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 ZleAbs : forall (z : Z) (n : nat), (- Z_of_nat n <= z)%Z -> (z <= Z_of_nat n)%Z -> Zabs_nat z <= n.