(*Recursive Definition addZ : Z -> Z -> Z :=
      OZ   y => y
    | (pos O)   y => (succZ y)
    | (pos (S n1))   y => (succZ (addZ (pos n1) y))
    | (neg O)   y => (predZ y)
    | (neg (S n1))   y => (predZ (addZ (neg n1) y)).
*)

Fixpoint add1 (x2 : Z) (n : nat) {struct n} : Z :=
  match n with
  | O => succZ x2
  | S n0 => succZ (add1 x2 n0)
  end
 
 with add2 (x2 : Z) (n : nat) {struct n} : Z :=
  match n with
  | O => predZ x2
  | S n0 => predZ (add2 x2 n0)
  end.Definition addZ (x1 x2 : Z) : Z :=
  match x1 with
  | OZ => x2
  | pos n => add1 x2 n
  | neg n => add2 x2 n
  end.

Lemma pred_addZ_r : forall x y : Z, addZ x (predZ y) = predZ (addZ x y).