Proof using name_map_bijective multi_map_lb_congr multi_map_congr.
move => net net' failed failed' lb tr H_step.
invcs H_step => //=.
- apply (@LabeledStepOrderedFailure_deliver _ _ _ _ _ _ (@tot_map_msg _ _ _ _ msg_map m) (List.map (@tot_map_msg _ _ _ _ msg_map) ms) (List.map tot_map_output out) (tot_map_data d) (@tot_map_name_msgs _ _ _ _ _ msg_map l) (@tot_map_name _ _ _ _ name_map from) (@tot_map_name _ _ _ _ name_map to)) => //=.
* rewrite /tot_map_onet /=.
rewrite 2!tot_map_name_inv_inverse.
by find_rewrite.
* exact: not_in_failed_not_in.
* rewrite /tot_map_onet /= tot_map_name_inv_inverse.
have H_q := @tot_net_handlers_eq _ _ _ _ _ _ _ multi_map_congr to from m (onwState net to).
rewrite /tot_mapped_net_handlers /net_handlers /= /unlabeled_net_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
have H_q' := @tot_lb_net_handlers_eq _ _ _ _ _ _ _ _ multi_map_lb_congr _ _ _ _ _ _ _ _ Heqp1.
rewrite /tot_mapped_lb_net_handlers_label in H_q'.
repeat break_let.
by repeat tuple_inversion.
* rewrite /tot_map_onet /=.
rewrite (@collate_tot_map_update2_eq _ _ _ _ _ _ name_map_bijective).
set f1 := fun _ => tot_map_data _.
set f2 := update _ _ _ _.
have H_eq_f: f1 = f2.
rewrite /f1 /f2.
apply functional_extensionality => n.
rewrite /update.
break_if; break_if => //; first by rewrite -e tot_map_name_inverse_inv in n0.
by rewrite e tot_map_name_inv_inverse in n0.
by rewrite H_eq_f.
* by rewrite (@map_tot_map_trace_eq _ _ _ _ _ name_map).
- rewrite /tot_map_onet /=.
apply (@LabeledStepOrderedFailure_input _ _ (@tot_map_name _ _ _ _ name_map h) _ _ _ _ (List.map tot_map_output out) (tot_map_input inp) (tot_map_data d) (@tot_map_name_msgs _ _ _ _ _ msg_map l)).
* exact: not_in_failed_not_in.
* rewrite /tot_map_onet /= tot_map_name_inv_inverse.
have H_q := @tot_input_handlers_eq _ _ _ _ _ _ _ multi_map_congr h inp (onwState net h).
rewrite /tot_mapped_input_handlers /= /unlabeled_input_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
have H_q' := @tot_lb_input_handlers_eq _ _ _ _ _ _ _ _ multi_map_lb_congr _ _ _ _ _ _ _ Heqp1.
rewrite /tot_mapped_lb_input_handlers_label in H_q'.
repeat break_let.
by repeat tuple_inversion.
* rewrite /tot_map_onet /=.
rewrite (@collate_tot_map_eq _ _ _ _ _ _ name_map_bijective).
set f1 := fun _ => tot_map_data _.
set f2 := update _ _ _ _.
have H_eq_f: f1 = f2.
rewrite /f1 /f2.
apply functional_extensionality => n.
rewrite /update.
break_if; break_if => //; first by rewrite -e tot_map_name_inverse_inv in n0.
by rewrite e tot_map_name_inv_inverse in n0.
by rewrite H_eq_f.
* by rewrite (@map_tot_map_trace_eq _ _ _ _ _ name_map).
- rewrite tot_lb_label_silent_fst_snd.
exact: LabeledStepOrderedFailure_stutter.
Qed.
Definition tot_map_onet_event e :=
{| evt_a := (List.map tot_map_name (fst e.(evt_a)), tot_map_onet (snd e.(evt_a))) ;
evt_l := tot_map_label e.(evt_l) ;
evt_trace := List.map tot_map_trace e.(evt_trace) |}.

(* Specifications: *)
Lemma tot_map_onet_event_map_unfold : forall s,
Cons (tot_map_onet_event (hd s)) (map tot_map_onet_event (tl s)) = map tot_map_onet_event s.