Major Section: MISCELLANEOUS

Before ACL2 can apply a rule with hypotheses, it must establish that
the hypotheses are true. (We ignore the relaxing of this
requirement afforded by `case-split`

s and `force`

ed hypotheses.) ACL2
typically establishes each hypothesis by backchaining --
instantiating the hypothesis and then rewriting it recursively.
Here we describe how ACL2 allows the user to limit backchaining.

Each hypothesis of a `rewrite`

or `linear`

rule is assigned a
backchain-limit when the rule is stored. By default, this limit is
`nil`

, denoting infinity (no limit). However, the value used for
the default may be set to a non-negative integer (or to `nil`

) by
the user; see set-default-backchain-limit. The default is
overridden when a `:backchain-limit`

is supplied explicitly with
the rule; see rule-classes. The number of recursive
applications of backchaining starting with the hypothesis of a rule
is limited to the backchain-limit associated with that hypothesis.

Moreover, the user may set a global backchain-limit that limits the
total backchaining depth. See set-backchain-limit. Below we lay
out the precise sense in which this global backchain-limit interacts
with the backchain-limits of individual rules in order to limit
backchaining. But first we note that when further backchaining is
disallowed, ACL2 can still prove a hypothesis in a given context by
using that contextual information. This process is sometimes called
``type-set reasoning,'' an appropriate term since rules of class
`:`

`type-prescription`

may be used. So, for example, the
relieving of hypotheses may be limited to the use of contextual
information (without backchaining, i.e., without recursively
rewriting hypotheses) by executing `:set-backchain-limit 0`

.

Recall that there are two sorts of backchain limits: those applied
to hypotheses of individual rules, as assigned by their
`:`

`rule-classes`

or else taken from the default
(see set-default-backchain-limit); and the global limit, initially
`nil`

(no limit) but settable with `:`

`set-backchain-limit`

.
Here is how these two types of limits interact to limit backchaining,
i.e., recursive rewriting of hypotheses. ACL2 maintains a current
backchain limit, which is the limit on the depth of recursive calls
to the rewriter, as well as a current backchain depth, which is
initially 0 and is incremented each time ACL2 backchains (and is
decremented when a backchain completes). When ACL2 begins to
rewrite a literal (crudely, one of the ``top-level'' terms of the
goal currently being worked on), it sets the current backchain-limit
to the global value, which is initially `nil`

but can be set using
`:`

`set-backchain-limit`

. When ACL2 is preparing to relieve a
hypothesis by backchaining (hence, after it has already tried
type-set reasoning), it first makes sure that the current backchain
limit is greater than the current backchain depth. If not, then it
refuses to relieve that hypothesis. Otherwise, it increments the
current backchain depth and calculates a new current backchain-limit
by taking the minimum of two values: the existing current
backchain-limit, and the sum of the current backchain depth and the
the backchain-limit associated with the hypothesis. Thus, ACL2 only
modifies the current backchain-limit if it is necessary to decrease
that limit in order to respect the backchain limit associated with
the hypothesis.

We illustrate with the following examples.

; We stub out some functions so that we can reason about them.(defstub p0 (x) t) (defstub p1 (x) t) (defstub p2 (x) t) (defstub p3 (x) t)

; Initially, the default-backchain-limit is nil, or infinite.

(defaxiom p2-implies-p1-limitless (implies (p2 x) (p1 x)))

; The following rule will have a backchain-limit of 0.

(defaxiom p1-implies-p0-limit-0 (implies (p1 x) (p0 x)) :rule-classes ((:rewrite :backchain-limit 0)))

; We have (p2 x) ==> (p1 x) ==> (p0 x). We wish to establish that ; (p2 x) ==> (p0 x). Normally, this would be no problem, but here ; we fail because ACL2 cannot establish (p0 x) by type-set reasoning ; alone.

(thm (implies (p2 x) (p0 x)))

; We set the default-backchain-limit to 1.

:set-default-backchain-limit 1

; The following is more powerful than p1-implies-p0-limit-0 ; because it can use rewrite rules to establish (p1 x).

(defaxiom p1-implies-p0-limit-1 (implies (p1 x) (p0 x)))

; This theorem will succeed:

(thm (implies (p2 x) (p0 x)))

; We return the default-backchain-limit to its initial value.

:set-default-backchain-limit nil

; Here is our last axiom.

(defaxiom p3-implies-p2-limitless (implies (p3 x) (p2 x)))

; We now have (p3 x) ==> (p2 x) ==> (p1 x) ==> (p0 x). However the ; rule p1-implies-p0-limit-1 has a backchain-limit of 1; hence we ; are not allowed to backchain far enough back to use ; p3-implies-p2-limitless. We therefore lose.

(defthm will-fail (implies (p3 x) (p0 x)))