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.

Categorical combinators have been derived from the study of categorical semantics of lambda calculus, and it has been found that they may be used in implementation of functional languages. In this paper categorical combinators are extended so that functions with multiple arguments can be directly handled, thus making them more suitable for practical computation. A rewriting system named ${\operatorname{CCLM}}\beta $ is formulated for these combinators. This system naturally corresponds to the type-free $\lambda \beta $-calculus. The relationship between these two systems is established, and as a result of this, the Church–Rosser property of ${\operatorname{CCLM}}\beta $ is proved. A similar relationship is also established between the original ${\operatorname{CCL}}\beta $ by Curien and the type-free $\lambda \beta $-calculus with product. Finally the embedding theorem of ${\operatorname{CCLM}}\beta $ into ${\operatorname{CCL}}\beta $ is shown.