Major Section: MISCELLANEOUS
Lemma instances are the objects one provides via :use and :by hints
(see hints) to bring to the theorem prover's attention some
previously proved or easily provable fact. A typical use of the
:use hint is given below. The value specified is a list of five
lemma instances.
:use (reverse-reverse
(:type-prescription app)
(:instance assoc-of-app
(x a) (y b) (z c))
(:functional-instance p-f
(p consp) (f flatten))
(:instance (:theorem (equal x x))
(x (flatten a))))
Observe that an event name can be a lemma instance. The :use hint
allows a single lemma instance to be provided in lieu of a list, as
in:
:use reverse-reverseor
:use (:instance assoc-of-app (x a) (y b) (z c))
A lemma instance denotes a formula which is either known to be a theorem or which must be proved to be a theorem before it can be used. To use a lemma instance in a particular subgoal, the theorem prover adds the formula as a hypothesis to the subgoal before the normal theorem proving heuristics are applied.
A lemma instance, or lmi, is of one of the following five forms:
(1) name, where name names a previously proved theorem, axiom, or
definition and denotes the formula (theorem) of that name.
(2) rune, where rune is a rune (see rune) denoting the
:corollary justifying the rule named by the rune.
(3) (:theorem term), where term is any term alleged to be a theorem.
Such a lemma instance denotes the formula term. But before using
such a lemma instance the system will undertake to prove term.
(4) (:instance lmi (v1 t1) ... (vn tn)), where lmi is recursively a
lemma instance, the vi's are distinct variables and the ti's are
terms. Such a lemma instance denotes the formula obtained by
instantiating the formula denoted by lmi, replacing each vi by ti.
(5) (:functional-instance lmi (f1 g1) ... (fn gn)), where lmi
is recursively a lemma instance and each fi is an
``instantiable'' function symbol of arity ni and gi is a
function symbol or a pseudo-lambda expression of arity ni. An
instantiable function symbol is any defined or constrained function
symbol except the primitives not, member, implies, and
o<, and a few others, as listed by the constant
*non-instantiable-primitives*. These are built-in in such a way
that we cannot recover the constraints on them. A pseudo-lambda
expression is an expression of the form (lambda (v1 ... vn) body)
where the vi are distinct variable symbols and body is any
term. No a priori relation is imposed between the vi and the
variables of body, i.e., body may ignore some vi's and may
contain ``free'' variables. However, we do not permit v to occur
freely in body if the functional substitution is to be applied to
any formula (lmi or the constraints to be satisfied) that
contains v as a variable. This is our draconian restriction to
avoid capture. If you happen to violate this restriction by
choosing a v that does occur, say in one of the relevant
constraints, an informative error message will be printed. That
message will list for you the illegal choices for v in the
context in which the functional substitution is offered. A
:functional-instance lemma instance denotes the formula obtained by
functionally instantiating the formula denoted by lmi, replacing fi
by gi. However, before such a lemma instance can be used, the system
will generate proof obligations arising from the replacement of the fi's
by the gi's in constraints that ``support'' the lemma to be functionally
instantiated; see constraint. One might expect that if the same
instantiated constraint were generated on behalf of multiple events, then
each of those instances would have to be proved. However, for the sake of
efficiency, ACL2 stores the fact that such an instantiated constraint has
been proved and avoids it in future events.
See functional-instantiation-example for an example of the use
of :functional-instance (so-called ``functional instantiation).''