Simulating expansions without expansions

Abstract

We add extensional equalities for the functional and product types to the typed λ-calculus with, in addition to products and terminal object, sums and bounded recursion (a version of recursion that does not allow recursive calls of infinite length). We provide a confluent and strongly normalizing (thus decidable) rewriting system for the calculus that stays confluent when allowing unbounded recursion. To do this, we turn the extensional equalities into expansion rules, and not into contractions as is done traditionally. We first prove the calculus to be weakly confluent, which is a more complex and interesting task than for the usual λ-calculus. Then we provide an effective 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 confluence of the full calculus, but we also show how to deduce confluence directly form our simulation technique without using the weak confluence property.