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<html>
<head><title>DEFUN.html -- ACL2 Version 3.1</title></head>
<body text=#000000 bgcolor="#FFFFFF">
<h2>DEFUN</h2>define a function symbol
<pre>Major Section: <a href="EVENTS.html">EVENTS</a>
</pre><p>
<pre>
Examples:
(defun app (x y)
(if (consp x)
(cons (car x) (app (cdr x) y))
y))<p>
(defun fact (n)
(declare (xargs :guard (and (integerp n)
(>= n 0))))
(if (zp n)
1
(* n (fact (1- n)))))
<p>
General Form:
(defun fn (var1 ... varn) doc-string dcl ... dcl body),
</pre>
where <code>fn</code> is the symbol you wish to define and is a new symbolic
name (see <a href="NAME.html">name</a>), <code>(var1 ... varn)</code> is its list of formal
parameters (see <a href="NAME.html">name</a>), and <code>body</code> is its body. The definitional
axiom is logically admissible provided certain restrictions are met.
These are sketched below.<p>
Note that ACL2 does not support the use of <code>lambda-list</code> keywords
(such as <code>&optional</code>) 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 <a href="DEFMACRO.html">defmacro</a>. The <a href="DOCUMENTATION.html">documentation</a> string,
<code><a href="DOC-STRING.html">doc-string</a></code>, is optional; for a description of its form,
see <a href="DOC-STRING.html">doc-string</a>.<p>
The <em>declarations</em> (see <a href="DECLARE.html">declare</a>), <code>dcl</code>, are also optional.
If more than one <code>dcl</code> form appears, they are effectively grouped together
as one. Perhaps the most commonly used ACL2 specific declaration is
of the form <code>(declare (xargs :guard g :measure m))</code>. This declaration
in the <code>defun</code> of some function <code>fn</code> has the effect of making the
``<a href="GUARD.html">guard</a>'' for <code>fn</code> be the term <code>g</code> and the ``measure'' be the term <code>m</code>.
The notion of ``measure'' is crucial to ACL2's definitional
principle. The notion of ``guard'' is not, and is discussed elsewhere;
see <a href="VERIFY-GUARDS.html">verify-guards</a> and see <a href="SET-VERIFY-GUARDS-EAGERNESS.html">set-verify-guards-eagerness</a>. Note that the
<code>:measure</code> is ignored in <code>:</code><code><a href="PROGRAM.html">program</a></code> mode; see <a href="DEFUN-MODE.html">defun-mode</a>.<p>
We now briefly discuss the ACL2 definitional principle, using the
following definition form which is offered as a more or less generic
example.
<pre>
(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))))
</pre>
Note that in our generic example, <code>fn</code> has just two arguments, <code>x</code> and
<code>y</code>, the <a href="GUARD.html">guard</a> and measure terms involve both of them, and the body is
a simple case split on <code>(test x y)</code> leading to a ``non-recursive''
branch, <code>(stop x y)</code>, and a ``recursive'' branch. In the recursive
branch, <code>fn</code> is called after ``decrementing'' <code>x</code> to <code>(d x)</code> 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.<p>
Provided this definition is admissible under the logic, as outlined
below, it adds the following axiom to the logic.
<pre>
Defining Axiom:
(fn x y)
=
(if (test x y)
(stop x y)
(step (fn (d x) y)))
</pre>
Note that the <a href="GUARD.html">guard</a> of <code>fn</code> has no bearing on this logical axiom.<p>
This defining axiom is actually implemented in the ACL2 system by a
<code>:</code><code><a href="DEFINITION.html">definition</a></code> rule, namely
<pre>
(equal (fn x y)
(if (test a b)
(stop a b)
(step (fn (d a) b)))).
