Ltac omega_bal := match goal with
  | H:avl ?l, H':avl ?r |- context [ bal ?l ?x ?e ?r ] =>
     generalize (bal_height_1 x e H H') (bal_height_2 x e H H');
     omega_max
  end.

Lemma remove_avl_1 : forall m x, avl m -> avl (remove x m) /\ 0 <= height m - height (remove x m) <= 1.