Theorem Proving with Lemmas

The concern here is with proof procedures which are generalizations of input or unit deduction. The author's generalizations of input deduction involve lemmas, whereas those of unit deduction involve longer clauses and are akin to Robinson's P1 deduction. Chang's theorem, which establishes the equivalence of input and unit refutation, is extended to these generalizations. Completeness results of Henschen, Wos, and Kuehner for input or unit deduction applied to Horn sets are generalized to apply also to non-Horn sets. A key result is that any unsatisfiable set can be refuted by a lock linear resolution procedure in which the only lemmas are positive clauses composed entirely of instances of a small set of literals which can be specified in advance. In an implementation such lemmas would be generated only infrequently, thus allowing one to periodically gather the lemmas, discard other generated clauses, and restart the proof procedure.