Definition prefix_history {ora: RandomOracle} (h1 h2: RandomHistory): Prop :=
  forall n, h1 n = None \/ h1 n = h2 n.Definition strict_prefix_history {ora: RandomOracle} (h1 h2: RandomHistory): Prop :=
  prefix_history h1 h2 /\ exists n, h1 n <> h2 n.Definition n_bounded_prefix_history {ora: RandomOracle} (m: nat) (h1 h2: RandomHistory): Prop :=
  forall n, (h1 n = None /\ h2 (n + m) = None) \/ h1 n = h2 n.Definition n_conflict_history {ora: RandomOracle} (n: nat) (h1 h2: RandomHistory): Prop :=
  match h1 n, h2 n with
  | Some a1, Some a2 => projT1 a1 <> projT1 a2
  | Some _, None => True
  | None, Some _ => True
  | None, None => False
  end.Definition conflict_history {ora: RandomOracle} (h1 h2: RandomHistory): Prop :=
  exists n, history_coincide n h1 h2 /\ n_conflict_history n h1 h2.Definition strict_conflict_history {ora: RandomOracle} (h1 h2: RandomHistory): Prop :=
  exists n,
    history_coincide n h1 h2 /\
    match h1 n, h2 n with
    | Some a1, Some a2 => projT1 a1 <> projT1 a2
    | _, _ => False
    end.

Lemma is_at_least_n_history_fstn_history {ora: RandomOracle}: forall n m h, n <= m -> (is_at_least_n_history n (fstn_history m h) <-> is_at_least_n_history n h).