Non-elementary Speedups between Different Versions of Tableaux

We have shown that the use of the \(\delta ^{ + ^ + }\)-rule instead of the δ-rule and the (newly introduced) δ✶-rule instead of the \(\delta ^{ + ^ + }\)-rule, respectively, may lead to nonelementary shortenings of tableau-based derivations, without increasing the size of the tableaux in any case. These speedups are based on restrictions of the sets of variables on which the Skolem-functions depend, that are introduced by the quantifier elimination rules. The effect corresponds to the result of reducing the scope of quantifiers by shifting them over subformulae that do not contain the variable concerned.