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

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

Pappus’s Hexagon Theorem in Real Projective Plane

In this article we prove, using Mizar [2], [1], the Pappus’s hexagon theorem in the real projective plane: “Given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear”2. More precisely, we prove that the structure ProjectiveSpace TOP-REAL3 [10] (where TOP-REAL3 is a metric space defined in [5]) satisfies the Pappus’s axiom defined in [11] by Wojciech Leonczuk and Krzysztof Prazmowski. Eugeniusz Kusak andWojciech Leonczuk formalized the Hessenberg theorem early in the MML [9]. With this result, the real projective plane is Desarguesian. For proving the Pappus’s theorem, two different proofs are given. First, we use the techniques developed in the section “Projective Proofs of Pappus’s Theorem” in the chapter “Pappos’s Theorem: Nine proofs and three variations” [12]. Secondly, Pascal’s theorem [4] is used. In both cases, to prove some lemmas, we use Prover93, the successor of the Otter prover and ott2miz by Josef Urban4 [13], [8], [7]. In Coq, the Pappus’s theorem is proved as the application of Grassmann-Cayley algebra [6] and more recently in Tarski’s geometry [3].This work has been supported by the “Centre autonome de formation et de recherche en mathématiques et sciences avec assistants de preuve” ASBL (non-profit organization). Enterprise number: 0777.779.751. Belgium.Rue de la Brasserie 5, 7100 La Louvière, BelgiumGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: Stateof-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.Gabriel Braun and Julien Narboux. A synthetic proof of Pappus’ theorem in Tarski’s geometry. Journal of Automated Reasoning, 58(2):23, 2017. doi:10.1007/s10817-016-9374-4.Roland Coghetto. Pascal’s theorem in real projective plane. Formalized Mathematics, 25(2):107–119, 2017. doi:10.1515/forma-2017-0011.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.Laurent Fuchs and Laurent Thery. A formalization of Grassmann-Cayley algebra in Coq and its application to theorem proving in projective geometry. In Automated Deduction in Geometry, pages 51–67. Springer, 2010.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. Solving two problems in general topology via types. In Types for Proofs and Programs, International Workshop, TYPES 2004, Jouyen-Josas, France, December 15-18, 2004, Revised Selected Papers, pages 138–153, 2004. doi:10.1007/11617990 9. http://dblp.uni-trier.de/rec/bib/conf/types/Grabowski04.Eugeniusz Kusak and Wojciech Leonczuk. Hessenberg theorem. Formalized Mathematics, 2(2):217–219, 1991.Wojciech Leonczuk and Krzysztof Prazmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761–766, 1990.Wojciech Leonczuk and Krzysztof Prazmowski. Projective spaces – part I. Formalized Mathematics, 1(4):767–776, 1990.Jürgen Richter-Gebert. Pappos’s Theorem: Nine Proofs and Three Variations, pages 3–31. Springer Berlin Heidelberg, 2011. ISBN 978-3-642-17286-1. doi:10.1007/978-3-642-17286-1_1.Piotr Rudnicki and Josef Urban. Escape to ATP for Mizar. In First International Workshop on Proof eXchange for Theorem Proving-PxTP 2011, 2011.292697

A Case Study of Transporting Urysohn’s Lemma from Topology via Open Sets into Topology via Neighborhoods

