Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations

TL;DR

This paper introduces a formal verification tool using Taylor interval approximations within HOL Light to efficiently verify complex multivariate nonlinear inequalities, significantly advancing formal proof capabilities in high-dimensional cases.

Contribution

The paper presents a novel formal verification method for nonlinear inequalities using Taylor interval approximations, implemented in HOL Light, capable of handling high-dimensional problems efficiently.

Findings

Verified over 100 Flyspeck inequalities successfully.

Achieved verification speed about 3000 times slower than informal methods.

Demonstrated efficiency in proving high-dimensional inequalities in seconds.

Abstract

We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds. We developed the verification tool as a part of the Flyspeck project (a formal proof of the Kepler conjecture). The Flyspeck project includes about 1000 nonlinear inequalities. We successfully tested our method on more than 100 Flyspeck inequalities and estimated that the formal verification procedure is about 3000 times slower than an informal verification method implemented in C++. We also describe future work and prospective optimizations for our method.