some properties of NLE

[helm.git] / helm / software / matita / contribs / RELATIONAL / NLE / props.ma

diff --git a/helm/software/matita/contribs/RELATIONAL/NLE/props.ma b/helm/software/matita/contribs/RELATIONAL/NLE/props.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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/RELATIONAL/NLE/props".
+
+include "NLE/fwd.ma".
+
+theorem nle_refl: \forall x. x <= x.
+ intros 1; elim x; clear x; intros; auto.
+qed.
+
+theorem nle_trans_succ: \forall x,y. x <= y \to x <= succ y.
+ intros. elim H; clear H x y; intros; auto.
+qed.
+
+theorem nle_lt_or_eq: \forall y,x. x <= y \to x < y \lor x = y.
+ intros 1. elim y; clear y; intros;
+ [ lapply linear nle_gen_zero_2 to H. auto
+ | lapply linear nle_gen_succ_2 to H1. decompose;
+   [ rewrite > H1. clear H1. auto
+   | lapply linear H to H3 as H0. decompose;
+     [ rewrite > H1. clear H1 x. auto
+     | rewrite < H. clear H n. auto
+     ]
+   ]
+ ].
+qed.