Proof using.
elim => //=.
move => o out IH to.
by rewrite IH.
Qed.

Theorem step_ordered_failure_tot_mapped_simulation_1 :
forall net net' failed failed' tr,
@step_ordered_failure _ _ overlay_fst fail_msg_fst (failed, net) (failed', net') tr ->
@step_ordered_failure _ _ overlay_snd fail_msg_snd (map tot_map_name failed, tot_map_onet net) (map tot_map_name failed', tot_map_onet net') (map tot_map_trace tr).