Major Section: MISCELLANEOUS
We describe the procedure very roughly here.
Fundamental to the procedure is the notion of a linear polynomial
inequality. A ``linear polynomial'' is a sum of terms, each of
which is the product of a rational constant and an ``unknown.'' The
``unknown'' is permitted to be
1 simply to allow a term in the sum
to be constant. Thus, an example linear polynomial is
3*x + 7*a + 2; here
a are the (interesting) unknowns.
However, the unknowns need not be variable symbols. For
(length x) might be used as an unknown in a linear
polynomial. Thus, another linear polynomial is
3*(length x) + 7*a.
A ``linear polynomial inequality'' is an inequality
relation between two linear polynomials. Note that an equality may
be considered as a pair of inequalities; e.q.,
3*x + 7*a + 2 = 0
is the same as the conjunction of
3*x + 7*a + 2 <= 0 and
0 <= 3*x + 7*a + 2.
Certain linear polynomial inequalities can be combined by cross-multiplication and addition to permit the deduction of a third inequality with fewer unknowns. If this deduced inequality is manifestly false, a contradiction has been deduced from the assumed inequalities.
For example, suppose we have two assumptions
p1: 3*x + 7*a < 4 p2: 3 < 2*xand we wish to prove that, given
a < 0. As suggested above, we proceed by assuming the negation of our goal
p3: 0 <= a.and looking for a contradiction.
By cross-multiplying and adding the first two inequalities, (that is,
3 and adding the respective
sides), we deduce the intermediate result
p4: 6*x + 14*a + 9 < 8 + 6*xwhich, after cancellation, is:
p4: 14*a + 1 < 0.If we then cross-multiply and add
p4, we get
p5: 1 <= 0,a contradiction. Thus, we have proved that
p2imply the negation of
All of the unknowns of an inequality must be eliminated by
cancellation in order to produce a constant inequality. We can
choose to eliminate the unknowns in any order, but we eliminate them in
term-order, largest unknowns first. (See term-order.) That is, two
polys are cancelled against each other only when they have the same
largest unknown. For instance, in the above example we see that
is the largest unknown in each of
Now suppose that this procedure does not produce a contradiction but
instead yields a set of nontrivial inequalities. A contradiction
might still be deduced if we could add to the set some additional
inequalities allowing further cancellations. That is where
:linear lemmas come in. When the set of inequalities has stabilized
under cross-multiplication and addition and no contradiction is
produced, we search the data base of
linear rules for rules about
the unknowns that are candidates for cancellation (i.e., are the
largest unknowns in their respective inequalities). See linear
for a description of how
linear rules are used.
non-linear-arithmetic for a description of an extension
to the linear-arithmetic procedure described here.