Lemma doGenericServer_key_in_output_list : forall net h os st' ms id' client id, raft_intermediate_reachable net -> doGenericServer h (nwState net h) = (os, st', ms) -> key_in_output_list client id os -> id < id' -> before_func (has_key client id) (has_key client id') (applied_entries (update name_eq_dec (nwState net) h st')).