</pre>
See <a href="DEFINITION.html">definition</a> for a discussion of how definition rules are
applied. Roughly speaking, the rule causes certain instances of
<code>(fn x y)</code> to be replaced by the corresponding instances of the
body above. This is called ``opening up'' <code>(fn x y)</code>. The
instances of <code>(fn x y)</code> opened are chosen primarily by heuristics
which determine that the recursive calls of <code>fn</code> in the opened body
(after simplification) are more desirable than the unopened call of
<code>fn</code>.<p>
This discussion has assumed that the definition of <code>fn</code> was
admissible. Exactly what does that mean? First, <code>fn</code> must be a
previously unaxiomatized function symbol (however,
see <a href="LD-REDEFINITION-ACTION.html">ld-redefinition-action</a>). Second, the formal parameters
must be distinct variable names. Third, the <a href="GUARD.html">guard</a>, 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.<p>
The final conditions on admissibility concern the termination of the
recursion. Roughly put, all applications of <code>fn</code> must terminate.
In particular, there must exist a binary relation, <code>rel</code>, and some
unary predicate <code>mp</code> such that <code>rel</code> is well-founded on objects
satisfying <code>mp</code>, the measure term <code>m</code> must always produce
something satisfying <code>mp</code>, and the measure term must decrease
according to <code>rel</code> in each recursive call, under the hypothesis
that all the tests governing the call are satisfied. By the meaning
of well-foundedness, we know there are no infinitely descending
chains of successively <code>rel</code>-smaller <code>mp</code>-objects. Thus, the
recursion must terminate.<p>
The only primitive well-founded relation in ACL2 is <code><a href="O_lt_.html">o<</a></code>
(see <a href="O_lt_.html">o<</a>), which is known to be well-founded on the
<code><a href="O-P.html">o-p</a></code>s (see <a href="O-P.html">o-p</a>). For the proof of
well-foundedness, see <a href="PROOF-OF-WELL-FOUNDEDNESS.html">proof-of-well-foundedness</a>. However it is
possible to add new well-founded relations. For details,
see <a href="WELL-FOUNDED-RELATION.html">well-founded-relation</a>. We discuss later how to specify
which well-founded relation is selected by <code>defun</code> and in the
present discussion we assume, without loss of generality, that it is
<code><a href="O_lt_.html">o<</a></code> on the <code><a href="O-P.html">o-p</a></code>s.<p>
For example, for our generic definition of <code>fn</code> above, with measure
term <code>(m x y)</code>, two theorems must be proved. The first establishes
that <code>m</code> produces an ordinal:
<pre>
(o-p (m x y)).
</pre>
The second shows that <code>m</code> decreases in the (only) recursive call of
<code>fn</code>:
<pre>
(implies (not (test x y))
(o< (m (d x) y) (m x y))).
</pre>
Observe that in the latter formula we must show that the
``<code>m</code>-size'' of <code>(d x)</code> and <code>y</code> is ``smaller than'' the <code>m</code>-size of <code>x</code> and <code>y</code>,
provided the test, <code>(test x y)</code>, in the body fails, thus leading to
the recursive call <code>(fn (d x) y)</code>.<p>
See <a href="O_lt_.html">o<</a> 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, <code><a href="O_lt_.html">o<</a></code> is just
the familiar ``less than'' relation (<code><a href="_lt_.html"><</a></code>). Thus, it is very common
to use a measure <code>m</code> that returns a nonnegative integer, for then
<code>(o-p (m x y))</code> becomes a simple conjecture about the type of
<code>m</code> and the second formula above becomes a conjecture about the less-than
relationship of nonnegative integer arithmetic.<p>
The most commonly used measure function is <code><a href="ACL2-COUNT.html">acl2-count</a></code>, which
computes a nonnegative integer size for all ACL2 objects.
