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## 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.