Advice to the theorem proving process

Examples: The following :hints value is nonsensical. Nevertheless, it illustrates all of the available hint keywords except the ``custom keywords'' (see custom-keyword-hints) definable by the user. :hints (("Goal" :do-not-induct t :do-not '(generalize fertilize) :expand ((assoc x a) :lambdas (:free (y) (:with member (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)) :clause-processor (:function cl-proc :hint (my-hint clause)) :instructions (:x :prove) :cases ((true-listp a) (consp a)) :by (:instance rev-rev (x (cdr z))) :nonlinearp t :backchain-limit-rw 3 :reorder (4 7 2) :case-split-limitations (20 10) :no-op t :no-thanks t :error ("Bad value ~x0." 123) :or (hint-kwd-alist-1 ... hint-kwd-alist-k) :rw-cache-state nil :backtrack (my-computed-hint clause processor clause-list)))

Many of these hints affect the how the prover operates not only on the goal
to which they are applied but also on its subgoals (and its subgoals'
subgoals, etc.; for a deeper explanation see hints-and-the-waterfall).
The following hints, however, have a specific effect only on the goal to which
they are applied:

A very common hint is the

; Attach :use hint to the top-level goal G, which is named "Goal", ; replacing it by (implies P G) where P is the statement of lemma23: :hints (("Goal" :use lemma23)) ; Equivalent to the above, using the trivial instance (i.e., with the empty ; substitution) of lemma23: :hints (("Goal" :use ((:instance lemma23)))) ; Attach :use hint to the named subgoal, where the indicated lemma is used ; with the substitution that maps x to 17 and y to (foo z): :hints (("[1]Subgoal *1/1.2'" :use ((:instance lemma23 (x 17) (y (foo z)))))) ; Equivalent to the above: ACL2 allows you to omit the outer parentheses if ; there is only one lemma used. :hints (("[1]Subgoal *1/1.2'" :use (:instance lemma23 (x 17) (y (foo z)))))

ACL2 also provides ``custom keyword'' hints (see custom-keyword-hints) and even more general ``computed hints'' for the
advanced user (see computed-hints). Not documented in this topic are
such hints implemented in books; for an example of so-called

When the ACL2 prover encounters a goal

Duplicate hint keywords are prohibited in a hint; for example, the hint

- It is illegal to use two or more keywords from the following list, except
that
:use and:cases may be used together:

(:induct :use :cases :by :bdd :clause-processor :or) . - It is illegal to use
:reorder with either:or or:induct .

Background: `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 `defun`'s `xargs`
declaration. To pass a hint to the theorem prover during the guard
verification portion of admitting a `defun`, use 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

General Form of Common :hints: ((goal-spec :key1 val1 ... :keyn valn) ... (goal-spec :key1 val1 ... :keyn valn))

where `goal-spec` is as described elsewhere (see 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. The hint keywords below are considered
in alphabetical order.

:backchain-limit-rw Value is a natural number ornil , indicating the level of backchaining for rewrite, meta, and linear rules. This overrides, for the current goal and (as with: `in-theory`hints) descendant goals, the default backchain-limit (see set-backchain-limit).:backtrack This is an advanced hint. You can probably accomplish its effect by the use of ordinary computed hints; see computed-hints. But if you are an expert, read on. (See hints-and-the-waterfall for some relevant background.)

Value is a computed hint, which is an expression that evaluates either tonil — indicating that the:backtrack hint is to have no effect — or to a non-empty alternating list of:keyi vali pairs, as expected for a hint. However, unlike ordinary computed hints,:backtrack hints are evaluated**after**a goal has been processed to yield zero or more subgoals, not before. Moreover, variablesPROCESSOR andCLAUSE-LIST are allowed, but variableSTABLE-UNDER-SIMPLIFICATIONP is not. We explain in more detail below, but first consider the following simple example. First we define a standard list reversal function:(defun rev (x) (if (consp x) (append (rev (cdr x)) (cons (car x) nil)) nil))

Now we prove:

(thm (true-listp (rev x)))

The successful proof includes the following output.

