Lemma applyEntries_output_correct : forall l c i o h st os st' es, applyEntries h st l = (os, st') -> in_output_list c i o os -> (stateMachine st = snd (execute_log (deduplicate_log es))) -> (forall c i o, getLastId st c = Some (i, o) -> output_correct c i o es) -> (forall c i o e', getLastId st c = Some (i, o) -> In e' es -> eClient e' = c -> eId e' <= i) -> (forall e', In e' es -> exists i o, getLastId st (eClient e') = Some (i, o) /\ eId e' <= i) -> output_correct c i o (es ++ l).