  Faugere's F5 algorithm is one of the fastest known algorithms for the
computation of Grobner bases. So far only the F5 Criterion is proved, whereas
the second powerful criterion, the Rewritten Criterion, is not understood very
well until now. We give a proof of both, the F5 Criterion and the Rewritten
Criterion showing their connection to syzygies, i.e. the relations between the
S-Polynomials to be investigated by the algorithm. Using the example of a
Grobner basis computation stated in Faugere's F5 paper we show how the criteria
work, and discuss the possibility of improving the F5 Criterion. An
introduction to a SINGULAR implementation of F5 is given in the end.
