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.A very common hint is the: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)) :cases ((true-listp a) (consp a)) :by (:instance rev-rev (x (cdr z))) :nonlinearp t :reorder (4 7 2) :no-op t :error ("Bad value ~x0." 123) :or (hint-kwd-alist-1 ... hint-kwd-alist-k)))

`: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.

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 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`

, `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. 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`

, `:otf-flg-override`

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: A `:do-not-induct`

hint is ignored for any
goal on which an `:induct`

hint is supplied.

`: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. **Notes**:
(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. `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, `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.

`:`

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)`

.

`: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))`

.
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.

`: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.

`: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.

`: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
(custom-keyword-hints) and computed hints (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.

`: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-typin,
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))).

`: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))))... and so on. This example illustrates how ACL2 processes[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).

`: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.