We have previously introduced the static dependency pair method that proves termination by analyzing the static recursive structure of various extensions of term rewriting systems for handling higher-order functions. The key is to succeed with the formalization of recursive structures based on the notion of strong computability, which is introduced for the termination of typed λ-calculi. To bring the static dependency pair method close to existing functional programs, we also extend the method to term rewriting models in which functional abstractions with patterns are permitted. Since the static dependency pair method is not sound in general, we formulate a class; namely, accessibility, in which the method works well. The static dependency pair method is a very natural reasoning; therefore, our extension differs only slightly from previous results. On the other hand, a soundness proof is dramatically difficult.

For simply-typed term rewriting systems (STRSs) and higher-order rewrite systems (HRSs) a la Nipkow, we proposed a method for proving termination, namely the static dependency pair method. The method combines the dependency pair method introduced for first-order rewrite systems with the notion of strong computability introduced for typed λ-calculi. This method analyzes a static recursive structure based on definition dependency. By solving suitable constraints generated by the analysis, we can prove termination. In this paper, we extend the method to rewriting systems for functional programs (RFPs) with product, algebraic data, and ML-polymorphic types. Although the type system in STRSs contains only product and simple types and the type system in HRSs contains only simple types, our RFPs allow product types, type constructors (algebraic data types), and type variables (ML-polymorphic types). Hence, our RFPs are more representative of existing functional programs than STRSs and HRSs. Therefore, our result makes a large contribution to applying theoretical rewriting techniques to actual problems, that is, to proving the termination of existing functional programs.

A static dependency pair method, proposed by us, can effectively prove termination of simply-typed term rewriting systems (STRSs). The theoretical basis is given by the notion of strong computability. This method analyzes a static recursive structure based on definition dependency. By solving suitable constraints generated by the analysis result, we can prove the termination. Since this method is not applicable to every system, we proposed a class, namely, plain function-passing, as a restriction. In this paper, we first propose the class of safe function-passing, which relaxes the restriction by plain function-passing. To solve constraints, we often use the notion of reduction pairs, which is designed from a reduction order by the argument filtering method. Next, we improve the argument filtering method for STRSs. Our argument filtering method does not destroy type structure unlike the existing method for STRSs. Hence, our method can effectively apply reduction orders which make use of type information. To reduce constraints, the notion of usable rules is proposed. Finally, we enhance the effectiveness of reducing constraints by incorporating argument filtering into usable rules for STRSs.