  We show that the types of the witnesses in the Herbrand functional
interpretation can be simplified, avoiding the use of "sets of functionals" in
the interpretation of implication and universal quantification. This is done by
presenting an alternative formulation of the Herbrand functional
interpretation, which we show to be equivalent to the original presentation. As
a result of this investigation we also strengthen the monotonicity property of
the original presentation, and prove a monotonicity property for our
alternative definition.
