This paper is concerned with proof methods for the temporal propertyof eventuality (a type of liveness) in systems of polynomial ordinary differential equations (ODEs) evolving under constraints. This problem is of a more general interest to hybrid system verification, where reasoning about temporal properties in the continuous fragment is often a bottleneck. Much of the difficulty in handling continuous systems stems from the fact that closed-form solutions to non-linear ODEs are rarely available. We present a general method for provingeventuality properties that works with the differential equations directly, without the need to compute their solutions. Our method is intuitively simple, yet much less conservative than previously reported approaches, making it highly amenable to use as a rule of inference in a formal proof calculus for hybrid systems.
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