The justification of the use of Remove Redundant requires the following proposition.

**Propostion 12.14***
Let
= (V,E) be a Gröbner graph, V be a finite set,
and X be the set which is generated by RemoveRedundant when it is
applied to G and
.
The ideal generated by
equals the ideal generated by X.
*

Proof.
For each , let be defined as
in Lemma 12.13. Since ,
. To show that
, it suffices to show
that for .
Let . Since ,
is not the empty set. Let
be as in Lemma 12.13.
is empty for each starting vertex *w*
of .
By Proposition
12.8,
*v* lies in the ideal generated by the starting
vertices of and so *v* lies in the ideal
generated by *X*.
This completes the proof of Proposition 12.14.

RemoveRedundant can be a very fast operation since it requires only the searching of graphs (which is fast in comparison to GBA which is what the previous two idealized operations used).

Wed Jul 3 10:27:42 PDT 1996