Proof using name_map_bijective multi_map_lb_congr multi_map_congr.
move => net net' failed failed' lb tr H_neq H_step.
have H_neq': lb <> label_silent.
rewrite -pt_lb_label_silent_fst_snd in H_neq.
move => H_eq.
by rewrite H_eq in H_neq.
invcs H_step => //=.
- rewrite {2}/pt_map_odnet /=.
case H_m: (@pt_map_msg _ _ _ _ msg_map m) => [m'|]; last first.
have H_q := @pt_lb_net_handlers_none _ _ _ _ _ _ _ _ multi_map_lb_congr to from _ d H_m.
rewrite /tot_mapped_lb_net_handlers_label in H_q.
repeat break_let.
by tuple_inversion.
apply (@LabeledStepOrderedDynamicFailure_deliver _ _ _ _ _ _ m' (filterMap (@pt_map_msg _ _ _ _ msg_map) ms) (filterMap pt_map_output out) (pt_map_data d) (pt_map_data d') (filterMap (@pt_map_name_msg _ _ _ _ _ msg_map) l) (@tot_map_name _ _ _ _ name_map from) (@tot_map_name _ _ _ _ name_map to)).
* exact: not_in_failed_not_in.
* exact: in_failed_in.
* by rewrite /pt_map_odnet /= tot_map_name_inv_inverse H5.
* by rewrite /pt_map_odnet /= 2!tot_map_name_inv_inverse H6 /= H_m.
* have H_q := @pt_net_handlers_some _ _ _ _ _ _ _ multi_map_congr to from m d _ H_m.
rewrite /pt_mapped_net_handlers /net_handlers /= /unlabeled_net_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
have H_q' := @pt_lb_net_handlers_some _ _ _ _ _ _ _ _ multi_map_lb_congr _ _ _ _ _ _ _ _ _ H_m Heqp1.
rewrite /tot_mapped_lb_net_handlers_label in H_q'.
repeat break_let.
break_and.
by repeat tuple_inversion.
* rewrite {2}/pt_map_odnet /=.
rewrite (@collate_pt_map_update2_eq _ _ _ _ name_map).
set f1 := fun _ => match _ with _ => _ end.
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 (@filterMap_pt_map_trace_ev_outputs_eq _ _ _ _ _ name_map).
- rewrite {2}/pt_map_odnet /=.
case H_i: pt_map_input => [inp'|]; last first.
have H_q := @pt_lb_input_handlers_none _ _ _ _ _ _ _ _ multi_map_lb_congr h inp d H_i.
rewrite /tot_mapped_lb_input_handlers_label in H_q.
repeat break_let.
by tuple_inversion.
apply (@LabeledStepOrderedDynamicFailure_input _ _ (@tot_map_name _ _ _ _ name_map h) _ _ _ _ (filterMap pt_map_output out) inp' (pt_map_data d) (pt_map_data d') (filterMap (@pt_map_name_msg _ _ _ _ _ msg_map) l)).
* exact: not_in_failed_not_in.
* exact: in_failed_in.
* by rewrite /pt_map_odnet /= tot_map_name_inv_inverse H5.
* have H_q := @pt_input_handlers_some _ _ _ _ _ _ _ multi_map_congr h inp d _ H_i.
rewrite /pt_mapped_input_handlers /input_handlers /= /unlabeled_input_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
have H_q' := @pt_lb_input_handlers_some _ _ _ _ _ _ _ _ multi_map_lb_congr _ _ _ _ _ _ _ _ H_i Heqp1.
rewrite /tot_mapped_lb_input_handlers_label in H_q'.
repeat break_let.
break_and.
by repeat tuple_inversion.
* rewrite {2}/pt_map_odnet /=.
rewrite (@collate_pt_map_eq _ _ _ _ name_map).
set f1 := fun _ => match _ with _ => _ end.
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 (@filterMap_pt_map_trace_ev_outputs_eq _ _ _ _ _ name_map).
Qed.

Theorem lb_step_ordered_dynamic_failure_pt_mapped_simulation_1_silent :
forall net net' failed failed' lb tr,
tot_map_label lb = label_silent ->
@lb_step_ordered_dynamic_failure _ labeled_multi_fst (failed, net) lb (failed', net') tr ->
@lb_step_ordered_dynamic_failure _ labeled_multi_snd (List.map tot_map_name failed, pt_map_odnet net) label_silent (List.map tot_map_name failed', pt_map_odnet net') [] /\ filterMap pt_map_trace_ev tr = [].