`NTH`

/`UPDATE-NTH`

expressions
Major Section: MISCELLANEOUS

The rewriter contains special provisions for rewriting expressions
composed of `nth`

, `update-nth`

, `update-nth-array`

, together
with `let`

expressions and other applications of non-recursive
functions or `lambda`

expressions. For details see the paper
``Rewriting for Symbolic Execution of State Machine Models'' by J
Strother Moore. Also see set-nu-rewriter-mode.

The ``nu-rewriter'' is a recent addition to the main rewrite engine in ACL2. Consider the expression

(let ((s (update-nth 1 (new-a x s) s))) (let ((s (update-nth 2 (new-b x s) s))) (let ((s (update-nth 3 (new-c x s) s))) s)))If the

`let`

s in this expression are expanded, a very large
expression results because of the duplicate occurrences of `s`

:
(update-nth 3 (new-c x (update-nth 2 (new-b x (update-nth 1 (new-a x s) s)) (update-nth 1 (new-a x s) s))) (update-nth 2 (new-b x (update-nth 1 (new-a x s) s)) (update-nth 1 (new-a x s) s))).This expansion of the

`let`

expression can be very expensive in space
and time. In particular, the `(new-a x s)`

expression might be
rewritten many times.Now imagine asking what 2nd component of the structure is. That is, consider

(nth 2 (let ((s (update-nth 1 (new-a x s) s))) (let ((s (update-nth 2 (new-b x s) s))) (let ((s (update-nth 3 (new-c x s) s))) s))))The normal ACL2 rewrite engine would answer this question by first rewriting the arguments to the

`nth`

expression; in particular, it would
expand the nested `let`

expression to the large nested `update-nth`

expression and then, using rules such as
(defthm nth-update-nth (equal (nth m (update-nth n val l)) (if (equal (nfix m) (nfix n)) val (nth m l))))would reduce the expression to

`(new-b x (update-nth 1 (new-a x s) s))`

.
The purpose of the nu-rewriter is to allow simplifications like this
without first expanding the `let`

s. The ``nu'' in the name is an
acronym for ```nth/update-nth`

''. The nu-rewriter knows how to
move an `nth`

into a `let`

without expanding the `let`

and how
to simplify it if it nestles up against an `update-nth`

.
There are four characteristics of this problem: the presence of
`nth`

, the presence of `update-nth`

, the use of `let`

to
provide ``sequential'' updates, and the use of constant indices.
`Nth`

and `update-nth`

need not occur explicitly; they may be
used inside of definitions of ``wrapper'' functions.

The because the nu-rewriter changes the order in which things are rewritten,
its routine use can make ACL2 unable to reproduce old proofs. It is
therefore switched off by default. If your application exhibits the
characteristics above, you might wish to switch the nu-rewriter on
using `set-nu-rewriter-mode`

.

More will eventually be written about the nu-rewriter. However, it
is described in detail in the paper cited at the beginning of this
documentation topic.