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  3. Formalized Mathematics
  4. Formalized Mathematics, 2025, Volume 33, Issue 1

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    • Proszę używać tego identyfikatora do cytowań lub wstaw link do tej pozycji: http://hdl.handle.net/11320/19594
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    Pole DCWartośćJęzyk
    dc.contributor.authorPąk, Karol-
    dc.date.accessioned2026-01-09T11:44:20Z-
    dc.date.available2026-01-09T11:44:20Z-
    dc.date.issued2025-
    dc.identifier.citationFormalized Mathematics, Volume 33, Issue 1, Pages 25-41pl
    dc.identifier.issn1426-2630-
    dc.identifier.urihttp://hdl.handle.net/11320/19594-
    dc.description.abstractThis paper presents a formal definition of the Conway normal form, a structured representation uniquely suited to characterising surreal numbers by expressing them as sums within a hierarchically ordered group. To this end, we formalise the first sections of the chapter The Structure of the General Surreal Number in Conway’s book. In particular, we define omega maps and prove the existence and uniqueness of the Conway name for surreal numbers.pl
    dc.language.isoenpl
    dc.publisherUniversity of Białystokpl
    dc.rightsAttribution-ShareAlike 4.0 International (CC BY-SA 4.0)pl
    dc.rights.urihttps://creativecommons.org/licenses/by-sa/4.0/pl
    dc.subjectsurreal numberpl
    dc.subjectConway’s gamepl
    dc.subjectnormal formpl
    dc.titleConway’s Normal Form in the Mizar Systempl
    dc.typeArticlepl
    dc.rights.holder© 2025 The Author(s)pl
    dc.rights.holderCC BY-SA 4.0 licensepl
    dc.identifier.doi10.2478/forma-2025-0003-
    dc.description.AffiliationFaculty of Computer Science, University of Białystok, Polandpl
    dc.description.referencesVincent Bagayoko and Joris van der Hoeven. Surreal substructures. Fundamenta Mathe maticae, 266(1):25–96, 2024. doi:10.4064/fm231020-23-2.pl
    dc.description.referencesAlessandro Berarducci, Salma Kuhlmann, Vincenzo Mantova, and Mickael Matusinski. Exponential fields and Conway’s omega-map. Proceedings of the American Mathematical Society, 151(6):2655–2669, 2023. doi:10.1090/proc/14577.pl
    dc.description.referencesWolfgang Bertram. On Conway’s numbers and games, the von Neumann universe, and pure set theory. ArXiv preprint arXiv:2501.04412, 2025.pl
    dc.description.referencesJohn Horton Conway. On Numbers and Games. A K Peters Ltd., Natick, MA, second edition, 2001. ISBN 1-56881-127-6.pl
    dc.description.referencesOvidiu Costin and Philip Ehrlich. Integration on the surreals. Advances in Mathematics, 452:109823, 2024. doi:10.1016/j.aim.2024.109823.pl
    dc.description.referencesPeter Dybjer. A general formulation of simultaneous inductive-recursive definitions in type theory. The Journal of Symbolic Logic, 65(2):525–549, 2000. doi:10.2307/2586554.pl
    dc.description.referencesPhilip Ehrlich. Conway names, the simplicity hierarchy and the surreal number tree. Journal of Logic and Analysis, 3(1):1–26, 2011. doi:10.4115/jla.2011.3.1.pl
    dc.description.referencesPhilip Ehrlich. The absolute arithmetic continuum and the unification of all numbers great and small. The Bulletin of Symbolic Logic, 18(1):1–45, 2012. doi:10.2178/bsl/1327328438.pl
    dc.description.referencesPhilp Ehrlich. Number systems with simplicity hierarchies: A generalization of Conway’s theory of surreal numbers. Journal of Symbolic Logic, 66(3):1231–1258, 2001. doi:10.2307/2695104.pl
    dc.description.referencesHarry Gonshor. An Introduction to the Theory of Surreal Numbers. London Mathematical Society Lecture Note Series, 110. Cambridge University Press, 1986. ISBN 0521312051.pl
    dc.description.referencesAdam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.pl
    dc.description.referencesDonald E. Knuth. Surreal Numbers: How Two Ex-students Turned on to Pure Mathematics and Found Total Happiness. Addison-Wesley, 1974.pl
    dc.description.referencesLionel Elie Mamane. Surreal numbers in Coq. In Jean-Christophe Filliˆatre, Christine Paulin-Mohring, and Benjamin Werner, editors, Types for Proofs and Programs, TYPES 2004, volume 3839 of LNCS, pages 170–185. Springer, 2004. doi:10.1007/11617990_11.pl
    dc.description.referencesSteven Obua. Partizan games in Isabelle/HOLZF. In Kamel Barkaoui, Ana Cavalcanti, and Antonio Cerone, editors, Theoretical Aspects of Computing – ICTAC 2006, volume 4281 of LNCS, pages 272–286. Springer, 2006.pl
    dc.description.referencesKarol Pąk. Conway numbers – formal introduction. Formalized Mathematics, 31(1): 193–203, 2023. doi:10.2478/forma-2023-0018.pl
    dc.description.referencesKarol Pąk. Surreal dyadic and real numbers: A formal construction. Formalized Mathematics, 33(1):11–23, 2025. doi:10.2478/forma-2025-0002.pl
    dc.description.referencesKarol Pąk. Integration of game theoretic and tree theoretic approaches to Conway numbers. Formalized Mathematics, 31(1):205–213, 2023. doi:10.2478/forma-2023-0019.pl
    dc.description.referencesKarol Pąk and Cezary Kaliszyk. Conway normal form: Bridging approaches for comprehensive formalization of surreal numbers. In Yves Bertot, Temur Kutsia, and Michael Norrish, editors, 15th International Conference on Interactive Theorem Proving, ITP 2024, September 9-14, 2024, Tbilisi, Georgia, volume 309 of LIPIcs, pages 29:1–29:18. Schloss Dagstuhl – Leibniz-Zentrum f¨ur Informatik, 2024. doi:10.4230/LIPICS.ITP.2024.29.pl
    dc.description.referencesAlex Ryba, Philip Ehrlich, Richard Kenyon, Jeffrey Lagarias, James Propp, and Louis Kauffman. Conway’s mathematics after Conway. Notices of the American Mathematical Society, 69:1145–1155, 2022. doi:10.1090/noti2513.pl
    dc.description.referencesMakarius Wenzel, Lawrence C. Paulson, and Tobias Nipkow. The Isabelle framework. In Otmane Ait Mohamed, C´esar Mu˜noz, and Sofi`ene Tahar, editors, Theorem Proving in Higher Order Logics, pages 33–38. Springer Berlin Heidelberg, 2008.pl
    dc.identifier.eissn1898-9934-
    dc.description.volume33pl
    dc.description.issue1pl
    dc.description.firstpage25pl
    dc.description.lastpage41pl
    dc.identifier.citation2Formalized Mathematicspl
    dc.identifier.orcid0000-0002-7099-1669-
    Występuje w kolekcji(ach):Formalized Mathematics, 2025, Volume 33, Issue 1

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