theorem eq_gen_S_S: \forall m,n. (S m) = (S n) \to m = n.
intros. cut ((pred (S m)) = (pred (S n))).
-assumption. elim H. auto paramodulation.
+assumption. elim H. autobatch.
qed.
theorem eq_gen_S_S_cc: \forall m,n. m = n \to (S m) = (S n).
-intros. elim H. auto paramodulation.
+intros. elim H. autobatch.
qed.
inductive le: nat \to nat \to Prop \def
| le_succ: \forall m, n. (le m n) \to (le (S m) (S n)).
theorem le_refl: \forall x. (le x x).
-intros. elim x; auto new.
+intros. elim x; autobatch.
qed.
theorem le_gen_x_O_aux: \forall x, y. (le x y) \to (y =O) \to
(x = O).
-intros 3. elim H. auto paramodulation. apply eq_gen_S_O. exact n1. auto paramodulation.
+intros 3. elim H. autobatch. apply eq_gen_S_O. exact n1. autobatch.
qed.
theorem le_gen_x_O: \forall x. (le x O) \to (x = O).
-intros. apply le_gen_x_O_aux. exact O. auto paramodulation. auto paramodulation.
+intros. apply le_gen_x_O_aux. exact O. autobatch. autobatch.
qed.
theorem le_gen_x_O_cc: \forall x. (x = O) \to (le x O).
-intros. elim H. auto new.
+intros. elim H. autobatch.
qed.
theorem le_gen_S_x_aux: \forall m,x,y. (le y x) \to (y = S m) \to
(\exists n. x = (S n) \land (le m n)).
intros 4. elim H; clear H x y.
-apply eq_gen_S_O. exact m. elim H1. auto paramodulation.
+apply eq_gen_S_O. exact m. elim H1. autobatch.
clear H2. cut (n = m).
elim Hcut. apply ex_intro. exact n1. split; autobatch.
apply eq_gen_S_S. autobatch.
theorem le_gen_S_x: \forall m,x. (le (S m) x) \to
(\exists n. x = (S n) \land (le m n)).
-intros. apply le_gen_S_x_aux. exact (S m). auto paramodulation. auto paramodulation.
+intros. apply le_gen_S_x_aux. exact (S m). autobatch. autobatch.
qed.
theorem le_gen_S_x_cc: \forall m,x. (\exists n. x = (S n) \land (le m n)) \to
(le (S m) x).
-intros. elim H. elim H1. cut ((S x1) = x). elim Hcut. auto new.
-elim H2. auto paramodulation.
+intros. elim H. elim H1. cut ((S x1) = x). elim Hcut. autobatch.
+elim H2. autobatch.
qed.
theorem le_gen_S_S: \forall m,n. (le (S m) (S n)) \to (le m n).
qed.
theorem le_gen_S_S_cc: \forall m,n. (le m n) \to (le (S m) (S n)).
-intros. auto new.
+intros. autobatch.
qed.
(*
theorem le_trans: \forall x,y. (le x y) \to \forall z. (le y z) \to (le x z).
intros 1. elim x; clear H. clear x.
-auto paramodulation.
+autobatch.
fwd H1 [H]. decompose.
*)