Hierarchic superposition: Completeness without compactness

Abstract. Many applications of automated deduction and verification require reasoning in combinations of theories, such as, on the one hand (some fragment of) first-order logic, and on the other hand a background theory, such as some form of arithmetic. Unfortunately, due to the high expressivity of the full logic, complete reasoning is impossible in general. It is a realistic goal, however, to devise theorem provers that are “reasonably complete ” in practice, and the hier-archic superposition calculus has been designed as a theoretical basis for that. In a recent paper we introduced an extension of hierarchic superposition and proved its completeness for the fragment where every term of the background sort is ground. In this paper, we extend this result and obtain completeness for a larger fragment that admits variables in certain places. 1 Hierarchic Superposition Many applications of automated deduction and verification require reasoning in combi-nations of theories, such as, on the one hand (some fragment of) first-order logic and on the other hand some form of arithmetic. In hierarchic superposition [2, 3] we conside