milestone update in ground, partial commit

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_plus_pred.ma b/matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_plus_pred.ma
new file mode 100644 (file)
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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_pred_succ.ma".
+include "ground/arith/ynat_plus.ma".
+include "ground/arith/ynat_lt_pred_succ.ma".
+
+(* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
+
+(* Inversions with yplus and ypred ******************************************)
+
+(*** yplus_inv_succ_lt_dx *)
+lemma eq_inv_succ_yplus_lt_dx (z) (x) (y):  𝟎 < y → ↑z = x + y → z = x + ↓y.
+#z #x #y #Hy >(ylt_des_gen_dx … Hy) -Hy
+<yplus_succ_dx <ypred_succ
+/2 width=1 by eq_inv_ysucc_bi/
+qed-.
+
+(*** yplus_inv_succ_lt_sn *)
+lemma eq_inv_succ_yplus_lt_sn (z) (x) (y): 𝟎 < x → ↑z = x + y → z = ↓x + y.
+/2 width=1 by eq_inv_succ_yplus_lt_dx/ qed-.
+
+(* Destructions with yplus and ypred ****************************************)
+
+(*** yplus_pred1 *)
+lemma yplus_pred_sn (x) (y): 𝟎 < x → ↓(x+y) = ↓x + y.
+#x #y #Hx >(ylt_des_gen_dx … Hx) -Hx
+<yplus_succ_sn <ypred_succ <ypred_succ //
+qed-.
+
+(*** yplus_pred2 *)
+lemma yplus_pred_dx (x) (y): 𝟎 < y → x + ↓y = ↓(x+y).
+/2 width=1 by yplus_pred_sn/ qed-.
+