## HINTS

advice to the theorem proving process
```Major Section:  MISCELLANEOUS
```

```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
:or (hint-kwd-alist-1 ... hint-kwd-alist-k)
:rw-cache-state nil
:backtrack (my-computed-hint clause processor clause-list)))
```
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 ``custom keyword'' hints (see custom-keyword-hints) and even more general ``computed hints'' for the advanced user (see computed-hints).

Only the first hint applicable to a goal, as specified in the user-supplied list of `:hints` followed by the default hints (see default-hints-table), will be applied to that goal. For an advanced exception, see override-hints. For a detailed discussion of how hints fit into the ACL2 waterfall, see hints-and-the-waterfall. For examples of the sophisticated use of hints, primarily for experts, see distributed book `books/hints/basic-tests.lisp`.

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 portion of admitting a `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 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.)

`:DO-NOT-INDUCT`
`Value` is `t`, `:otf-flg-override`, `:otf`, `name` or `nil`, indicating whether induction is permitted under the specified goal. If `value` is `t` 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` is specified (see otf-flg) and `value` is `t`, then the proof will continue until the time at which the goal pushed for induction is finally considered. The latter behavior is also what occurs if `value` is `:otf`. Thus, any non-`nil` value requires the indicated goal to 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`, `:otf-flg-override`, `:otf` 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`. Note that a `:do-not-induct` hint is ignored for any goal on which an `:induct` hint is supplied. For an advanced example of the use of value `:otf` with override-hints, see distributed book books/hints/basic-tests.lisp.

`: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)`.

`: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)`, 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. Remarks: (1) 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) Also allowed are ``terms'' of the form `(:with name term)`, where `name` is a function symbol, a macro name that denotes a function symbol (see macro-aliases-table), or a rune. The corresponding rule of class `:rewrite`, which is often a definition rule but need not be, is then used in place of the current body for the function symbol of `term`; see show-bodies and see set-body. If the rule is of the form `(implies hyp (equiv lhs rhs))`, then after matching `lhs` to the current term in a context that is maintaining equivalence relation `equiv`, ACL2 will replace the current term with `(if hyp rhs (hide term))`, or just `rhs` 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 distributed book `books/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 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.

`:INDUCT`
`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.

`:USE`
`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.

Note that if the value is the name of a function symbol introduced by `defun`, then the ``normalized'' body of that definition is used, for which ACL2 has propagated `IF` tests upward. This behavior differs from that provided by a `:by` hint, where the original body of the definition is used.

`:`BDD
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.

`: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-clause-processor-table)`.

`:INSTRUCTIONS`

`Value` is a list of proof-checker 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.

`: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.

`: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 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))`.

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, where the so-called ``normalized'' body, for which ACL2 has propagated `IF` tests upward.

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.

`:RESTRICT`
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.)

```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 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.

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

`:BACKCHAIN-LIMIT-RW`
`Value` is a natural number or `nil`, indicating the level of backchaining for rewrite, meta, and linear rules. This overrides, for the current goal and (as with `:``in-theory` hints) descendent goals, the default backchain-limit (see set-backchain-limit).

`:REORDER`
`Value` is a list of positive integers without duplicates, corresponding to the numbering of subgoals generated for the `goal-spec`, say `"G.k"` down to `"G.1"`. Those subgoals are reordered so that if `value` 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 of `ni`, after which the goals not yet printed are printed in their original order.

`: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 to `set-case-split-limitations`; see set-case-split-limitations. This behavior will persist through subgoals unless overridden by another `:CASE-SPLIT-LIMITATIONS` hint.

`:NO-OP`
`Value` is any object and is irrelevant. This hint does nothing. But empty hints, such as `("Goal")`, are illegal and there are occasions, especially when writing custom keyword hints (see custom-keyword-hints) and computed hints (see computed-hints) where it is convenient to be able to generate a non-empty no-op hint. The standard idiom is `("Goal" :NO-OP T)` but the `T` is completely ignored. 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 if `value` is non-`nil` then the usual ``[Note: A hint was supplied... Thanks!]'' is not printed.

`: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 (custom-keyword-hints) and computed hints (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 about `val` 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 value `123`, 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.

`:OR`
`Value` is a list `(kwd-val-listp-1 ... kwd-val-listp-k)`, where each `kwd-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 hints kwd-val-listp-1 in turn, 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 an enabled
:REWRITE or :DEFINITION rule, so you may want to consider disabling
(:REWRITE CDR-CONS).

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 each `i` from 1 to `k`, a so-called ``disjunctive'' subgoal is created by splicing `kwd-val-listp-i` into the other hint values (if any) supplied for the given goal, in order. A corresponding subgoal is created for each `i`, numbered in the usual manner (hence, counting down) except that the ```D`'' is prefixed to each resulting goal.

`: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 descendents, to set the so-called ``rw-cache-state'' to the indicated value; see set-rw-cache-state.

`: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 to `nil` -- 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, variables `PROCESSOR` and `CLAUSE-LIST` are allowed, but variable `STABLE-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 not `nil`, 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, then `form` 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 variable `STABLE-UNDER-SIMPLIFICATIONP` is not allowed to occur free in `form`, 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, and `CLAUSE-LIST` is bound to the list of clauses (each a list of literals that is implicitly disjoined) returned by that clause processor. Second, the variables `HIST` and `PSPV` 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 returns `nil`, then the `:backtrack` hint has no effect, and the goal is replaced by the list of goals (the value of `CLAUSE-LIST` described above), as usual. Otherwise, the clause processor is deemed to have failed, and the goal clause is tried again after applying 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: override-hints (if any) are applied in their processing. See override-hints.