Complete Inference Rules for the Cancellation Laws

1 . I n t r o d u c t i o n In many cases, specialiied inference rules which incorporate certain axioms into the inference mechanism can produce fewer redundant consequences and more efficient proofs. The most notable example is paramodulation. Inference rules for inequalities, partial orerings, special binary relations have also been found ([BIH80], [S1N73], [MaW85]). Recently there has also been considerable interests in inference rules for the cancellation law. Stickel ((Sti84j) used i t , wi th the Knuth-Bendix method, to prove that implies xy =yx in ring theory. [W0M86] introduced an inference rule called negative paramodulation to find useful consequences resulting from cancellation. However, these methods provide only ad hoc treatments of cancellation. These inference rules are not complete and cannot eliminate the cancellation axiom from the input set of clauses. In this paper we present some complete sets of inference rules, which can replace the cancellation axioms. Due to space l imitat ion, we only outline the inference rules and state the main theorem wi thout proofs. Only simple examples are given to i l lustrate how the rules are used. The proof and more examples wi l l be given in the ful l paper.