Fixpoint alist_sorted (l:alist A) : bool :=
    match l with
    | nil => true
    | (a, _) :: l' =>
        match l' with
        | nil => true
        | (a', y') :: l'' => andb (Nless a a') (alist_sorted l')
        end
    end.Fixpoint alist_nth_ad (n:nat) (l:alist A) {struct l} : ad :=
    match l with
    | nil => N0 (* dummy *)
    | (a, y) :: l' => match n with
                      | O => a
                      | S n' => alist_nth_ad n' l'
                      end
    end.Definition alist_sorted_1 (l:alist A) :=
    forall n:nat,
      S (S n) <= length l ->
      Nless (alist_nth_ad n l) (alist_nth_ad (S n) l) = true.

Lemma alist_sorted_imp_1 : forall l:alist A, alist_sorted l = true -> alist_sorted_1 l.