A linearization of the Lambda-calculus and consequences

We embed the standard λ-calculus, denoted Λ, into two larger λ-calculi, denoted Λ∧ and &Λ∧. The standard notion of β-reduction for Λ corresponds to two new notions of reduction, β∧ for Λ∧ and &β∧ for &Λ∧. A distinctive feature of our new calculus Λ∧ (resp., &Λ∧) is that, in every function application, an argument is used at most once (resp., exactly once) in the body of the function. We establish various connections between the three notions of reduction, β, β∧ and &β∧. As a consequence, we provide an alternative framework to study the relationship between β-weak normalization and β-strong normalization, and give a new proof of the oft-mentioned equivalence between β-strong normalization of standard λ-terms and typability in a system of “intersection types”.