Józef Białas and Yatsuka Nakamura has completely formalized a proof of Urysohn’s lemma in the article [4], in the context of a topological space defined via open sets. In the Mizar Mathematical Library (MML), the topological space is defined in this way by Beata Padlewska and Agata Darmochwał in the article [18]. In [7] the topological space is defined via neighborhoods. It is well known that these definitions are equivalent [5, 6]. In the definitions, an abstract structure (i.e. the article [17, STRUCT 0] and its descendants, all of them directly or indirectly using Mizar structures [3]) have been used (see [10], [9]). The first topological definition is based on the Mizar structure TopStruct and the topological space defined via neighborhoods with the Mizar structure: FMT Space Str. To emphasize the notion of a neighborhood, we rename FMT TopSpace (topology from neighbourhoods) to NTopSpace (a neighborhood topological space). Using Mizar [2], we transport the Urysohn’s lemma from TopSpace to NTop-Space. In some cases, Mizar allows certain techniques for transporting proofs, definitions or theorems. Generally speaking, there is no such automatic translating. In Coq, Isabelle/HOL or homotopy type theory transport is also studied, sometimes with a more systematic aim [14], [21], [11], [12], [8], [19]. In [1], two co-existing Isabelle libraries: Isabelle/HOL and Isabelle/Mizar, have been aligned in a single foundation in the Isabelle logical framework. In the MML, they have been used since the beginning: reconsider, registration, cluster, others were later implemented [13]: identify. In some proofs, it is possible to define particular functors between different structures, mainly useful when results are already obtained in a given structure. This technique is used, for example, in [15] to define two functors MXR2MXF and MXF2MXF between Matrix of REAL and Matrix of F-Real and to transport the definition of the addition from one structure to the other: [...] A + B -> Matrix of REAL equals MXF2MXR ((MXR2MXF A) + (MXR2MXF B)) [...]. In this paper, first we align the necessary topological concepts. For the formalization, we were inspired by the works of Claude Wagschal [20]. It allows us to transport more naturally the Urysohn’s lemma ([4, URYSOHN3:20]) to the topological space defined via neighborhoods. Nakasho and Shidama have developed a solution to explore the notions introduced in various ways https://mimosa-project.github.io/mmlreference/current/ [16]. The definitions can be directly linked in the HTML version of the Mizar library (example: Urysohn’s lemma http://mizar.org/version/current/html/urysohn3.html#T20).Higher-Order Tarski Grothendieck as a Foundation for Formal Proof, Sep. 2019. Zenodo. doi:10.4230/lipics.itp.2019.9.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.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.Józef Białas and Yatsuka Nakamura. The Urysohn lemma. Formalized Mathematics, 9 (3):631–636, 2001.Nicolas Bourbaki. General Topology: Chapters 1–4. Springer Science and Business Media, 2013.Nicolas Bourbaki. Topologie générale: Chapitres 1à 4. Eléments de mathématique. Springer Science & Business Media, 2007.Roland Coghetto. Topology from neighbourhoods. Formalized Mathematics, 23(4):289–296, 2015. doi:10.1515/forma-2015-0023.Thierry Coquand. Théorie des types dépendants et axiome d’univalence. Séminaire Bourbaki, 66:1085, 2014.Adam Grabowski and Roland Coghetto. Extending Formal Topology in Mizar by Uniform Spaces, pages 77–105. Springer, Cham, 2020. ISBN 978-3-030-41425-2. doi:10.1007/978-3-030-41425-2_2.Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. Refining Algebraic Hierarchy in Mathematical Repository of Mizar, pages 49–75. Springer, Cham, 2020. ISBN 978-3-030-41425-2. doi:10.1007/978-3-030-41425-2_2.Brian Huffman and Ondřej Kunčar. Lifting and transfer: A modular design for quotients in Isabelle/HOL. In International Conference on Certified Programs and Proofs, pages 131–146. Springer, 2013.Einar Broch Johnsen and Christoph Lüth. Theorem reuse by proof term transformation. In International Conference on Theorem Proving in Higher Order Logics, pages 152–167. Springer, 2004.Artur Korniłowicz. How to define terms in Mizar effectively. Studies in Logic, Grammar and Rhetoric, 18:67–77, 2009.Nicolas Magaud. Changing data representation within the Coq system. In International Conference on Theorem Proving in Higher Order Logics, pages 87–102. Springer, 2003.Yatsuka Nakamura, Nobuyuki Tamura, and Wenpai Chang. A theory of matrices of real elements. Formalized Mathematics, 14(1):21–28, 2006. doi:10.2478/v10037-006-0004-1.Kazuhisa Nakasho and Yasunari Shidama. Documentation generator focusing on symbols for the HTML-ized Mizar library. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, CICM 2015, volume 9150 of Lecture Notes in Computer Science, pages 343–347. Springer, Cham, 2015. doi:10.1007/978-3-319-20615-8_25.Library Committee of the Association of Mizar Users. Preliminaries to structures. Mizar Mathematical Library, 1995.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.Nicolas Tabareau, Éric Tanter, and Matthieu Sozeau. Equivalences for free: univalent parametricity for effective transport. Proceedings of the ACM on Programming Languages, 2(ICFP):1–29, 2018.Claude Wagschal. Topologie et analyse fonctionnelle. Hermann, 1995.Theo Zimmermann and Hugo Herbelin. Automatic and transparent transfer of theorems along isomorphisms in the Coq proof assistant. arXiv preprint arXiv:1505.05028, 2015.28322723

