## Formal proofs of transcendence for e and pi as an application of multivariate and symmetric polynomials

**Mon 18 Jan 2016 16:30 - 17:00 at Room St Petersburg II**- Session 4: Mathematics

We describe the formalisation in Coq of a proof that the numbers e and pi are transcendental. This is the first formalized proof of transcendance for pi. 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 formalized proof originate in the more ancient library for real numbers included in the Coq distribution. The case of pi relied 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. In the paper, we describe the various original aspects of the work: orchestrating a development based on several libraries, designing a new library for multivariate polynomials, and specialized developments performed solely for this proof. Apart from achieving a landmark result, this work also teaches lessons about the evolution of modern formalized mathematics.