Refutational Theorem Proving for Hierarchic First-Order Theories

We extend previous results on theorem proving for first-order clauses with equality to hierarchic first-order theories. Semantically such theories are confined to conservative extensions of the base models. It is shown that superposition together with variable abstraction and constraint refutation is refutationally complete for theories that are sufficiently complete with respect to simple instances. For the proof we introduce a concept of approximation between theorem proving systems, which makes it possible to reduce the problem to the known case of (flat) first-order theories. These results allow the modular combination of a superposition-based theorem prover with an arbitrary refutational prover for the primitive base theory, whose axiomatic representation in some logic may remain hidden. Furthermore they can be used to eliminate existentially quantified predicate symbols from certain second-order formulae