Proof using.
intros n; elim n; simpl in |- *; auto.
intros l t H H0; elim H0.
intros n0 H l t H0 H1.
case in_flat_map_ex with (1 := H1); clear H1.
intros x; case x; clear x.
simpl in |- *; intros (H1, H2); case H2.
intros b x; case x; clear x.
simpl in |- *; intros (H1, [H2| H2]).
rewrite <- H2.
rewrite permutation_one_inv with (a := b) (l := l); auto.
apply all_permutations_permutation; auto.
case H2.
intros b1 l1 (H1, H2).
apply cover_node with (l2 := l1) (t1 := b) (t2 := b1); auto.
apply permutation_sym; apply all_permutations_permutation; auto.
apply H; auto.
apply eq_add_S; apply trans_equal with (1 := H0).
apply trans_equal with (length (b :: b1 :: l1)); auto.
apply permutation_length.
apply permutation_sym; apply all_permutations_permutation; auto.
Qed.

(* The list of all covers contain only covers *)
Theorem all_cover_cover : forall l t, In t (all_cover l) -> cover l t.
Proof using.