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).