qed.
theorem symmetric_times : symmetric nat times.
-simplify.
+unfold symmetric.
intros.elim x.
simplify.apply times_n_O.
simplify.rewrite > H.apply times_n_Sm.
symmetric_times.
theorem distributive_times_plus : distributive nat times plus.
-simplify.
+unfold distributive.
intros.elim x.
simplify.reflexivity.
simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
\def distributive_times_plus.
theorem associative_times: associative nat times.
-simplify.intros.
+unfold associative.intros.
elim x.simplify.apply refl_eq.
simplify.rewrite < sym_times.
rewrite > distr_times_plus.