A formal proof of Hensel's lemma over the p-adic integers

The field of p-adic numbers _p and the ring of p-adic integers _p are essential constructions of modern number theory. Hensels lemma, described by Gouva as the most important algebraic property of the p-adic numbers, shows the existence of roots of polynomials over _p provided an initial seed point. The theorem can be proved for the p-adics with significantly weaker hypotheses than for general rings. We construct _p and _p in the Lean proof assistant, with various associated algebraic properties, and formally prove a strong form of Hensels lemma. The proof lies at the intersection of algebraic and analytic reasoning and demonstrates how the Lean mathematical library handles such a heterogeneous topic.