi: 10.15439/2016F520.Nathan Jacobson. Lecture Notes in Abstract Algebra, III. Theory of Fields and Galois Theory. Springer-Verlag, 1964.Manfred Knebusch and Claus Scheiderer. Einfšuhrung in die reelle Algebra. Vieweg-Verlag, 1989.Alexander Prestel. Lectures on Formally Real Fields. Springer-Verlag, 1984.Knut Radbruch. Geordnete Kšorper. Lecture Notes, University of Kaiserslautern, Germany, 1991.Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559-564, 2001.Christoph Schwarzweller. Ordered rings and fields. Formalized Mathematics, 25(1):63-72, 2017. doi: 10.1515/forma-2017-0006.Christoph Schwarzweller. On roots of polynomials and algebraically closed fields. Formalized Mathematics, 25(3):185-195, 2017. doi: 10.1515/forma-2017-0018.Christoph Schwarzweller and Artur KorniĆowicz. Characteristic of rings. Prime fields. Formalized Mathematics, 23(4):333-349, 2015. doi: 10.1515/forma-2015-0027.25424925
Propositional Linear Temporal Logic with Initial Validity Semantics
In the article [10] a formal system for Propositional Linear Temporal Logic (in short LTLB) with normal semantics is introduced. The language of this logic consists of âuntilâ operator in a very strict version. The very strict âuntilâ operator enables to express all other temporal operators.In this article we construct a formal system for LTLB with the initial semantics [12]. Initial semantics means that we define the validity of the formula in a model as satisfaction in the initial state of model while normal semantics means that we define the validity as satisfaction in all states of model. We prove the Deduction Theorem, and the soundness and completeness of the introduced formal system. We also prove some theorems to compare both formal systems, i.e., the one introduced in the article [10] and the one introduced in this article.Formal systems for temporal logics are applied in the verification of computer programs. In order to carry out the verification one has to derive an appropriate formula within a selected formal system. The formal systems introduced in [10] and in this article can be used to carry out such verifications in Mizar [4].This work was supported by the University of Bialystok grants: BST447 Formalization of temporal logics in a proof-assistant. Application to System Verification , and BST225 Database of mathematical texts checked by computer.Faculty of Economics and Informatics, University of BiaĆystok, Kalvariju 135, LT-08221 Vilnius, Lithuaniais work was supported by the University of Bialystok grants: BST447 Formalization of temporal logics in a proof-assistant. Application to System Verification, and BST225 Database of mathematical texts checked by computer.â©Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41â46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91â96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107â114, 1990.Grzegorz 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. [Crossref]CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1): 55â65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153â164, 1990.CzesĆaw ByliĆski. Partial functions. Formalized Mathematics, 1(2):357â367, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47â53, 1990.Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165â167, 1990.Mariusz Giero. The axiomatization of propositional linear time temporal logic. Formalized Mathematics, 19(2):113â119, 2011. doi:10.2478/v10037-011-0018-1. [Crossref]Adam Grabowski. Hilbert positive propositional calculus. Formalized Mathematics, 8(1): 69â72, 1999.Fred Kröger and Stephan Merz. Temporal Logic and State Systems. Springer-Verlag, 2008.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115â122, 1990.Andrzej Trybulec. Defining by structural induction in the positive propositional language. Formalized Mathematics, 8(1):133â137, 1999.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67â71, 1990.Edmund Woronowicz. Many argument relations. Formalized Mathematics, 1(4):733â737, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73â83, 1990
Riemann-Stieltjes Integral
In this article, the definitions and basic properties of Riemann-Stieltjes integral are formalized in Mizar [1]. In the first section, we showed the preliminary definition. We proved also some properties of finite sequences of real numbers. In Sec. 2, we defined variation. Using the definition, we also defined bounded variation and total variation, and proved theorems about related properties.In Sec. 3, we defined Riemann-Stieltjes integral. Referring to the way of the article [7], we described the definitions. In the last section, we proved theorems about linearity of Riemann-Stieltjes integral. Because there are two types of linearity in Riemann-Stieltjes integral, we proved linearity in two ways. We showed the proof of theorems based on the description of the article [7]. These formalizations are based on [8], [5], [3], and [4].Narita Keiko - Hirosaki-city Aomori, JapanNakasho