The Multiset-Theoretic Collapse of the Sequential Intersection Type System is Surjective

We present a type system (that we refer to as System S) that features sequences as intersection types, in contrast with Gardner/de Carvalho’s System R, that uses multisets to represent intersection. Systems S and R are both linear and forbid contraction/duplication. We explain why any reduction choice in System R (i.e. any way to perform Subject Reduction) may be encoded in System S by resorting to suitable isomorphisms of labelled trees, that we call interfaces. The main theorem of this article is the following: every reduction choice may be encoded by a trivial interface. This means that not only the syntactical equality between labelled trees used in System S can represent the far more relaxed equality used in System R (relying upon nested permutations in multisets), but that it is also enough to capture convoluted reduction choices. We actually consider coinductive type grammars. Contrary to most inductive cases, typability does not ensure here any form of normalizability. From the technical perspective, the chief contribution of this paper is the introduction of a method, based upon the study on a suitable first order theory, that does not rely on any notion of productive reduction, which is pivotal to prove the Representation Theorem. We also briefly discuss the applications of this work and of coinductive intersection type grammars on the finitary or infinitary λ-calculus and its semantics.