On Termination and Confluence Properties of Disjoint and Constructor-Sharing Conditional Rewrite Systems

We investigate the modularity behaviour of termination and connuence properties of (join) conditional term rewriting systems. We give counterexamples showing that the properties weak termination, weak innermost termination and (strong) innermost termination are not preserved in general under signature extensions. Then we develop suucient conditions for the preservation of these properties under signature extensions, and more generally, for their modularity in the disjoint union case. This leads to new criteria for modularity of termination and completeness generalizing known results for unconditional systems. Finally, combining our analysis with recent related results on the preservation of semi-completeness, we show how to cover the (non-disjoint) case of combined conditional rewrite systems with shared constructors, too.