The following theorem further illustrates the usefulness of the notion of
dimension. An interesting alternative view of this theorem will be found at the
end of the section on anti-unification and most general solutions.
THEOREM 2
Suppose the Herbrand universe is infinite. Let £ be a set of equations of
dimension d and E
v
E
2
be sets of equations of dimension strictly less
than d. Then there is a solution of Ε which is not a solution of any E
r
Proof:
We can assume, without loss of generality, that Ε and the E
i
are in
solved form. Note that, since Ε has strictly greater dimension (and so strictly
lesser rank ...
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