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 Zle_Zmult_comp_l : forall x y z : Z, (0 <= z)%Z -> (x <= y)%Z -> (z * x <= z * y)%Z.