Proof using.
intros net tr_m H.
find_apply_lem_hyp refl_trans_1n_n1_trace.
prep_induction H.
induction H; intros; subst.
- simpl in *. rewrite inputs_m_on_nil' in H3.
destruct tr_1; try discriminate.
invc H.
+ simpl. apply correspond_init.
+ invc H1. discriminate.
- repeat concludes.
invc H1; simpl in *.
+ find_apply_lem_hyp PB_net_defn.
intuition; subst;
try rewrite inputs_m_app in *;
try rewrite inputs_m_inr_singleton in *;
try rewrite app_nil_r in *;
try break_exists;
try break_let;
try break_and;
subst; repeat find_rewrite;
eauto using
correspond_preserved_primary_same_no_outputs,
correspond_preserved_primary_apply_entry,
update_nop,
update_eq,
update_diff,
outputs_m_inr_nil_singleton.
+ find_apply_lem_hyp PB_input_handler_defn.
intuition; subst;
repeat rewrite snoc_assoc in *;
repeat rewrite inputs_m_app in *.
* break_exists. break_and. subst.
rewrite inputs_m_inr_singleton in *. rewrite app_nil_r in *.
rewrite inputs_m_primary_inl_request_singleton in *.
find_apply_lem_hyp inputs_1_invert_app. break_exists. break_and.
subst. simpl in *. find_inversion. destruct x1.
find_apply_lem_hyp step_1_snoc_inv. break_exists. break_and.
{ eapply correspond_preserved_primary_same_no_outputs; eauto.
eapply correspond_preserved_snoc; eauto.
- eauto using step_1_singleton_inversion.
- rewrite update_eq by auto. auto.
- rewrite update_eq by auto. auto.
}
* rewrite inputs_m_inr_singleton in *. rewrite app_nil_r in *.
rewrite inputs_m_inl_read_singleton in *. rewrite app_nil_r in *.
eauto using
correspond_preserved_primary_same_no_outputs,
update_nop,
outputs_m_inl_read_singleton,
outputs_m_read_response_singleton.
* rewrite inputs_m_inr_singleton in *.
rewrite inputs_m_backup_singleton in *. repeat rewrite app_nil_r in *.
eauto using correspond_preserved_primary_same_no_outputs, update_diff.
Qed.

Lemma correspond_inductive :
forall net net' tr_m',
step_async net net' tr_m' ->
forall st st' tr_m tr_1 tr_1',
correspond st (nwState net) tr_1 tr_m ->
step_1_star st st' tr_1' ->
inputs_1 tr_1' = inputs_m tr_m' ->
correspond st' (nwState net') (tr_1 ++ tr_1') (tr_m ++ tr_m').