Topology from Neighbourhoods

Using Mizar [9], and the formal topological space structure (FMT_Space_Str) [19], we introduce the three U-FMT conditions (U-FMT filter, U-FMT with point and U-FMT local) similar to those VI, VII, VIII and VIV of the proposition 2 in [10]: If to each element x of a set X there corresponds a set B(x) of subsets of X such that the properties VI, VII, VIII and VIV are satisfied, then there is a unique topological structure on X such that, for each x ∈ X, B(x) is the set of neighborhoods of x in this topology.We present a correspondence between a topological space and a space defined with the formal topological space structure with the three U-FMT conditions called the topology from neighbourhoods. For the formalization, we were inspired by the works of Bourbaki [11] and Claude Wagschal [31].Rue de la Brasserie 5, 7100 La Louvière, BelgiumGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. Complete lattices. Formalized Mathematics, 2(5):719–725, 1991.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. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93–107, 1997.Grzegorz Bancerek. Prime ideals and filters. Formalized Mathematics, 6(2):241–247, 1997.Grzegorz Bancerek. Bases and refinements of topologies. Formalized Mathematics, 7(1): 35–43, 1998.Grzegorz Bancerek, Noboru Endou, and Yuji Sakai. On the characterizations of compactness. Formalized Mathematics, 9(4):733–738, 2001.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]Nicolas Bourbaki. General Topology: Chapters 1–4. Springer Science and Business Media, 2013.Nicolas Bourbaki. Topologie générale: Chapitres 1 à 4. Eléments de mathématique. Springer Science & Business Media, 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. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Roland Coghetto. Convergent filter bases. Formalized Mathematics, 23(3):189–203, 2015. doi:10.1515/forma-2015-0016. [Crossref]Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Adam Grabowski and Robert Milewski. Boolean posets, posets under inclusion and products of relational structures. Formalized Mathematics, 6(1):117–121, 1997.Gang Liu, Yasushi Fuwa, and Masayoshi Eguchi. Formal topological spaces. Formalized Mathematics, 9(3):537–543, 2001.Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Formalized Mathematics, 5(2):167–172, 1996.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. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341–347, 2003.Andrzej Trybulec. Moore-Smith convergence. Formalized Mathematics, 6(2):213–225, 1997.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski – Zorn lemma. Formalized Mathematics, 1(2):387–393, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Josef Urban. Basic facts about inaccessible and measurable cardinals. Formalized Mathematics, 9(2):323–329, 2001.Claude Wagschal. Topologie et analyse fonctionnelle. Hermann, 1995.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.Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215–222, 1990

Summable Family in a Commutative Group

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.Rue de la Brasserie 5, 7100 La Louvière, BelgiumGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 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 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.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.Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481–485, 1991.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 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. 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. Convergent filter bases. Formalized Mathematics, 23(3):189–203, 2015. doi:10.1515/forma-2015-0016. [Crossref]Roland Coghetto. Groups – additive notation. Formalized Mathematics, 23(2):127–160, 2015. doi:10.1515/forma-2015-0013. [Crossref]Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23–28, 2003.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]Adam Grabowski and Robert Milewski. Boolean posets, posets under inclusion and products of relational structures. Formalized Mathematics, 6(1):117–121, 1997.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.Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.Artur Korniłowicz. The definition and basic properties of topological groups. Formalized Mathematics, 7(2):217–225, 1998.Artur Korniłowicz. Introduction to meet-continuous topological lattices. Formalized Mathematics, 7(2):279–283, 1998.Michał Muzalewski and Wojciech Skaba. From loops to Abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93–96, 1991.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.Bartłomiej Skorulski. First-countable, sequential, and Frechet spaces. Formalized Mathematics, 7(1):81–86, 1998.Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1 (2):369–376, 1990.Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1 (1):187–190, 1990.Wojciech A. Trybulec. 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Klein-Beltrami model. Part III

