A Connuent Reduction for the Extensional Typed ?calculus with Pairs, Sums, Recursion and Terminal Object

We add extensional equalities for the functional and product types to the typed-calculus with not only products and terminal object, but also sums and bounded recursion (a version of recursion that does not allow recursive calls of innnite length). We provide a connuent and strongly normalizing (thus decidable) rewriting system for the calculus, that stays connuent when allowing unbounded recursion. For that, we turn the extensional equalities into expansion rules, and not into contractions as is done traditionally. We rst prove the calculus to be weakly connuent, which is a more complex and interesting task than for the usual-calculus. Then we provide an eeective mechanism to simulate expansions without expansion rules, so that the strong normalization of the calculus can be derived from that of the underlying, traditional, non extensional system. These results give us the connuence of the full calculus, but we also show how to deduce connuence directly form our simulation technique, without the weak connuence property.