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 => //=.
- destruct (pt_map_packet p) eqn:?; last first.
destruct p.
simpl in *.
break_match => //.
have H_q := @pt_lb_net_handlers_none _ _ _ _ _ _ _ _ multi_map_lb_congr pDst pSrc _ (nwState net pDst) Heqo0.
rewrite /tot_mapped_lb_net_handlers_label in H_q.
repeat break_let.
by tuple_inversion.
have H_eq_n: tot_map_name (pDst p) = pDst p0.
destruct p.
simpl in *.
break_match => //.
by find_injection.
rewrite H_eq_n.
apply (@LabeledStepFailure_deliver _ _ _ _ _ _ (filterMap pt_map_packet xs) (filterMap pt_map_packet ys) (filterMap pt_map_output out) (pt_map_data d) (filterMap (@pt_map_name_msg _ _ _ _ _ msg_map) l)).
* rewrite /pt_map_net /=.
find_rewrite.
by rewrite filterMap_app /= Heqo.
* rewrite -H_eq_n.
exact: not_in_failed_not_in.
* rewrite /pt_map_net /= -{2}H_eq_n tot_map_name_inv_inverse.
destruct p, p0.
simpl in *.
break_match => //.
find_injection.
clean.
have H_q := @pt_net_handlers_some _ _ _ _ _ _ _ multi_map_congr pDst pSrc pBody (nwState net pDst) _ Heqo0.
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 _ _ _ _ _ _ _ _ _ Heqo0 Heqp1.
rewrite /tot_mapped_lb_net_handlers_label in H_q'.
repeat break_let.
break_and.
by repeat tuple_inversion.
* rewrite /pt_map_net /= 2!filterMap_app.
by rewrite (filterMap_pt_map_packet_map_eq_some _ _ Heqo) (pt_map_update_eq_some _ _ _ Heqo).
- case H_i: pt_map_input => [inp'|]; last first.
have H_q := @pt_lb_input_handlers_none _ _ _ _ _ _ _ _ multi_map_lb_congr h _ (nwState net h) H_i.
rewrite /tot_mapped_lb_input_handlers_label /= in H_q.
repeat break_let.
by tuple_inversion.
apply (@LabeledStepFailure_input _ _ _ _ _ _ _ _ (pt_map_data d) (filterMap (@pt_map_name_msg _ _ _ _ _ msg_map) l)).
* exact: not_in_failed_not_in.
* have H_q := @pt_input_handlers_some _ _ _ _ _ _ _ multi_map_congr h _ (nwState net h) _ 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.
break_and.
unfold tot_mapped_lb_input_handlers_label in *.
repeat break_let.
repeat tuple_inversion.
by rewrite /pt_map_net /= tot_map_name_inv_inverse.
* rewrite /pt_map_net /=.
rewrite filterMap_app filterMap_pt_map_packet_map_eq.
by rewrite -(@pt_map_update_eq  _ _ _ _ _ _ name_map_bijective).
Qed.

(* Specifications: *)
Theorem lb_step_failure_pt_mapped_simulation_1_silent :
forall net net' failed failed' lb tr,
tot_map_label lb = label_silent ->
@lb_step_failure _ labeled_multi_fst (failed, net) lb (failed', net') tr ->
@lb_step_failure _ labeled_multi_snd (List.map tot_map_name failed, pt_map_net net) label_silent (List.map tot_map_name failed', pt_map_net net') [] /\ filterMap trace_non_empty_out (filterMap pt_map_trace_occ tr) = [].