Exponential Objects

In the first part of this article we formalize the concepts of terminal and initial object, categorical product [4] and natural transformation within a free-object category [1]. In particular, we show that this definition of natural transformation is equivalent to the standard definition [13]. Then we introduce the exponential object using its universal property and we show the isomorphism between the exponential object of categories and the functor category [12].Via del Pero 102, 54038 Montignoso, ItalyJiri Adamek, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: The Joy of Cats. Dover Publication, New York, 2009.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Francis Borceaux. Handbook of Categorical Algebra I. Basic Category Theory, volume 50 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994.Czesław Byliński. Introduction to categories and functors. Formalized Mathematics, 1 (2):409–420, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Krzysztof Hryniewiecki. Graphs. Formalized Mathematics, 2(3):365–370, 1991.F. William Lawvere. Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories. Reprints in Theory and Applications of Categories, 5:1–121, 2004.Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer Verlag, New York, Heidelberg, Berlin, 1971.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.Marco Riccardi. Object-free definition of categories. Formalized Mathematics, 21(3): 193–205, 2013. doi:10.2478/forma-2013-0021. [Crossref]Marco Riccardi. Categorical pullbacks. Formalized Mathematics, 23(1):1–14, 2015. doi:10.2478/forma-2015-0001. [Crossref]Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.Andrzej Trybulec. Isomorphisms of categories. Formalized Mathematics, 2(5):629–634, 1991.Andrzej Trybulec. Natural transformations. Discrete categories. Formalized Mathematics, 2(4):467–474, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990

Finite Product of Semiring of Sets

AbstractWe formalize that the image of a semiring of sets [17] by an injective function is a semiring of sets.We offer a non-trivial example of a semiring of sets in a topological space [21]. Finally, we show that the finite product of a semiring of sets is also a semiring of sets [21] and that the finite product of a classical semiring of sets [8] is a classical semiring of sets. In this case, we use here the notation from the book of Aliprantis and Border [1].Rue de la Brasserie 5 7100 La Louvière, BelgiumCharalambos D. Aliprantis and Kim C. Border. Infinite dimensional analysis. Springer- Verlag, Berlin, Heidelberg, 2006.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Roland Coghetto. Semiring of sets. Formalized Mathematics, 22(1):79-84, 2014. doi:10.2478/forma-2014-0008. [Crossref]Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Noboru Endou, Kazuhisa Nakasho, and Yasunari Shidama. σ-ring and σ-algebra of sets. Formalized Mathematics, 23(1):51-57, 2015. doi:10.2478/forma-2015-0004. [Crossref]D.F. Goguadze. About the notion of semiring of sets. Mathematical Notes, 74:346-351, 2003. ISSN 0001-4346. doi:10.1023/A:1026102701631. [Crossref]Zbigniew Karno. On discrete and almost discrete topological spaces. Formalized Mathematics, 3(2):305-310, 1992.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jean Schmets. Théorie de la mesure. Notes de cours, Université de Liège, 146 pages, 2004.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1 (1):187-190, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Extended Real-Valued Double Sequence and Its Convergence

AbstractIn this article we introduce the convergence of extended realvalued double sequences [16], [17]. It is similar to our previous articles [15], [10]. In addition, we also prove Fatou’s lemma and the monotone convergence theorem for double sequences.This work was supported by JSPS KAKENHI 2350002Gifu National College of Technology, Gifu, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991.Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Noboru Endou. Double series and sums. Formalized Mathematics, 22(1):57-68, 2014. doi:10.2478/forma-2014-0006. [Crossref]Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006. doi:10.2478/v10037-006-0008-x. [Crossref]Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491-494, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008. doi:10.2478/v10037-008-0023-1. [Crossref]Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Double sequences and limits. Formalized Mathematics, 21(3):163-170, 2013. doi:10.2478/forma-2013-0018. [Crossref]Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2 edition, 1999.D.J.H. Garling. A Course in Mathematical Analysis: Volume 1, Foundations and Elementary Real Analysis, volume 1. Cambridge University Press, 2013.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1 (3):471-475, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers. Formalized Mathematics, 15(4):231-236, 2007. doi:10.2478/v10037-007-0026-3. [Crossref

