Editing Subtyping (section)

=== Width and depth subtyping ===

Types of [[Record (computer science)|records]] give rise to the concepts of ''width'' and ''depth'' subtyping. These express two different ways of obtaining a new type of record that allows the same operations as the original record type.

Recall that a record is a collection of (named) fields. Since a subtype is a type which allows all operations allowed on the original type, a record subtype should support the same operations on the fields as the original type supported.

One kind of way to achieve such support, called ''width subtyping'', adds more fields to the record. More formally, every (named) field appearing in the width supertype will appear in the width subtype. Thus, any operation feasible on the supertype will be supported by the subtype.

The second method, called ''depth subtyping'', replaces the various fields with their subtypes. That is, the fields of the subtype are subtypes of the fields of the supertype. Since any operation supported for a field in the supertype is supported for its subtype, any operation feasible on the record supertype is supported by the record subtype. Depth subtyping only makes sense for immutable records: for example, you can assign 1.5 to the 'x' field of a real point (a record with two real fields), but you can't do the same to the 'x' field of an integer point (which, however, is a deep subtype of the real point type) because 1.5 is not an integer (see [[Covariance and contravariance (computer science)|Variance]]).

Subtyping of records can be defined in [[System F-sub|System F<sub><:</sub>]], which combines [[parametric polymorphism]] with subtyping of record types and is a theoretical basis for many [[functional programming languages]] that support both features.

Some systems also support subtyping of labeled [[disjoint union]] types (such as [[algebraic data type]]s). The rule for width subtyping is reversed: every tag appearing in the width subtype must appear in the width supertype.