Proof using.
intros l1; elim l1 using list_length_ind.
intros l0 H t; case t.
intros a H1; rewrite cover_inv_leaf with (1 := H1).
exists (leaf a :: nil); auto.
intros t1 t2 H1; case cover_inv_app with (1 := H1).
intros H2; exists l0; split; auto; rewrite H2; auto.
intros (l2, (l3, ((HH1, HH2), HH3))).
case H with (2 := HH1); auto.
rewrite permutation_length with (1 := HH3).
generalize HH2; rewrite length_app; case l3; simpl in |- *; auto with arith.
intros HH4; case cover_not_nil with (1 := HH4); auto.
intros; rewrite plus_comm; simpl in |- *; auto with arith.
intros l4 (HP1, HP2).
case H with (2 := HH2); auto.
rewrite permutation_length with (1 := HH3).
generalize HH1; rewrite length_app; case l2; simpl in |- *; auto with arith.
intros HH4; case cover_not_nil with (1 := HH4); auto.
intros l5 (HP3, HP4).
exists (l4 ++ l5); split; auto.
apply permutation_trans with (1 := HH3); auto.
Qed.

(* If the ordered cover is composed of only leaves, they are the exact leaves of the tree *)
Theorem ulist_ordered_cover :
forall l1 l2 t,
ordered_cover l1 t ->
ulist l2 -> l1 = map (fun x : A => leaf x) l2 -> all_leaves t = l2.