Categorical Pullbacks

The main purpose of this article is to introduce the categorical concept of pullback in Mizar. In the first part of this article we redefine homsets, monomorphisms, epimorpshisms and isomorphisms [7] within a free-object category [1] and it is shown there that ordinal numbers can be considered as categories. Then the pullback is introduced in terms of its universal property and the Pullback Lemma is formalized [15]. In the last part of the article we formalize the pullback of functors [14] and it is also shown that it is not possible to write an equivalent definition in the context of the previous Mizar formalization of category theory [8].Via del Pero 102, 54038 Montignoso, ItalyJiri Adamek, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: The Joy of Cats. Dover Publication, New York, 2009.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Grzegorz Bancerek. The well ordering relations. Formalized Mathematics, 1(1):123–129, 1990.Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1 (2):265–267, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Francis Borceaux. Handbook of Categorical Algebra I. Basic Category Theory, volume 50 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994.Czesław Byliński. Introduction to categories and functors. Formalized Mathematics, 1 (2):409–420, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.F. William Lawvere. Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories. Reprints in Theory and Applications of Categories, 5:1–121, 2004.Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer Verlag, New York, Heidelberg, Berlin, 1971.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.Marco Riccardi. Object-free definition of categories. Formalized Mathematics, 21(3): 193–205, 2013. doi:10.2478/forma-2013-0021.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990

Product Pre-Measure

In this article we formalize in Mizar [5] product pre-measure on product sets of measurable sets. Although there are some approaches to construct product measure [22], [6], [9], [21], [25], we start it from σ-measure because existence of σ-measure on any semialgebras has been proved in [15]. In this approach, we use some theorems for integrals.EndouGifu Noboru - Gifu National College of Technology Gifu, JapanGrzegorz Bancerek. Towards the construction of a model of Mizar concepts. Formalized Mathematics, 16(2):207-230, 2008. doi:10.2478/v10037-008-0027-x.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, 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.Heinz Bauer. Measure and Integration Theory. Walter de Gruyter Inc.Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245-254, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Noboru Endou. Construction of measure from semialgebra of sets. Formalized Mathematics, 23(4):309-323, 2015. doi:10.1515/forma-2015-0025.Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006. doi:10.2478/v10037-006-0008-x.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. The measurability of extended real valued functions. Formalized Mathematics, 9(3):525-529, 2001.Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008. doi:10.2478/v10037-008-0023-1.Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Hopf extension theorem of measure. Formalized Mathematics, 17(2):157-162, 2009. doi:10.2478/v10037-009-0018-6.Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2 edition, 1999.P. R. Halmos. Measure Theory. Springer-Verlag, 1974.Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.M.M. Rao. Measure Theory and Integration. Marcel Dekker, 2nd edition, 2004.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers. Formalized Mathematics, 15(4):231-236, 2007. doi:10.2478/v10037-007-0026-3

Cousin’s Lemma

We formalize, in two different ways, that “the n-dimensional Euclidean metric space is a complete metric space” (version 1. with the results obtained in [13], [26], [25] and version 2., the results obtained in [13], [14], (registrations) [24]).With the Cantor’s theorem - in complete metric space (proof by Karol Pąk in [22]), we formalize “The Nested Intervals Theorem in 1-dimensional Euclidean metric space”.Pierre Cousin’s proof in 1892 [18] the lemma, published in 1895 [9] states that: “Soit, sur le plan YOX, une aire connexe S limitée par un contour fermé simple ou complexe; on suppose qu’à chaque point de S ou de son périmètre correspond un cercle, de rayon non nul, ayant ce point pour centre : il est alors toujours possible de subdiviser S en régions, en nombre fini et assez petites pour que chacune d’elles soit complétement intérieure au cercle correspondant à un point convenablement choisi dans S ou sur son périmètre.” (In the plane YOX let S be a connected area bounded by a closed contour, simple or complex; one supposes that at each point of S or its perimeter there is a circle, of non-zero radius, having this point as its centre; it is then always possible to subdivide S into regions, finite in number and sufficiently small for each one of them to be entirely inside a circle corresponding to a suitably chosen point in S or on its perimeter) [23].Cousin’s Lemma, used in Henstock and Kurzweil integral [29] (generalized Riemann integral), state that: “for any gauge δ, there exists at least one δ-fine tagged partition”. In the last section, we formalize this theorem. We use the suggestions given to the Cousin’s Theorem p.11 in [5] and with notations: [4], [29], [19], [28] and [12].Coghetto Roland - Rue de la Brasserie 5, 7100 La Louvière, BelgiumGrzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 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.Robert G. Bartle. Return to the Riemann integral. American Mathematical Monthly, pages 625–632, 1996.Robert G. Bartle. A modern theory of integration, volume 32. American Mathematical Society Providence, 2001.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Czesław Byliński. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241–245, 1996.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Pierre Cousin. Sur les fonctions de n variables complexes. Acta Mathematica, 19(1):1–61, 1895. doi:10.1007/BF02402869.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces – fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93–102, 1999.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577–580, 2005.Noboru Endou, Yasunari Shidama, and Katsumasa Okamura. Baire’s category theorem and some spaces generated from real normed space. Formalized Mathematics, 14(4): 213–219, 2006. doi:10.2478/v10037-006-0024-x.Adam Grabowski and Yatsuka Nakamura. Some properties of real maps. Formalized Mathematics, 6(4):455–459, 1997.Artur Korniłowicz. Properties of connected subsets of the real line. Formalized Mathematics, 13(2):315–323, 2005.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.Bernard Maurey and Jean-Pierre Tacchi. La genèse du théorème de recouvrement de Borel. Revue d’histoire des mathématiques, 11(2):163–204, 2005.Jean Mawhin. L’éternel retour des sommes de Riemann-Stieltjes dans l’évolution du calcul intégral. Bulletin de la Société Royale des Sciences de Liège, 70(4–6):345–364, 2001.Yatsuka Nakamura and Andrzej Trybulec. A decomposition of a simple closed curves and the order of their points. Formalized Mathematics, 6(4):563–572, 1997.Robin Nittka. Conway’s games and some of their basic properties. Formalized Mathematics, 19(2):73–81, 2011. doi:10.2478/v10037-011-0013-6.Karol Pąk. Complete spaces. Formalized Mathematics, 16(1):35–43, 2008. doi:10.2478/v10037-008-0006-2.Manya Raman-Sundström. A pedagogical history of compactness. The American Mathematical Monthly, 122(7):619–635, 2015.Hideki Sakurai, Hisayoshi Kunimune, and Yasunari Shidama. Uniform boundedness principle. Formalized Mathematics, 16(1):19–21, 2008. doi:10.2478/v10037-008-0003-5.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2004.Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377–380, 2003.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Lee Peng Yee. The integral à la Henstock. Scientiae Mathematicae Japonicae, 67(1): 13–21, 2008.Lee Peng Yee and Rudolf Vyborny. Integral: an easy approach after Kurzweil and Henstock, volume 14. Cambridge University Press, 2000

