Completion Light for HOL Light

For simplification to have good theoretical properties, the set of simp rules must be terminating and confluent. Completion is the process of transforming a set of simp rules, and attempting to make them terminating and confluent. We give a brief introduction to the theory of completion, and then describe an implementation of (basic) completion in HOL Light. Our notation follows that in [BN98]. Rewriting is one of the main techniques used in theorem proving. When using a set of rewrite rules, one desires that they are terminating (so that one does not have to spend time waiting for a run of rewriting that never terminates). Additionally, it is helpful if the rules are confluent. Confluence ensures that normal forms for equal terms are the same. This gives rise to an easy method for proving equalities: simply reduce each side to the normal form, and check that the normal forms are the same. The question then arises: how can one ensure that a set of rules are terminating and confluent? Termination is handled in a straightforward way: one provides a termination ordering (essentially a well founded ordering on terms obeying some further properties to ensure the termination measure interacts well with rewriting), such that each rule l → r is a reduction wrt. the ordering. Confluence is trickier, and completion is the process of taking a set of rules and adding further derived rules in an attempt to produce an equivalent set of confluent rules. Completion takes a set of rewrite rules