Kazuhisa - Akita Prefectural University Akita, JapanShidama Yasunari - Shinshu University Nagano, JapanGrzegorz 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 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1.CzesĆaw Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.S.L. Gupta and Nisha Rani. Fundamental Real Analysis. Vikas Pub., 1986.Einar Hille. Methods in classical and functional analysis. Addison-Wesley Publishing Co., Halsted Press, 1974.H. Kestelman. Modern theories of integration. Dover Publications, 2nd edition, 1960.JarosĆaw Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Keiichi Miyajima, Takahiro Kato, and Yasunari Shidama. Riemann integral of functions from â into real normed space. Formalized Mathematics, 19(1):17-22, 2011.Daniel W. Stroock. A Concise Introduction to the Theory of Integration. Springer Science & Business Media, 1999
Algebraic Extensions
In this article we further develop field theory in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension E of a field F is algebraic, if every element of E is algebraic over F. We prove amongst others that finite extensions are algebraic and that field extensions generated by a finite set of algebraic elements are finite. From this immediately follows that field extensions generated by roots of a polynomial over F are both finite and algebraic. We also define the field of algebraic elements of E over F and show that this field is an intermediate field of E|F.Christoph Schwarzweller - Institute of Informatics, University of Gdansk, 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.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.Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985.Serge Lang. Algebra. Springer, 3rd edition, 2005.Christoph Schwarzweller. Ring and field adjunctions, algebraic elements and minimal polynomials. Formalized Mathematics, 28(3):251â261, 2020. doi:10.2478/forma-2020-0022.291394
Grothendieck Universes
The foundation of the Mizar Mathematical Library [2], is first-order Tarski-Grothendieck set theory. However, the foundation explicitly refers only to Tarskiâs Axiom A, which states that for every set X there is a Tarski universe U such that X â U. In this article, we prove, using the Mizar [3] formalism, that the Grothendieck name is justified. We show the relationship between Tarski and Grothendieck universe.This work has been supported by the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473.Institute of Informatics, University of BiaĆystok, PolandGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91â96, 1990.Grzegorz 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.Chad E. Brown and Karol PÄ
k. A tale of two set theories. In Cezary Kaliszyk, Edwin Brady, Andrea Kohlhase, and Claudio Sacerdoti Coen, editors, Intelligent Computer Mathematics â 12th International Conference, CICM 2019, CIIRC, Prague, Czech Republic, July 8-12, 2019, Proceedings, volume 11617 of Lecture Notes in Computer Science, pages 44â60. Springer, 2019. doi:10.1007/978-3-030-23250-4_4.N. H. Williams. On Grothendieck universes. Compositio Mathematica, 21(1):1â3, 1969.28221121
Quadratic Extensions
In this article we further develop field theory [6], [7], [12] in Mizar [1], [2], [3]: we deal with quadratic polynomials and quadratic extensions [5], [4]. First we introduce quadratic polynomials, their discriminants and prove the midnight formula. Then we show that - in case the discriminant of p being non square - adjoining a root of pâs discriminant results in a splitting field of p. Finally we prove that these are the only field extensions of degree 2, e.g. that an extension E of F is quadratic if and only if there is a non square Element a â F such that E and F(âa) are isomorphic over F.Christoph Schwarzweller - Institute of Informatics, University of GdaĆsk, PolandAgnieszka RowiĆska-Schwarzweller - Sopot, 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.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.Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985.Serge Lang. Algebra. Springer Verlag, 2002 (Revised Third Edition).Heinz Luneburg. Gruppen, Ringe, Kšorper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1999.Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991.Christoph Schwarzweller. Ring and field adjunctions, algebraic elements and minimal polynomials. Formalized Mathematics, 28(3):251â261, 2020. doi:10.2478/forma-2020-0022.Christoph Schwarzweller. Formally real fields. Formalized Mathematics, 25(4):249â259, 2017. doi:10.1515/forma-2017-0024.Christoph Schwarzweller. On roots of polynomials and algebraically closed fields. Formalized Mathematics, 25(3):185â195, 2017. doi:10.1515/forma-2017-0018.Christoph Schwarzweller and Artur KorniĆowicz. Characteristic of rings. Prime fields. Formalized Mathematics, 23(4):333â349, 2015. doi:10.1515/forma-2015-0027.Steven H. Weintraub. Galois Theory. Springer-Verlag, 2 edition, 2009.29422924
- âŠ