Uniqueness typing for intersection types

TL;DR

This paper proves that in a specific intersection type system, every strongly normalizing term has a unique typing, ensuring a one-to-one correspondence between terms and their types under certain conditions.

Contribution

It introduces the concept of uniqueness typing for strongly normalizing terms within an intersection type system, establishing a novel property of type assignment.

Findings

Every strongly normalizing term admits a unique intersection type

The paper discusses various presentations of intersection type algebras

Type assignment rules influence the structure of intersection types

Abstract

Working in a variant of the intersection type assignment system of Coppo, Dezani-Ciancaglini and Veneri [1981], we prove several facts about sets of terms having a given intersection type. Our main result is that every strongly normalizing term $M$ admits a *uniqueness typing*, which is a pair $(\Gamma,A)$ such that 1) $\Gamma \vdash M : A$ 2) $\Gamma \vdash N : A \Longrightarrow M =_{\beta\eta} N$ We also discuss several presentations of intersection type algebras, and the corresponding choices of type assignment rules.