:rewrite rules are
used may be found in the discussion of ``Replacement Rules'' on page
279 of A Computational Logic Handbook (second edition).
:rewrite rules (possibly conditional ones)
Major Section: RULE-CLASSES
See rule-classes for a general discussion of rule classes and
how they are used to build rules from formulas. Example :corollary
formulas from which :rewrite rules might be built are:
Example:
(equal (+ x y) (+ y x)) replace (+ a b) by (+ b a) provided
certain heuristics approve the
permutation.
(implies (true-listp x) replace (append a nil) by a, if
(equal (append x nil) x)) (true-listp a) rewrites to t
(implies replace (member a (append b c)) by
(and (eqlablep e) (member a (append c b)) in contexts
(true-listp x) in which propositional equivalence
(true-listp y)) is sufficient, provided (eqlablep a)
(iff (member e (append x y)) (true-listp b) and (true-listp c)
(member e (append y x)))) rewrite to t and the permutative
heuristics approve
General Form:
(and ...
(implies (and ...hi...)
(implies (and ...hk...)
(and ...
(equiv lhs rhs)
...)))
...)
Note: One :rewrite rule class object might create many rewrite
rules from the :corollary formula. To create the rules, we first
translate the formula (expanding all macros; also see trans).
Next, we eliminate all lambdas; one may think of this step as
simply substituting away every let, let*, and mv-let in the
formula. We then flatten the and and implies structure of the
formula, transforming it into a conjunction of formulas, each of the
form
(implies (and h1 ... hn) concl)where no hypothesis is a conjunction and
concl is neither a
conjunction nor an implication. If necessary, the hypothesis of
such a conjunct may be vacuous. We then further coerce each concl
into the form (equiv lhs rhs), where equiv is a known equivalence
relation, by replacing any concl not of that form by (iff concl t).
A concl of the form (not term) is considered to be of the form
(iff term nil). By these steps we reduce the given :corollary to
a sequence of conjuncts, each of which is of the form
(implies (and h1 ... hn)
(equiv lhs rhs))
where equiv is a known equivalence relation. See equivalence
for a general discussion of the introduction of new equivalence
relations.
We create a :rewrite rule for each such conjunct, if possible, and
otherwise cause an error. It is possible to create a rewrite rule
from such a conjunct provided lhs is not a variable, a quoted
constant, a let-expression, a lambda application, or an
if-expression.
A :rewrite rule is used when any instance of the lhs occurs in a
context in which the equivalence relation is operative. First, we
find a substitution that makes lhs equal to the target term. Then
we attempt to relieve the instantiated hypotheses of the rule.
Hypotheses that are fully instantiated are relieved by recursive
rewriting. Hypotheses that contain ``free variables'' (variables
not assigned by the unifying substitution) are relieved by
attempting to guess a suitable instance so as to make the hypothesis
equal to some known assumption in the context of the target. If the
hypotheses are relieved, and certain restrictions that prevent some
forms of infinite regress are met (see loop-stopper), the
target is replaced by the instantiated rhs, which is then
recursively rewritten.
At the moment, the best description of how ACL2 :rewrite rules are
used may be found in the discussion of ``Replacement Rules'' on page
279 of A Computational Logic Handbook (second edition).