Fixpoint PdivlessAux (bound p base length : positive) {struct length}
 : Option positive * Option positive * nat :=
  match Pcompare bound p Datatypes.Eq with
  | Datatypes.Gt => (Some _ p, None _, 0)
  | _ =>
      match PdivAux p base with
      | (None, None) => (None _, None _, 1)
      | (None, Some r1) => (None _, Some _ r1, 1)
      | (Some q1, None) =>
          match length with
          | xH => (Some _ q1, None _, 0)
          | xO length' =>
              match PdivlessAux bound q1 base length' with
              | (s2, None, n) => (s2, None _, S n)
              | (s2, Some r2, n) =>
                  (s2,
                  Some _
                    ((fun (x : positive)
                        (_ : positive -> positive)
                        (y : positive) => (x * y)%positive) r2
                       (fun x => x) base), S n)
              end
          | xI length' =>
              match PdivlessAux bound q1 base length' with
              | (s2, None, n) => (s2, None _, S n)
              | (s2, Some r2, n) =>
                  (s2,
                  Some _
                    ((fun (x : positive)
                        (_ : positive -> positive)
                        (y : positive) => (x * y)%positive) r2
                       (fun x => x) base), S n)
              end
          end
      | (Some q1, Some r1) =>
          match length with
          | xH => (Some _ q1, None _, 0)
          | xO length' =>
              match PdivlessAux bound q1 base length' with
              | (s2, None, n) => (s2, Some _ r1, S n)
              | (s2, Some r2, n) =>
                  (s2,
                  Some _
                    ((fun (x : positive)
                        (_ : positive -> positive)
                        (y : positive) => x * y) r2 (
                       fun x => x) base + r1)%positive, 
                  S n)
              end
          | xI length' =>
              match PdivlessAux bound q1 base length' with
              | (s2, None, n) => (s2, Some _ r1, S n)
              | (s2, Some r2, n) =>
                  (s2,
                  Some _
                    ((fun (x : positive)
                        (_ : positive -> positive)
                        (y : positive) => x * y) r2 (
                       fun x => x) base + r1)%positive, 
                  S n)
              end
          end
      end
  end.Definition Pdivless := PdivlessAux Pdiv.

Theorem odd_even_lem : forall p q, 2 * p + 1 <> 2 * q.