Lemma applied_implies_input_update_split : forall client id i net h d ps, applied_implies_input_state client id i (mkNetwork ps (update name_eq_dec (nwState net) h d)) -> exists e, correct_entry client id i e /\ (In e (log d) \/ (exists h, In e (log (nwState net h))) \/ (exists p es, In p ps /\ mEntries (pBody p) = Some es /\ In e es)).