Deep-Embedded Unification

We present some experiments with CiME/COCCINELLE about standard first-order unification of deep-embedded terms in the Coq proof assistant. We have modelled parts of the usual inference rules for unification, namely Decompose, Merge and Coalesce as Coq functions and we proved their soundness and completeness. Regarding the Generalized Occur-Check, we use the CiME rewriting toolbox as an oracle either to provide a total ordering compatible with Occur-Check, or to exhibit a cycle. Then, using these informations, it is possible to prove in Coq that there is no most general unifier (mgu), or to compute it and prove that it actually enjoys the mgu properties. All together, it means that it is possible to compute an mgu for deep-embedded terms in Coq. This is a first step towards formal proofs of confluence for TRS.