Proof using name_map_bijective.
move => n.
elim => //=.
move => n' failed IH H_in H_in'.
case: H_in' => H_in'.
case: H_in.
left.
rewrite -(tot_map_name_inv_inverse n').
rewrite H_in'.
exact: tot_map_name_inv_inverse.
contradict H_in'.
apply: IH.
move => H_in'.
case: H_in.
by right.
Qed.

(* Specifications: *)
Lemma in_failed_in :
forall n failed,
In n failed ->
In (tot_map_name n) (map tot_map_name failed).