ground_2 released and permanently renamed as ground

[helm.git] / matita / matita / contribs / lambdadelta / ground / etc / ynat / ynat_max.etc

diff --git a/matita/matita/contribs/lambdadelta/ground/etc/ynat/ynat_max.etc b/matita/matita/contribs/lambdadelta/ground/etc/ynat/ynat_max.etc
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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_2/ynat/ynat_plus.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+lemma ymax_pre_dx: ∀x,y. x ≤ y → x - y + y = y.
+#x #y * -x -y //
+#x #y #Hxy >yminus_inj >(eq_minus_O … Hxy) -Hxy //
+qed-.
+
+lemma ymax_pre_sn: ∀x,y. y ≤ x → x - y + y = x.
+#x #y * -x -y
+[ #x #y #Hxy >yminus_inj /3 width=3 by plus_minus, eq_f/
+| * //
+]
+qed-.
+
+lemma ymax_pre_i_dx: ∀y,x. y ≤ x - y + y.
+// qed.
+
+lemma ymax_pre_i_sn: ∀y,x. x ≤ x - y + y.
+* // #y * /2 width=1 by yle_inj/
+qed.
+
+lemma ymax_pre_e: ∀x,z. x ≤ z → ∀y. y ≤ z → x - y + y ≤ z.
+#x #z #Hxz #y #Hyz elim (yle_split x y)
+[ #Hxy >(ymax_pre_dx … Hxy) -x //
+| #Hyx >(ymax_pre_sn … Hyx) -y //
+]
+qed.
+
+lemma ymax_pre_dx_comm: ∀x,y. x ≤ y → y + (x - y) = y.
+/2 width=1 by ymax_pre_dx/ qed-.
+
+lemma ymax_pre_sn_comm: ∀x,y. y ≤ x → y + (x - y) = x.
+/2 width=1 by ymax_pre_sn/ qed-.
+
+lemma ymax_pre_i_dx_comm: ∀y,x. y ≤ y + (x - y).
+// qed.
+
+lemma ymax_pre_i_sn_comm: ∀y,x. x ≤ y + (x - y).
+/2 width=1 by ymax_pre_i_sn/ qed.
+
+lemma ymax_pre_e_comm: ∀x,z. x ≤ z → ∀y. y ≤ z → y + (x - y) ≤ z.
+/2 width=1 by ymax_pre_e/ qed.