Proof using.
intros a1 l l1; generalize a1 l; elim l1; simpl in |- *; auto; clear a1 l l1.
intros a1 l H H0; elim H0; auto.
intros a; case a.
intros l H a1 l0; case l0.
intros H0; elim H0; auto.
intros b; case b; simpl in |- *; auto.
intros l1 H0 H1; apply pbfree_right2.
apply H; auto.
intros l H a1 l0; case l0; simpl in |- *; auto.
intros H0; elim H0; auto.
intros b; case b; simpl in |- *; auto.
intros l1 H0 H1; apply pbfree_left2.
apply H; auto.
Qed.

(* Specifications: *)
(* The free positions now include the ones of the added tree *)
Theorem pbfree_pbadd_prop2 :
forall a l1 l2 l3,
pbfree l1 l3 ->
~ is_prefix l2 l1 -> ~ is_prefix l1 l2 -> pbfree l1 (pbadd a l3 l2).