milestone update in ground, partial commit

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_le_succ.ma b/matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_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_lt_succ.ma".
+include "ground/arith/ynat_lt_le.ma".
+
+(* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
+
+(* Constructions with ysucc *************************************************)
+
+(*** yle_lt yle_succ1_inj *)
+lemma ylt_le_succ_sn (x) (y):
+      x < ∞ → ↑x ≤ y → x < y.
+/3 width=3 by ylt_yle_trans, ylt_succ_dx_refl/ qed.
+
+(* Inversions and destructions on yle ***************************************)
+
+(*** ylt_inv_le *)
+lemma ylt_inv_le_succ_sn (x) (y):
+      x < y → ∧∧ x < ∞ & ↑x ≤ y.
+#x #y * -x -y
+/3 width=1 by yle_inj, conj/
+qed-.
+
+(* Destructions with yle and ysucc ******************************************)
+
+(*** ylt_fwd_le_succ1 *)
+lemma ylt_des_le_succ_sn (x) (y): x < y → ↑x ≤ y.
+#x #y #H
+elim (ylt_inv_le_succ_sn … H) -H #_ //
+qed-.
+
+(*** ylt_fwd_succ2 *)
+lemma ylt_des_succ_dx (x) (y): x < ↑y → x ≤ y.
+#x #y @(ynat_split_nat_inf … y) -y //
+#n <ysucc_inj #H
+elim (ylt_inv_inj_dx … H) -H #m #Hm #H destruct
+/3 width=1 by yle_inj, nlt_inv_succ_dx/
+qed-.