Argument Filterings and Usable Rules for Simply Typed Dependency Pairs

Simply typed term rewriting [Yam01] is a framework of higher-order term rewriting without bound variables. The authors extended the first-order dependency pair approach [AG00] to the case of simply typed term rewriting [AY05]. They gave a characterization of minimal non-terminating simply typed terms and incorporated the notions of dependency pairs, dependency graphs, and estimated dependency graphs into the simply typed framework. They extended the subterm criterion [HM04] of first-order dependency pairs and introduced the head instantiation technique to make the simply typed dependency pair method effectively applicable even in the presence of function variables. In this paper, we incorporate termination criteria using reduction pairs and related refinements into the simply typed dependency pair framework. In particular, we extend the notions of argument filterings [AG00] and usable rules [HM04, TGSK04] of first-order dependency pairs to the case of simply typed term rewriting. Refinements of dependency pair technique for higher-order systems with bound variables are studied in [Bla06, SK05], and an approach to deal within the framework of first-order dependency pairs is studied in [GTSK05]. In our framework the presence of simple types and higher-order variables/rules are reflected in more specific way comparing with [GTSK05]. On the other hand, since bound variables are not included in our framework, our dependency pair framework is simpler and thus easy to automate compared to the methods in [Bla06, SK05].