CONGRUENCE

the relations to maintain while simplifying arguments 
Major Section:  RULE-CLASSES

See rule-classes for a general discussion of rule classes and how they are used to build rules from formulas. An example :corollary formula from which a :congruence rule might be built is: 

Example:
(implies (set-equal x y)
         (iff (memb e x) (memb e y))).
Also see defcong and see equivalence. 

General Form:
(implies (equiv1 xk xk-equiv)
         (equiv2 (fn x1... xk       ...xn)
                 (fn x1... xk-equiv ...xn)))
where equiv1 and equiv2 are known equivalence relations, fn is an n-ary function symbol and the xi and xk-equiv are all distinct variables. The effect of such a rule is to record that the equiv2-equivalence of fn-expressions can be maintained if, while rewriting the kth argument position, equiv1-equivalence is maintained. See equivalence for a general discussion of the issues. We say that equiv2, above, is the ``outside equivalence'' in the rule and equiv1 is the ``inside equivalence for the kth argument''

The macro form (defcong equiv1 equiv2 (fn x1 ... x1) k) is an abbreviation for a defthm of rule-class :congruence that attempts to establish that equiv2 is maintained by maintaining equiv1 in fn's kth argument. The defcong macro automatically generates the general formula shown above. See defcong.

The memb example above tells us that (memb e x) is propositionally equivalent to (memb e y), provided x and y are set-equal. The outside equivalence is iff and the inside equivalence for the second argument is set-equal. If we see a memb expression in a propositional context, e.g., as a literal of a clause or test of an if (but not, for example, as an argument to cons), we can rewrite its second argument maintaining set-equality. For example, a rule stating the commutativity of append (modulo set-equality) could be applied in this context. Since equality is a refinement of all equivalence relations, all equality rules are always available. See refinement.

All known :congruence rules about a given outside equivalence and fn can be used independently. That is, consider two :congruence rules with the same outside equivalence, equiv, and about the same function fn. Suppose one says that equiv1 is the inside equivalence for the first argument and the other says equiv2 is the inside equivalence for the second argument. Then (fn a b) is equiv (fn a' b') provided a is equiv1 to a' and b is equiv2 to b'. This is an easy consequence of the transitivity of equiv. It permits you to think independently about the inside equivalences.

Furthermore, it is possible that more than one inside equivalence for a given argument slot will maintain a given outside equivalence. For example, (length a) is equal to (length a') if a and a' are related either by list-equal or by string-equal. You may prove two (or more) :congruence rules for the same slot of a function. The result is that the system uses a new, ``generated'' equivalence relation for that slot with the result that rules of both (or all) kinds are available while rewriting.

:Congruence rules can be disabled. For example, if you have two different inside equivalences for a given argument position and you find that the :rewrite rules for one are unexpectedly preventing the application of the desired rule, you can disable the rule that introduced the unwanted inside equivalence.

Remark on Replacing IFF by EQUAL. You may encounter a warning suggesting that a congruence rule ``can be strengthened by replacing the second equivalence relation, IFF, by EQUAL.'' Suppose for example that this warning occurs when you submit the following rule:

(defcong equiv1 iff (fn x y) 2)
which is shorthand for the following: 
(defthm equiv1-implies-iff-fn-2
       (implies (equiv1 y y-equiv)
                (iff (fn x y) (fn x y-equiv)))
       :rule-classes (:congruence))
The warning is telling you that ACL2 was able to deduce that fn always returns a Boolean, and hence a trivial but useful consequence is obtained by replacing iff by equal -- 
(defcong equiv1 equal (fn x y) 2)
-- which is shorthand for the following: 
(defthm equiv1-implies-equal-fn-2
       (implies (equiv1 y y-equiv)
                (equal (fn x y) (fn x y-equiv)))
       :rule-classes (:congruence))
If you have difficulty proving the latter directly, you can derive it from the former by giving a suitable hint, minimally as follows. 
(defcong equiv1 equal (fn x y) 2
  :hints (("Goal"
           :use equiv1-implies-iff-fn-2
           :in-theory
           (union-theories '((:type-prescription fn))
                           (theory 'minimal-theory)))))
By heeding this warning, you may avoid unnecessary double-rewrite warnings later. We now explain why, but see double-rewrite for relevant background material.

For example, suppose you have proved the ``iff'' version of the congruence rule above, and later you submit the following rewrite rule. 

(defthm equal-list-perm
  (implies (equiv1 x y)
           (fn x y)))
Since fn is known to return a Boolean, ACL2 performs an optimization that stores this rule as though it were the following. 
(defthm equal-list-perm
  (implies (equiv1 x y)
           (equal (fn x y) t)))
Thus, if ACL2's rewriter sees a term (fn a b) in a context where the equivalence relation iff is not being maintained, then it cannot use rule equiv1-implies-iff-fn-2, so it rewrites argument a without the benefit of knowing that it suffices to maintain equiv1; and then it caches the result. When ACL2 subsequently attempts to relieve the hypothesis (equiv1 x y), it will rewrite x simply by returning the rewritten value of a from the result cache. This is unfortunate if a could have been rewritten more completely under maintainance of the equivalence relation equiv1 -- which is legal in the hypothesis since a is an argument of equiv1, which is an equivalence relation. The user who observes the warning from rule equiv1-implies-iff-fn-2, and replaces it with equiv1-implies-equal-fn-2, will avoid this unfortunate case.