Normalization Results for Typeable Rewrite Systems

In this paper we introduce Curryfied term rewriting systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts andω. Three operations on types—substitution, expansion, and lifting—are used to define type assignment and are proved to be sound. With this result the system is proved closed for reduction. Using a more liberal approach to recursion, we define a general scheme for recursive definitions and prove that, for all systems that satisfy this scheme, every term typeable without using the type-constantωis strongly normalizable. We also show that, under certain restrictions, all typeable terms have a (weak) head-normal form, and that terms whose type does not containωare normalizable.