An insertion operator preserving infinite reduction sequences

A common way to show the termination of the union of two abstract reduction systems, provided both systems terminate, is to prove they enjoy a specific property (some sort of “commutation” for instance). This specific property is actually used to show that, for the union not to terminate, one out of both systems must itself be non-terminating, which leads to a contradiction. Unfortunately, the property may be impossible to prove because some of the objects that are reduced do not enjoy an adequate form. The purpose of this article is then threefold. It first introduces an operator enabling to insert a reduction step on such an object, and therefore to change its shape, while still preserving the ability to use the property. Of course, some new properties should be verified. Secondly, as an instance of our technique, the operator is applied to relax a well-known lemma stating the termination of the union of two termination abstract reduction systems. Finally, this last lemma is applied in a peculiar and in a more general way to show the termination of some lambda-calculi with inductive types augmented with specific reductions dealing with: (1) copies of inductive types; and (2) with the representation of symmetric groups.