A type s is said to be a subtype of a type t if for every expression $e$ the test type(e, s) evaluates to true, then the expression type(e, t) will also evaluate to true. If a type is identified with its extension, then the ``subtype'' relation is the relation of inclusion.

The subtype(s, t) function attempts to determine if the type s is a subtype of the type t.

If subtype can prove that s is a subtype of t, then the value true is returned. In the same manner, if subtype can prove that s is not a subtype of t, then the value false is returned. Otherwise, if it is not possible to compute whether one type is a subtype of another, the value FAIL is returned.

In general, it is not possible to compute whether one type is a subtype of another.

Note: Not all pairs of types are comparable. For example, the types list and set are disjoint types; no expression is both a list and a set. Thus, both subtype( 'set', 'list' ) and subtype( 'list', 'set' ) return false.

$\mathrm{subtype}\left('\mathrm{integer}'\,'\mathrm{rational}'\right)$

$\mathrm{true}$

$\mathrm{subtype}\left('\mathrm{polynom}'\,'\left\{\mathrm{string}\,\mathrm{algebraic}\right\}'\right)$

$\mathrm{true}$

$\mathrm{subtype}\left('\mathrm{And}\left(\mathrm{name}\,\mathrm{algebraic}\right)'\,'\mathrm{name}'\right)$

$\mathrm{true}$

$\mathrm{subtype}\left('\mathrm{Vector}\left(\mathrm{integer}\right)'\,'\mathrm{Vector}\left(\mathrm{rational}\right)'\right)$

$\mathrm{true}$

$\mathrm{subtype}\left('\mathrm{specfunc}\left(\mathrm{integer}\,\sin \right)'\,'\mathrm{typefunc}\left(\mathrm{rational}\,\mathrm{name}\right)'\right)$

$\mathrm{true}$

$\mathrm{subtype}\left('\left[\mathrm{integer}\,\mathrm{integer}\right]'\,'\mathrm{list}\left(\mathrm{rational}\right)'\right)$

$\mathrm{true}$

$\mathrm{subtype}\left('\left[\mathrm{integer}\,\mathrm{integer}\right]'\,'\left[\mathrm{anything}\,\mathrm{anything}\right]'\right)$

$\mathrm{true}$