Double Sequences and Iterated Limits in Regular Space

First, we define in Mizar [5], the Cartesian product of two filters bases and the Cartesian product of two filters. After comparing the product of two Fréchet filters on ℕ (F1) with the Fréchet filter on ℕ × ℕ (F2), we compare limF₁ and limF₂ for all double sequences in a non empty topological space.Endou, Okazaki and Shidama formalized in [14] the “convergence in Pringsheim’s sense” for double sequence of real numbers. We show some basic correspondences between the p-convergence and the filter convergence in a topological space. Then we formalize that the double sequence converges in “Pringsheim’s sense” but not in Frechet filter on ℕ × ℕ sense.In the next section, we generalize some definitions: “is convergent in the first coordinate”, “is convergent in the second coordinate”, “the lim in the first coordinate of”, “the lim in the second coordinate of” according to [14], in Hausdorff space.Finally, we generalize two theorems: (3) and (4) from [14] in the case of double sequences and we formalize the “iterated limit” theorem (“Double limit” [7], p. 81, par. 8.5 “Double limite” [6] (TG I,57)), all in regular space. We were inspired by the exercises (2.11.4), (2.17.5) [17] and the corrections B.10 [18].Coghetto Roland - Rue de la Brasserie 5 7100 La Louvière, BelgiumGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93-107, 1997.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek, Noboru Endou, and Yuji Sakai. On the characterizations of compactness. Formalized Mathematics, 9(4):733-738, 2001.Grzegorz 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.Nicolas Bourbaki. Topologie générale: Chapitres 1 à 4. Eléments de mathématique. Springer Science & Business Media, 2007.Nicolas Bourbaki. General Topology: Chapters 1-4. Springer Science and Business Media, 2013.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Roland Coghetto. Convergent filter bases. Formalized Mathematics, 23(3):189-203, 2015.Roland Coghetto. Summable family in a commutative group. Formalized Mathematics, 23(4):279-288, 2015.Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Double sequences and limits. Formalized Mathematics, 21(3):163-170, 2013.Andrzej Owsiejczuk. Combinatorial Grassmannians. Formalized Mathematics, 15(2):27-33, 2007.Karol Pak. Stirling numbers of the second kind. Formalized Mathematics, 13(2):337-345, 2005.Claude Wagschal. Topologie et analyse fonctionnelle. Hermann, 1995.Claude Wagschal. Topologie: Exercices et problémes corrigés. Hermann, 1995.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990

Algebraic Numbers

This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”. Definitions for an integral closure, an algebraic integer and a transcendental numbers [14], [1], [10] and [7] are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field ℚ induced by substitution of an algebraic number to the polynomial ring of ℚ[x] turns to be a field.Suginami-ku Matsunoki 6, 3-21 Tokyo, JapanMichael Francis Atiyah and Ian Grant Macdonald. Introduction to Commutative Algebra, volume 2. Addison-Wesley Reading, 1969.Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.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.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.Hideyuki Matsumura. Commutative Ring Theory. Cambridge University Press, 2nd edition, 1989. Cambridge Studies in Advanced Mathematics.Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339–346, 2001.Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391–395, 2001.Masayoshi Nagata. Theory of Commutative Fields, volume 125. American Mathematical Society, 1985. Translations of Mathematical Monographs.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.Oscar Zariski and Pierre Samuel. Commutative Algebra I. Springer, 2nd edition, 1975

