Fixpoint Qhomographic_Qpositive_to_Qpositive (a b c d : Z) 
 (p : Qpositive) (hyp : homographicAcc a b c d p) {struct hyp} : Qpositive :=
  match Qpositive_dec_One p with
  | left H_p_is_One =>
      let H :=
        eq_ind p (fun p : Qpositive => homographicAcc a b c d p) hyp One
          H_p_is_One in
      (fun hyp0 : homographicAcc a b c d One =>
       (fun (Hab : (0 < a + b)%Z) (Hcd : (0 < c + d)%Z) =>
        positive_fraction_encoding (a + b) (c + d) Hab Hcd)
         (homographicacc_0_num a b c d hyp0)
         (homographicacc_0_denom a b c d hyp0)) H
  | right H_p_not_One =>
      match top_more_informative a b c d with
      | left H_abcd =>
          nR
            (Qhomographic_Qpositive_to_Qpositive (a - c)%Z 
               (b - d)%Z c d p
               (homographicacc_1 a b c d p hyp H_p_not_One H_abcd))
      | right H_abcd =>
          match top_more_informative c d a b with
          | left H_cdab =>
              dL
                (Qhomographic_Qpositive_to_Qpositive a b 
                   (c - a)%Z (d - b)%Z p
                   (homographicacc_2 a b c d p hyp H_p_not_One H_abcd H_cdab))
          | right H_cdab =>
              match p as q return (p = q -> Qpositive) with
              | nR q =>
                  fun H : p = nR q =>
                  Qhomographic_Qpositive_to_Qpositive a 
                    (a + b)%Z c (c + d)%Z q
                    (homographicacc_3 a b c d p hyp q H H_abcd H_cdab)
              | dL q =>
                  fun H : p = dL q =>
                  Qhomographic_Qpositive_to_Qpositive 
                    (a + b)%Z b (c + d)%Z d q
                    (homographicacc_3' a b c d p hyp q H H_abcd H_cdab)
              | One =>
                  fun q : p = One =>
                  False_rec Qpositive (False_ind False (H_p_not_One q))
              end (refl_equal p)
          end
      end
  end.(** some sort of lexicographical order on binary lists *)

Fixpoint Qpositive_length (qp : Qpositive) : nat :=
  match qp with
  | One => 0
  | dL qp1 => S (Qpositive_length qp1)
  | nR qp1 => S (Qpositive_length qp1)
  end.Definition bin_lt (qp1 qp2 : Qpositive) : Prop :=
  Qpositive_length qp1 < Qpositive_length qp2.Definition bin_eq (x y : Qpositive) := x = y.

Lemma compare_nR : forall x : Qpositive, bin_lt x (nR x).