Proof using name_map_bijective.
elim => /=; first by move => H_nd; exact: NoDup_nil.
move => n ns IH H_nd.
inversion H_nd.
find_apply_lem_hyp IH.
by apply NoDup_cons; first exact: not_in_failed_not_in.
Qed.

Lemma permutation_nodes :
Permutation nodes (map tot_map_name nodes).