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 fstn_app_inf_fin {ora: RandomOracle}: forall l h n, n >= length l -> fstn_history n (inf_history (app_fin_inf l h)) = fin_history (l ++ fisrtn_list_from_fun h (n - length l)).