(**************************************************************************)
set "baseuri" "cic:/matita/tests/fguidi/".
-include "legacy/coq.ma".
+include "../legacy/coq.ma".
alias id "O" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/1)".
alias id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)".
].
theorem eq_gen_S_O: \forall x. (S x = O) \to \forall P:Prop. P.
-intros. apply False_ind. cut (is_S O). auto new. elim H. exact I.
+intros. apply False_ind. cut (is_S O). elim Hcut. rewrite < H. apply I.
qed.
theorem eq_gen_S_O_cc: (\forall P:Prop. P) \to \forall x. (S x = O).
-intros. auto new.
+intros. apply H.
qed.
theorem eq_gen_S_S: \forall m,n. (S m) = (S n) \to m = n.
| 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 paramodulation. auto paramodulation.
+intros. elim x; auto new.
qed.
theorem le_gen_x_O_aux: \forall x, y. (le x y) \to (y =O) \to
qed.
theorem le_gen_x_O_cc: \forall x. (x = O) \to (le x O).
-intros. elim H. auto paramodulation.
+intros. elim H. auto new.
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.
+intros 4. elim H; clear H x y.
apply eq_gen_S_O. exact m. elim H1. auto paramodulation.
-cut (n = m). elim Hcut. apply ex_intro. exact n1. auto paramodulation. auto new. (* paramodulation non trova la prova *)
+clear H2. cut (n = m).
+elim Hcut. apply ex_intro. exact n1. split; auto.
+apply eq_gen_S_S. auto.
qed.
theorem le_gen_S_x: \forall m,x. (le (S m) x) \to
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 paramodulation. elim H2. auto paramodulation.
+intros. elim H. elim H1. cut ((S x1) = x). elim Hcut. auto new.
+elim H2. auto paramodulation.
qed.
theorem le_gen_S_S: \forall m,n. (le (S m) (S n)) \to (le m n).
intros.
-lapply le_gen_S_x to H using H0. elim H0. elim H1.
-lapply eq_gen_S_S to H2 using H4. rewrite > H4. assumption.
+lapply le_gen_S_x to H as H0. elim H0. elim H1.
+lapply eq_gen_S_S to H2 as H4. rewrite > H4. assumption.
qed.
theorem le_gen_S_S_cc: \forall m,n. (le m n) \to (le (S m) (S n)).
-intros. auto paramodulation.
+intros. auto new.
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.
-fwd H1 [H]. decompose H.
+fwd H1 [H]. decompose.
*)