Proof using name_map_bijective multi_map_congr.
move => net tr H_step.
remember step_async_init as y in *.
move: Heqy.
induction H_step using refl_trans_1n_trace_n1_ind => H_init /=.
rewrite H_init.
rewrite /step_async_init /= /pt_map_net /=.
rewrite pt_init_handlers_fun_eq.
exists [].
split => //.
exact: RT1nTBase.
concludes.
rewrite H_init in H_step2 H_step1.
apply step_async_pt_mapped_simulation_1 in H.
case: H => H.
move: IHH_step1 => [tr' [H_star H_eq_tr]].
exists (tr' ++ filterMap pt_map_trace_occ tr2).
split.
* have H_trans := refl_trans_1n_trace_trans H_star.
apply: H_trans.
rewrite (app_nil_end (filterMap pt_map_trace_occ _)).
apply: (@RT1nTStep _ _ _ _ (pt_map_net x'')) => //.
exact: RT1nTBase.
* by rewrite 2!filterMap_app H_eq_tr filterMap_app.
move: H => [H_eq H_eq'].
rewrite H_eq.
move: IHH_step1 => [tr' [H_star H_tr]].
exists tr'.
split => //.
rewrite 2!filterMap_app.
by rewrite H_eq' -app_nil_end.
Qed.
Context {fail_fst : FailureParams multi_fst}.
Context {fail_snd : FailureParams multi_snd}.
Context {fail_map_congr : FailureParamsPartialMapCongruency fail_fst fail_snd base_map}.

Theorem step_failure_pt_mapped_simulation_1 :
forall net net' failed failed' tr,
@step_failure _ _ fail_fst (failed, net) (failed', net') tr ->
@step_failure _ _ fail_snd (map tot_map_name failed, pt_map_net net) (map tot_map_name failed', pt_map_net net') (filterMap pt_map_trace_occ tr) \/
(pt_map_net net' = pt_map_net net /\ failed = failed' /\ filterMap trace_non_empty_out (filterMap pt_map_trace_occ tr) = []).