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15-424 : Foundations of Cyber-Physical Systems Lecture Notes on Virtual Substitution & Real Arithmetic

Reasoning about cyber-physical systems and hybrid systems requires understanding and handling their real arithmetic, which can be challenging, because cyber-physical systems can have complex behavior. Differential dynamic logic and its proof calculus [Pla08, Pla10, Pla12] reduce the verification of hybrid systems to real arithmetic. How arithmetic interfaces with proofs has already been discussed in Lecture 9 on Proofs & Arithmetic. How real arithmetic with linear and quadratic equations can be handled by virtual substitution has been shown in Lecture 18 on Virtual Substitution & Real Equations. Today’s lecture shows how virtual substitution for quantifier elimination in real arithmetic extends to the case of linear and quadratic inequalities. These lecture notes are loosely based on [Wei97, Pla10, Appendix D]. They add substantial intuition and motivation that is helpful for following the technical development. More information about virtual substitution can be found in the literature [Wei97]. See, e.g., [PQR09, Pas11] for an overview of other techniques for real arithmetic. The most important learning goals of this lecture are: