Factoring Derivation Spaces via Intersection Types

Abstract

In typical non-idempotent intersection type systems, proof normalization is not confluent. In this paper we introduce a confluent non-idempotent intersection type system for the \(\lambda \)-calculus. Typing derivations are presented using proof term syntax. The system enjoys good properties: subject reduction, strong normalization, and a very regular theory of residuals. A correspondence with the \(\lambda \)-calculus is established by simulation theorems. The machinery of non-idempotent intersection types allows us to track the usage of resources required to obtain an answer. In particular, it induces a notion of garbage: a computation is garbage if it does not contribute to obtaining an answer. Using these notions, we show that the derivation space of a \(\lambda \)-term may be factorized using a variant of the Grothendieck construction for semilattices. This means, in particular, that any derivation in the \(\lambda \)-calculus can be uniquely written as a garbage-free prefix followed by garbage.