Combining Deduction Modulo and Logics of Fixed-Point Definitions

Abstract—Inductive and coinductive specifications are widely used in formalizing computational systems. Such specifications have a natural rendition in logics that support fixed-point defi-nitions. Another useful formalization device is that of recursive specifications. These specifications are not directly complemented by fixed-point reasoning techniques and, correspondingly, do not have to satisfy strong monotonicity restrictions. We show how to incorporate a rewriting capability into logics of fixed-point defini-tions towards additionally supporting recursive specifications. In particular, we describe a natural deduction calculus that adds a form of “closed-world ” equality—a key ingredient to supporting fixed-point definitions—to deduction modulo, a framework for extending a logic with a rewriting layer operating on formulas. We show that our calculus enjoys strong normalizability when the rewrite system satisfies general properties and we demonstrate its usefulness in specifying and reasoning about syntax-based descriptions. The integration of closed-world equality into deduc-tion modulo leads us to reconfigure the elimination principle for this form of equality in a way that resolves long-standing issues concerning the stability of finite proofs under proof reduction. I