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Region stability allows one to formalize hybrid systems whose trajectories may oscillate (within a given allowance) even after having `stabilized'. Unfortunately, until today no proof rule (giving necessary and sufficient conditions for the purpose of verifying region stability) has been available. This paper fills the gap. Our (sound and complete) proof rule connects region stability with the finiteness of specific state sequences and thus with the emerging set of verification methods for program termination

We present a method and tool (and implementation) for automatic proofs of region stability for hybrid systems. The formal basis of our approach is the new notion of \emph{snapshot sequences}. We use snapshot sequences for a characterization of region stability. Our abstraction-based algorithm checks the conditions in this characterization. A number of experiments demonstrate the practical potential of our approach

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Abstract. Procedure summaries are an approximation of the effect of a procedure call. They have been used to prove partial correctness and safety properties. In this paper, we introduce a generalized notion of procedure summaries and present a framework to verify total correctness and liveness properties of a general class of while programs with recursion. We provide a fixpoint system for computing summaries, and a proof rule for total correctness of a program given a summary. With suitable abstraction methods and algorithms for efficient summary computation, the results presented here can be used for the automatic verification of termination and liveness properties for while programs with recursion.