Define a function symbol
(defun app (x y)
(if (consp x)
(cons (car x) (app (cdr x) y))
(defun fact (n)
(declare (xargs :guard (and (integerp n)
(>= n 0))))
(if (zp n)
(* n (fact (1- n)))))
(defun fn (var1 ... varn) doc-string dcl ... dcl body),
where fn is the symbol you wish to define and is a new symbolic name
(see name), (var1 ... varn) is its list of formal parameters (see
name), and body is its body. The definitional axiom is logically
admissible provided certain restrictions are met. These are sketched
See also mutual-recursion for how to use defun to make mutually
recursive definitions, including discussion of how the xargs declarations in one defun may affect the other definitions.
Note that ACL2 does not support the use of lambda-list keywords (such
as &optional) in the formals list of functions. We do support some such
keywords in macros and often you can achieve the desired syntax by defining a
macro in addition to the general version of your function. See defmacro. Doc-string, if non-nil, is an optional string that can
provide documentation but is essentially ignored by ACL2.
The declarations (see declare), dcl, are also optional.
If more than one dcl form appears, they are effectively grouped together
as one. Perhaps the most commonly used ACL2 specific declaration is of the
form (declare (xargs :guard g :measure m)). This declaration in the
defun of some function fn has the effect of making the ``guard'' for fn be the term g and the ``measure'' be the term
m. The notion of ``measure'' is crucial to ACL2's definitional
principle. The notion of ``guard'' is not, and is discussed elsewhere; see
verify-guards and see set-verify-guards-eagerness. Note that a
:measure is not allowed for a non-recursive definition unless it is part
of a mutual-recursion (exception: a measure of nil is treated as
though the measure was omitted); moreover, if a :measure is
supplied, then it must be a legal term. Apart from these restrictions, the
:measure is ignored in :program mode; see defun-mode.
We now briefly discuss the ACL2 definitional principle, using the following
definition form which is offered as a more or less generic example.
(defun fn (x y)
(declare (xargs :guard (g x y)
:measure (m x y)))
(if (test x y)
(stop x y)
(step (fn (d x) y))))
Note that in our generic example, fn has just two arguments, x
and y, the guard and measure terms involve both of them, and the
body is a simple case split on (test x y) leading to a ``non-recursive''
branch, (stop x y), and a ``recursive'' branch. In the recursive branch,
fn is called after ``decrementing'' x to (d x) and some step
function is applied to the result. Of course, this generic example is quite
specific in form but is intended to illustrate the more general case.
Provided this definition is admissible under the logic, as outlined below,
it adds the following axiom to the logic.
(fn x y)
(if (test x y)
(stop x y)
(step (fn (d x) y)))
Note that the guard of fn has no bearing on this logical
This defining axiom is actually implemented in the ACL2 system by a
:definition rule, namely
(equal (fn x y)
(if (test a b)
(stop a b)
(step (fn (d a) b)))).
See definition for a discussion of how definition rules are applied.
Roughly speaking, the rule causes certain instances of (fn x y) to be
replaced by the corresponding instances of the body above. This is called
``opening up'' (fn x y). The instances of (fn x y) opened are
chosen primarily by heuristics which determine that the recursive calls of
fn in the opened body (after simplification) are more desirable than the
unopened call of fn.
This discussion has assumed that the definition of fn was admissible.
Exactly what does that mean? First, fn must be a previously
unaxiomatized function symbol (however, see ld-redefinition-action).
Second, the formal parameters must be distinct variable names. Third, the
guard, measure, and body should all be terms and should mention no free
variables except the formal parameters. Thus, for example, body may not
contain references to ``global'' or ``special'' variables; ACL2 constants or
additional formals should be used instead.
The final conditions on admissibility concern the termination of the
recursion. Roughly put, all applications of fn must terminate. In
particular, there must exist a binary relation, rel, and some unary
predicate mp such that rel is well-founded on objects satisfying
mp, the measure term m must always produce something satisfying
mp, and the measure term must decrease according to rel in each
recursive call, under the hypothesis that all the tests ruling the call are
satisfied (see rulers). By the meaning of well-foundedness, we know
there are no infinitely descending chains of successively rel-smaller
mp-objects. Thus, the recursion must terminate.
The default well-founded relation is o<, an ``ordinal less-than''
relation (discussed further below) that reduces to ordinary < on the
natural numbers. The default measure term is (acl2-count var), where
var is a formal parameter that is chosen heuristically: roughly speaking,
it is the first formal that is tested along every branch and changed in each
The only primitive well-founded relation in ACL2 is o<, which is
known to be well-founded on the o-ps. For the proof of
well-foundedness, see proof-of-well-foundedness. However it is
possible to add new well-founded relations. For details, see well-founded-relation. We discuss later how to specify which well-founded
relation is selected by defun and in the present discussion we assume,
without loss of generality, that it is o< on the o-ps.
