(primitive) apply a rewrite rule

(rewrite reverse-reverse)
   -- apply the rewrite rule `reverse-reverse'
(rewrite (:rewrite reverse-reverse))
   -- same as above
(rewrite 2)
   -- apply the second rewrite rule, as displayed by show-rewrites
   -- apply the first rewrite rule, as displayed by show-rewrites
(rewrite transitivity-of-< ((y 7)))
   -- apply the rewrite rule transitivity-of-< with the substitution
      that associates 7 to the ``free variable'' y
(rewrite foo ((x 2) (y 3)) t)
   -- apply the rewrite rule foo by substituting 2 and 3 for free
      variables x and y, respectively, and also binding all other
      free variables possible by using the current context
      (hypotheses and governors)

General Form: (rewrite &optional rule-id substitution instantiate-free)

Replace the current subterm with a new term by applying a rewrite rule. The replacement will be done according to the information provided by the show-rewrites (sr) command.

If rule-id is a positive integer n, then the nth rewrite rule as displayed by show-rewrites is the one that is applied. If rule-id is nil or is not supplied, then it is treated as the number 1. Otherwise, rule-id should be either a rune of or name of a rewrite rule. If a name is supplied, then any rule of that name may be used. We say more about this, and describe the other optional arguments, below.

Consider first the following example. Suppose that the current subterm is (reverse (reverse y)) and that there is a rewrite rule called reverse-reverse of the form

(implies (true-listp x)
         (equal (reverse (reverse x)) x)) .
Then the instruction (rewrite reverse-reverse) would cause the current subterm to be replaced by y and would create a new goal with conclusion (true-listp y). An exception is that if the top-level hypotheses imply (true-listp y) using only ``trivial reasoning'' (more on this below), then no new goal is created.

If the rule-id argument is a number or is not supplied, then the system will store an instruction of the form (rewrite name ...), where name is the name of a rewrite rule; this is in order to make it easier to replay instructions when there have been changes to the history. Actually, instead of the name (whether the name is supplied or calculated), the system stores the rune if there is any chance of ambiguity. (Formally, ``ambiguity'' here means that the rune being applied is of the form (:rewrite name . index), where index is not nil.)

Speaking in general, then, a rewrite instruction works as follows:

First, a rewrite rule is selected according to the arguments of the rewrite instruction. The selection is made as explained under ``General Form'' above.

Next, the left-hand side of the rule is matched with the current subterm, i.e., a substitution unify-subst is found such that if one instantiates the left-hand side of the rule with unify-subst, then one obtains the current subterm. If this match fails, then the instruction fails.

Next, an attempt is made to relieve (discharge) the hypotheses, much as the theorem prover relieves hypotheses except that there is no call to the rewriter. First, the substitution unify-subst is extended with the substitution argument, which may bind free variables (see free-variables). Each hypothesis of the rewrite rule is then considered in turn, from first to last. For each hypothesis, first the current substitution is applied, and then the system checks whether the hypothesis is ``clearly'' true in the current context. If there are variables in the hypotheses of the rewrite rule that are not bound by the current substitution, then a weak attempt is made to extend that substitution so that the hypothesis is present in the current context (see the documentation for the proof-checker hyps command under proof-checker-commands), much as would be done by the theorem prover's rewriter.

If in the process above there are free variables, but the proof-checker can see how to bind them to relieve all hypotheses, then it will do so in both the show-rewrites (sr) and rewrite commands. But normally, if even one hypothesis remains unrelieved, then no automatic extension of the substitution is made. Except, if instantiate-free is not nil, then that extension to the substitution is kept.

Finally, the instruction is applied as follows. The current subterm is replaced by applying the final substitution described above to the right-hand side of the selected rewrite rule. And, one new subgoal is created for each unrelieved hypothesis of the rule, whose top-level hypotheses are the governors and top-level hypotheses of the current goal and whose conclusion and current subterm are the instance, by that same final substitution, of that unrelieved hypothesis.

Note: The substitution argument should be a list whose elements have the form (variable term), where term may contain abbreviations.