See <a href="ACL2-COUNT.html">acl2-count</a>.<p>
Probably the most common recursive scheme in Lisp <a href="PROGRAMMING.html">programming</a> is
when some formal is supposed to be a list and in the recursive call
it is replaced by its <code><a href="CDR.html">cdr</a></code>. For example, <code>(test x y)</code> might be simply
<code>(atom x)</code> and <code>(d x)</code> might be <code>(cdr x)</code>. In that case, <code>(acl2-count x)</code>
is a suitable measure because the <code><a href="ACL2-COUNT.html">acl2-count</a></code> of a <code><a href="CONS.html">cons</a></code> is strictly
larger than the <code><a href="ACL2-COUNT.html">acl2-count</a></code>s of its <code><a href="CAR.html">car</a></code> and <code><a href="CDR.html">cdr</a></code>. Thus, ``recursion
by <code><a href="CAR.html">car</a></code>'' and ``recursion by <code><a href="CDR.html">cdr</a></code>'' are trivially admitted if
<code><a href="ACL2-COUNT.html">acl2-count</a></code> is used as the measure and the definition protects every
recursive call by a test insuring that the decremented argument is a
<code><a href="CONSP.html">consp</a></code>. Similarly, ``recursion by <code><a href="1-.html">1-</a></code>'' in which a positive integer
formal is decremented by one in recursion, is also trivially
admissible. See <a href="BUILT-IN-CLAUSES.html">built-in-clauses</a> to extend the class of
trivially admissible recursive schemes.<p>
We now turn to the question of which well-founded relation <code>defun</code>
uses. It should first be observed that <code>defun</code> must actually select
both a relation (e.g., <code><a href="O_lt_.html">o<</a></code>) and a domain predicate (e.g.,
<code><a href="O-P.html">o-p</a></code>) on which that relation is known to be well-founded.
But, as noted elsewhere (see <a href="WELL-FOUNDED-RELATION.html">well-founded-relation</a>), every
known well-founded relation has a unique domain predicate associated
with it and so it suffices to identify simply the relation here.<p>
The <code><a href="XARGS.html">xargs</a></code> field of a <code><a href="DECLARE.html">declare</a></code> permits the explicit specification of
any known well-founded relation with the keyword
<code>:</code><code><a href="WELL-FOUNDED-RELATION.html">well-founded-relation</a></code>. An example is given below. If the <code><a href="XARGS.html">xargs</a></code>
for a <code>defun</code> specifies a well-founded relation, that relation and its
associated domain predicate are used in generating the termination
conditions for the definition.<p>
If no <code>:</code><code><a href="WELL-FOUNDED-RELATION.html">well-founded-relation</a></code> is specified, <code>defun</code> uses the
<code>:</code><code><a href="WELL-FOUNDED-RELATION.html">well-founded-relation</a></code> specified in the <code><a href="ACL2-DEFAULTS-TABLE.html">acl2-defaults-table</a></code>.
See <a href="SET-WELL-FOUNDED-RELATION.html">set-well-founded-relation</a> to see how to set the default
well-founded relation (and, implicitly, its domain predicate). The
initial default well-founded relation is <code><a href="O_lt_.html">o<</a></code> (with domain
predicate <code><a href="O-P.html">o-p</a></code>).<p>
This completes the brief sketch of the ACL2 definitional principle.<p>
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 <code><a href="IF.html">if</a></code> (or which macroexpand to such an
expression). In those cases you may wish to supply the following to
<code><a href="XARGS.html">xargs</a></code>: <code>:normalize nil</code>. This is an advanced feature that turns
off ACL2's usual propagation upward of <code>if</code> tests.<p>
The following example illustrates all of the available declarations,
but is completely nonsensical. For documentation, see <a href="XARGS.html">xargs</a> and
see <a href="HINTS.html">hints</a>.
<pre>
(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)
:well-founded-relation my-wfr
:hints (("Goal"
:do-not-induct t
:do-not '(generalize fertilize)
:expand ((assoc x a) (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))
:non-executable t
:use ((:instance assoc-of-append
(x a) (y b) (z c))
(:functional-instance
(:instance p-f (x a) (y b))
(p consp)
(f assoc)))))
:guard-hints (("Subgoal *1/3'"
:use ((:instance assoc-of-append
(x a) (y b) (z c)))))
:mode :logic
:normalize nil
:otf-flg t))
(example-body x y z i j))
</pre>
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