PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2015 | 23 | 4 | 279-288
Tytuł artykułu

Summable Family in a Commutative Group

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [22], [7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space [30], a real normed linear space [29] and a linear topological space [14] with the concept of the limit of an image filter [16]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commutative group (“additive notation” in [17]), using the notion of filters.
Twórcy
  • Rue de la Brasserie 5, 7100 La Louvière, Belgium
Bibliografia
  • [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.
  • [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.
  • [3] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93–107, 1997.
  • [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.
  • [5] Grzegorz Bancerek, Noboru Endou, and Yuji Sakai. On the characterizations of compactness. Formalized Mathematics, 9(4):733–738, 2001.
  • [6] 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.
  • [7] 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.
  • [8] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481–485, 1991.
  • [9] Nicolas Bourbaki. Topologie générale: Chapitres 1 à 4. Eléments de mathématique. Springer Science & Business Media, 2007.
  • [10] Nicolas Bourbaki. General Topology: Chapters 1–4. Springer Science and Business Media, 2013.
  • [11] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.
  • [12] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.
  • [13] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.
  • [14] Czesław Byliński. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99–107, 2005.
  • [15] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.
  • [16] Roland Coghetto. Convergent filter bases. Formalized Mathematics, 23(3):189–203, 2015. doi:10.1515/forma-2015-0016.[Crossref]
  • [17] Roland Coghetto. Groups – additive notation. Formalized Mathematics, 23(2):127–160, 2015. doi:10.1515/forma-2015-0013.[Crossref]
  • [18] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.
  • [19] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23–28, 2003.
  • [20] 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.[Crossref]
  • [21] Adam Grabowski and Robert Milewski. Boolean posets, posets under inclusion and products of relational structures. Formalized Mathematics, 6(1):117–121, 1997.
  • [22] 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.
  • [23] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.
  • [24] Artur Korniłowicz. The definition and basic properties of topological groups. Formalized Mathematics, 7(2):217–225, 1998.
  • [25] Artur Korniłowicz. Introduction to meet-continuous topological lattices. Formalized Mathematics, 7(2):279–283, 1998.
  • [26] Michał Muzalewski and Wojciech Skaba. From loops to Abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.
  • [27] Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93–96, 1991.
  • [28] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.
  • [29] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.
  • [30] Bartłomiej Skorulski. First-countable, sequential, and Frechet spaces. Formalized Mathematics, 7(1):81–86, 1998.
  • [31] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1 (2):369–376, 1990.
  • [32] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1 (1):187–190, 1990.
  • [33] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.
  • [34] Wojciech A. Trybulec. Binary operations on finite sequences. Formalized Mathematics, 1 (5):979–981, 1990.
  • [35] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.
  • [36] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.
  • [37] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski – Zorn lemma. Formalized Mathematics, 1(2):387–393, 1990.
  • [38] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.
  • [39] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.
Typ dokumentu
Bibliografia
Identyfikatory
bwmeta1.id-class.MML
CARDFIL3
Identyfikator YADDA
bwmeta1.element.doi-10_1515_forma-2015-0022
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.