Definition RandomQA {ora: RandomOracle}: Type := {q: ro_question & ro_answer q}.
Definition RandomHistory {ora: RandomOracle}: Type := {h: nat -> option RandomQA | forall x y, x < y -> h x = None -> h y = None}.
Definition history_get {ora: RandomOracle} (h: RandomHistory) (n: nat) := proj1_sig h n.
Coercion history_get: RandomHistory >-> Funclass.

(* Specifications: *)
Lemma history_sound1: forall {ora: RandomOracle} (h: RandomHistory) (x y: nat), x <= y -> h x = None -> h y = None.
Lemma history_sound2: forall {ora: RandomOracle} (h: RandomHistory) (x y: nat), x <= y -> (exists a, h y = Some a) -> (exists a, h x = Some a).
Lemma history_extensionality {ora: RandomOracle}: forall h1 h2: RandomHistory, (forall n, h1 n = h2 n) -> h1 = h2.
Tactic Notation "history_extensionality" ident(x) :=
match goal with
[ |- ?X = ?Y ] =>
apply history_extensionality; intro x
end.