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DOI: 10.4230/LIPIcs.ITP.2024.6

Towards Solid Abelian Groups: A Formal Proof of Nöbeling’s Theorem

Authors: Dagur Asgeirsson

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract

Condensed mathematics, developed by Clausen and Scholze over the last few years, is a new way of studying the interplay between algebra and geometry. It replaces the concept of a topological space by a more sophisticated but better-behaved idea, namely that of a condensed set. Central to the theory are solid abelian groups and liquid vector spaces, analogues of complete topological groups. Nöbeling’s theorem, a surprising result from the 1960s about the structure of the abelian group of continuous maps from a profinite space to the integers, is a crucial ingredient in the theory of solid abelian groups; without it one cannot give any nonzero examples of solid abelian groups. We discuss a recently completed formalisation of this result in the Lean theorem prover, and give a more detailed proof than those previously available in the literature. The proof is somewhat unusual in that it requires induction over ordinals - a technique which has not previously been used to a great extent in formalised mathematics.

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Dagur Asgeirsson. Towards Solid Abelian Groups: A Formal Proof of Nöbeling’s Theorem. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{asgeirsson:LIPIcs.ITP.2024.6,
  author =	{Asgeirsson, Dagur},
  title =	{{Towards Solid Abelian Groups: A Formal Proof of N\"{o}beling’s Theorem}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.6},
  URN =		{urn:nbn:de:0030-drops-207347},
  doi =		{10.4230/LIPIcs.ITP.2024.6},
  annote =	{Keywords: Condensed mathematics, N\"{o}beling’s theorem, Lean, Mathlib, Interactive theorem proving}
}

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Towards Solid Abelian Groups: A Formal Proof of Nöbeling’s Theorem

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