Proof using name_map_bijective multi_map_lb_congr multi_map_congr.
move => net net' failed failed' lb tr H_step.
invcs H_step => //=.
- have ->: tot_map_name (pDst p) = pDst (tot_map_packet p) by destruct p.
apply: (@LabeledStepFailure_deliver _ _ _ _ _ _ (List.map tot_map_packet xs) (List.map tot_map_packet ys) (List.map tot_map_output out) (tot_map_data d) (@tot_map_name_msgs _ _ _ _ _ msg_map l)).
* rewrite /tot_map_net /=.
find_rewrite.
by rewrite map_app.
* destruct p.
simpl in *.
exact: not_in_failed_not_in.
* destruct p.
simpl in *.
rewrite tot_map_name_inv_inverse.
have H_q := @tot_net_handlers_eq _ _ _ _ _ _ _ multi_map_congr pDst pSrc pBody (nwState net pDst).
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_net /= 2!map_app -(@tot_map_update_packet_eq _ _ _ _ _ _ _ name_map_bijective).
destruct p.
by rewrite tot_map_packet_map_eq.
- apply: (@LabeledStepFailure_input _ _ _ _ _ _ _ _ (tot_map_data d) (tot_map_name_msgs l)).
* exact: not_in_failed_not_in.
* rewrite /tot_map_net /= tot_map_name_inv_inverse.
have H_q := @tot_input_handlers_eq _ _ _ _ _ _ _ multi_map_congr h inp (nwState 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.
* by rewrite /tot_map_net /= map_app tot_map_packet_map_eq -(@tot_map_update_eq _ _ _ _ _ _ name_map_bijective).
- rewrite tot_lb_label_silent_fst_snd.
exact: LabeledStepFailure_stutter.
Qed.
Definition tot_map_net_event e :=
{| evt_a := (List.map tot_map_name (fst e.(evt_a)), tot_map_net (snd e.(evt_a))) ;
evt_l := tot_map_label e.(evt_l) ;
evt_trace := List.map tot_map_trace_occ e.(evt_trace) |}.

Lemma tot_map_net_event_map_unfold : forall s,
Cons (tot_map_net_event (hd s)) (map tot_map_net_event (tl s)) = map tot_map_net_event s.