INTERSECTION$.html -- ACL2 Version 6.3

### INTERSECTION$

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 `member`s of `x` that are also `member`s 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`.