Lemma in_applied_entries_step_applied_implies_input_state : forall (s : list name * network) s' tr o, refl_trans_1n_trace step_failure step_failure_init s tr -> ~ in_applied_entries client id (snd s) -> step_failure s s' o -> in_applied_entries client id (snd s') -> (exists e, eClient e = client /\ eId e = id /\ applied_implies_input_state client id (eInput e) (snd s)) \/ exists h o' inp, o = (h, inl (ClientRequest client id inp)) :: o'.