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 nth_error_firstn_list_from_fun: forall {A} (f: nat -> A) n m, m < n -> nth_error (fisrtn_list_from_fun f n) m = Some (f m).