propagating the arithmetics library, partial commit

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/ynat_le_lminus.ma b/matita/matita/contribs/lambdadelta/ground/arith/ynat_le_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_le_minus.ma".
+include "ground/arith/ynat_lminus.ma".
+include "ground/arith/ynat_le.ma".
+
+(* ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY ****************************)
+
+(* Constructions with ylminus ***********************************************)
+
+(*** yle_minus_sn *)
+lemma yle_lminus_sn_refl_sn (x) (n): x - n ≤ x.
+#x @(ynat_split_nat_inf … x) -x //
+#m #n /2 width=1 by yle_inj/
+qed.
+
+(*** monotonic_yle_minus_dx *)
+lemma yle_lminus_bi_dx (o) (x) (y):
+      x ≤ y → x - o ≤ y - o.
+#o #x #y *
+/3 width=1 by nle_minus_bi_dx, yle_inj, yle_inf/
+qed.
+
+(*** yminus_to_le *)
+lemma yle_eq_zero_lminus (x) (n): 𝟎 = x - n → x ≤ yinj_nat n.
+#x @(ynat_split_nat_inf … x) -x
+[ #m #n <ylminus_inj_sn >yinj_nat_zero #H
+  /4 width=1 by nle_eq_zero_minus, yle_inj, eq_inv_yinj_nat_bi/
+| #n <ylminus_inf_sn #H destruct
+]
+qed.
+
+(* Inversions with ylminus **************************************************)
+
+(*** yle_to_minus *)
+lemma yle_inv_eq_zero_lminus (x) (n):
+      x ≤ yinj_nat n → 𝟎 = x - n.
+#x @(ynat_split_nat_inf … x) -x
+[ #m #n #H <ylminus_inj_sn
+  <nle_inv_eq_zero_minus /2 width=1 by yle_inv_inj_bi/
+| #n #H
+  lapply (yle_inv_inf_sn … H) -H #H
+  elim (eq_inv_inf_yinj_nat … H)
+]
+qed-.