Definition ordered_Rlist (l:list R) : Prop :=
  forall i:nat, (i < pred (length l))%nat -> pos_Rl l i <= pos_Rl l (S i).Fixpoint insert (l:list R) (x:R) : list R :=
  match l with
    | nil => x :: nil
    | a :: l' =>
      match Rle_dec a x with
        | left _ => a :: (insert l' x)
        | right _ => x :: l
      end
  end.Fixpoint cons_ORlist (k l:list R) : list R :=
  match k with
    | nil => l
    | a :: k' => cons_ORlist k' (insert l a)
  end.Fixpoint mid_Rlist (l:list R) (x:R) : list R :=
  match l with
    | nil => nil
    | a :: l' => ((x + a) / 2) :: (mid_Rlist l' a)
  end.Definition Rtail (l:list R) : list R :=
  match l with
    | nil => nil
    |  a :: l' => l'
  end.Definition FF (l:list R) (f:R -> R) : list R :=
  match l with
    | nil => nil
    | a :: l' => map f (mid_Rlist l' a)
  end.

Lemma RList_P12 : forall (l:list R) (i:nat) (f:R -> R), (i < length l)%nat -> pos_Rl (map f l) i = f (pos_Rl l i).