Subgoal *1/1' (IMPLIES (AND (CONSP X) (TRUE-LISTP (REV (CDR X)))) (TRUE-LISTP (APPEND (REV (CDR X)) (LIST (CAR X))))). The destructor terms (CAR X) and (CDR X) can be eliminated by using CAR-CDR-ELIM to replace X by (CONS X1 X2), (CAR X) by X1 and (CDR X) by X2. This produces the following goal. Subgoal *1/1'' (IMPLIES (AND (CONSP (CONS X1 X2)) (TRUE-LISTP (REV X2))) (TRUE-LISTP (APPEND (REV X2) (LIST X1)))).

But suppose that we attach a

:backtrack hint to the goal above at which destructor elimination was applied:(thm (true-listp (rev x)) :hints (("Subgoal *1/1'" :backtrack (quote (:do-not '(eliminate-destructors))))))

Then when ACL2 applies destructor elimination as displayed above, this time the

:backtrack hint applies, evaluating to(:do-not '(eliminate-destructors)) . Since this list is notnil , the prover decides not to keep the new subgoal, and instead supplies this:do-not hint before attacking the goal again. In this example, ACL2 happens to use a technique later in its ``waterfall'' arsenal than destructor elimination, namely, generalization:Subgoal *1/1' (IMPLIES (AND (CONSP X) (TRUE-LISTP (REV (CDR X)))) (TRUE-LISTP (APPEND (REV (CDR X)) (LIST (CAR X))))). [Note: A hint was supplied for our processing of the goal above, because of a :backtrack hint that is preventing destructor elimination. Thanks!] We generalize this conjecture, replacing (REV (CDR X)) by RV. This produces Subgoal *1/1'' (IMPLIES (AND (CONSP X) (TRUE-LISTP RV)) (TRUE-LISTP (APPEND RV (LIST (CAR X))))).

We now provide a careful explanation of how

:backtrack hints work, but we suggest that you keep the example above in mind. If ``:backtrack form '' is part of the hint that has been selected for a goal, thenform is evaluated when one of ACL2's clause processors successfully applies to the current goal to produce a list of subgoals. This evaluation takes place in an environment just like that for any computed hint (see computed-hints), with the following exceptions. First, the variableSTABLE-UNDER-SIMPLIFICATIONP is not allowed to occur free inform , but instead the following new variables are allowed to occur free and are bound for this evaluation as follows:PROCESSOR is bound to the processor in the list*preprocess-clause-ledge* that has applied to the goal, andCLAUSE-LIST is bound to the list of clauses (each a list of literals that is implicitly disjoined) returned by that clause processor. Second, the variablesHIST andPSPV are bound to the history and pspv returned by the clause processor,**not**the ones that were passed to the clause processor. If this evaluation returns an error, then the proof aborts, as for any computed hint whose evaluation returns an error. If this evaluation returnsnil , then the:backtrack hint has no effect, and the goal is replaced by the list of goals (the value ofCLAUSE-LIST described above), as usual. Otherwise, the clause processor is deemed to have failed, and the goal clause is tried again starting at the top of the waterfall after selecting the hint returned by the above evaluation. That hint will normally be an alternating list of hint keywords and their values, but if it is a custom keyword hint (see custom-keyword-hints), then it will be handled in the usual manner but with the first three variables above bound to the symbol:OMITTED . Of course, if the new hint includes a value for:BACKTRACK then this process can loop; care should be taken to keep that from happening.A final note about

:BACKTRACK hints: since these are a form of computed hints, override-hints (if any) are applied to their evaluation result just as with any computed hint. That is, the backtrack hint is successively modified with each override-hint, to produce a final hint that is actually used (or, ignored if that final hint isnil ). See override-hints.: `bdd`This hint indicates that ACL2's built-in 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 oft forvars . This means the same as anil 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 bet ornil ; it ist by default. When the goal is not proved and this value ist , the entire proof will abort. Use the valuenil if you are happy to the proof to go on with the simplified term.:by 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 clausec2 iff some instance ofc2 is a subset ofc1 . For example, the clause above is subsumed by{~(h1 x) (c x)} , which when viewed as a formula is(implies (h1 x) (c x)) .Note that if the value is the name of a function symbol introduced by

