Higher-Order Rewriting via Conditional First-Order Rewriting in the Open Calculus of Constructions

Although higher-order rewrite systems (HRS) seem to have a first-order flavor, the direct translation into first-order rewrite systems, using e.g. explicit substitutions, is by no means trivial. In this paper, we explore a two-stage approach, by showing how higher-order pattern rewrite systems, and in fact a somewhat more general class, can be expressed by conditional first-order rewriting in the open calculus of constructions (OCC), which itself has been presented and implemented using explicit substitutions. The key feature of OCC that we exploit is that conditions are allowed to contain quantifiers and equations which can be solved using first-order matching. The way we express HRS works in spite of the fact that structural equality of OCC does not subsume α-conversion. Another topic that we touch upon in this paper is the use of higher-order abstract syntax in a classical framework like OCC, because it is often used in connection with higher-order rewriting.