:rewrite, :meta, or :linear rule
Major Section: MISCELLANEOUS
Example:
Consider the :REWRITE rule created from
(IMPLIES (SYNTAXP (NOT (AND (CONSP X)
(EQ (CAR X) 'NORM))))
(EQUAL (LXD X)
(LXD (NORM X)))).
The syntaxp hypothesis in this rule will allow the rule to be
applied to (lxd (trn a b)) but will not allow it to be applied to
(lxd (norm a)).
General Form: (SYNTAXP test)
Syntaxp always returns t and so may be added as a vacuous
hypothesis. However, when relieving the hypothesis, the test
``inside'' the syntaxp form is actually treated as a meta-level
proposition about the proposed instantiation of the rule's variables
and that proposition must evaluate to true (non-nil) to
``establish'' the syntaxp hypothesis.Note that the test of a syntaxp hypothesis does not, in general,
deal with the meaning or semantics or values of the terms, but rather
with their syntactic forms. In the example above, the syntaxp
hypothesis allows the rule to be applied to every target of the form
(lxd u), provided u is not of the form (norm v).
Observe that without this syntactic restriction the rule above could
loop producing a sequence of increasingly complex targets (lxd a),
(lxd (norm a)), (lxd (norm (norm a))), etc. An intuitive reading
of the rule might be ``norm the argument of lxd unless it has
already been normed.''
Note also that a syntaxp hypothesis deals with the syntactic
form used internally by ACL2, rather than that seen by the user. In
some cases these are the same, but there can be subtle differences
with which the writer of a syntaxp hypothesis must be aware.
You can use :trans to display this internal representation.
There are two types of syntaxp hypotheses. The simpler type of
syntaxp hypothesis may be used as the nth hypothesis in a
:rewrite or :linear rule whose :corollary is
(implies (and hyp1 ... hypn ... hypk) (equiv lhs rhs))
provided test is a term, test contains at least one variable, and
every variable occuring freely in test occurs freely in lhs or in
some hypi, i<n. In addition, test must not use any
stobjs. The case of :meta rules is similar to the above, except
that it applies to the result of applying the hypothesis metafunction.
(Later below we will describe the second type, an extended syntaxp
hypothesis, which may use state.)
We illustrate the use of simple syntaxp hypotheses by slightly
elaborating the example given above. Consider a :rewrite
rule whose :corollary is:
(IMPLIES (AND (RATIONALP X)
(SYNTAXP (NOT (AND (CONSP X)
(EQ (CAR X) 'NORM)))))
(EQUAL (LXD X)
(LXD (NORM X))))
How is this rule applied to (lxd (trn a b))? First, we form a
substitution that instantiates the left-hand side of the conclusion
of the rule so that it is identical to the target term. In the
present case, the substitution replaces x with (trn a b).
(LXD X) ==> (LXD (trn a b)).Then we backchain to establish the hypotheses, in order. Ordinarily this means that we instantiate each hypothesis with our substitution and then attempt to rewrite the resulting instance to true. Thus, in order to relieve the first hypothesis above, we rewrite
(RATIONALP (trn a b)).If this rewrites to true, we continue.
Of course, most users are aware of some exceptions to this general
description of the way we relieve hypotheses. For
example, if a hypothesis contains a ``free-variable'' -- one not
bound by the current substitution -- we attempt to extend the
substitution by searching for an instance of the hypothesis among
known truths. See free-variables. Forced hypotheses are another exception to the
general rule of how hypotheses are relieved.
Hypotheses marked with syntaxp, as in (syntaxp test), are
also exceptions. We instantiate such a hypothesis; but instead of
rewriting the instantiated instance, we evaluate the instantiated
test. More precisely, we evaluate test in an environment in
which its variable symbols are bound to the quotations of the terms
to which those variables are bound in the instantiating
substitution. So in the case in point, we (in essence) evaluate
(NOT (AND (CONSP '(trn a b)) (EQ (CAR '(trn a b)) 'NORM))).This clearly evaluates to
t. When a syntaxp test evaluates
to true, we consider the syntaxp hypothesis to have been
established; this is sound because logically (syntaxp test) is
t regardless of test. If the test evaluates to nil (or
fails to evaluate because of guard violations) we act as though
we cannot establish the hypothesis and abandon the attempt to apply
the rule; it is always sound to give up.The acute reader will have noticed something odd about the form
(NOT (AND (CONSP '(trn a b)) (EQ (CAR '(trn a b)) 'NORM))).When relieving the first hypothesis,
(RATIONALP X), we substituted
(trn a b) for X; but when relieving the second hypothesis,
(SYNTAXP (NOT (AND (CONSP X) (EQ (CAR X) 'NORM)))), we substituted the
quotation of (trn a b) for X. Why the difference? Remember
that in the first hypothesis we are talking about the value of
(trn a b) -- is it rational -- while in the second one we are
talking about its syntactic form. Remember also that Lisp, and hence
ACL2, evaluates the arguments to a function before applying the function
to the resulting values. Thus, we are asking ``Is the list (trn a b)
a consp and if so, is its car the symbol NORM?'' The
quotes on both (trn a b) and NORM are therefore necessary.
One can verify this by defining trn to be, say cons, and then
evaluating forms such as
(AND (CONSP '(trn a b)) (EQ (CAR '(trn a b)) 'NORM)) (AND (CONSP (trn a b)) (EQ (CAR (trn a b)) NORM)) (AND (CONSP (trn 'a 'b)) (EQ (CAR (trn 'a 'b)) NORM)) (AND (CONSP '(trn a b)) (EQ '(CAR (trn a b)) ''NORM))at the top-level ACL2 prompt.
See syntaxp-examples for more examples of the use of syntaxp.
An extended syntaxp hypothesis is similar to the simple type
described above, but it uses two additional variables, mfc and state,
which must not be bound by the left hand side or an earlier hypothesis
of the rule. They must be the last two variables mentioned by form;
first mfc, then state. These two variables give access to
the functions mfc-xxx; see extended-metafunctions. As
described there, mfc is bound to the so-called
metafunction-context and state to ACL2's state.
See syntaxp-examples for an example of the use of these extended
syntaxp hypotheses.
