A Coq formal proof of the Lax–Milgram theorem

Sylvie Boldo (1) , François Clément (2) , Florian Faissole (1) , Vincent Martin (3) , Micaela Mayero (4)

1 TOCCATA - Formally Verified Programs, Certified Tools and Numerical Computations
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
3 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne
4 LIPN - Laboratoire d'Informatique de Paris-Nord

The Finite Element Method is a widely-used method to solve numerical problems coming for instance from physics or biology. To obtain the highest confidence on the correction of numerical simulation programs implementing the Finite Element Method, one has to formalize the mathematical notions and results that allow to establish the sound-ness of the method. The Lax–Milgram theorem may be seen as one of those theoretical cornerstones: under some completeness and coercivity assumptions, it states existence and uniqueness of the solution to the weak formulation of some boundary value problems. This article presents the full formal proof of the Lax–Milgram theorem in Coq. It requires many results from linear algebra, geometry, functional analysis , and Hilbert spaces.

Mots clés: Lax-Milgram theorem; formal proof; functional analysis; finite element method; Coq

Domaines: Logique en informatique [cs.LO]; Analyse fonctionnelle [math.FA]; Analyse numérique [math.NA]

Sylvie Boldo, François Clément, Florian Faissole, Vincent Martin, Micaela Mayero. A Coq formal proof of the Lax–Milgram theorem. 6th ACM SIGPLAN Conference on Certified Programs and Proofs, Jan 2017, Paris, France. ⟨10.1145/3018610.3018625⟩. ⟨hal-01391578⟩