Basic Superposition is Complete

We define equality constrained equations and clauses and use them to prove the completeness of what we have called basic superposition: a restricted form of superposition in which only the subterms not created in previous inferences is superposed upon. We first apply our results to the equational case and define an unfailing Knuth-Bendix completion procedure that uses basic superposition as inference rule. Second, we extend the techniques to completion of full first-order clauses with equality. Moreover, we prove the refutational completeness of a new simple inference system.