Major Section: EVENTS
Examples:
(defun app (x y)
(if (consp x)
(cons (car x) (app (cdr x) y))
y))
(defun fact (n)
(declare (xargs :guard (and (integerp n)
(>= n 0))))
(if (zp n)
1
(* n (fact (1- n)))))
General Form:
(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 below.
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. The documentation string,
doc-string, is optional; for a description of its form,
see doc-string.
The declarations (see declare), dcl, are also optional.
If multiple dcl forms appear, 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.
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.
Defining Axiom:
(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 axiom.
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 governing the call are satisfied. 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 only primitive well-founded relation in ACL2 is o<
(see o<), which is known to be well-founded on the
o-ps (see o-p). 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
fn:
(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 arithmetic.
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-clauses 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), 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.
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 ACL2's usual propagation upward of if tests.
The following example illustrates all of the available declarations, 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)
: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))
