Higher-order rewrite systems and their confluence

We study higher-order rewrite systems (HRSs) which extend term rewriting to λ-terms. HRSs can describe computations over terms with bound variables. We show that rewriting with HRSs is closely related to undirected equational reasoning. We define pattern rewrite systems (PRSs) as a special case of HRSs and extend three confluence results from term rewriting to PRSs: the critical pair lemma by Knuth and Bendix, confluence of rewriting modulo equations à la Huet, and confluence of orthogonal PRSs.