`defun`, then the original form of the body of that definition is used. This behavior differs from that provided by a:use hint, which uses the normalized (simplified) body; see normalize.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.'' Here is an example (admittedly contrived for illustration purposes).ACL2 !>(thm (equal (append (append x y) z) (append x y z)) :hints (("Subgoal *1/2'" :by nil))) Name the formula above *1. [[... output omitted here ...]] [Note: A hint was supplied for our processing of the goal below. Thanks!] Subgoal *1/2' (IMPLIES (AND (CONSP X) (EQUAL (APPEND (APPEND (CDR X) Y) Z) (APPEND (CDR X) Y Z))) (EQUAL (APPEND (APPEND X Y) Z) (APPEND X Y Z))). But we have been asked to pretend that this goal is subsumed by the yet-to-be-proved |THM Subgoal *1/2'|. Subgoal *1/1 [[... proof goes on; further output omitted here ...]]

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.: `case-split-limitations`Value is the same as for`set-case-split-limitations`. The simplifier will behave as though the value had instead been supplied toset-case-split-limitations ; see set-case-split-limitations. This behavior will persist through subgoals unless overridden by another:CASE-SPLIT-LIMITATIONS hint.:cases 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.: `clause-processor`Value specifies the application of a user-defined simplifier to the current goal. See clause-processor, which provides necessary background and hint syntax. Also see define-trusted-clause-processor for a discussion of ``trusted clause-processors'': goal-level simplifiers that may be external to ACL2 and do not need to be proved correct in ACL2.You can see all current

:clause-processor rules by issuing the command(print-clause-processor-rules) , and you can see the names of all trusted clause-processors by issuing the command(table trusted-cl-proc-table) .:do-not 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 eliminate-destructors fertilize ;use of equalities generalize eliminate-irrelevance).

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) .See also do-not-hint for a way to automatically provide

:do-not hints across several theorems.:do-not-induct Value indicates whether induction is permitted under the specified goal. The legal values aret ,:otf-flg-override ,:otf ,nil , or a non-keyword symbol other thant ornil . The default isnil , meaning that induction is permitted as usual. A non-nil value prohibits the use of induction to prove the indicated goal or any of its subgoals, as described below.If

value ist or:otf-flg-override , 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, except that if:otf-flg t is specified (see otf-flg) andvalue ist , then the proof will continue until the time at which the goal pushed for induction is finally encountered and causes failure. The latter behavior is also what occurs ifvalue is:otf . See however the:induct hint below. Ifvalue is a non-keyword symbol other thant ornil , the theorem prover will skip every subgoal under the indicated goal (giving it a ``bye'', as with a ``:by '' hint) that would otherwise be attacked with induction. This will cause the theorem prover to fail eventually, printing every subgoal thus skipped in the form of an event to prove, each with a name based on the value of the:do-not-induct hint that caused that subgoal to be skipped.**Remarks.**(1) An

:induct hint is applied to a goal even if a:do-not-induct hint is in effect for that goal. Consider the following examples.(thm (equal (append (append x y) z) (append x y z)) :hints (("Goal" :induct t :do-not-induct t))) (thm (and (equal (append (append x y) z) (append x y z)) (equal (append (append u v) w) (append u v w))) :hints (("Goal" :do-not-induct t) ("Subgoal 2" :induct t)))

In the first of these, the

:do-not-induct hint has no effect on the proof; instead, the:induct hint forces an induction that allows the proof to succeed (without any sub-inductions). The second of these illustrates that even though:do-not-induct can stop sub-inductions, its effect is overridden by:induct . For the proof of that second example, ACL2 immediately splits into two subgoals. Then in spite of the top-level:do-not-induct hint, the proof is allowed to proceed past Subgoal 2, which requires induction, because of the hint:induct t . However, the proof halts after Subgoal 1 because of the:do-not-induct hint that has been established ``above'' it, at"Goal" . (For more about the way hints are processed, see hints-and-the-waterfall.)(2) For an advanced example of the use of value

