5 Types and Classes

Union Types

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)
  Þ  #f

Given

• x is an object.

• s1...sm and t1...tn are non-union types.

• The notation type-union*(t1...tn) 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...tn)) will be true if and only if instance?(x, t) is true for some t in t1...tn.

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

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

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

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