Proof using.
intros l; elim l; simpl in |- *; auto.
intros H; case H; auto.
intros a l0; case l0; auto.
intros H H0; exists a; auto.
intros b l1 H H0; case H.
red in |- *; intros; discriminate.
intros x H1; exists (node a x).
apply cover_cons_l; auto.
Qed.

(* Subtrees of the cover are subtrees of the tree *)
Theorem cover_in_inb_inb :
forall l t1 t2 t3, cover l t1 -> In t2 l -> inb t3 t2 -> inb t3 t1.