Strong Conjunction and Intersection Types

Provable realizability with untyped λ-terms as realizers of formulas of the propositional language with implication and the so called "strong conjunction" is proved to be equivalent to type inference in the system with intersection types of Coppo and Dezani. A similar equivalence is proved if realizers are terms of Combinatory Logic. These results are used to derive properties of models of λ-calculus and Combinatory Logic.