  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.
