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\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 | |