Definition BDDvar := ad.Definition BDDcompare (x y : BDDvar) :=
  match x, y with
  | N0, N0 => Datatypes.Eq
  | N0, Npos _ => Datatypes.Lt
  | Npos _, N0 => Datatypes.Gt
  | Npos p1, Npos p2 => Pcompare p1 p2 Datatypes.Eq
  end.Definition ad_S (a : ad) :=
  match a with
  | N0 => Npos 1
  | Npos p => Npos (Psucc p)
  end.Definition max (m n : nat) := if leb m n then n else m.Definition BDDvar_max (x y : BDDvar) := if Nleb x y then y else x.Inductive no_dup_list (A : Set) : list A -> Prop :=
  | no_dup_nil : no_dup_list A nil
  | no_dup_cons :
      forall (a : A) (l : list A),
      ~ In a l -> no_dup_list _ l -> no_dup_list _ (a :: l).Inductive no_dup_list (A : Set) : list A -> Prop :=
  | no_dup_nil : no_dup_list A nil
  | no_dup_cons :
      forall (a : A) (l : list A),
      ~ In a l -> no_dup_list _ l -> no_dup_list _ (a :: l).Inductive no_dup_list (A : Set) : list A -> Prop :=
  | no_dup_nil : no_dup_list A nil
  | no_dup_cons :
      forall (a : A) (l : list A),
      ~ In a l -> no_dup_list _ l -> no_dup_list _ (a :: l).

Lemma ad_S_compare : forall x y : ad, BDDcompare x y = BDDcompare (ad_S x) (ad_S y).