# Intersection Subtyping with Constructors

## TL;DR

This paper extends the BCD intersection type system with product types and a generic subtyping relation, ensuring type preservation during reduction and expansion, and introduces a unified framework for these extensions.

## Contribution

It introduces a novel extension of the BCD intersection type system with product types and a generic subtyping relation, maintaining key properties.

## Key findings

- Unified subtyping relation for intersection, omega, arrow, and product types.
- Preservation of typing during reduction and expansion.
- Framework based on a subformula property.

## Abstract

We study the question of extending the BCD intersection type system with additional type constructors. On the typing side, we focus on adding the usual rules for product types. On the subtyping side, we consider a generic way of defining a subtyping relation on families of types which include intersection types. We find back the BCD subtyping relation by considering the particular case where the type constructors are intersection, omega and arrow. We obtain an extension of BCD subtyping to product types as another instance. We show how the preservation of typing by both reduction and expansion is satisfied in all the considered cases. Our approach takes benefits from a "subformula property" of the proposed presentation of the subtyping relation.