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Conway Normal Form: Bridging Approaches for Comprehensive Formalization of Surreal Numbers

Authors Karol Pąk , Cezary Kaliszyk

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Author Details

Karol Pąk
  • University of Białystok, Poland
Cezary Kaliszyk
  • University of Melbourne, Australia
  • University of Innsbruck, Austria

Funding

Supported by ERC PoC grant no. 101156734 FormalWeb3 and Cost action CA20111 EPN.

Cite As Get BibTex

Karol Pąk and Cezary Kaliszyk. Conway Normal Form: Bridging Approaches for Comprehensive Formalization of Surreal Numbers. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITP.2024.29

Abstract

The proper class of Conway’s surreal numbers forms a rich totally ordered algebraically closed field with many arithmetic and algebraic properties close to those of real numbers, the ordinals, and infinitesimal numbers. In this paper, we formalize the construction of Conway’s numbers in Mizar using two approaches and propose a bridge between them, aiming to combine their advantages for efficient formalization. By replacing transfinite induction-recursion with transfinite induction, we streamline their construction. Additionally, we introduce a method to merge proofs from both approaches using global choice, facilitating formal proof. We demonstrate that surreal numbers form a field, including the square root, and that they encompass subsets such as reals, ordinals, and powers of ω. We combined Conway’s work with Ehrlich’s generalization to formally prove Conway’s Normal Form, paving the way for many formal developments in surreal number theory.

Subject Classification

ACM Subject Classification
  • Theory of computation → Interactive proof systems
Keywords
  • Surreal numbers
  • Conway normal form
  • Mizar

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Supplementary Materials

References

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