Fixpoint rgopher (s: tree) : tree :=
 match s with
 | EE => EE
 | T _ sl k sr => match rgopher sr with
                          | T _ srl j srr => T Red  (T Red sl k srl) j srr
                          | EE => s
                          end
  end.

Lemma searchtree_rgopher: forall a lo hi, searchtree lo hi a <-> searchtree lo hi (rgopher a).