apply assoc_plus.
qed.
+theorem times_O_to_O: \forall n,m:nat.n*m = O \to n = O \lor m= O.
+apply nat_elim2;intros
+ [left.reflexivity
+ |right.reflexivity
+ |apply False_ind.
+ simplify in H1.
+ apply (not_eq_O_S ? (sym_eq ? ? ? H1))
+ ]
+qed.
+
theorem times_n_SO : \forall n:nat. n = n * S O.
intros.
rewrite < times_n_Sm.
reflexivity.
qed.
+theorem times_SSO_n : \forall n:nat. n + n = S (S O) * n.
+intros.
+simplify.
+rewrite < plus_n_O.
+reflexivity.
+qed.
+
theorem symmetric_times : symmetric nat times.
unfold symmetric.
intros.elim x.