\def injective_S.
theorem not_eq_S : \forall n,m:nat.
-Not (n=m) \to Not (S n = S m).
+\lnot n=m \to \lnot (S n = S m).
intros. simplify. intros.
apply H. apply injective_S. assumption.
qed.
[ O \Rightarrow False
| (S p) \Rightarrow True ].
-theorem not_eq_O_S : \forall n:nat. Not (O=S n).
+theorem not_eq_O_S : \forall n:nat. \lnot O=S n.
intros. simplify. intros.
cut (not_zero O).
exact Hcut.
rewrite > H.exact I.
qed.
-theorem not_eq_n_Sn : \forall n:nat. Not (n=S n).
+theorem not_eq_n_Sn : \forall n:nat. \lnot n=S n.
intros.elim n.
apply not_eq_O_S.
apply not_eq_S.assumption.