Proof using name_map_bijective multi_map_congr fail_map_congr.
move => net net' failed failed' tr H_step.
invcs H_step.
- rewrite /=.
case H_m: (pt_map_packet p) => [p'|].
destruct p, p'.
simpl in *.
move: H_m.
case H_m: pt_map_msg => [m|] //.
move => H_eq.
find_inversion.
left.
rewrite /pt_map_net /= H3 filterMap_app /= H_m /= 2!filterMap_app {H3}.
set p' := {| pSrc := _ ; pDst := _ ; pBody := _ |}.
have ->: tot_map_name pDst = Net.pDst p' by [].
apply (@StepFailure_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 l)) => //=.
* exact: not_in_failed_not_in.
* rewrite /= -(pt_net_handlers_some _ _ _ _ H_m)  /pt_mapped_net_handlers /= tot_map_name_inv_inverse.
repeat break_let.
by find_inversion.
* pose p := {| pSrc := pSrc ; pDst := pDst ; pBody := pBody |}.
have H_p: pt_map_packet p = Some p'.
rewrite /pt_map_packet.
break_match.
rewrite /p in Heqp0.
find_inversion.
by rewrite H_m.
by rewrite (filterMap_pt_map_packet_map_eq_some _ _ H_p) (pt_map_update_eq_some _ _ _ H_p).
right.
rewrite /pt_map_net /=.
destruct p.
simpl in *.
move: H_m.
case H_m: pt_map_msg => [m|] //.
move => H_eq.
have H_m' := @pt_net_handlers_none _ _ _ _ _ name_map msg_map multi_map_congr _ _ _ _ _ _ _ H_m H6.
break_and.
rewrite H3 3!filterMap_app H1.
rewrite (filterMap_pt_map_name_msg_empty_eq _ pDst H0) /= H_m.
set nwS1 := fun _ => _.
set nwS2 := fun _ => _.
have H_eq_s: nwS1 = nwS2.
rewrite /nwS1 /nwS2 /=.
apply functional_extensionality => n.
rewrite /update /=.
case (name_eq_dec _ _) => H_dec //.
by rewrite H_dec.
by rewrite H_eq_s.
- rewrite /= /pt_map_net /=.
case H_i: pt_map_input => [inp'|].
left.
apply (@StepFailure_input _ _ _ _ _ _ _ _ _ (pt_map_data d) (filterMap pt_map_name_msg l)).
* exact: not_in_failed_not_in.
* rewrite /= -(pt_input_handlers_some _ _ _ H_i)  /pt_mapped_input_handlers /= tot_map_name_inv_inverse.
repeat break_let.
by tuple_inversion.
* rewrite /= -filterMap_pt_map_packet_map_eq -filterMap_app.
by rewrite pt_map_update_eq.
right.
rewrite /=.
have [H_d [H_l H_o]] := pt_input_handlers_none _ _ _ H_i H5.
rewrite H_o.
split => //.
rewrite filterMap_app.
rewrite (filterMap_pt_map_name_msg_empty_eq _ h H_l) /=.
set nwS1 := fun _ => _.
set nwS2 := fun _ => _.
have H_eq_s: nwS1 = nwS2.
rewrite /nwS1 /nwS2 /=.
apply functional_extensionality => n.
rewrite /update /=.
case (name_eq_dec _ _) => H_dec //.
by rewrite H_dec H_d.
by rewrite H_eq_s.
- case H_m: (pt_map_packet p) => [p'|].
left.
destruct p, p'.
simpl in *.
move: H_m.
case H_m: pt_map_msg => [m|] //.
move => H_eq.
find_inversion.
rewrite /pt_map_net /=.
find_rewrite.
rewrite filterMap_app /= H_m filterMap_app.
move: H4.
set p := {| pSrc := _ ; pDst := _ ; pBody := _ |}.
move => H_eq.
set p' := {| pSrc := _ ; pDst := _ ; pBody := _ |}.
exact: (@StepFailure_drop _ _ _ _ _ _ p' (filterMap pt_map_packet xs) (filterMap pt_map_packet ys)).
right.
split => //.
rewrite /pt_map_net /=.
find_rewrite.
by rewrite 2!filterMap_app /= H_m.
- rewrite /pt_map_net /=.
find_rewrite.
case H_p: pt_map_packet => [p'|]; last by right.
left.
rewrite filterMap_app /= H_p.
exact: (@StepFailure_dup _ _ _ _ _ _ p' (filterMap pt_map_packet xs) (filterMap pt_map_packet ys)).
- left.
exact: StepFailure_fail.
- rewrite remove_tot_map_eq /=.
left.
rewrite {2}/pt_map_net /=.
eapply (@StepFailure_reboot _ _ _ (tot_map_name h)) => //; first exact: in_failed_in.
by rewrite pt_map_update_eq /pt_map_net pt_reboot_eq /= tot_map_name_inv_inverse.
Qed.

Corollary step_failure_pt_mapped_simulation_star_1 :
forall net failed tr,
@step_failure_star _ _ fail_fst step_failure_init (failed, net) tr ->
exists tr', @step_failure_star _ _ fail_snd step_failure_init (map tot_map_name failed, pt_map_net net) tr' /\
filterMap trace_non_empty_out (filterMap pt_map_trace_occ tr) = filterMap trace_non_empty_out tr'.