Parametrised dependency pairs for a general form of termination in annotated rewriting

We define a general form of termination, α-termination, for term rewrite systems with an annotation α. Standard termination and top-termination are particular instances. We present a variation on the dependency pair criterion of Arts and Giesl and prove that an α-annotated term rewrite system is α-terminating if and only there are no chains of α-dependency pairs. The α-dependency graphs can be analysed in very much the same way as standard dependency graphs, thanks to Hirokawa and Middeldorp’s Subterm Criterion as we demonstrate on some examples. The α-dependency pair criterion generalises both the dependency pair criterion for termination and the strong dependency pair criterion for strong convergence.