Proof with try tauto.
intros.
apply height_zero_eq in H0.
destruct H0;subst.
rewrite decompose_top in H. inv H...
rewrite decompose_bot in H. inv H...
Qed.

Lemma decompose_equal: forall a b aL aR bL bR,
decompose a = (aL,aR) ->
decompose b = (bL,bR) ->
(a = b <-> aL = bL /\ aR = bR).