Proof using.
intros.
eapply NoDup_map_partition.
- apply NoDup_deduplicate_log.
- eauto.
- eauto.
- simpl. congruence.
Qed.

Lemma applied_entries_applied_implies_input_state :
forall net e,
In e (applied_entries (nwState net)) ->
applied_implies_input_state (eClient e) (eId e) (eInput e) net.