:otf for:do-not-induct combined with override-hints, see community bookbooks/hints/basic-tests.lisp .:error Value is typically a ``fmt message'' to be printed by the`fmt`tilde-directive ~@ but may be any object. The effect of this hint is to cause an error when the hint is translated. There is no reason to include an:ERROR hint in any user-typein, since it will only cause an error when the form is evaluated.:ERROR hints are useful in the definition of functions that generate custom keyword hints (see custom-keyword-hints) and computed hints (see computed-hints). For example, if you wish to define a custom keyword hint:my-hint val and you wish the hint to signal an error if there is something inappropriate aboutval in the context of the hint, use the following code to generate the hint(list :ERROR (cons "Your specified value, ~x0, is inappropriate" (list (cons #0 val))))

which is equivalent to

(list :ERROR (msg "Your specified value, ~x0, is inappropriate" val))

which, if

val has the value123 , would evaluate to the hint(:ERROR ("Your specified value, ~x0, is inappropriate" (#0 . 123))).

Note that any time an

:ERROR keyword is produced during hint processing, including iterations of the expansions of custom keyword hints or of override-hints, an error will occur.:expand Value is a true list of terms, each of which is of one of the forms(let ((v1 t1)...) b) or(fn t1 ... tn) , wherefn is a defined function symbol with formalsv1, ..., vn, andbody b . Such a term is said to be ``expandable:'' it can be replaced by the result of substituting theti 's for thevi 's inb . 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. (There is no separate ``expand'' process.) We permitvalue to be a single such term instead of a singleton list.**Remarks**: (0) Note that in the event that a:definition rule has been admitted forfn , then by default, the bodyb is determined by the (most recently admitted such) rule rather than the original definition offn ; see definition. (1) Allowed are ``terms'' of the form(:free (var1 var2 ... varn) pattern) where the indicated variables are distinct andpattern 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 frompattern by instantiating the variables(var1 var2 ... varn) , then it expands that term. (2) Also allowed are ``terms'' of the form(:with name term) , wherename is a function symbol, a macro name that denotes a function symbol (see macro-aliases-table), or a rune. The corresponding definition rule or (less often)`rewrite`rule is then used in place of the current body for the function symbol ofterm ; see show-bodies and see set-body. If the rule is of the form(implies hyp (equiv lhs rhs)) , then after matchinglhs to the current term in a context that is maintaining equivalence relationequiv , ACL2 will replace the current term with(if hyp rhs (hide term)) , or justrhs if the rule is just(equal lhs rhs) . (3) A combination of both:free and:with , as described above, is legal. (4) 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).:hands-off 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. Note however that subterms will still be rewritten; see hide if that is not what is intended. (The community bookbooks/clause-processors/autohide.lisp from Jared Davis may also be helpful in that case.)Value may also be a single function symbol or lambda expression instead of a list.: `in-theory`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 enablesg relative to the`current-theory`of the current logical world,*not*relative to the theory produced by the hint atGoal . Thus, the`disable`off on behalf of the hint atGoal will be lost atSubgoal 3 , andf will be enabled atSubgoal 3 if was enabled globally whenprop was submitted.:induct Value is eithert or a term containing at least one recursively defined function symbol; ift , this hint indicates that the system should proceed to apply its induction heuristic to the specified goal produced (without trying simplification, etc.); ifvalue is a term other thant , then not only should the system apply induction immediately, but it should analyzevalue rather than the goal to generate its induction scheme. Merging and the other induction heuristics are applied. Thus, ifvalue 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 , anda .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.: `instructions`Value is a list of interactive proof-builder instructions; see instructions. Unlike other hint keywords described here, this one is actually a custom keyword hint (see custom-keyword-hints) that generates a suitable: `clause-processor`hint.:no-op Value is any object and is irrelevant. This hint has no effect, although unlike an empty hint such as("Goal") , it is not dropped. Thus,("Goal") :do-not t will shadow any later (or default) hint on"Goal" , but("Goal") will not. Unlike other hint keywords, multiple occurrences of the keyword:no-op are tolerated.:no-thanks Value is any object. This hint does nothing, except that ifvalue is non-nil then the usual ``[Note: A hint was supplied... Thanks!]'' is not printed.:nonlinearp Value ist ornil , indicating whether non-linear-arithmetic is active. The default value isnil . See non-linear-arithmetic.:or Value is a list(kwd-val-listp-1 ... kwd-val-listp-k) , where eachkwd-val-listp-i is a list satisfying`keyword-value-listp`, i.e., an alternating list of keywords and values. This hint causes an attempt to prove the specified goal using hintskwd-val-listp-i in sequence (firstkwd-val-listp-1 , thenkwd-val-listp-2 , and so on), until the first of these succeeds. If none succeeds, then the prover proceeds after heuristically choosing the ``best'' result, taking into account the goals pushed in each case for proof by induction.The following (contrived but illustrative example illustrates how

