A de Bruijn Notation for Higher-Order Rewriting

Expression Reduction Systems with Patterns

“…Then, the metasubstitution sigma = {X plus(0, 0), M 0, N 0} is not l-constructor for β PM , but the metasubstitution theta = {X a 0, s(x) , M a 0, x , N plus (3,4)} is l-constructor for it. As explained in the introduction of this section, ERSPs need to be l-constructor in order to guarantee confluence.…”

“…This formalism can be seen as an extension of ERSs [16] and SERSs [4] to the case of patterns, and an extension of [15] to the case of nonfunctional rewrite rules. Many simple notions in the mentioned previous works do not trivially extend to our case: on the one hand, the complexity of ERSPs does not only appear at the level of metaterms but also at the level of terms; on the other hand, binders are not always so simple as in the case of λ-calculus.…”

“…A position is a word over the alphabet IN; we use to denote the empty word. The set of positions of a metaterm t, denoted POS(t), is defined as usual [1] except for the term s{ p by u} for which we define 1.1.q ∈ POS(s{ p by u}) if q ∈ POS(s) and 2.q ∈ POS(s{ p by u}) if q ∈ POS(u) (see also [4]). The justification for this case comes from the fact that, when reasoning about positions, the metaterm s{ p by u} can be seen as "app(μp.s, u)," where app is a function symbol and μ is a binder.…”

“…Our calculus constitutes an extension of ERSs [16] and SERSs [4] to the case of patterns and a generalization of the Pattern-Matching Calculus to the case of general higher-order rewriting (and not only functional rewriting). ERSP patterns are defined as combinations of standard algebraic structures with special choice constructors used to denote different possible syntactic forms for any abstracted argument.…”

“…Many higher-order rewrite systems exist in the literature [4,16,20,23,25] starting with the seminal work by Klop [17]. The theory of higher-order rewriting is considerably more involved than that of first-order rewriting; many articles were devoted to the study of its foundations, applications, semantics, and implementation.…”