Abstract. The hierarchical superposition based theorem proving calculus of Bachmair, Ganzinger, and Waldmann enables the hierarchic combination of a theory with full first-order logic. If a clause set of the combination enjoys a sufficient completeness criterion, the calculus is even complete. We instantiate and refine the calculus for the theory of linear arithmetic. In particular, we develop new effective versions for the standard superposition redundancy criteria taking the linear arithmetic theory into account. The resulting calculus is implemented in SPASS(LA) and extends the state of the art in proving properties of first-order formulas over linear arithmetic.
The first-order theory over non-linear arithmetic including transcendental functions (NLA) is undecidable. Nevertheless, in this paper we show that a particular combination with superposition leads to a sound and complete calculus that is useful in practice. We follow basically the ideas of the SUP(LA) combination, but have to take care of undecidability, resulting in ``unknown'' answers by the NLA reasoning procedure. A pipeline of NLA constraint simplification techniques related to the SUP(NLA) framework significantly decreases the number of ``unknown'' answers. The resulting approach is implemented as SUP(NLA) by a system combination of SPASS and iSAT. Applied to various scenarios of traffic collision avoidance protocols, we show by experiments that SPASS(iSAT) can fully automatically proof and disproof safety properties of such protocols using the very same formalization
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