We present a Coq library that allows for readily proving that a function is computable in polynomial time. It is based on quasi-interpretations that, in combination with termination ordering, provide a characterisation of the class fp of functions computable in polynomial time. At the heart of this formalisation is a proof of soundness and extensional completeness. Compared to the original paper proof, we had to fill a lot of not so trivial details that were left to the reader and fix a few glitches. To demonstrate the usability of our library, we apply it to the modular exponentiation.

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 soundness 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.