## Andrew Polonsky ; Richard Statman - On sets of terms having a given intersection type

lmcs:4849 - Logical Methods in Computer Science, September 21, 2022, Volume 18, Issue 3 - https://doi.org/10.46298/lmcs-18(3:35)2022
On sets of terms having a given intersection type

Authors: Andrew Polonsky ; Richard Statman

Working in a variant of the intersection type assignment system of Coppo, Dezani-Ciancaglini and Venneri [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. Moreover, we show that the set of closed terms with a given type is uniformly separable, and, if infinite, forms an adequate numeral system. The proof of this fact uses an internal version of the Böhm-out technique, adapted to terms of a given intersection type.

Volume: Volume 18, Issue 3
Published on: September 21, 2022
Accepted on: April 3, 2022
Submitted on: September 24, 2018
Keywords: Computer Science - Logic in Computer Science,03B38,F.4.1