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 lt_Zlt_inv : forall n m : nat, (Z_of_nat n < Z_of_nat m)%Z -> n < m.