Timothy Makarios (with Isabelle/HOL1) and John Harrison (with HOL-Light2) shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom” [2],[3],[4],[5]. With the Mizar system [1] we use some ideas taken from Tim Makarios’s MSc thesis [10] to formalize some definitions (like the absolute) and lemmas necessary for the verification of the independence of the parallel postulate. In this article we prove that our constructed model (we prefer “Beltrami-Klein” name over “Klein-Beltrami”, which can be seen in the naming convention for Mizar functors, and even MML identifiers) satisfies the congruence symmetry, the congruence equivalence relation, and the congruence identity axioms formulated by Tarski (and formalized in Mizar as described briefly in [8]).Rue de la Brasserie 5, 7100 La Louvière, BelgiumGrzegorz 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.Eugenio Beltrami. Saggio di interpetrazione della geometria non-euclidea. Giornale di Matematiche, 6:284–322, 1868.Eugenio Beltrami. Essai d’interprétation de la géométrie non-euclidéenne. In Annales scientifiques de l’École Normale Supérieure. Trad. par J. Hoüel, volume 6, pages 251–288. Elsevier, 1869.Karol Borsuk and Wanda Szmielew. Foundations of Geometry. North Holland, 1960.Karol Borsuk and Wanda Szmielew. Podstawy geometrii. Panstwowe Wydawnictwo Naukowe, Warszawa, 1955 (in Polish).Roland Coghetto. Group of homography in real projective plane. Formalized Mathematics, 25(1):55–62, 2017. doi:10.1515/forma-2017-0005.Roland Coghetto. Klein-Beltrami model. Part II. Formalized Mathematics, 26(1):33–48, 2018. doi:10.2478/forma-2018-0004.Adam Grabowski and Roland Coghetto. Tarski’s geometry and the Euclidean plane in Mizar. In Joint Proceedings of the FM4M, MathUI, and ThEdu Workshops, Doctoral Program, and Work in Progress at the Conference on Intelligent Computer Mathematics 2016 co-located with the 9th Conference on Intelligent Computer Mathematics (CICM 2016), Białystok, Poland, July 25–29, 2016, volume 1785 of CEUR-WS, pages 4–9. CEURWS.org, 2016.Wojciech Leonczuk and Krzysztof Prazmowski. Incidence projective spaces. Formalized Mathematics, 2(2):225–232, 1991.Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean Axiom. Victoria University ofWellington, New Zealand, 2012. Master’s thesis.2811

Klein-Beltrami model. Part IV

Timothy Makarios (with Isabelle/HOL1) and John Harrison (with HOL-Light2) shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom” [2],[3],[4, 5]. With the Mizar system [1] we use some ideas taken from Tim Makarios’s MSc thesis [10] to formalize some definitions and lemmas necessary for the verification of the independence of the parallel postulate. In this article, which is the continuation of [8], we prove that our constructed model satisfies the axioms of segment construction, the axiom of betweenness identity, and the axiom of Pasch due to Tarski, as formalized in [11] and related Mizar articles.Rue de la Brasserie 5, 7100 La Louvière, BelgiumGrzegorz 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.Eugenio Beltrami. Saggio di interpetrazione della geometria non-euclidea. Giornale di Matematiche, 6:284–322, 1868.Eugenio Beltrami. Essai d’interprétation de la géométrie non-euclidéenne. In Annales scientifiques de l’École Normale Supérieure. Trad. par J. Hoüel, volume 6, pages 251–288. Elsevier, 1869.Karol Borsuk and Wanda Szmielew. Foundations of Geometry. North Holland, 1960.Karol Borsuk and Wanda Szmielew. Podstawy geometrii. Panstwowe Wydawnictwo Naukowe, Warszawa, 1955 (in Polish).Roland Coghetto. Homography in RP2. Formalized Mathematics, 24(4):239–251, 2016.doi:10.1515/forma-2016-0020.Roland Coghetto. Klein-Beltrami model. Part I. Formalized Mathematics, 26(1):21–32, 2018. doi:10.2478/forma-2018-0003.Roland Coghetto. Klein-Beltrami model. Part III. Formalized Mathematics, 28(1):1–7, 2020. doi:10.2478/forma-2020-0001.Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381–383, 2003.Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean Axiom. Victoria University ofWellington, New Zealand, 2012. Master’s thesis.William Richter, Adam Grabowski, and Jesse Alama. Tarski geometry axioms. Formalized Mathematics, 22(2):167–176, 2014. doi:10.2478/forma-2014-0017.92