Confluence of untyped lambda calculus via simple types

Abstract

We present a new proof of confluence of the untyped lambda calculus by reducing the confluence of β-reduction in the untyped lambda calculus to the confluence of β-reduction in the simply typed lambda calculus. This is achieved by embedding typed lambda terms into simply typed lambda terms. Using this embedding, an auxiliary reduction, and β-reduction on simply typed lambda terms we define a new reduction on all lambda terms. The transitive closure of the reduction defined is β-reduction on all lambda terms. This embedding allows us to use the confluence of β-reduction on simply typed lambda terms and thus prove the confluence of the reduction defined. As a consequence we obtain the confluence of β-reduction in the untyped lambda calculus.