induction s; intros; auto.
simpl fold'.
remember (fold' (list key) (@cons key)) as g.
rewrite <- Heqg.
rewrite <- IHs2.
f_equal. simpl.
f_equal.
apply IHs1.
Qed.

Lemma tree_similar_lemma':
forall c c' k k' s s' lo hi lo' hi',
searchtree lo hi (T c EE k s) ->
searchtree lo' hi' (T c' EE k' s') ->
(forall x : key,
interp (T c EE k s) x <-> interp (T c' EE k' s') x) ->
K.eq k k' /\ (forall x, interp s x -> interp s' x).