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This paper presents a new kind of abstraction, which has been developed for the purpose of proofplanning. The basic idea of this paper is to abstract a given theorem and to find an abstractproof of it. Once an abstract proof has been found, this proof has to be refined to a real proofof the original theorem. We present a goal oriented abstraction for the purpose of equality proofplanning, which is parameterized by common parts of the left- and right-hand sides of the givenequality. Therefore, this abstraction technique provides an abstract equality problem which ismore adequate than those generated by the abstractions known so far. The presented abstractionalso supports the heuristic search process based on the difference reduction paradigm. We give aformal definition of the abstract space including the objects and their manipulation. Furthermore,we prove some properties in order to allow an efficient implementation of the presented abstraction.

Simultaneous quantifier elimination in sequent calculus is an improvement over the well-known skolemization. It allows a lazy handling of instantiations as well as of the order of certain reductions. We prove the soundness of a sequent calculus which incorporates a rule for simultaneous quantifier elimination. The proof is performed by semantical arguments and provides some insights into the dependencies between various formulas in a sequent.