The dependency pair technique is a powerful modular method for automated termination proofs of term rewrite systems (TRSs). We present two important extensions of this technique: First, we show how to prove termination of higher-order functions using dependency pairs. To this end, the dependency pair technique is extended to handle (untyped) applicative TRSs. Second, we introduce a method to prove non-termination with dependency pairs, while up to now dependency pairs were only used to verify termination. Our results lead to a framework for combining termination and non-termination techniques for firstand higher-order functions in a very flexible way. We implemented and evaluated our results in the automated termination prover AProVE.

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

Abstract. Many functional programs and higher order term rewrite systems contain, besides higher order rules, also a significant first order part. We discuss how an automatic termination prover can split a rewrite system into a first order and a higher order part. The results are applicable to all common styles of higher order rewriting with simple types, although some dependency pair approach is needed to use them. Key words: higher order rewriting, termination, dependency pairs, modularity 1

The notion of inductive theorems is well-established in first-order term rewriting. In higherorder term rewriting, in contrast, it is not straightforward to extend this notion because of extensionality (Meinke, 1992). When extending the term rewriting based program transformation of Chiba et al. (2005) to higher-order term rewriting, we need extensibility, a property stating that inductive theorems are preserved by adding new functions via macros. In this paper, we propose and study a new notion of inductive theorems for higher-order rewriting, natural inductive theorems. This allows to incorporate properties such as extensionality and extensibility, based on simply typed S-expression rewriting (Yamada, 2001).