For example, for our generic definition of fn above, with measure term
(m x y), two theorems must be proved. The first establishes that m
produces an ordinal:
(o-p (m x y)).
The second shows that m decreases in the (only) recursive call of
(implies (not (test x y))
(o< (m (d x) y) (m x y))).
Observe that in the latter formula we must show that the ``m-size'' of
(d x) and y is ``smaller than'' the m-size of x and
y, provided the test, (test x y), in the body fails, thus leading to
the recursive call (fn (d x) y).
See o< for a discussion of this notion of ``smaller than.'' It
should be noted that the most commonly used ordinals are the natural numbers
and that on natural numbers, o< is just the familiar ``less than''
relation (<). Thus, it is very common to use a measure m that
returns a nonnegative integer, for then (o-p (m x y)) becomes a simple
conjecture about the type of m and the second formula above becomes a
conjecture about the less-than relationship of nonnegative integer
The most commonly used measure function is ACL2-count, which
computes a nonnegative integer size for all ACL2 objects. See ACL2-count.
Probably the most common recursive scheme in Lisp programming is
when some formal is supposed to be a list and in the recursive call it is
replaced by its cdr. For example, (test x y) might be simply
(atom x) and (d x) might be (cdr x). In that case,
(acl2-count x) is a suitable measure because the ACL2-count of a
cons is strictly larger than the ACL2-counts of its car and cdr. Thus, ``recursion by car'' and ``recursion by
cdr'' are trivially admitted if ACL2-count is used as the
measure and the definition protects every recursive call by a test insuring
that the decremented argument is a consp. Similarly, ``recursion by
1-'' in which a positive integer formal is decremented by one in
recursion, is also trivially admissible. See built-in-clause to extend
the class of trivially admissible recursive schemes.
We now turn to the question of which well-founded relation defun uses.
It should first be observed that defun must actually select both a
relation (e.g., o<) and a domain predicate (e.g., o-p) on
which that relation is known to be well-founded. But, as noted elsewhere (see
well-founded-relation-rule), every known well-founded relation has a
unique domain predicate associated with it and so it suffices to identify
simply the relation here.
The xargs field of a declare permits the explicit
specification of any known well-founded relation with the keyword
:well-founded-relation. An example is given below. If the xargs
for a defun specifies a well-founded relation, that relation and its
associated domain predicate are used in generating the termination conditions
for the definition.
If no :well-founded-relation is specified, defun uses the
:well-founded-relation specified in the ACL2-defaults-table. See
set-well-founded-relation to see how to set the default well-founded
relation (and, implicitly, its domain predicate). The initial default
well-founded relation is o< (with domain predicate o-p).
This completes the brief sketch of the ACL2 definitional principle.
Optionally, see rulers for a more detailed discussion of the
termination analysis and resulting proof obligations for admissibility, as
well as a discussion of the relation to how ACL2 stores induction schemes.
On very rare occasions ACL2 will seem to "hang" when processing a
definition, especially if there are many subexpressions of the body whose
function symbol is if (or which macroexpand to such an expression).
In those cases you may wish to supply the following to xargs:
:normalize nil. This is an advanced feature that turns off certain
simplification of definition bodies and guards; normalize.
When a defun form is submitted, ACL2 sometimes computes and stores a
type-prescription rule for the function. See type-prescription-debugging for relevant discussion.
The following example illustrates all of the available declarations and
most hint keywords, but is completely nonsensical. For documentation, see
xargs and see hints.
(defun example (x y z a b c i j)
(declare (ignore a b c)
(type integer i j)
(xargs :guard (symbolp x)
:measure (- i j)
:do-not '(generalize fertilize)
:expand ((assoc x a) (member y z))
:restrict ((<-trans ((x x) (y (foo x)))))
:hands-off (length binary-append)
:induct (and (nth n a) (nth n b))
:use ((:instance assoc-of-append
(x a) (y b) (z c))
(:instance p-f (x a) (y b))
:guard-hints (("Subgoal *1/3'"
:use ((:instance assoc-of-append
(x a) (y b) (z c)))))
:type-prescription (natp (example x y z a b c i j))
(example-body x y z i j))
- Extra arguments, for example to give hints to defun
- Determines whether a function definition is a logical act
- Define some mutually recursive functions
- Define a potentially inlined function symbol and associated macro
- Control for ACL2's termination and induction analyses
- Define a non-executable function symbol
- Define a function symbol and then disable it
- Allow unused formals and locals without an ignore or
- Set the default well-founded relation
- Set the default measure function symbol
- Allow irrelevant formals in definitions
- Define a not-to-be-inlined function symbol and associated macro
- Allow unnecessary (xargs :hints ...) and measures.
- Define a disabled non-executable function symbol
- Define a function symbol and generate a warrant
- Define a disabled, not-to-be-inlined function symbol and associated macro
- disabled definition with guard t
- Definition with guard t
- Define a potentially disabled, inlined function symbol and associated macro
- Allow unnecessary measures and (xargs :hints ...).