Certified Subterm Criterion and Certified Usable Rules

In this paper we present our formalization of two important termination techniques for term rewrite systems: the subterm criterion and the reduction pair processor in combination with usable rules. For both techniques we developed executable check functions in the theorem prover Isabelle/HOL which can certify the correct application of these techniques in some given termination proof. As there are several variants of usable rules we designed our check function in such a way that it accepts all known variants, even those which are not explicitly spelled out in previous papers. We integrated our formalization in the publicly available IsaFoR-library. This led to a significant increase in the power of CeTA, the corresponding certified termination proof checker that is extracted from IsaFoR.