\DOC AC

\TYPE {AC : thm -> term -> thm}

\SYNOPSIS
Proves equality of terms using associative, commutative, and optionally
idempotence laws.

\KEYWORDS
conversion, associative, commutative.

\DESCRIBE
Suppose {_} is a function, which is assumed to be infix in the following
syntax, and {acth} is a theorem expressing associativity and commutativity in
the particular canonical form:
{
   acth = |- m _ n = n _ m /\
             (m _ n) _ p = m _ n _ p /\
             m _ n _ p = n _ m _ p
}
\noindent Then {AC acth} will prove equations whose left and right sides can be
made identical using these associative and commutative laws. If the input
theorem also has idempotence property in this canonical form:
{
  |- (p _ q = q _ p) /\
     ((p _ q) _ r = p _ q _ r) /\
     (p _ q _ r = q _ p _ r) /\
     (p _ p = p) /\
     (p _ p _ q = p _ q)
}
then idempotence will also be applied.

\FAILURE
Fails if the terms are not proved equivalent under the appropriate laws. This
may happen because the input theorem does not have the correct canonical form.
The latter problem will not in itself cause failure until it is applied to the
term.

\EXAMPLE
{
  # AC ADD_AC `1 + 2 + 3 = 2 + 1 + 3`;;
  val it : thm = |- 1 + 2 + 3 = 2 + 1 + 3
  # AC CONJ_ACI `p /\ (q /\ p) <=> (p /\ q) /\ (p /\ q)`;;
  val it : thm = |- p /\ q /\ p <=> (p /\ q) /\ p /\ q
}

\COMMENTS
Note that pre-proved theorems in the correct canonical form for {AC} are
already present for many standard operators, e.g. {ADD_AC}, {MULT_AC},
{INT_ADD_AC}, {INT_MUL_AC}, {REAL_ADD_AC}, {REAL_MUL_AC}, {CONJ_ACI},
{DISJ_ACI} and {INSERT_AC}. The underlying algorithm is not particularly
delicate, and normalization under the associative/commutative/idempotent laws
can be achieved by direct rewriting with the same canonical theorems. For some
cases, specially optimized rules are available such as {CONJ_ACI_RULE} and
{DISJ_ACI_RULE}.

\SEEALSO
ASSOC_CONV, CONJ_ACI_RULE, DISJ_ACI_RULE, SYM_CONV.

\ENDDOC