apply assoc_plus.
qed.
-theorem times_n_SO : \forall n:nat. eq nat n (times n (S O)).
+theorem times_n_SO : \forall n:nat. n = n * S O.
intros.
rewrite < times_n_Sm.
rewrite < times_n_O.
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.
-apply eq_f.rewrite < assoc_plus. rewrite < sym_plus ? z.
+apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
rewrite > assoc_plus.reflexivity.
qed.
\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.
rewrite < sym_times.
-rewrite < sym_times (times n y) z.
+rewrite < (sym_times (times n y) z).
rewrite < H.apply refl_eq.
qed.