A strategy for a rewriting system is a sub-rewriting system with the same objects and same normal forms.
Definition (from Terese "Term Rewriting Systems"). Let N be a superset of the normal forms of a rewriting system $\to$. A strategy S for $\to$ is normalizing if the restriction of S (as a relation) to steps which do not have their sources in N is terminating.
I understand that the authors want a more general notion of normalizing such as the strategy which finds a (weak) head normal form. However, take N to be be exactly the set of normal forms. Then the definition should express that the strategy always finds a normal form provided one exists. I am having trouble bridging the gap here... It seems that N should have to also include the terms which do not have a normal form.