  We study constructively the relations between the finite cases of Dickson's
lemma. Although there are many constructive proofs of them, the novel aspect of
our proofs is the extraction of a corresponding bound. We provide some new
one-step unprovability results i.e., results of the form "a finite case of
Dickson's lemma does not prove in one step a stronger case of it". Moreover, we
study the infinite cases of Dickson's lemma from the point of view of
constructive reverse mathematics. We work within Bishop's informal system of
constructive mathematics BISH.
