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Term-order

The ordering relation on terms used by ACL2

ACL2 must occasionally choose which of two terms is syntactically smaller. The need for such a choice arises, for example, when using equality hypotheses in conjectures (the smaller term is substituted for the larger elsewhere in the formula), in stopping loops in permutative rewrite rules (see loop-stopper), and in choosing the order in which to try to cancel the addends in linear arithmetic inequalities. When this notion of syntactic size is needed, ACL2 uses ``term order.'' Popularly speaking, term order is just a lexicographic ordering on terms. But the situation is actually more complicated.

We define term order only with respect to terms in translated form. See trans. Constants are viewed as built up by pseudo-function applications, as described at the end of this documentation.

Term1 comes before term2 in the term order iff

(a) the number of variable occurrences in term1 is less than that in term2, or

(b) the numbers of variable occurrences in the two terms are equal but the number of function applications in term1 is less than that in term2, or

(c) the numbers of variable occurrences in the two terms are equal, the numbers of functions applications in the two terms are equal, but pseudo-function application count for term1 is less than that for term2, or

(d) the numbers of variable occurrences in the two terms are equal, the numbers of functions applications in the two terms are equal, the pseudo-function application counts for the two terms are equal, and term1 comes before term2 in a lexicographic ordering, lexorder, based their structure as Lisp objects: see lexorder.

The function term-order, when applied to the translations of two ACL2 terms, returns t iff the first is ``less than or equal'' to the second in the term order.

By ``number of variable occurrences'' we do not mean ``number of distinct variables'' but ``number of times a variable symbol is mentioned.'' (Cons x x) has two variable occurrences, not one. Thus, perhaps counterintuitively, a large term that contains only one variable occurrence, e.g., (standard-char-p (car (reverse x))) comes before (cons x x) in the term order.

Since constants contain no variable occurrences and non-constant expressions must contain at least one variable occurrence, constants come before non-constants in the term order, no matter how large the constants. For example, the list constant

'(monday tuesday wednesday thursday friday)

comes before x in the term order. Because term order is involved in the control of permutative rewrite rules and used to shift smaller terms to the left, a set of permutative rules designed to allow the permutation of any two tips in a tree representing the nested application of some function will always move the constants into the left-most tips. Thus,

(+ x 3 (dx i j) (car (reverse klst))) ,

which in translated form is

(binary-+ x
          (binary-+ '3
                    (binary-+ (dx i j)
                              (car (reverse klst))))),

will be permuted under the built-in commutativity rules to

(binary-+ '3
          (binary-+ x
                    (binary-+ (car (reverse klst))
                              (dx i j))))

(+ 3 x (car (reverse klst)) (dx i j)).

Two terms with the same numbers of variable occurrences and function applications and the same pseudo-function application count are ordered by lexicographic means, based on their structures. See lexorder. Thus, if two terms (member ...) and (reverse ...) contain the same numbers of variable occurrences and function applications, and no quoted constants, then the member term is first in the term order because member comes before reverse in the term order (which is here reduced to alphabetic ordering).

It remains to discuss the notion of pseudo-function application count.

Clearly, two constants are ordered using cases (c) and (d) of term order, since they each contain 0 variable occurrences and no function calls. This raises the question ``How many function applications are in a constant?'' Because we regard the number of function applications as a more fundamental measure of the size of a constant than lexicographic considerations, we decided that for the purposes of term order, constants would be seen as being built by primitive constructor functions. These constructor functions are not actually defined in ACL2 but merely imagined for the purposes of term order. We here use suggestive names for these imagined functions, ignoring entirely the prior use of these names within ACL2. The imagined applications of these functions are what we refer to as pseudo-function applications.

The constant function z constructs 0. Positive integers are constructed from (z) by the successor function, s. Thus 2 is (s (s (z))) and contains three function applications. 100 contains one hundred and one applications. Negative integers are constructed from their positive counterparts by -. Thus, -2 is (- (s (s (z)))) and has four applications. Ratios are constructed by the dyadic function /. Thus, -1/2 is

(/ (- (s (z))) (s (s (z))))

and contains seven applications. Complex rationals are similarly constructed from rationals. All character objects are considered primitive and are constructed by constant functions of the same name. Thus #\a and #\b both contain one application. Strings are built from the empty string, (o) by the ``string-cons'' function written cs. Thus "AB" is (cs (#\a) (cs (#\b) (o))) and contains five applications. Symbols are obtained from strings by ``packing'' the symbol-name with the unary function p. Thus 'ab is

(p (cs (#\a) (cs (#\b) (o))))

and has six applications. Note that packages are here ignored and thus 'acl2::ab and 'my-package::ab each contain just six applications. Finally, conses are built with cons, as usual. So '(1 . 2) is (cons '1 '2) and contains six applications, since '1 contains two and '2 contains three. This, for better or worse, answers the question ``How many function applications are in a constant?''

Finally, when we refer to the ``pseudo-function application count'', we mean the number of pseudo-function applications as described above, except that we bound this number by the constant (fn-count-evg-max-val). (This bound is important for efficiency, so that constants that are very large cons structures do not cause significant slowdown as ACL2 attempts to walk through them while computing their pseudo-function application count.)