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`

. Normally
ACL2 enforces the requirement that every variable `vi`

must be bound in the
formula denoted by `lmi`

. However, the keyword `:extra-bindings-ok`

may
be inserted immediately after the lemma instance in order to remove that
requirement: `(:instance lmi :extra-bindings-ok (v1 t1) ... (vn tn))`

.

(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, a macro alias for a
function symbol `gi'`

(see macro-aliases-table) in which case we treat
`gi`

as `gi'`

, 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. (Special case: a function
introduced in the `:partial-theory`

of a dependent clause-processor is not
instantiable; see define-trusted-clause-processor.) 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) in a way that inserts `v`

into the scope
of a binding of `v`

by `let`

or `mv-let`

(or, `lambda`

). If
you happen to violate this restriction, an informative error message will be
printed. That message will list for you the potentially 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 several 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.

Note that ACL2(r) (see real) imposes additional requirements for functional instantiation. See functional-instantiation-in-acl2r.

Obscure case for definitions. If the lemma instance refers to a
`:definition`

rune, then it refers to the `corollary`

formula of
that rune, which can be a simplified (``normalized'') form of the original
formula. However, if the hint is a `:by`

hint and the lemma instance is
based on a name (i.e., a symbol), rather than a rune, then the formula is the
original formula of the event, as shown by `:`

`pe`

, rather than the
normalized version, as shown by `:`

`pf`

. This is as one would expect:
If you supply the name of an event, you expect it to refer to the original
event. For `:use`

hints we use the simplified (normalized) form instead,
which is reasonable since one would expect simplification during the proof
that re-traces the normalization done at the time the rule was created.

See functional-instantiation-example for an example of the use
of `:functional-instance`

(so-called ``functional instantiation).''