advice to the theorem proving process

The following :hints value is nonsensical.  Nevertheless, it
illustrates all of the available hint keywords:

:hints (("Goal" :do-not-induct t :do-not '(generalize fertilize) :expand ((assoc x a) :lambdas (:free (y) (member y z))) :restrict ((<-trans ((x x) (y (foo x))))) :hands-off (length binary-append) :in-theory (set-difference-theories (current-theory :here) '(assoc)) :induct (and (nth n a) (nth n b)) :use ((:instance assoc-of-append (x a) (y b) (z c)) (:functional-instance (:instance p-f (x a) (y b)) (p consp) (f assoc))) :bdd (:vars (c a0 b0 a1 b1) :prove nil :bdd-constructors (cons)) :cases ((true-listp a) (consp a)) :by (:instance rev-rev (x (cdr z))) :nonlinearp t))

A very common hint is the :use hint, which in general takes as its value a list of ``lemma instances'' (see lemma-instance) but which allows a single lemma name as a special case:
:hints (("[1]Subgoal *1/1.2'" :use lemma23))

ACL2 also provides ``computed hints'' for the advanced user. See computed-hints

Background: Hints are allowed in all events that use the theorem prover. During defun events there are two different uses of the theorem prover: one to prove termination and another to verify the guards. To pass a hint to the theorem prover during termination proofs, use the :hints keyword in the defun's xargs declaration. To pass a hint to the theorem prover during the guard verification of defun, use the :guard-hints keyword in the defun's xargs declaration. The verify-guards event and the defthm event also use the theorem prover. To pass hints to them, use the :hints keyword argument to the event.

General Form of Common :hints:
  ((goal-spec :key1 val1 ... :keyn valn)
   (goal-spec :key1 val1 ... :keyn valn))
where goal-spec is as described in the documentation for goal-spec and the keys and their respective values are shown below with their interpretations. (We also provide ``computed hints'' but discuss them separately; see computed-hints.)

Value is t, name or nil, indicating whether induction is permitted under the specified goal. If value is t, then the attempt to apply induction to the indicated goal or any subgoal under the indicated goal will immediately cause the theorem prover to report failure. Thus, the indicated goal must be proved entirely by simplification, destructor elimination, and the other ``waterfall'' processes. Induction to prove the indicated goal (or any subgoal) is not permitted. See however the :induct hint below. If value is a symbol other than t or nil, the theorem prover will give a ``bye'' to any subgoal that would otherwise be attacked with induction. This will cause the theorem prover to fail eventually but will collect the necessary subgoals. If value is nil, this hint means induction is permitted. Since that is the default, there is no reason to use the value nil.

Value is a term having at most the single free variable world, which when evaluated (with world bound to the current ACL2 logical world) produces a list of symbols that is a subset of the list

(preprocess ;propositional logic, simple rules
 simplify   ;as above plus rewriting, linear arithmetic
 fertilize  ;use of equalities
The hint indicates that the ``processes'' named should not be used at or below the goal in question. Thus, to prevent generalization and fertilization, say, include the hint
:do-not '(generalize fertilize)
If value is a single symbol, as in
:do-not generalize,
it is taken to be '(value).

Value is a true list of terms, each of which is of one of the forms (let ((v1 t1)...) b) or (fn t1 ... tn), where fn is a defined function symbol with formals v1, ..., vn, and body b. Such a term is said to be ``expandable:'' it can be replaced by the result of substituting the ti's for the vi's in b. The terms listed in the :expand hint are expanded when they are encountered by the simplifier while working on the specified goal or any of its subgoals. We permit value to be a single such term instead of a singleton list. Notes: (1) Also allowed are ``terms'' of the form (:free (var1 var2 ... varn) pattern) where the indicated variables are distinct and pattern is a term. Such ``terms'' indicate that we consider the indicated variables to be instantiatable, in the following sense: whenever the simplifier encounters a term that can be obtained from pattern by instantiating the variables (var1 var2 ... varn), then it expands that term. (2) The term :LAMBDAS is treated specially. It denotes the list of all lambda applications (i.e., let expressions) encountered during the proof. Conceptually, this use of :LAMBDAS tells ACL2 to treat lambda applications as a notation for substitutions, rather than as function calls whose opening is subject to the ACL2 rewriter's heuristics (specifically, not allowing lambda applications to open when they introduce ``too many'' if terms).

Value is a true list of function symbols or lambda expressions, indicating that under the specified goal applications of these functions are not to be rewritten. Value may also be a single function symbol or lambda expression instead of a list.

