Corollary step_ordered_pt_ext_mapped_simulation_star_1 :
  forall net tr,
    @step_ordered_star _ multi_fst step_ordered_init net tr ->
    exists tr', @step_ordered_star _ multi_snd step_ordered_init (pt_ext_map_onet net) tr'.

Theorem step_ordered_failure_pt_ext_mapped_simulation_1 : forall net net' failed failed' tr, @step_ordered_failure _ _ overlay_fst fail_msg_fst (failed, net) (failed', net') tr -> (exists tr', @step_ordered_failure _ _ overlay_snd fail_msg_snd (map tot_map_name failed, pt_ext_map_onet net) (map tot_map_name failed', pt_ext_map_onet net') tr') \/ pt_ext_map_onet net' = pt_ext_map_onet net /\ failed = failed'.