Reasoning and Rewriting with Set-Relations I: Ground Completeness

The paper investigates reasoning with set-relations: intersection, inclusion and identity of 1-element sets. A language is introduced which, interpreted in a multi-algebraic semantics, allows one to specify such relations. An inference system is given and shown sound and refutationally ground-complete for a particular proof strategy which selects only maximal literals from the premise clauses. Each of the introduced set-relations satisfies only two among the three properties of the equivalence relations — we study rewriting with such non-equivalence relations and point out differences from the equational case. As a corollary of the main ground-completeness theorem we obtain ground-completeness of the introduced rewriting technique.