Proof using.
unfold applied_implies_input_state.
intros.
break_exists_exists.
intuition.
break_exists.
simpl in *.
destruct (name_eq_dec h x0); rewrite_update; eauto.
Qed.

Lemma aiis_intro_state :
forall client id i net e h,
In e (log (nwState net h)) ->
correct_entry client id i e ->
applied_implies_input_state client id i net.