Lemma in_applied_entries_step_applied_implies_input_state' : forall (failed : list name) net failed' net' o, raft_intermediate_reachable net -> step_failure (failed, net) (failed', net') o -> ~ in_applied_entries client id net -> in_applied_entries client id net' -> (exists e, eClient e = client /\ eId e = id /\ applied_implies_input_state client id (eInput e) net) \/ exists h o' inp, o = (h, inl (ClientRequest client id inp)) :: o'.