milestone update in ground, partial commit

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/ynat_le_pred_succ.ma b/matita/matita/contribs/lambdadelta/ground/arith/ynat_le_pred_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_pred.ma".
+
+(* ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY ****************************)
+
+(* Constructions with ypred and ysucc ***************************************)
+
+(*** yle_refl_SP_dx *)
+lemma yle_succ_pred_dx_refl (x): x ≤ ↑↓x.
+#x @(ynat_split_nat_inf … x) -x
+/2 width=1 by yle_inj/
+qed.
+
+(*** yle_inv_succ2 *)
+lemma yle_pred_sn (x) (y): x ≤ ↑y → ↓x ≤ y.
+#x #y0 @(insert_eq_1 … (↑y0))
+#y * -x -y
+[ #m #n0 #Hmn #H
+  elim (eq_inv_ysucc_inj … H) -H #n #H1 #H2 destruct
+  /3 width=1 by yle_inj, nle_pred_sn/
+| #x0 #H <(eq_inv_ysucc_inf … H) -y0 //
+]
+qed.
+
+(* Inversions with ypred and ysucc ******************************************)
+
+(*** yle_succ2 *)
+lemma yle_inv_pred_sn (x) (y): ↓x ≤ y → x ≤ ↑y.
+#x0 #y @(insert_eq_1 … (↓x0))
+#x * -x -y // #m0 #n #Hmn #H
+elim (eq_inv_ypred_inj … H) -H #m #H1 #H2 destruct
+/3 width=1 by yle_inj, nle_inv_pred_sn/
+qed-.
+
+(*** yle_inv_succ1 *)
+lemma yle_inv_succ_sn (x) (y):
+      ↑x ≤ y → ∧∧ x ≤ ↓y & y = ↑↓y.
+#x0 #y @(insert_eq_1 … (↑x0))
+#x * -x -y
+[ #m0 #n #Hmn #H
+  elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct
+  elim (nle_inv_succ_sn … Hmn) -Hmn #Hmn #Hn
+  /3 width=1 by yle_inj, conj/
+| /2 width=1 by yle_inf, conj/
+]
+qed-.