theorem injective_S : \forall n,m:nat.
(eq nat (S n) (S m)) \to (eq nat n m).
intros;
-rewrite > pred_Sn n;
-rewrite > pred_Sn m;
+rewrite > pred_Sn;
+rewrite > pred_Sn m.
apply f_equal; assumption.
qed.
theorem sym_times :
\forall n,m:nat. eq nat (times n m) (times m n).
intros.elim n.simplify.apply times_n_O.
-simplify.rewrite < sym_eq ? ? ? H.apply times_n_Sm.
+simplify.rewrite > H.apply times_n_Sm.
qed.
let rec minus n m \def