Semi-completeness of hierarchical and super-hierarchical combinations of term rewriting systems

In this paper, we study modular aspects of hierarchical and super hierarchical combinations of term rewriting systems. In particular, a sufficient condition for modularity of semi-completeness of hierarchical and super hierarchical combinations is proposed. We first establish modularity of weak normalization for this class (defined by the sufficient condition) and modularity of semi-completeness for a class of crosswise independent unions. From these results, we obtain modularity of semicompleteness for a class of hierarchical and super hierarchical combinations. Our results generalize the semi-completeness results of Ohlebusch [14] and Middeldorp and Toyama [13]. The notion of crosswise independent unions is a generalization of both constructor sharing unions as well as Plump's crosswise disjoint unions.