intros t1 t2 l1 H; elim H; clear t1 l1 H; simpl in |- *; auto.
intros t; apply cover_node with (l2 := nil (A:=btree A)) (t1 := t2) (t2 := t);
auto.
intros l1 l2 t1 t0 t3 H H0 H1.
apply cover_node with (l2 := t2 :: l2) (t1 := t1) (t2 := t0); auto.
apply permutation_trans with (t2 :: t1 :: t0 :: l2); auto.
apply permutation_trans with (t1 :: t2 :: t0 :: l2); auto.
apply cover_permutation with (1 := H1); auto.
Qed.

(* A cover cannot be empty *)
Theorem cover_not_nil : forall l t, cover l t -> l <> nil.
Proof using.