Groups – Additive Notation

AbstractWe translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25]. In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented. Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.Rue de la Brasserie 5 7100 La Louvière, BelgiumJonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565-582, 2001.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1 (2):265-267, 1990.Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.Richard E. Blahut. Cryptography and Secure Communication. Cambridge University Press, 2014.Sylvie Boldo, Catherine Lelay, and Guillaume Melquiond. Formalization of real analysis: A survey of proof assistants and libraries. Mathematical Structures in Computer Science, pages 1-38, 2014.Nicolas Bourbaki. General Topology: Chapters 1-4. Springer Science and Business Media, 2013.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383-386, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Z-modules. Formalized Mathematics, 20(1):47-59, 2012. doi:10.2478/v10037-012-0007-z. [Crossref]Edwin Hewitt and Kenneth A. Ross. Abstract Harmonic Analysis: Volume I. Structure of Topological Groups. Integration. Theory Group Representations, volume 115. Springer Science and Business Media, 2012.Johannes Hölzl, Fabian Immler, and Brian Huffman. Type classes and filters for mathematical analysis in Isabelle/HOL. In Interactive Theorem Proving, pages 279-294. Springer, 2013.Teturo Inui, Yukito Tanabe, and Yositaka Onodera. Group theory and its applications in physics, volume 78. Springer Science and Business Media, 2012.Artur Korniłowicz. The definition and basic properties of topological groups. Formalized Mathematics, 7(2):217-225, 1998.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.Christopher Norman. Basic theory of additive Abelian groups. In Finitely Generated Abelian Groups and Similarity of Matrices over a Field, Springer Undergraduate Mathematics Series, pages 47-96. Springer, 2012. ISBN 978-1-4471-2729-1. doi:10.1007/978-1-4471-2730-7 2. [Crossref]Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93-96, 1991.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics, 5(2):233-236, 1996.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4): 535-545, 1991.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1 (2):369-376, 1990.Andrzej Trybulec. Baire spaces, Sober spaces. Formalized Mathematics, 6(2):289-294, 1997.Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1 (1):187-190, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855-864, 1990.Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990

Torsion Part of ℤ-module

In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational numbers as an example of torsion-free non free ℤ-modules. We also formalize the rank-nullity theorem over finite-rank free ℤ-modules (previously formalized in [1]). ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [23] and cryptographic systems with lattices [24].Yuichi Futa - Japan Advanced Institute of Science and Technology, Ishikawa, JapanHiroyuki Okazaki - Shinshu University, Nagano, JapanYasunari Shidama - Shinshu University, Nagano, JapanJesse Alama. The rank+nullity theorem. Formalized Mathematics, 15(3):137–142, 2007. doi:10.2478/v10037-007-0015-6. [Crossref]Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.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. [Crossref]Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. ℤ-modules. Formalized Mathematics, 20(1):47–59, 2012. doi:10.2478/v10037-012-0007-z. [Crossref]Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of ℤ-module. Formalized Mathematics, 20(3):205–214, 2012. doi:10.2478/v10037-012-0024-y. [Crossref]Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Free ℤ-module. Formalized Mathematics, 20(4):275–280, 2012. doi:10.2478/v10037-012-0033-x. [Crossref]Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, and Yasunari Shidama. Gaussian integers. Formalized Mathematics, 21(2):115–125, 2013. doi:10.2478/forma-2013-0013. [Crossref]Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Submodule of free ℤ-module. Formalized Mathematics, 21(4):273–282, 2013. doi:10.2478/forma-2013-0029. [Crossref]Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, and Yasunari Shidama. Torsion ℤ-module and torsion-free ℤ-module. Formalized Mathematics, 22(4):277–289, 2014. doi:10.2478/forma-2014-0028. [Crossref]Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829–832, 1990.A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász. 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.Michał Muzalewski. Rings and modules – part II. Formalized Mathematics, 2(4):579–585, 1991.Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Rank of submodule, linear transformations and linearly independent subsets of ℤ-module. Formalized Mathematics, 22(3):189–198, 2014. doi:10.2478/forma-2014-0021. [Crossref]Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559–564, 2001.Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29–34, 1999.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Wojciech A. Trybulec. Operations on subspaces in real linear space. Formalized Mathematics, 1(2):395–399, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865–870, 1990.Wojciech A. Trybulec. Operations on subspaces in vector space. Formalized Mathematics, 1(5):871–876, 1990.Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1 (5):877–882, 1990.Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883–885, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990