+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_2A/ynat/ynat_plus.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-fact ymin_pre_dx_aux: ∀x,y. y ≤ x → x - (x - y) ≤ y.
-#x #y * -x -y
-[ #x #y #Hxy >yminus_inj
- /3 width=4 by yle_inj, monotonic_le_minus_l/
-| * //
-]
-qed-.
-
-lemma ymin_pre_sn: ∀x,y. x ≤ y → x - (x - y) = x.
-#x #y * -x -y //
-#x #y #Hxy >yminus_inj >(eq_minus_O … Hxy) -Hxy //
-qed-.
-
-lemma ymin_pre_i_dx: ∀x,y. x - (x - y) ≤ y.
-#x #y elim (yle_split x y) /2 width=1 by ymin_pre_dx_aux/
-#Hxy >(ymin_pre_sn … Hxy) //
-qed.
-
-lemma ymin_pre_i_sn: ∀x,y. x - (x - y) ≤ x.
-// qed.
-
-lemma ymin_pre_dx: ∀x,y. y ≤ yinj x → yinj x - (yinj x - y) = y.
-#x #y #H elim (yle_inv_inj2 … H) -H
-#z #Hzx #H destruct >yminus_inj
-/3 width=4 by minus_le_minus_minus_comm, eq_f/
-qed-.
-
-lemma ymin_pre_e: ∀z,x. z ≤ yinj x → ∀y. z ≤ y →
- z ≤ yinj x - (yinj x - y).
-#z #x #Hzx #y #Hzy elim (yle_split x y)
-[ #H >(ymin_pre_sn … H) -y //
-| #H >(ymin_pre_dx … H) -x //
-]
-qed.