2 Search Results for "Dillies, Yaël"

Document
DOI: 10.4230/LIPIcs.ITP.2023.13
Formalizing Norm Extensions and Applications to Number Theory

Authors: María Inés de Frutos-Fernández

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)

Abstract
The field ℝ of real numbers is obtained from the rational numbers ℚ by taking the completion with respect to the usual absolute value. We then define the complex numbers ℂ as an algebraic closure of ℝ. The p-adic analogue of the real numbers is the field ℚ_p of p-adic numbers, obtained by completing ℚ with respect to the p-adic norm. In this paper, we formalize in Lean 3 the definition of the p-adic analogue of the complex numbers, which is the field ℂ_p of p-adic complex numbers, a field extension of ℚ_p which is both algebraically closed and complete with respect to the extension of the p-adic norm. More generally, given a field K complete with respect to a nonarchimedean real-valued norm, and an algebraic field extension L/K, we show that there is a unique norm on L extending the given norm on K, with an explicit description. Building on the definition of ℂ_p, we formalize the definition of the Fontaine period ring B_{HT} and discuss some applications to the theory of Galois representations and to p-adic Hodge theory. The results formalized in this paper are a prerequisite to formalize Local Class Field Theory, which is a fundamental ingredient of the proof of Fermat’s Last Theorem.
Cite as

María Inés de Frutos-Fernández. Formalizing Norm Extensions and Applications to Number Theory. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{defrutosfernandez:LIPIcs.ITP.2023.13,
  author =	{de Frutos-Fern\'{a}ndez, Mar{\'\i}a In\'{e}s},
  title =	{{Formalizing Norm Extensions and Applications to Number Theory}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{13:1--13:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.13},
  URN =		{urn:nbn:de:0030-drops-183880},
  doi =		{10.4230/LIPIcs.ITP.2023.13},
  annote =	{Keywords: formal mathematics, Lean, mathlib, algebraic number theory, p-adic analysis, Galois representations, p-adic Hodge theory}
}
Document
DOI: 10.4230/LIPIcs.ITP.2022.9
Formalising Szemerédi’s Regularity Lemma in Lean

Authors: Yaël Dillies and Bhavik Mehta

Published in: LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

Abstract
Szemerédi’s Regularity Lemma is a fundamental result in graph theory with extensive applications to combinatorics and number theory. In essence, it says that all graphs can be approximated by well-behaved unions of random bipartite graphs. We present a formalisation in the Lean theorem prover of a strong version of this lemma in which each part of the union must be approximately the same size. This stronger version has not been formalised previously in any theorem prover. Our proof closely follows the pen-and-paper method, allowing our formalisation to provide an explicit upper bound on the number of parts. An application of this lemma is also formalised, namely Roth’s theorem on arithmetic progressions in qualitative form via the triangle removal lemma.
Cite as

Yaël Dillies and Bhavik Mehta. Formalising Szemerédi’s Regularity Lemma in Lean. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{dillies_et_al:LIPIcs.ITP.2022.9,
  author =	{Dillies, Ya\"{e}l and Mehta, Bhavik},
  title =	{{Formalising Szemer\'{e}di’s Regularity Lemma in Lean}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-252-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{237},
  editor =	{Andronick, June and de Moura, Leonardo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.9},
  URN =		{urn:nbn:de:0030-drops-167185},
  doi =		{10.4230/LIPIcs.ITP.2022.9},
  annote =	{Keywords: Lean, formalisation, formal proof, graph theory, combinatorics, additive combinatorics, Szemer\'{e}di’s Regularity Lemma, Roth’s Theorem}
}
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