Combining Algebraic Rewriting, Extensional Lambda Calculi, and Fixpoints

It is well known that confluence and strong normalization are preserved when combining algebraic rewriting systems with the simply typed lambda calculus. It is equally well known that confluence fails when adding either the usual contraction rule for η, or recursion together with the usual contraction rule for surjective pairing. We show that confluence and strong normalization are modular properties for the combination of algebraic rewriting systems with typed lambda calculi enriched with expansive extensional rules for η and surjective pairing. We also show how to preserve confluence in a modular way when adding fixpoints to different rewriting systems. This result is also obtained by a simple translation technique allowing to simulate bounded recursion.