A de bruijn notation for higher-order rewriting

Abstract

We propose a formalism for higher-order rewriting in de Bruijn notation. This notation not only is used for terms (as usually done in the literature) but also for metaterms, which are the syntactical objects used to express general higher-order rewrite systems. We give formal translations from higher-order rewriting with names to higher-order rewriting with de Bruijn indices, and vice-versa. These translations can be viewed as an interface in programming languages based on higher-order rewrite systems, and they are also used to show some properties, namely, that both formalisms are operationally equivalent, and that confluence is preserved when translating one formalism into the other.