].
theorem eq_gen_S_O: \forall x. (S x = O) \to \forall P:Prop. P.
-intros. apply False_ind. cut (is_S O). auto paramodulation. elim H. exact I.
+intros. apply False_ind. cut (is_S O). auto new. elim H. exact I.
qed.
theorem eq_gen_S_O_cc: (\forall P:Prop. P) \to \forall x. (S x = O).
-intros. auto.
+intros. auto new.
qed.
theorem eq_gen_S_S: \forall m,n. (S m) = (S n) \to m = n.
(\exists n. x = (S n) \land (le m n)).
intros 4. elim H.
apply eq_gen_S_O. exact m. elim H1. auto paramodulation.
-cut (n = m). elim Hcut. apply ex_intro. exact n1. auto paramodulation. auto. (* paramodulation non trova la prova *)
+cut (n = m). elim Hcut. apply ex_intro. exact n1. auto paramodulation. auto new. (* paramodulation non trova la prova *)
qed.
theorem le_gen_S_x: \forall m,x. (le (S m) x) \to