`:rewrite`

rules (possibly conditional ones)
Major Section: RULE-CLASSES

This doc topic discusses the rule-class `:rewrite`

. If you want a general
discussion of how rewriting works in ACL2 and some guidance on how to
construct effective rewrite rules, see introduction-to-rewrite-rules-part-1
and then see introduction-to-rewrite-rules-part-2.

See rule-classes for a general discussion of rule classes and
how they are used to build rules from formulas. Example `:`

`corollary`

formulas from which `:rewrite`

rules might be built are:

Example: (equal (+ x y) (+ y x)) replace (+ a b) by (+ b a) provided certain heuristics approve the permutation.(implies (true-listp x) replace (append a nil) by a, if (equal (append x nil) x)) (true-listp a) rewrites to t

(implies replace (member a (append b c)) by (and (eqlablep e) (member a (append c b)) in contexts (true-listp x) in which propositional equivalence (true-listp y)) is sufficient, provided (eqlablep a) (iff (member e (append x y)) (true-listp b) and (true-listp c) (member e (append y x)))) rewrite to t and the permutative heuristics approve

## Some Related Topics

### BIND-FREE -- to bind free variables of a rewrite or linear rule

### CASE-SPLIT -- like force but immediately splits the top-level goal on the hypothesis

### DOUBLE-REWRITE -- cause a term to be rewritten twice

### FORCE -- identity function used to force a hypothesis

### FREE-VARIABLES -- free variables in rules

### SYNTAXP -- attach a heuristic filter on a

`:`

`rewrite`

,`:`

`meta`

, or`:`

`linear`

rule

`:rewrite`

rule class object might create many rewrite
rules from the `:`

`corollary`

formula. To create the rules, we first
translate the formula (expanding all macros; also see trans).
Next, we eliminate all `lambda`

s; one may think of this step as
simply substituting away every `let`

, `let*`

, and `mv-let`

in the
formula. We then flatten the `AND`

and `IMPLIES`

structure of the
formula; for example, if the hypothesis or conclusion is of the form
`(and (and term1 term2) term3)`

, then we replace that by the ``flat'' term
`(and term1 term2 term3)`

. (The latter is actually an abbreviation for the
right-associated term `(and term1 (and term2 term3))`

.) The result is a
conjunction of formulas, each of the form
(implies (and h1 ... hn) concl)where no hypothesis is a conjunction and

`concl`

is neither a conjunction
nor an implication. If necessary, the hypothesis of such a conjunct may be
vacuous. We then further coerce each `concl`

into the form
`(equiv lhs rhs)`

, where `equiv`

is a known equivalence relation, by
replacing any `concl`

not of that form by `(iff concl t)`

. A `concl`

of the form `(not term)`

is considered to be of the form
`(iff term nil)`

. By these steps we reduce the given `:`

`corollary`

to a sequence of conjuncts, each of which is of the form
(implies (and h1 ... hn) (equiv lhs rhs))where

`equiv`

is a known equivalence relation. See equivalence for a
general discussion of the introduction of new equivalence relations.
We create a `:rewrite`

rule for each such conjunct, if possible, and
otherwise cause an error. It is possible to create a rewrite rule from such
a conjunct provided `lhs`

is not a variable, a quoted constant, a
`let`

-expression, a `lambda`

application, or an `if`

-expression.

A `:rewrite`

rule is used when any instance of the `lhs`

occurs in a
context in which the equivalence relation is an admissible congruence
relation. First, we find a substitution that makes `lhs`

equal to the
target term. Then we attempt to relieve the instantiated hypotheses of the
rule. Hypotheses that are fully instantiated are relieved by recursive
rewriting. Hypotheses that contain ``free variables'' (variables not
assigned by the unifying substitution) are relieved by attempting to guess a
suitable instance so as to make the hypothesis equal to some known assumption
in the context of the target. If the hypotheses are relieved, and certain
restrictions that prevent some forms of infinite regress are
met (see loop-stopper), the target is replaced by the instantiated `rhs`

,
which is then recursively rewritten.

ACL2's rewriting process has undergone some optimization. In particular,
when a term `t1`

is rewritten to a new term `t2`

, the rewriter is then
immediately applied to `t2`

. On rare occasions you may find that you do
not want this behavior, in which case you may wish to use a trick involving
`hide`

; see meta, near the end of that documentation.

In another optimization, when the hypotheses and right-hand side are rewritten,
ACL2 does not really first apply the substitution and then rewrite; instead, it
as it rewrites those terms it looks up the *already rewritten* values of the
bound variables. Sometimes you may want those bindings rewritten again, e.g.,
because the variables occur in slots that admit additional equivalence
relations. See *double-rewrite*.

See introduction-to-rewrite-rules-part-1 and
see introduction-to-rewrite-rules-part-2 for an extended discussion of how
to create effective rewrite rules.