Transforming orthogonal inductive definition sets into confluent term rewrite systems

In this paper, we transform an orthogonal inductive definition set, which is a set of productions for inductive predicates, into a confluent term rewrite system such that a quantifier-free sequent is valid w.r.t. the inductive definition set if and only if an equation representing the sequent is an inductive theorem of the term rewrite system. To this end, we first propose a transformation of an orthogonal inductive definition set into a confluent term rewrite system that is equivalent to the inductive definition set in the sense of evaluating ground formulas. Then, we show that termination of the inductive definition set is proved by the generalized subterm criterion if and only if termination of the transformed term rewrite system is so. Finally, we show that the transformed term rewrite system with some rewrite rules for sequents has the expected property. In addition, we show a termination criterion for the union of term rewrite systems whose termination is proved by the generalized subterm criterion.