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_mult_simpl_l : forall a b c : Z, (0 < c)%Z -> (c * a < c * b)%Z -> (a < b)%Z.