+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/RELATIONAL/NLE/fwd".
+
+include "logic/connectives.ma".
+
+include "Nat/fwd.ma".
+include "NLE/defs.ma".
+
+theorem nle_gen_succ_1: \forall x,y. x < y \to
+ \exists z. y = succ z \land x <= z.
+ intros. inversion H; clear H; intros;
+ [ apply (eq_gen_succ_zero ? ? H)
+ | lapply linear eq_gen_succ_succ to H2 as H0.
+ rewrite > H0. clear H0.
+ apply ex_intro; [|auto] (**)
+ ].
+qed.
+
+theorem nle_gen_succ_succ: \forall x,y. x < succ y \to x <= y.
+ intros; inversion H; clear H; intros;
+ [ apply (eq_gen_succ_zero ? ? H)
+ | lapply linear eq_gen_succ_succ to H2 as H0.
+ lapply linear eq_gen_succ_succ to H3 as H2.
+ rewrite > H0. rewrite > H2. clear H0 H2.
+ auto
+ ].
+qed.
+
+theorem nle_gen_succ_zero: \forall (P:Prop). \forall x. x < zero \to P.
+ intros.
+ lapply linear nle_gen_succ_1 to H. decompose.
+ apply (eq_gen_zero_succ ? ? H1).
+qed.
+
+theorem nle_gen_zero_2: \forall x. x <= zero \to x = zero.
+ intros 1. elim x; clear x; intros;
+ [ auto
+ | apply (nle_gen_succ_zero ? ? H1)
+ ].
+qed.
+
+theorem nle_gen_succ_2: \forall y,x. x <= succ y \to
+ x = zero \lor \exists z. x = succ z \land z <= y.
+ intros 2; elim x; clear x; intros;
+ [ auto
+ | lapply linear nle_gen_succ_succ to H1.
+ right. apply ex_intro; [|auto] (**)
+ ].
+qed.