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