The Logic of Intersection Subtyping

The subtyping relation of programming languages can be analysed as an entailment relation by means of proof theory. We are interested in two main families of systems: intersection types and polymorphic subtyping. They share the fact that implication has some distributivity property: over intersection in the first case and over universal quantification in the second one.

We introduce a restriction of the second-order (full) Lambek calculus which is stable under cut-elimination and conservatively extends these two subtyping relations. This new system IS is an intuitionistic non-commutative linear sequent calculus which provides a natural logical setting for the study of subtyping relations.

We recover sequent calculi from the literature (as well as new variants) as restrictions of IS (thanks to the proof-theoretical analysis of the system: admissible rules, invertibility, focusing, etc.), so that IS appears as a unifying logic for subtyping. We also develop translations relating IS with relevant logic, the (unconstrained) Lambek calculus or cyclic linear logic.