milestone update in ground, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / ynat_le_succ.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/ynat_le_succ.ma b/matita/matita/contribs/lambdadelta/ground/arith/ynat_le_succ.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 "ground/arith/ynat_succ.ma".
+include "ground/arith/ynat_le.ma".
+
+(* ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY ****************************)
+
+(* Constructions with ysucc *************************************************)
+
+(*** yle_succ *)
+lemma yle_succ_bi (x) (y): x ≤ y → ↑x ≤ ↑y.
+#x #y * -x -y
+/3 width=1 by yle_inj, yle_inf, nle_succ_bi/
+qed.
+
+(*** yle_succ_dx *)
+lemma yle_succ_dx (x) (y): x ≤ y → x ≤ ↑y.
+#x #y * -x -y
+/3 width=1 by yle_inj, yle_inf, nle_succ_dx/
+qed.
+
+(*** yle_refl_S_dx *)
+lemma yle_succ_dx_refl (x): x ≤ ↑x.
+/2 width=1 by yle_succ_dx/ qed.
+
+(* Inversions with ysucc ****************************************************)
+
+(*** yle_inv_succ *)
+lemma yle_inv_succ_bi (x) (y): ↑x ≤ ↑y → x ≤ y.
+#x #y @(ynat_split_nat_inf … y) -y //
+#n <ysucc_inj #H
+elim (yle_inv_inj_dx … H) -H #o #Hmn #H
+elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct
+/3 width=1 by yle_inj, nle_inv_succ_bi/
+qed-.