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_Rlt : forall z1 z2 : Z, (IZR z1 < IZR z2)%R -> (z1 < z2)%Z.