(** * A direct way to compute Z.modulo *)

Fixpoint Zmod_POS (a : positive) (b : Z) : Z  :=
  match a with
   | xI a' =>
      let r := Zmod_POS a' b in
      let r' := (2 * r + 1) in
      if r' <? b then r' else (r' - b)
   | xO a' =>
      let r := Zmod_POS a' b in
      let r' := (2 * r) in
      if r' <? b then r' else (r' - b)
   | xH => if 2 <=? b then 1 else 0
  end.Definition Zmod' a b :=
  match a with
   | Z0 => 0
   | Zpos a' =>
      match b with
       | Z0 => a
       | Zpos _ => Zmod_POS a' b
       | Zneg b' =>
          let r := Zmod_POS a' (Zpos b') in
          match r  with Z0 =>  0 |  _  =>   b + r end
      end
   | Zneg a' =>
      match b with
       | Z0 => a
       | Zpos _ =>
          let r := Zmod_POS a' b in
          match r with Z0 =>  0 | _  =>  b - r end
       | Zneg b' => - (Zmod_POS a' (Zpos b'))
      end
  end.

Theorem Zdiv_eucl_extended : forall b:Z, b <> 0 -> forall a:Z, {qr : Z * Z | let (q, r) := qr in a = b * q + r /\ 0 <= r < Z.abs b}.