Church-Rosser property and unique normal form property of non-duplicating term rewriting systems

We propose a new type of conditional term rewriting systems: left-right separated conditional term rewriting systems, in which the left-hand side and the right-hand side of a rewrite rule have separate variables. By developing a concept of weight decreasing joinability we first present a sufficient condition for the Church-Rosser property of left-right separated conditional term rewriting systems which may have overlapping rewrite rules. We next apply this result to show sufficient conditions for the unique normal form property and the Church-Rosser property of unconditional term rewriting systems which are non-duplicating, non-left-linear, and overlapping.