Conditional rewrite rules : confluence and termination

Algebraic specifications of abstract data types can often be viewed as systems of rewrite rules. Here we consider rewrite rules with conditions, such as they arise, e.g., from algebraic specifications with positive conditional equations. The conditional term rewriting systems thus obtained which we will study, are based upon the well-known class of left-linear, nonambiguous TRSs. A large part of the theory for such TRSs can be generalized to the conditional case. Our approach is non-hierarchical: the conditions are to be evaluated in the same rewriting system. We prove confluence results and termination results for some well-known reduction strategies