Proof with try tauto;try omega.
intros. icase n.
apply tree_avg_avg2round in H.
destruct H.
destruct (le_dec (height s) (S n))...
rewrite tree_round_left_None in H...
inv H.
Qed.

(* Specifications: *)
Lemma Lsh_recompose: Lsh = recompose (top, bot).
Lemma Rsh_recompose: Rsh = recompose (bot,top).
Lemma decompose_Rsh: forall sh,
unrel Rsh sh = snd (decompose sh).
Lemma decompose_Lsh: forall sh,
unrel Lsh sh = fst (decompose sh).
Lemma rel_Lsh: forall sh,
rel Lsh sh = recompose (sh,bot).
Lemma rel_Rsh: forall sh,
rel Rsh sh = recompose (bot,sh).