elements of one list that are not elements of another
Major Section:  ACL2-BUILT-INS

General Forms:
(intersection$ l1 l2 ... lk)
(intersection$ l1 l2 ... lk :test 'eql) ; same as above
(intersection$ l1 l2 ... lk :test 'eq)    ; same, but eq is equality test
(intersection$ l1 l2 ... lk :test 'equal) ; same, but equal is equality test

(Intersection$ x y) equals a list that contains the members of x that are also members of y. More precisely, the resulting list is the result of deleting from x those members that that are not members of y. The optional keyword, :TEST, has no effect logically, but provides the test (default eql) used for comparing members of the two lists.

Intersection$ need not take exactly two arguments, though it must take at least one argument: (intersection$ x) is x, (intersection$ x y z ... :test test) is (intersection$ x (intersection$ y z ... :test test) :test test), and if :TEST is not supplied, then (intersection$ x y z ...) is (intersection$ x (intersection$ y z ...)). For the discussion below we restrict ourselves, then, to the cases (intersection$ x y) and (intersection$ x y :test test).

The guard for a call of intersection$ (in the two cases just above) depends on the test. In all cases, both arguments must satisfy true-listp. If the test is eql, then one of the arguments must satisfy eqlable-listp. If the test is eq, then one of the arguments must satisfy symbol-listp.

See equality-variants for a discussion of the relation between intersection$ and its variants:

(intersection-eq x lst) is equivalent to (intersection$ x lst :test 'eq);

(intersection-equal x lst) is equivalent to (intersection$ x lst :test 'equal).

In particular, reasoning about any of these primitives reduces to reasoning about the function intersection-equal.

Note that intersection-eq can take any positive number of arguments, in analogy to intersection$; indeed, (intersection-eq ...) expands to (intersection$ ... :test 'eq). However, intersection-equal is a function, not a macro, and takes exactly two arguments.

Intersection$ is similar to the Common Lisp primitive intersection. However, Common Lisp does not specify the order of elements in the result of a call of intersection.