Section inner.
Variable client : clientId.
Variable id : nat.
Variable i : input.
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.
Proof using.
unfold applied_implies_input_state.
eauto 10.
Qed.

Lemma aiis_intro_packet :
forall client id i net e p es,
mEntries (pBody p) = Some es ->
In p (nwPackets net) ->
correct_entry client id i e ->
In e es ->
applied_implies_input_state client id i net.