D ec 2 01 5 Formal Proofs of Transcendence for e and π as an Application of Multivariate and Symmetric Polynomials ∗

We describe the formalisation in Coq of a proof that the numberse andπ are transcendental. This proof lies at the interface of two domains of mathematics that are often considered separa t ly: calculus (real and elementary complex analysis) and algebr a. For the work on calculus, we rely on the Coquelicot library and fo r the work on algebra, we rely on the Mathematical Components libr ary. Moreover, some of the elements of our formalized proof origi nate in the more ancient library for real numbers included in the C oq distribution. The case of π relies extensively on properties of multivariate polynomials and this experiment was also an occas ion to put to test a newly developed library for these multivariate polynomials.