2026-08-18T03:58:43Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20734 2024-09-02T10:03:24Z ddc:004 open_access

Towards Solid Abelian Groups: A Formal Proof of Nöbeling’s Theorem Asgeirsson, Dagur Condensed mathematics Nöbeling’s theorem Lean Mathlib Interactive theorem proving 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Dagur Asgeirsson 2024 Is Part Of LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITP.2024.6 urn:nbn:de:0030-drops-207347 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.6 eng https://creativecommons.org/licenses/by/4.0/legalcode