Value is a ``theory expression,'' i.e., a term having at most the single free variable world which when evaluated (with world bound to the current ACL2 logical world (see world)) will produce a theory to use as the current theory for the goal specified. See theories.

Note that an :IN-THEORY hint will always be evaluated relative to the current ACL2 logical world, not relative to the theory of a previous goal. Consider the following example.

(defthm prop
  (p (f (g x)))
  :hints (("Goal"      :in-theory (disable f))
          ("Subgoal 3" :in-theory (enable  g))))
Consider in particular the theory in effect at Subgoal 3. This call of the enable macro enables g relative to the current-theory of the current logical world, not relative to the theory produced by the hint at Goal. Thus, the disable of f on behalf of the hint at Goal will be lost at Subgoal 3, and f will be enabled at Subgoal 3 if was enabled globally when prop was submitted.

Value is either t or a term containing at least one recursively defined function symbol; if t, this hint indicates that the system should proceed to apply its induction heuristic to the specified goal produced (without trying simplification, etc.); if value is a term other than t, then not only should the system apply induction immediately, but it should analyze value rather than the goal to generate its induction scheme. Merging and the other induction heuristics are applied. Thus, if value contains several mergeable inductions, the ``best'' will be created and chosen. E.g., the :induct hint

 (and (nth i a) (nth j a))
suggests simultaneous induction on i, j, and a.

If both an :induct and a :do-not-induct hint are supplied for a given goal then the indicated induction is applied to the goal and the :do-not-induct hint is inherited by all subgoals generated.

Value is a lemma-instance or a true list of lemma-instances, indicating that the propositions denoted by the instances be added as hypotheses to the specified goal. See lemma-instance. Note that :use makes the given instances available as ordinary hypotheses of the formula to be proved. The :instance form of a lemma-instance permits you to instantiate the free variables of previously proved theorems any way you wish; but it is up to you to provide the appropriate instantiations because once the instances are added as hypotheses their variables are no longer instantiable. These new hypotheses participate fully in all subsequent rewriting, etc. If the goal in question is in fact an instance of a previously proved theorem, you may wish to use :by below. Note that theories may be helpful when employing :use hints; see minimal-theory.

This hint indicates that ordered binary decision diagrams (BDDs) with rewriting are to be used to prove or simplify the goal. See bdd for an introduction to the ACL2 BDD algorithm.

Value is a list of even length, such that every other element, starting with the first, is one of the keywords :vars, :bdd-constructors, :prove, or :literal. Each keyword that is supplied should be followed by a value of the appropriate form, as shown below; for others, a default is used. Although :vars must always be supplied, we expect that most users will be content with the defaults used for the other values.

:vars -- A list of ACL2 variables, which are to be treated as Boolean variables. The prover must be able to check, using trivial reasoning (see type-set), that each of these variables is Boolean in the context of the current goal. Note that the prover will use very simple heuristics to order any variables that do not occur in :vars (so that they are ``greater than'' the variables that do occur in :vars), and these heuristics are often far from optimal. In addition, any variables not listed may fail to be assumed Boolean by the prover, which is likely to seriously impede the effectiveness of ACL2's BDD algorithm. Thus, users are encouraged not to rely on the default order, but to supply a list of variables instead. Finally, it is allowed to use a value of t for vars. This means the same as a nil value, except that the BDD algorithm is directed to fail unless it can guarantee that all variables in the input term are known to be Boolean (in a sense discussed elsewhere; see bdd-algorithm).

:literal -- An indication of which part of the current goal should receive BDD processing. Possible values are:

  :all     treat entire goal as a single literal (the default)
  :conc    process the conclusion
  n        process the hypothesis with index n (1, 2, ...)

:bdd-constructors -- When supplied, this value should be a list of function symbols in the current ACL2 world; it is (cons) by default, unless :bdd-constructors has a value in the acl2-defaults-table by default, in which case that value is the default. We expect that most users will be content with the default. See bdd-algorithm for information about how this value is used.

:prove -- When supplied, this value should be t or nil; it is t by default. When the goal is not proved and this value is t, the entire proof will abort. Use the value nil if you are happy to the proof to go on with the simplified term.

Value is a non-empty list of terms. For each term in the list, a new goal is created from the current goal by assuming that term; and also, in essence, one additional new goal is created by assuming all the terms in the list false. We say ``in essence'' because if the disjunction of the terms supplied is a tautology, then that final goal will be a tautology and hence will in fact never actually be created.

