Definition Zmax : forall x_ x_ : Z, Z :=
  fun n m : Z =>
  match (n ?= m)%Z with
  | Datatypes.Eq => m
  | Datatypes.Lt => m
  | Datatypes.Gt => n
  end.

Theorem inj_pred : forall n : nat, n <> 0 -> Z_of_nat (pred n) = Zpred (Z_of_nat n).