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 Zmax_le1 : forall z1 z2 : Z, (z2 <= z1)%Z -> Zmax z1 z2 = z1.