Major Section: THEORIES
Examples: (:rewrite assoc-of-app) (:linear delta-aref . 2) (:definition length) (:executable-counterpart length)
Background: The theorem prover is driven from a data base of rules.
The most common rules are
rewrite rules, which cause the simplifier
to replace one term with another. Events introduce rules into the
data base. For example, a
defun event may introduce as many as four
rules: one for symbolically replacing a function call by its
instantiated body, one for evaluating the function on constants, and
two for determining the type of a call of the function (one of which
is deduced before guard verification and the other of which is
deduced after guard verification).
Defthm may introduce several
rules, one for each of the
rule-classes specified (where one
rule class is specified if
:ilc[rule-classes] is omitted, namely,
Every rule in the system has a name. Each name is a structured
object called a ``rune,'' which is short for ``rule name''. Runes
are always of the form
(:token symbol . x), where
:token is some
keyword symbol indicating what kind of rule is named,
symbol is the
event name that created the rule (and is called the ``base symbol''
of the rune), and
x is either
nil or a natural number that makes the
rule name distinct from that of rules generated by other events or
rule-classes within the same event.
For example, an event of the form
(defthm name thm :rule-classes ((:REWRITE :COROLLARY term1) (:REWRITE :COROLLARY term2) (:ELIM :COROLLARY term3)))typically creates three rules, each with a unique rune. The runes are
(:REWRITE name . 1), (:REWRITE name . 2), and (:ELIM name).However, a given formula may create more than one rule, and all rules generated by the same
:corollaryformula will share the same rune. Consider the following example.
(defthm my-thm (and (equal (foo (bar x)) x) (equal (bar (foo x)) x)))This is treated identically to the following.
(defthm my-thm (and (equal (foo (bar x)) x) (equal (bar (foo x)) x)) :rule-classes ((:rewrite :corollary (and (equal (foo (bar x)) x) (equal (bar (foo x)) x)))))In either case, two rules are created: one rewriting
(foo (bar x))to
x, and one rewriting
(bar (foo x))to
x. However, only a single rune is created,
(:REWRITE MY-THM), because there is only one rule class. But now consider the following example.
(defthm my-thm2 (and (equal (foo (bar x)) x) (equal (bar (foo x)) x)) :rule-classes ((:rewrite :corollary (and (equal (foo (bar x)) x) (equal (bar (foo x)) x))) (:rewrite :corollary (and (equal (foo (bar (foo x))) (foo x)) (equal (bar (foo (bar x))) (bar x))))))This time there are four rules created. The first two rules are as before, and are assigned the rune
(:REWRITE MY-THM . 1). The other two rules are similarly generated for the second
:corollary, and are assigned the rune
(:REWRITE MY-THM . 2).
corollary will return the corollary term associated
with a given rune in a given world. Example:
(corollary '(:TYPE-PRESCRIPTION DIGIT-TO-CHAR) (w state))However, the preferred way to see the corollary term associated with a rune or a name is to use
:pf; see pf.
defun event creates as many as four rules.
(:definition fn) is
the rune given to the equality axiom defining the function,
(:executable-counterpart fn) is the rune given to the rule for computing
fn on known arguments. A type prescription rule may be created under the
(:type-prescription fn), and an induction rule may be created under
Runes may be individually enabled and disabled, according to whether
they are included in the current theory. See theories. Thus,
it is permitted to disable
(:elim name), say, while enabling the
other rules derived from name. Similarly,
(:definition fn) may be
(:executable-counterpart fn) and the type
fn are enabled.
Associated with most runes is the formula justifying the rule named. This is
called the ``corollary formula'' of the rune and may be obtained via the
corollary, which takes as its argument a rune and a property
list world. Also see pf. The corollary formula for
(:rewrite name . 1)
defthm event above is
term1. The corollary formulas for
(:definition fn) and
(:executable-counterpart fn) are always
identical: the defining axiom. Some runes, e.g.,
(:definition car), do
not have corollary formulas.
nil on such runes.
In any case, the corollary formula of a rune, when it is non-
nil, is a
theorem and may be used in the
Note: The system has many built in rules that, for regularity, ought
to have names but don't because they can never be disabled. One
such rule is that implemented by the linear arithmetic package.
Because many of our subroutines are required by their calling
conventions to return the justifying rune, we have invented the
notion of ``fake runes.'' Fake runes always have the base symbol
nil, use a keyword token that includes the phrase ``fake-rune'', and
are always enabled. For example,
(:fake-rune-for-linear nil) is a
fake rune. Occasionally the system will print a fake rune where a
rune is expected. For example, when the linear arithmetic fake rune
is reported among the rules used in a proof, it is an indication
that the linear arithmetic package was used. However, fake runes
are not allowed in theories, they cannot be enabled or disabled, and
they do not have associated corollary formulas. In short, despite
the fact that the user may sometimes see fake runes printed, they
should never be typed.