2 Search Results for "Kudryashov, Yury"

Document
DOI: 10.4230/LIPIcs.ITP.2024.37
Integrals Within Integrals: A Formalization of the Gagliardo-Nirenberg-Sobolev Inequality

Authors: Floris van Doorn and Heather Macbeth

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

Abstract
We introduce an abstraction which allows arguments involving iterated integrals to be formalized conveniently in type-theory-based proof assistants. We call this abstraction the marginal construction, since it is connected to the marginal distribution in probability theory. The marginal construction gracefully handles permutations to the order of integration (Tonelli’s theorem in several variables), as well as arguments involving an induction over dimension. We implement the marginal construction and several applications in the language Lean. The most difficult of these applications, the Gagliardo-Nirenberg-Sobolev inequality, is a foundational result in the theory of elliptic partial differential equations and has not previously been formalized.
Cite as

Floris van Doorn and Heather Macbeth. Integrals Within Integrals: A Formalization of the Gagliardo-Nirenberg-Sobolev Inequality. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vandoorn_et_al:LIPIcs.ITP.2024.37,
  author =	{van Doorn, Floris and Macbeth, Heather},
  title =	{{Integrals Within Integrals: A Formalization of the Gagliardo-Nirenberg-Sobolev Inequality}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{37:1--37:18},
  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.2024.37},
  URN =		{urn:nbn:de:0030-drops-207657},
  doi =		{10.4230/LIPIcs.ITP.2024.37},
  annote =	{Keywords: Sobolev inequality, measure theory, Lean, formalized mathematics}
}
Document
DOI: 10.4230/LIPIcs.ITP.2022.23
Formalizing the Divergence Theorem and the Cauchy Integral Formula in Lean

Authors: Yury Kudryashov

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

Abstract
I formalize a version of the divergence theorem for a function on a rectangular box that does not assume regularity of individual partial derivatives, only Fréchet differentiability of the vector field and integrability of its divergence. Then I use this theorem to prove the Cauchy-Goursat theorem (for some simple domains) and bootstrap complex analysis in the Lean mathematical library. The main tool is the GP-integral, a version of the Henstock-Kurzweil integral introduced by J. Mawhin in 1981. The divergence theorem for this integral does not require integrability of the divergence.
Cite as

Yury Kudryashov. Formalizing the Divergence Theorem and the Cauchy Integral Formula in Lean. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kudryashov:LIPIcs.ITP.2022.23,
  author =	{Kudryashov, Yury},
  title =	{{Formalizing the Divergence Theorem and the Cauchy Integral Formula in Lean}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{23:1--23:19},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.23},
  URN =		{urn:nbn:de:0030-drops-167326},
  doi =		{10.4230/LIPIcs.ITP.2022.23},
  annote =	{Keywords: divergence theorem, Green’s theorem, Gauge integral, Cauchy integral formula, Cauchy-Goursat theorem, complex analysis}
}
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