Higher-Order Families

A redex family is a set of redexes which arècreated in the same way'. Families specify which redexes should be shared in any so-called optimal implementation of a rewriting system. We formalise the notion of family for orthogonal higher-order term rewriting systems (OHRSs). In order to comfort our formalisation of the intuitive concept of family, we actually provide three conceptually diierent formalisations, via labelling, extraction and zigzag and show them to be equivalent. This generalises the results known from literature and gives a rm theoretical basis for the optimal implementation of OHRSs.