Corollary add_items_empty_hist_length : forall lr,
  Forall (fun '(la, h') => add_items_trace empty_hists la h') lr ->
  Forall (fun h' => Zlength h' = size) (map snd lr).Definition hists_mono (h1 h2 : list (hist * hist)) := Forall2 (fun a b =>
  (exists l, fst b = fst a ++ l) /\ (exists l, snd b = snd a ++ l)) h1 h2.Instance hists_mono_refl : RelationClasses.Reflexive hists_mono.Instance hists_mono_trans : RelationClasses.Transitive hists_mono.

Lemma add_item_trace_mono : forall h k v i s h', add_item_trace h k v i s h' -> hists_mono h h'.