Proof using overlay_map_congr name_map_bijective multi_map_congr fail_msg_map_congr.
move => net failed tr H_step.
remember step_ordered_failure_init as y in *.
have H_eq_f: failed = fst (failed, net) by [].
have H_eq_n: net = snd (failed, net) by [].
rewrite H_eq_f {H_eq_f}.
rewrite {2}H_eq_n {H_eq_n}.
move: Heqy.
induction H_step using refl_trans_1n_trace_n1_ind => H_init /=.
rewrite H_init.
rewrite /step_ordered_failure_init /= /pt_ext_map_onet /=.
exists [].
rewrite -pt_ext_init_handlers_fun_eq.
exact: RT1nTBase.
concludes.
rewrite H_init {H_init x} in H_step2 H_step1.
case: x' H IHH_step1 H_step1 => failed' net'.
case: x'' H_step2 => failed'' net''.
rewrite /=.
move => H_step2 H IHH_step1 H_step1.
apply step_ordered_failure_pt_ext_mapped_simulation_1 in H.
case: H => H.
move: IHH_step1 => [tr' H_star].
move: H => [tr'' H].
exists (tr' ++ tr'').
have H_trans := refl_trans_1n_trace_trans H_star.
apply: H_trans.
have ->: tr'' = tr'' ++ [] by rewrite -app_nil_end.
apply: (@RT1nTStep _ _ _ _ (map tot_map_name failed'', pt_ext_map_onet net'')) => //.
exact: RT1nTBase.
move: H => [H_eq_n H_eq_f].
rewrite H_eq_n -H_eq_f.
move: IHH_step1 => [tr' H_star].
by exists tr'.
Qed.
Context {new_msg_fst : NewMsgParams multi_fst}.
Context {new_msg_snd : NewMsgParams multi_snd}.
Context {new_msg_map_congr : NewMsgParamsPartialMapCongruency new_msg_fst new_msg_snd msg_map}.
Definition pt_ext_map_odnet (net : @ordered_dynamic_network _ multi_fst) : @ordered_dynamic_network _ multi_snd :=
{| odnwNodes := map tot_map_name net.(odnwNodes) ;
odnwPackets := fun src dst => filterMap pt_map_msg (net.(odnwPackets) (tot_map_name_inv src) (tot_map_name_inv dst)) ;
odnwState := fun n => match net.(odnwState) (tot_map_name_inv n) with
| None => None
| Some d => Some (pt_ext_map_data d (tot_map_name_inv n))
end |}.

Theorem step_ordered_dynamic_failure_pt_ext_mapped_simulation_1 :
forall net net' failed failed' tr,
NoDup (odnwNodes net) ->
@step_ordered_dynamic_failure _ _ overlay_fst new_msg_fst fail_msg_fst (failed, net) (failed', net') tr ->
(exists tr', @step_ordered_dynamic_failure _ _ overlay_snd new_msg_snd fail_msg_snd (map tot_map_name failed, pt_ext_map_odnet net) (map tot_map_name failed', pt_ext_map_odnet net') tr') \/ (pt_ext_map_odnet net' = pt_ext_map_odnet net /\ failed = failed').