Conditional equations have been studied for their use in the specification of abstract data types and as a computational paradigm that combines logic and function programming in a clean way. In this paper we examine different formulations of conditional equations as rewrite systems, compare their expressive power and give sufficient conditions for rewrite systems to have the confluence property. We then examine a restriction of these systems using a decreasing ordering. With this restriction, most of the basic notions (like rewriting and computing normal forms) are decidable, the critical pair lemma holds, and some formulations preserve canonicity.