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[helm.git] / matita / contribs / RELATIONAL / NLE / order.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+
+
+include "NLE/inv.ma".
+
+theorem nle_refl: \forall x. x <= x.
+ intros; elim x; clear x; autobatch.
+qed.
+
+theorem nle_trans: \forall x,y. x <= y \to
+                   \forall z. y <= z \to x <= z.
+ intros 3. elim H; clear H x y;
+ [ autobatch
+ | lapply linear nle_inv_succ_1 to H3. decompose. destruct. 
+   autobatch
+ ].
+qed.
+
+theorem nle_false: \forall x,y. x <= y \to y < x \to False.
+ intros 3; elim H; clear H x y; autobatch.
+qed.
+
+theorem nle_irrefl: \forall x. x < x \to False.
+ intros. autobatch.
+qed.
+
+theorem nle_irrefl_ei: \forall x, z. z <= x \to z = succ x \to False.
+ intros 3. elim H; clear H x z; destruct. autobatch.
+qed.
+
+theorem nle_irrefl_smart: \forall x. x < x \to False.
+ intros 1. elim x; clear x; autobatch.
+qed.
+
+theorem nle_lt_or_eq: \forall y, x. x <= y \to x < y \lor x = y.
+ intros. elim H; clear H x y;
+ [ elim n; clear n
+ | decompose
+ ]; autobatch.
+qed.