Embedding a Second-Order Type System into an Intersection Type System

Abstract

This paper presents the relationship between a second-order type assignment system T∀ and an intersection type assignment system T∧. First we define a translation tr from intersection types to second-order types. Then we define a system T∧* obtained from T∧ by restricting the use of the intersection type introduction rule, and show that T∧* and T∀ are equivalent in the following senses: (a) if a λ-term M has a type σ in T∧*, then M has the type tr(σ) in T∀; and conversely, (b) if M has a type T in T∀, then M has a type σ in T∧* such that tr(σ) is equivalent to T. These two theorems mean that T∀ is embedded into T∧.