Formally Verified Approximations of Definite Integrals

International audienceFinding an elementary form for an antiderivative is often a difficult task, so numerical integration has become a common tool when it comes to making sense of a definite integral. Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. Yet numerical integration is still missing from the toolbox when performing formal proofs in analysis. This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system. Our approach is not based on traditional quadrature methods such as Newton-Cotes formulas. Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting. This work has been integrated to the CoqInterval library

Formally Verified Computation of Enclosures of Solutions of Ordinary Differential Equations

Ordinary differential equations (ODEs) are ubiquitous when mod-eling continuous dynamics. Classical numerical methods compute ap-proximations of the solution, however without any guarantees on the quality of the approximation. Nevertheless, methods have been devel-oped that are supposed to compute enclosures of the solution. In this paper, we demonstrate that enclosures of the solution can be verified with a high level of rigor: We implement a functional algo-rithm that computes enclosures of solutions of ODEs in the interactive theorem prover Isabelle/HOL, where we formally verify (and have me-chanically checked) the safety of the enclosures against the existing theory of ODEs in Isabelle/HOL. Our algorithm works with dyadic rational numbers with statically fixed precision and is based on the well-known Euler method. We ab-stract discretization and round-off errors in the domain of affine forms