Unique Normal Form Property of Higher-Order Rewriting Systems

. Within the framework of Higher-Order Rewriting Systems proposed by van Oostrom, a sufficient condition for the unique normal form property is presented. This requires neither left-linearity nor termination of the system. 1 Introduction Several frameworks of rewriting systems for higher-order expressions have been proposed [Klo80, Nip91, MN94, LS93, KvO95]. Van Oostrom and van Raamdonk proposed a framework of Higher-Order Rewriting Systems (HORSs) [vO94, vOvR94, vR96], capable of unifying the existing theory of rewriting, e.g., Combinatory Reduction Systems (CRSs) [Klo80], (another variation of) Higher-order Rewriting Systems (HRSs) by Nipkow [Nip91], and Term Rewriting Systems (TRSs). They also presented a sufficient condition for the Church-Rosser property of HORSs by introducing a notion corresponding to orthogonality (i.e., non-overlap and left-linearity) of TRSs. The framework of HORSs is characterised by the clear separation of replacement with rewrite rules and matching/subst..