Value is a lemma-instance, nil, or a new event name. If the value is a lemma-instance (see lemma-instance), then it indicates that the goal (when viewed as a clause) is either equal to the proposition denoted by the instance, or is subsumed by that proposition when both are viewed as clauses. To view a formula as a clause, union together the negations of the hypotheses and add the conclusion. For example,

(IMPLIES (AND (h1 t1) (h2 t2)) (c t1))
may be viewed as the clause
{~(h1 t1) ~(h2 t2) (c t1)}.
Clause c1 is ``subsumed'' by clause c2 iff some instance of c2 is a subset of c1. For example, the clause above is subsumed by {~(h1 x) (c x)}, which when viewed as a formula is (implies (h1 x) (c x)).

If the value is nil or a new name, the prover does not even attempt to prove the goal to which this hint is attached. Instead the goal is given a ``bye'', i.e., it is skipped and the proof attempt continues as though the goal had been proved. If the prover terminates without error then it reports that the proof would have succeeded had the indicated goals been proved and it prints an appropriate defthm form to define each of the :by names. The ``name'' nil means ``make up a name.''

The system does not attempt to check the uniqueness of the :by names (supplied or made up), since by the time those goals are proved the namespace will be cluttered still further. Therefore, the final list of ``appropriate'' defthm forms may be impossible to admit without some renaming by the user. If you must invent new names, remember to substitute the new ones for the old ones in the :by hints themselves.

Warning: This is a sophisticated hint, suggested by Bishop Brock, that is intended for advanced users. In particular, :restrict hints are ignored by the preprocessor, so you might find it useful to give the hint :do-not '(preprocess) when using any :restrict hints, at least if the rules in question are abbreviations (see simple).

Value is an association list. Its members are of the form (x subst1 subst2 ...), where: x is either (1) a rune whose car is :rewrite or :definition or (2) an event name corresponding to one or more such runes; and (subst1 subst2 ...) is a non-empty list of substitutions, i.e., of association lists pairing variables with terms. First consider the case that x is a :rewrite or :definition rune. Recall that without this hint, the rule named x is used by matching its left-hand side (call it lhs) against the term currently being considered by the rewriter, that is, by attempting to find a substitution s such that the instantiation of lhs using s is equal to that term. If however the :restrict hint contains (x subst1 subst2 ...), then this behavior will be modified by restricting s so that it must extend subst1; and if there is no such s, then s is restricted so that it must extend subst2; and so on, until the list of substitutions is exhausted. If no such s is found, then the rewrite or definition rule named x is not applied to that term. Finally, if x is an event name corresponding to one or more :rewrite or :definition runes (that is, x is the ``base symbol'' of such runes; see rune), say runes r1, ... rn, then the meaning is the same except that (x subst1 subst2 ...) is replaced by (ri subst1 subst2 ...) for each i. Once this replacement is complete, the hint may not contain two members whose car is the same rune.

Note that the substitutions in :restrict hints refer to the variables actually appearing in the goals, not to the variables appearing in the rule being restricted.

Here is an example, supplied by Bishop Brock. Suppose that the database includes the following rewrite rule, which is probably kept disabled. (We ignore the question of how to prove this rule.)

(implies (and (rationalp x)
              (rationalp y)
              (rationalp z))
         (equal (< y z)
                (if (< x 0)
                    (> (* x y) (* x z))
                  (if (> x 0)
                      (< (* x y) (* x z))
                    (hide (< y z))))))
Then ACL2 can prove the following theorem (unless other rules get in the way), essentially by multiplying both sides by x.
  (implies (and (rationalp x)
                (< 1 x))
           (< (/ x) 1))
    :in-theory (enable cancel-<-*$free)
    :restrict ((cancel-<-*$free ((x x) (y (/ x)) (z 1)))))))
The :restrict hint above says that the variables x, y, and z in the rewrite rule cancel-<-*$free above should be instantiated respectively by x, (/ x), and 1. Thus (< y z) becomes (< (/ x) 1), and this inequality is replaced by the corresponding instance of the right-hand-side of cancel-<-*$free. Since the current conjecture assumes (< 1 x), that instance of the right-hand side simplifies to
(< (* x (/ x)) (* x 1))
which in turn simplifies to (< 1 x), a hypothesis in the present theorem.

Value is t or nil, indicating whether non-linear-arithmetic is active. The default value is nil. See non-linear-arithmetic.