Fixpoint Pdiv (p q : positive) {struct p} :
 Option positive * Option positive :=
  match p with
  | xH =>
      match q with
      | xH => (Some _ 1%positive, None _)
      | xO r => (None _, Some _ p)
      | xI r => (None _, Some _ p)
      end
  | xI p' =>
      match Pdiv p' q with
      | (None, None) =>
          match (1 - Zpos q)%Z with
          | Z0 => (Some _ 1%positive, None _)
          | Zpos r' => (Some _ 1%positive, Some _ r')
          | Zneg r' => (None _, Some _ 1%positive)
          end
      | (None, Some r1) =>
          match (Zpos (xI r1) - Zpos q)%Z with
          | Z0 => (Some _ 1%positive, None _)
          | Zpos r' => (Some _ 1%positive, Some _ r')
          | Zneg r' => (None _, Some _ (xI r1))
          end
      | (Some q1, None) =>
          match (1 - Zpos q)%Z with
          | Z0 => (Some _ (xI q1), None _)
          | Zpos r' => (Some _ (xI q1), Some _ r')
          | Zneg r' => (Some _ (xO q1), Some _ 1%positive)
          end
      | (Some q1, Some r1) =>
          match (Zpos (xI r1) - Zpos q)%Z with
          | Z0 => (Some _ (xI q1), None _)
          | Zpos r' => (Some _ (xI q1), Some _ r')
          | Zneg r' => (Some _ (xO q1), Some _ (xI r1))
          end
      end
  | xO p' =>
      match Pdiv p' q with
      | (None, None) => (None _, None _)
      | (None, Some r1) =>
          match (Zpos (xO r1) - Zpos q)%Z with
          | Z0 => (Some _ 1%positive, None _)
          | Zpos r' => (Some _ 1%positive, Some _ r')
          | Zneg r' => (None _, Some _ (xO r1))
          end
      | (Some q1, None) => (Some _ (xO q1), None _)
      | (Some q1, Some r1) =>
          match (Zpos (xO r1) - Zpos q)%Z with
          | Z0 => (Some _ (xI q1), None _)
          | Zpos r' => (Some _ (xI q1), Some _ r')
          | Zneg r' => (Some _ (xO q1), Some _ (xO r1))
          end
      end
  end.Definition oZ h := match h with
                   | None => 0
                   | Some p => nat_of_P p
                   end.

(* Specifications: *)
Theorem Pdiv_correct : forall p q, nat_of_P p = oZ (fst (Pdiv p q)) * nat_of_P q + oZ (snd (Pdiv p q)) /\ oZ (snd (Pdiv p q)) < nat_of_P q.