DROPS

Document

DOI: 10.4230/LIPIcs.ITP.2025.4

A Formalization of Divided Powers in Lean

Authors: Antoine Chambert-Loir and María Inés de Frutos-Fernández

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract

Given an ideal I in a commutative ring A, a divided power structure on I is a collection of maps {γ_n : I → A}_{n ∈ ℕ}, subject to axioms that imply that it behaves like the family {x ↦ xⁿ/n!}_{n ∈ ℕ}, but which can be defined even when division by factorials is not possible in A. Divided power structures have important applications in diverse areas of mathematics, including algebraic topology, number theory and algebraic geometry. In this article we describe a formalization in Lean 4 of the basic theory of divided power structures, including divided power morphisms and sub-divided power ideals, and we provide several fundamental constructions, in particular quotients and sums. This constitutes the first formalization of this theory in any theorem prover. As a prerequisite of general interest, we expand the formalized theory of multivariate power series rings, endowing them with a topology and defining evaluation and substitution of power series.

Cite as

Antoine Chambert-Loir and María Inés de Frutos-Fernández. A Formalization of Divided Powers in Lean. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{chambertloir_et_al:LIPIcs.ITP.2025.4,
  author =	{Chambert-Loir, Antoine and de Frutos-Fern\'{a}ndez, Mar{\'\i}a In\'{e}s},
  title =	{{A Formalization of Divided Powers in Lean}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{4:1--4:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.4},
  URN =		{urn:nbn:de:0030-drops-246038},
  doi =		{10.4230/LIPIcs.ITP.2025.4},
  annote =	{Keywords: Formal mathematics, algebraic number theory, commutative algebra, divided powers, Lean, Mathlib}
}

Search Results

Documents authored by Chambert-Loir, Antoine

A Formalization of Divided Powers in Lean

Abstract

Cite as

Thanks for your feedback!

Could not send message