Union Types

Union types represent the union of the instances of two or more other types. Union types are created with the function type-union. They are not classes.

Union types are useful as slot specializers, and describe the return types of many common functions. For example, the return type of the collection method on size could be expressed as type-union(<integer>, singleton(#f)).

define constant <green-thing> = type-union(<frog>, <broccoli>);

define constant kermit = make(<frog>);

define method red? (x :: <green-thing>)
  #f
end method;

red?(kermit)
  ⇒  #f

The following rules govern subtype? and instance? for union types.

Given

x is an object.
s₁…s_m and t₁…t_n are nonunion types.
The notation type-union*(t₁…t_n) stands for any arrangement of nested calls to type-union, where none of the arguments is a subtype of any other, and none of the arguments forms an exhaustive partition of any other type.

Then

type-union(t₁, t₁) is type equivalent to t₁.

type-union(t₁, t₂) is type equivalent to type-union(t₂, t₁).

type-union(t₁, type-union(t₂, t₃)) is type equivalent to type-union(type-union(t₁, t₂), t₃).

type-union(t₁, t₂) is type equivalent to t₂ when subtype?(t₁, t₂).

instance?(x, type-union*(t₁…t_n)) will be true if and only if instance?(x, t) is true for some t in t₁…t_n.

subtype?(type-union*(t₁…t_n), s₁) will be true if and only if subtype?(t, s₁) is true for every t in t₁…t_n.

subtype?(s₁, type-union*(t₁…t_n)) will be true if and only if subtype?(s₁, t) is true for some t in t₁…t_n.

subtype?(type-union*(s₁…s_m), type-union*(t₁…t_n)) will be true if and only if every s in s₁…s_m is a subtype of some t in t₁…t_n.

Errata: In the published book, the comma between the arguments to subtype? is missing.