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[helm.git] / matitaB / 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: ∀x. x ≤ x.
+ intros; elim x; clear x; autobatch.
+qed.
+
+theorem nle_trans: ∀x,y. x ≤ y → ∀z. y ≤ z → 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: ∀x,y. x ≤ y → y < x → False.
+ intros 3; elim H; clear H x y; autobatch.
+qed.
+
+theorem nle_irrefl: ∀x. x < x → False.
+ intros. autobatch.
+qed.
+
+theorem nle_irrefl_ei: ∀x, z. z ≤ x → z = succ x → False.
+ intros 3; elim H; clear H x z; destruct; autobatch.
+qed.
+
+theorem nle_irrefl_smart: ∀x. x < x → False.
+ intros 1. elim x; clear x; autobatch.
+qed.
+
+theorem nle_lt_or_eq: ∀y, x. x ≤ y → x < y ∨ x = y.
+ intros; elim H; clear H x y;
+ [ elim n; clear n
+ | decompose
+ ]; autobatch.
+qed.