milestone update in ground, partial commit

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_succ.ma b/matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_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.ma".
+
+(* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
+
+(* Constructions with ysucc *************************************************)
+
+(*** ylt_O_succ *)
+lemma ylt_zero_succ (y): 𝟎 < ↑y.
+#y @(ynat_split_nat_inf … y) -y
+/2 width=1 by ylt_inj/
+qed.
+
+(*** ylt_succ *)
+lemma ylt_succ_bi (x) (y): x < y → ↑x < ↑y.
+#x #y * -x -y
+/3 width=1 by ylt_inj, ylt_inf, nlt_succ_bi/
+qed.
+
+(*** ylt_succ_Y *)
+lemma ylt_succ_inf (x): x < ∞ → ↑x < ∞.
+#x @(ynat_split_nat_inf … x) -x //
+qed.
+
+(*** ylt_succ2_refl *)
+lemma ylt_succ_dx_refl (x) (y): x < y → x < ↑x.
+#x #y #H
+elim (ylt_des_gen_sn … H) -y #n #H destruct
+/2 width=1 by ylt_inj/
+qed.
+
+(* Inversions with ysucc ****************************************************)
+
+lemma ylt_inv_succ_inf (x): ↑x < ∞ → x < ∞.
+#x #H
+elim (ylt_des_gen_sn … H) -H #m0 #H
+elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct //
+qed-.
+
+(*** ylt_inv_succ *)
+lemma ylt_inv_succ_bi (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 #m0 #Hmn #H
+  elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct
+  /3 width=1 by ylt_inj, nlt_inv_succ_bi/
+| /2 width=1 by ylt_inv_succ_inf/
+]
+qed-.