Formal Proofs of Transcendence for e and $\pi$ as an Application of Multivariate and Symmetric Polynomials

TL;DR

This paper presents a formal proof in Coq demonstrating that e and π are transcendental numbers, integrating calculus and algebra through libraries and testing new tools for multivariate polynomials.

Contribution

It is the first formal proof of transcendence for e and π combining calculus and algebra in Coq, utilizing and testing new multivariate polynomial libraries.

Findings

Successful formalization of transcendence proofs in Coq

Integration of calculus and algebra libraries in proof development

Validation of new multivariate polynomial library

Abstract

We describe the formalisation in Coq of a proof that the numbers e and $\pi$ are transcendental. This proof lies at the interface of two domains of mathematics that are often considered separately: calculus (real and elementary complex analysis) and algebra. For the work on calculus, we rely on the Coquelicot library and for the work on algebra, we rely on the Mathematical Components library. Moreover, some of the elements of our formalized proof originate in the more ancient library for real numbers included in the Coq distribution. The case of $\pi$ relies extensively on properties of multivariate polynomials and this experiment was also an occasion to put to test a newly developed library for these multivariate polynomials.