Major Section: MISCELLANEOUS
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
term1is less than that in
(b) the numbers of variable occurrences in the two terms are equal but the number of function applications in
term1is less than that in
(c) the numbers of variable occurrences in the two terms are equal, the numbers of functions applications in the two terms are equal, but the number of pseudo-function applications in
term1is less than that in
(d) the numbers of variable occurrences in the two terms are equal, the numbers of functions applications in the two terms are equal, the numbers of pseudo-function applications in the two terms are equal, and
term2in a lexicographic ordering,
lexorder, based their structure as Lisp objects: see lexorder.
term-order, when applied to the translations of two ACL2 terms, returns
tiff 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
(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)))
(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
xin 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 (car (reverse klst)) (dx i j)) ,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))))or
(+ 3 x (car (reverse klst)) (dx i j)).Two terms with the same numbers of variable occurrences, function applications, and pseudo-function applications are ordered by lexicographic means, based on their structures. See lexorder. Thus, if two terms
(reverse ...)contain the same numbers of variable occurrences and function applications, then the
memberterm is first in the term order because
reversein the term order (which is here reduced to alphabetic ordering).
It remains to discuss the notion of pseudo-function applications.
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
0. Positive integers are
(z) by the successor function,
(s (s (z))) and contains three function applications.
contains one hundred and one applications. Negative integers are
constructed from their positive counterparts by
(- (s (s (z)))) and has four applications. Ratios are
constructed by the dyadic function
(/ (- (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
#\bboth contain one application. Strings are built from the empty string,
(o)by the ``string-cons'' function written
(cs (#\a) (cs (#\b) (o)))and contains five applications. Symbols are obtained from strings by ``packing'' the
symbol-namewith the unary function
(p (cs (#\a) (cs (#\b) (o))))and has six applications. Note that packages are here ignored and thus
'my-package::abeach contain just six applications. Finally, conses are built with
cons, as usual. So
'(1 . 2)is
(cons '1 '2)and contains six applications, since
'1contains two and
'2contains three. This, for better or worse, answers the question ``How many function applications are in a constant?''