Fixpoint app_fin_inf {ora: RandomOracle} (l: list RandomQA) (f: nat -> RandomQA) :=
  match l with
  | nil => f
  | qa :: l0 => fun n =>
                match n with
                | 0 => qa
                | S m => app_fin_inf l0 f m
                end
  end.

Lemma inf_history_construction {ora: RandomOracle}: forall (P: RandomHistory -> Prop) (init: list RandomQA), P (fin_history init) -> (forall l, P (fin_history l) -> exists qa, P (fin_history (l ++ qa :: nil))) -> exists h, is_inf_history h /\ forall n, n >= length init -> P (fstn_history n h).