Integrating Decision Procedures in Reflective Rewriting-Based Theorem Provers ?

We propose a design for the integration of decision procedures in reflective rewritingbased equational theorem provers. Rewriting-based equational theorem provers use term rewriting as their basic proof engine; they are particularly well suited for proving properties of equational specifications. A reflective rewriting-based theorem prover is itself an executable equational specification, which has reflective access to the rewriting engine responsible of its execution to efficiently accomplish rewriting-based equational proofs. This reflective access means that the built-in rewriting engine can be called with different rewriting commands. This opens up the possibility of interpolating calls to appropriate decision procedures in the midst of a rewriting-based proof step —typically to solve a condition in the application of a conditional equation. To illustrate our proposal and show its feasibility, we explain how our reflective design ideas can be used to integrate a decision procedure for Presburger arithmetic in the ITP tool, a reflective rewriting-based equational theorem prover, written entirely in Maude, for proving properties of Maude equational specification.