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Abstract. We extend the higher-order termination method of dynamic dependency pairs to Algebraic Functional Systems (AFSs). In this setting, simply typed lambda-terms with algebraic reduction and separate β-steps are considered. For left-linear AFSs, the method is shown to be complete. For so-called local AFSs we define a variation of usable rules and an extension of argument filterings. All these techniques have been implemented in the higher-order termination tool WANDA.

In ~his pape~ we prove confluence for weakly orthogonal Higher-Order Rewriting Systems. This generalises all the known 'confluence by orthogonality' results.

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Abstract. In this paper two formats of higher-order rewriting are compared: Combinatory Reduction Systems introduced by Klop and Higherorder Rewrite Systems defined by Nipkow. Although it always has been obvious that both formats are closely related to each other, up to now the exact relationship between them has not been clear. This was an unsatisfying situation since it meant that proofs for much related frameworks were given twice. We present two translations, one from Combinatory Reduction Systems into Higher-Order Rewrite Systems and one vice versa, based on a detailed comparison of both formats. Since the translations are very 'neat' in the sense that the rewrite relation is preserved and (almost) reflected, we can conclude that as far as rewrite theory is concerned, Combinatory Reduction Systems and Higher-Order Rewrite Systems are equivalent, the only difference being that Combinatory Reduction Systems employ a more 'lazy' evaluation strategy. Moreover, due to this result it is the case that some syntactic properties derived for the one class also hold for the other.The research of the second author is supported by NWO/SION project 612-316-606.