:or hints work.ACL2 !>(thm (f x) :hints (("Goal" :expand ((nth x 3)) :or ((:in-theory (disable car-cons)) (:use cdr-cons :in-theory (enable append))) :do-not '(generalize)))) [Note: A hint was supplied for our processing of the goal above. Thanks!] The :OR hint for Goal gives rise to two disjunctive branches. Proving any one of these branches would suffice to prove Goal. We explore them in turn, describing their derivations as we go. --- Subgoal D2 ( same formula as Goal ). The first disjunctive branch (of 2) for Goal can be created by applying the hint: ("Subgoal D2" :EXPAND ((NTH X 3)) :IN-THEORY (DISABLE CAR-CONS) :DO-NOT '(GENERALIZE)). [Note: A hint was supplied for our processing of the goal above. Thanks!] Normally we would attempt to prove this formula by induction. However, we prefer in this instance to focus on the original input conjecture rather than this simplified special case. We therefore abandon our previous work on this conjecture and reassign the name *1 to the original conjecture. (See :DOC otf-flg.) [Note: Thanks again for the hint.] --- Subgoal D1 ( same formula as Goal ). The second disjunctive branch (of 2) for Goal can be created by applying the hint: ("Subgoal D1" :EXPAND ((NTH X 3)) :USE CDR-CONS :IN-THEORY (ENABLE APPEND) :DO-NOT '(GENERALIZE)). [Note: A hint was supplied for our processing of the goal above. Thanks!] ACL2 Warning [Use] in ( THM ...): It is unusual to :USE the formula of an enabled :REWRITE or :DEFINITION rule, so you may want to consider disabling (:REWRITE CDR-CONS) in the hint provided for Subgoal D1. See :DOC using-enabled-rules. We augment the goal with the hypothesis provided by the :USE hint. The hypothesis can be obtained from CDR-CONS. We are left with the following subgoal. Subgoal D1' (IMPLIES (EQUAL (CDR (CONS X Y)) Y) (F X)). By the simple :rewrite rule CDR-CONS we reduce the conjecture to Subgoal D1'' (F X).

... and so on. This example illustrates how ACL2 processes

:or hints in general. For eachi from 1 tok , a so-called ``disjunctive'' subgoal is created by splicingkwd-val-listp-i into the other hint values (if any) supplied for the given goal, in order. A corresponding subgoal is created for eachi , numbered in the usual manner (hence, counting down) except that the ``D '' is prefixed to each resulting goal.:reorder Value is a list of positive integers without duplicates, corresponding to the numbering of subgoals generated for the goal-spec"G" , say"G.k" down to"G.1" . Those subgoals are reordered so that ifvalue is(n1 n2 ... nk) , then the goal now numbered"G.k" will be the goal originally numbered"G.n1" ; the goal now numbered"G.k-1" will be the goal formerly numbered"G.n2" ; and so on, down the list ofni , after which the goals not yet printed are printed in their original order. Note that reordering for subgoals of a goal to be proved by induction, such as*1 , is not supported.:restrict This hint, originally suggested by Bishop Brock, sometimes allows rules with free variables (see free-variables) to be applied successfully by the rewriter, thus avoiding the clutter, case-splitting, and theory management (disabling) that can occur with

