Modularity of Strong Normalization and Confluence in the algebraic-lambda-Cube

elimination des coupures dans l'arithm etique d'ordre sup erieur. 24] Mitsuhiro Okada. Strong normalizability for the combined system of the types lambda calculus and an arbitrary convergent term rewrite system. higher order rules ((rst order rules are simply required to be non-duplicating). Connuence and strong normalization are essential properties of logical systems, since they ensure the consistence of the system. Proving these properties is in general a diicult task, so, it is important to study under which conditions these proofs are modular. Our results show that in order to prove strong normalization of any of the systems in the R-cube it is suucient to prove termination of the rst order rewrite rules in R on algebraic terms, provided that R satisses certain syntactical conditions, namely non-duplication for FOR and the general schema for HOR. As a consequence, we get the strong normal-ization of a restriction of CCI (with pattern-matching) where the inductive types are deened by structural induction. The restriction on rst order rules is not important in practice, since most implementations of rewriting use sharing, and shared reductions are always conservative. The general schema, however, limits the power of the higher order rules. The generalization of the proof of strong normalization to wider classes of higher order rules will be the subject of future work. Acknowledgements We wish to thank Jean-Pierre Jouannaud and Mar-iangiola Dezani for their scientiic support. The rst author is also grateful to Margherita Lombardi for her constant encouragement. References 1] F. Barbanera. Adding algebraic rewriting to the calculus of constructions: Strong normalization preserved. In Proc. of the 2nd Int. Workshop on Conditional and Typed Rewriting, 1990. 2] F. Barbanera and M. Fernn andez. Combining rst and higher order rewrite systems with type assignment systems.ity of termination and connuence in combinations of rewrite systems with !. Proceed-Ciancaglini. A lter-model and the completeness of type assignment.tension of the basic functionality theory for the-calculus.