Orthogonal Higher-Order Rewrite Systems are Confluent

Two important aspects of computation are reduction, hence the current infatuation with termrewriting systems (TI~Ss), and abstraction, hence the study of A-calculus. In his groundbreaking thesis, Klop [10] combined both areas in the framework of combinatory reduction systems (ClaSs) and lifted many results from the A-calculus to the CRS-level. More recently [15], we proposed a different although related approach to rewriting with abstractions, higher-order rewrite systems (HRSs). HRSs use the simply-typed A-calculus as a metaAanguage for the description of reductions of terms with bound variables. Furthermore we showed that the wellknown critical pairs can be extended to this setting, thus proving that confluence of terminating HItSs is decidable. This paper is a second step towards a general theory of higher-order rewrite systems. It considers the complementary case, when there are no critical pairs and all rules are left-llnear (no free variable appears twice on the left-hand side). Such systems are usually called orthogonal. The main result of this paper is that all orthogonal HKSs are confluent, irrespective of their termination. To a specialist in the field this will probably not come as a surprise because the same result holds for TRSs and CKSs, and the latter are close to HILSs. There are also many partial results in this direction [18, 1]. Hence the contribution of the paper is not so much the result itself but the combination of the following points: