- subtraction (and related notions) removed

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / ynat / ynat_max.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_max.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_max.ma
deleted file mode 100644 (file)
index acbd31d..0000000
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_max.ma
+++ /dev/null
@@ -1,58 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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.