## Section: New Results

### Formal proof that $e$ and $\pi $ are transcendental

Participants : Sophie Bernard, Laurence Rideau.

We constructed formal proofs that $\pi $ is irrational, $e$ is transcendental, and $\pi $ is transcendental. These proofs share a common initial pattern, where rationality or algebraicity of the mathematical constants are shown to imply the existence of a sequence of positive integers that must decrease indefinitely.

This proof development is an opportunity to study the interplay between several existing libraries about algebraic structures and analysis: the ssreflect library for algebra and the Coquelicot library for calculus. Moreover, the proof that $\pi $ is transcendental was an occasion to test the newly developed module on symmetric polynomials by P.-Y. Strub at IMDEA.