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DOI: 10.4230/LIPIcs.ITP.2026.21

Formalizing Abstract Simplicial Complexes & Stellar Subdivisions in Lean

Authors: Garett Cunningham, Daniel Zach, and Stefan Friedl

Published in: LIPIcs, Volume 382, 17th International Conference on Interactive Theorem Proving (ITP 2026)

Abstract

The theory of simplicial complexes is a cornerstone of topology, offering a sophisticated tool for computing invariants. We present a formalization of abstract simplicial complexes and stellar subdivisions in the Lean proof assistant. We adopt a purely combinatorial framework in order to provide a cohesive foundation for studying the theory of stellar subdivisions as seen in many contexts of combinatorial topology. In particular, we provide formalizations of morphisms between abstract simplicial complexes; several crucial constructions and operations on complexes, such as links and joins; and perform a comprehensive study of how stellar subdivisions interact with these operations. We state and prove a number of identities commonly used in the study of triangulated manifolds, such as deriving equivalences between links in an abstract simplicial complex K and in a stellar subdivision σ_s K, including results with no references in the standard literature. To our knowledge, this is the first formalization of stellar subdivisions in any proof assistant.

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Garett Cunningham, Daniel Zach, and Stefan Friedl. Formalizing Abstract Simplicial Complexes & Stellar Subdivisions in Lean. In 17th International Conference on Interactive Theorem Proving (ITP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 382, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cunningham_et_al:LIPIcs.ITP.2026.21,
  author =	{Cunningham, Garett and Zach, Daniel and Friedl, Stefan},
  title =	{{Formalizing Abstract Simplicial Complexes \& Stellar Subdivisions in Lean}},
  booktitle =	{17th International Conference on Interactive Theorem Proving (ITP 2026)},
  pages =	{21:1--21:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-436-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{382},
  editor =	{Komendantskaya, Ekaterina and Nipkow, Tobias},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2026.21},
  URN =		{urn:nbn:de:0030-drops-269956},
  doi =		{10.4230/LIPIcs.ITP.2026.21},
  annote =	{Keywords: Lean, mathlib, simplicial complex, stellar subdivision, combinatorial topology}
}

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Formalizing Abstract Simplicial Complexes & Stellar Subdivisions in Lean

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