Proof using name_map_bijective multi_map_congr fail_map_congr.
move => net failed tr H_step.
remember step_failure_init as y in *.
change failed with (fst (failed, net)).
change net with (snd (failed, net)) at 2.
move: Heqy.
induction H_step using refl_trans_1n_trace_n1_ind => H_init /=.
rewrite H_init /step_failure_init /= /step_async_init /tot_map_net /= tot_init_handlers_fun_eq.
exact: RT1nTBase.
concludes.
repeat find_rewrite.
destruct x'.
destruct x''.
rewrite /=.
find_apply_lem_hyp step_failure_tot_mapped_simulation_1.
rewrite map_app.
match goal with
| H : step_failure_star _ _ _ |- _ => apply: (refl_trans_1n_trace_trans H)
end.
rewrite (app_nil_end (map tot_map_trace_occ _)).
apply (@RT1nTStep _ _ _ _ (map tot_map_name l0, tot_map_net n0)) => //.
exact: RT1nTBase.
Qed.

(* Specifications: *)
Lemma map_eq_name_eq_eq :
forall l l', map tot_map_name l = map tot_map_name l' -> l = l'.