Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations

  • Alexey Solovyev
  • Thomas C. Hales
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7871)


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.


Interval Arithmetic Original Domain Taylor Approximation Polynomial Inequality Nonlinear Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexey Solovyev
    • 1
  • Thomas C. Hales
    • 1
  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

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