propagating the arithmetics library, partial commit

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_lminus.ma b/matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_lminus.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/nat_lt_minus.ma".
+include "ground/arith/ynat_lminus.ma".
+include "ground/arith/ynat_lt.ma".
+
+(* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
+
+(* Constructions with ylminus ***********************************************)
+
+(*** ylt_to_minus *)
+lemma ylt_zero_lminus (m) (y):
+      yinj_nat m < y → 𝟎 < y - m.
+#m #y @(ynat_split_nat_inf … y) -y //
+#n #Hmn <ylminus_inj_sn >yinj_nat_zero >(nminus_refl m)
+/4 width=1 by ylt_inj, ylt_inv_inj_bi, nlt_minus_bi_dx/
+qed.
+
+(* Inversions with ylminus **************************************************)
+
+(*** yminus_to_lt *)
+lemma ylt_inv_zero_lminus (m) (y):
+      (𝟎) < y - m → yinj_nat m < y.
+#m #y @(ynat_split_nat_inf … y) -y //
+#n <ylminus_inj_sn >yinj_nat_zero >(nminus_refl m) #Hmm
+/4 width=2 by ylt_inv_inj_bi, ylt_inj, nlt_inv_minus_bi_dx/
+qed-.
+
+(* Destructions with ylminus ************************************************)
+
+(*** yminus_pred *)
+lemma ylminus_pred_bi (x:ynat) (n):
+      (𝟎) < x → 𝟎 < n → x - n = ↓x - ↓n.
+#x @(ynat_split_nat_inf … x) -x //
+#m #n >yinj_nat_zero
+#Hm #Hn <ylminus_inj_sn <ypred_inj <ylminus_inj_sn
+<nminus_pred_bi /2 width=1 by ylt_inv_inj_bi/
+qed-.