Completeness of type assignment systems with intersection, union, and type quantifiers

This paper develops type assignment systems for intersection and union types, and type quantifiers. The known system for these types is not semantically complete. We introduce a certain class of typing statements, called stable statements, which include all statements without type quantifiers, and we show that the known system is complete for stable statements if we add two axiom schemas expressing the distributive laws of intersection over union and existential quantifier, respectively. All the results are obtained in a systematic way with sequent calculi for type assignment and the cut-elimination for them