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# Formalizing the Divergence Theorem and the Cauchy Integral Formula in Lean

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## Acknowledgements

I want to thank Patrick Massot for bringing up an idea of formalizing a sufficiently general version of the divergence theorem, Sébastien Gouëzel for fruitful discussions and peer review of most of the code, and my wife Nataliya Goncharuk for constant support. I also want to thank Anne Baanen, Johan Commelin, Nataliya Goncharuk, and Robert Y. Lewis for valuable comments on the draft versions of this paper and I thank my son Konstantin for finding lots of typos. I am also grateful to the anonymous referees for their valuable comments.

## 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)
https://doi.org/10.4230/LIPIcs.ITP.2022.23

## 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.

## Subject Classification

##### ACM Subject Classification
• Security and privacy → Logic and verification
• Mathematics of computing → Integral calculus
##### Keywords
• divergence theorem
• Green’s theorem
• Gauge integral
• Cauchy integral formula
• Cauchy-Goursat theorem
• complex analysis

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