Require Export Coq.Logic.Classical.
Require Export Coq.Logic.FunctionalExtensionality.
Require Export Coq.Classes.Equivalence.
Require Export Coq.Classes.Morphisms.
Class RandomOracle: Type := {
ro_question: Type;
ro_answer: ro_question -> Type;
ro_default_question: ro_question;
ro_default_answer: ro_answer ro_default_question
}.
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.

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).