Proof using name_map_bijective multi_map_congr.
move => net tr H_step.
remember step_async_init as y in *.
move: Heqy.
induction H_step using refl_trans_1n_trace_n1_ind => H_init /=.
rewrite H_init /step_async_init /= /tot_map_net /=.
rewrite tot_init_handlers_fun_eq.
exact: RT1nTBase.
concludes.
repeat find_rewrite.
find_apply_lem_hyp step_async_tot_mapped_simulation_1.
rewrite map_app.
match goal with
| H : step_async_star _ _ _ |- _ => apply: (refl_trans_1n_trace_trans H)
end.
rewrite (app_nil_end (map _ _)).
apply: (@RT1nTStep _ _ _ _ (tot_map_net x'')) => //.
exact: RT1nTBase.
Qed.
Hypothesis tot_map_output_injective :
forall o o', tot_map_output o = tot_map_output o' -> o = o'.

Theorem step_async_tot_mapped_simulation_2 :
forall net net' out mnet mout,
@step_async _ multi_snd net net' out ->
tot_map_net mnet = net ->
map tot_map_trace_occ mout = out ->
exists mnet', @step_async _ multi_fst mnet mnet' mout /\
tot_map_net mnet' = net'.