:use hints.Warning: This is a sophisticated hint that may be most appropriate for experienced ACL2 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 thatx is a: `rewrite`or: `definition`rune. Recall that without this hint, the rule namedx is used by matching its left-hand side (call itlhs ) against the term currently being considered by the rewriter, that is, by attempting to find a substitutions such that the instantiation oflhs usings is equal to that term. If however the:restrict hint contains(x subst1 subst2 ...) , then this behavior will be modified by restrictings so that it must extendsubst1 ; and if there is no suchs , thens is restricted so that it must extendsubst2 ; and so on, until the list of substitutions is exhausted. If no suchs is found, then the rewrite or definition rule namedx is not applied to that term. Finally, ifx 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 runesr1 , ...rn , then the meaning is the same except that(x subst1 subst2 ...) is replaced by(ri subst1 subst2 ...) for eachi . 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.The following example, supplied by Mihir Mehta, illustrates the use of

:restrict to handle free variables (in this case, a single free variabley ). The call of`thm`below fails without the indicated:restrict hint.(defthm subsetp-trans (implies (and (subsetp x y) (subsetp y z)) (subsetp x z))) (defthm subsetp-evens (subsetp-equal (evens l) l)) (thm (subsetp (evens (evens l)) l) :hints (("Goal" :restrict ((subsetp-trans ((y (evens l))))))))

Here is another example, this one 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.)

cancel-<-*$free: (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 .(thm (implies (and (rationalp x) (< 1 x)) (< (/ x) 1)) :hints (("Goal" :in-theory (enable cancel-<-*$free) :restrict ((cancel-<-*$free ((x x) (y (/ x)) (z 1)))))))

The

:restrict hint above says that the variablesx ,y , andz in the rewrite rulecancel-<-*$free above should be instantiated respectively byx ,(/ x) , and1 . Thus(< y z) becomes(< (/ x) 1) , and this inequality is replaced by the corresponding instance of the right-hand-side ofcancel-<-*$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.:rw-cache-state Value is an element of the list constant*legal-rw-cache-states* ::atom (the default),nil ,t , or:disabled . This hint applies to the indicated goal and all its descendants, to set the so-called ``rw-cache-state'' to the indicated value; see set-rw-cache-state.:use Examples of

:USE hints are shown near the top of this documentation topic.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: that is, the:use hint replaces a goal,G , by the new goal,(IMPLIES P G) , whereP is the theorem specified by the (conjunction of the) lemma instances provided. The:instance form of a lemma-instance permits you to instantiate the free variables of previously proved theorems any way you wish, even allowing for differences in packages; see lemma-instance for details. 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 (documented above). Sometimes theories are helpful when employing:use hints; see minimal-theory.If the value is the name of a function symbol introduced by

`defun`, then the normalized (simplified) body of that definition is used; see normalize. This behavior differs from that provided by a:by hint, where the original body of the definition is used.

- Lemma-instance
- An object denoting an instance of a theorem
- Computed-hints
- Computing advice to the theorem proving process
- Override-hints
- A list of hints given priority in every proof attempt
- Hints-and-the-waterfall
- How hints fit into the ACL2 proof waterfall
- Goal-spec
- To indicate where a hint is to be used
- Termination-theorem-example
- How to use a previously-proved measure theorem
- Consideration
- An object indicating that some instantiation is relevant.
- Hint-wrapper
- Supply hints in the statement of a theorem
- Default-hints
- A list of hints added to every proof attempt
- Guard-theorem-example
- How to use a previously-proved guard theorem
- Do-not-hint
- Give
:do-not hints automatically. - Guard-theorem
- Use a previously-proved guard theorem
- Using-computed-hints
- How to use computed hints
- Termination-theorem
- Use a (functional instance of a) previously-proved measure theorem
- Custom-keyword-hints
- User-defined hints
- Do-not
- Instruct the theorem prover not to do certain things.