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) 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.

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).''