Proof using.
intros t l1 l2 H; generalize l2; elim H; clear H t l1 l2; auto.
intros t l2 H; rewrite (permutation_one_inv _ _ _ H); auto.
intros l1 l2 t1 t2 t3 H H0 H1 l0 H2.
apply cover_node with (2 := H0).
apply permutation_trans with (2 := H).
apply permutation_sym; auto.
Qed.

(* A trivial way to cover a node *)
Theorem cover_cons_l :
forall t1 t2 l1, cover l1 t1 -> cover (t2 :: l1) (node t2 t1).