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 Zlt_absolu : forall (x : Z) (n : nat), Zabs_nat x < n -> (x < Z_of_nat n)%Z.