milestone update in ground, partial commit

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_pred_succ.ma b/matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_pred_succ.ma
new file mode 100644 (file)
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+++ b/matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_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_lt_pred.ma".
+
+(* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
+
+(* Constructions with ypred and ysucc ***************************************)
+
+(*** ylt_O1 *)
+lemma ylt_zero_sn (y): y = ↑↓y → 𝟎 < y.
+#y @(ynat_split_nat_inf … y) -y
+/4 width=1 by ylt_inj, eq_inv_yinj_nat_bi, nlt_zero_sn/
+qed.
+
+(* Destructions with ypred and ysucc ****************************************)
+
+(*** ylt_inv_O1 *)
+lemma ylt_des_gen_dx (x) (y): x < y → y = ↑↓y.
+#x #y * //
+#m #n #H
+lapply (nlt_des_gen … H) -H //
+qed-.
+
+lemma ylt_des_succ_sn (x) (y):
+      ↑x < y → x < ↓y.
+#x #y @(insert_eq_1 … (↑x))
+#x0 * -x0 -y
+[ #m0 #n #Hn #H
+  elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct
+  elim (nlt_inv_succ_sn … Hn) -Hn #Hm #_
+  /2 width=1 by ylt_inj/
+| #m0 #H
+  elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct //  
+]
+qed-.
+
+(* Inversions with ypred and ysucc ******************************************)
+
+(*** ylt_inv_succ1 *)
+lemma ylt_inv_succ_sn (x) (y):
+      ↑x < y → ∧∧ x < ↓y & y = ↑↓y.
+/3 width=2 by ylt_des_succ_sn, ylt_des_gen_dx, conj/ qed-.