Proof using name_map_bijective multi_map_lb_congr multi_map_congr.
move => net net' failed failed' lb tr H_eq H_step.
invcs H_step => //=.
- rewrite {2}/pt_map_onet /=.
case H_m: (@pt_map_msg _ _ _ _ msg_map m) => [m'|].
have H_q := @pt_net_handlers_some _ _ _ _ _ _ _ multi_map_congr to from m (onwState net to) _ 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.
break_and.
unfold tot_mapped_lb_net_handlers_label in *.
repeat break_let.
by repeat tuple_inversion.
have H_q := @pt_net_handlers_none _ _ _ _ _ _ _ multi_map_congr to from m (onwState net to) out d l H_m.
rewrite /net_handlers /= /unlabeled_net_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
concludes.
break_and.
have H_q' := @pt_lb_net_handlers_none _ _ _ _ _ _ _ _ multi_map_lb_congr to from _ (onwState net to) H_m.
rewrite /tot_mapped_lb_net_handlers_label in H_q'.
repeat break_let.
repeat tuple_inversion.
rewrite /pt_map_onet /=.
rewrite (@collate_pt_map_update2_eq _ _ _ _ name_map) /=.
rewrite H0 /=.
set p1 := fun _ _ => _.
set p2 := update2 _ _ _ _ _.
set s1 := fun _ => _.
set s2 := fun _ => _.
have H_eq_p: p1 = p2.
rewrite /p1 /p2 /update2.
apply functional_extensionality => src.
apply functional_extensionality => dst.
break_if => //.
break_and.
by rewrite -H2 -H5 2!tot_map_name_inv_inverse H3 /= H_m.
have H_eq_s: s1 = s2.
rewrite /s1 /s2 /update.
apply functional_extensionality => n.
break_if => //.
by rewrite H e.
rewrite H_eq_p H_eq_s.
split; first exact: LabeledStepOrderedFailure_stutter.
rewrite (@filterMap_pt_map_trace_ev_outputs_eq _ _ _ _ _ name_map out to).
by repeat find_rewrite.
- case H_i: (pt_map_input inp) => [inp'|].
have H_q := @pt_input_handlers_some _ _ _ _ _ _ _ multi_map_congr h _ (onwState 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.
by tuple_inversion.
have H_q := @pt_input_handlers_none _ _ _ _ _ _ _ multi_map_congr h _ (onwState net h) out d l H_i.
rewrite /input_handlers /= /unlabeled_input_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
concludes.
break_and.
have H_q' := @pt_lb_input_handlers_none _ _ _ _ _ _ _ _ multi_map_lb_congr h _ (onwState net h) H_i.
rewrite /tot_mapped_lb_input_handlers_label in H_q'.
repeat break_let.
repeat tuple_inversion.
rewrite /pt_map_onet /=.
rewrite (@collate_pt_map_eq _ _ _ _ name_map) H0 /=.
set s1 := fun _ => pt_map_data _.
set s2 := fun _ => pt_map_data _.
have H_eq_s: s1 = s2.
rewrite /s1 /s2.
apply functional_extensionality => n.
rewrite /update.
by break_if; first by rewrite H e.
rewrite -H_eq_s /s1 {s1 s2 H_eq_s}.
split; first exact: LabeledStepOrderedFailure_stutter.
rewrite (@filterMap_pt_map_trace_ev_outputs_eq _ _ _ _ _ name_map).
by repeat find_rewrite.
- by split => //; exact: LabeledStepOrderedFailure_stutter.
Qed.
Definition pt_map_onet_event e :=
{| evt_a := (List.map tot_map_name (fst e.(evt_a)), pt_map_onet (snd e.(evt_a))) ;
evt_l := tot_map_label e.(evt_l) ;
evt_trace := filterMap pt_map_trace_ev e.(evt_trace) |}.

Lemma pt_map_onet_event_Map_unfold : forall s,
Cons (pt_map_onet_event (hd s)) (map pt_map_onet_event (tl s)) = map pt_map_onet_event s.