Lemma handleInput_log : forall net h inp os st' ms h' e, raft_intermediate_reachable net -> handleInput h inp (nwState net h) = (os, st', ms) -> In e (log (update name_eq_dec (nwState net) h st' h')) -> (applied_implies_input_state (eClient e) (eId e) (eInput e) net \/ inp = ClientRequest (eClient e) (eId e) (eInput e)).