milestone update in ground_2 and basic_2A

[helm.git] / matita / matita / contribs / lambdadelta / ground_2A / ynat / ynat_minus.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_minus.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_minus.ma
deleted file mode 100644 (file)
index e4763f4..0000000
--- a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_minus.ma
+++ /dev/null
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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_2A/ynat/ynat_lt.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-(* subtraction *)
-definition yminus: ynat → ynat → ynat ≝ λx,y. match y with
-[ yinj n ⇒ ypred^n x
-| Y      ⇒ yinj 0
-].
-
-interpretation "ynat minus" 'minus x y = (yminus x y).
-
-(* Basic properties *********************************************************)
-
-lemma yminus_inj: ∀n,m. yinj m - yinj n = yinj (m - n).
-#n elim n -n /2 width=3 by trans_eq/
-qed.
-
-lemma yminus_Y_inj: ∀n. ∞ - yinj n = ∞.
-#n elim n -n // normalize
-#n #IHn >IHn //
-qed.
-
-lemma yminus_O1: ∀x:ynat. 0 - x = 0.
-* // qed.
-
-lemma yminus_refl: ∀x:ynat. x - x = 0.
-* // qed.
-
-lemma yminus_minus_comm: ∀y,z,x. x - y - z = x - z - y.
-* #y [ * #z [ * // ] ] >yminus_O1 //
-qed.
-
-(* Properties on predecessor ************************************************)
-
-lemma yminus_SO2: ∀m. m - 1 = ⫰m.
-* //
-qed.
-
-lemma yminus_pred: ∀n,m. 0 < m → 0 < n → ⫰m - ⫰n = m - n.
-* // #n *
-[ #m #Hm #Hn >yminus_inj >yminus_inj
-  /4 width=1 by ylt_inv_inj, minus_pred_pred, eq_f/
-| >yminus_Y_inj //
-]
-qed-.
-
-(* Properties on successor **************************************************)
-
-lemma yminus_succ: ∀n,m. ⫯m - ⫯n = m - n.
-* // #n * [2: >yminus_Y_inj // ]
-#m >yminus_inj //
-qed.
-
-lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m → ⫯m - n = ⫯(m - n).
-#n *
-[ #m #Hmn >yminus_inj >yminus_inj
-  /4 width=1 by yle_inv_inj, plus_minus, eq_f/
-| >yminus_Y_inj //
-]
-qed-.
-
-lemma yminus_succ2: ∀y,x. x - ⫯y = ⫰(x-y).
-* //
-qed.
-
-(* Properties on order ******************************************************)
-
-lemma yle_minus_sn: ∀n,m. m - n ≤ m.
-* // #n * /2 width=1 by yle_inj/
-qed.
-
-lemma yle_to_minus: ∀m:ynat. ∀n:ynat. m ≤ n → m - n = 0.
-#m #n * -m -n /3 width=3 by eq_minus_O, eq_f/
-qed-.
-
-lemma yminus_to_le: ∀n:ynat. ∀m:ynat. m - n = 0 → m ≤ n.
-* // #n *
-[ #m >yminus_inj #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
-  /2 width=1 by yle_inj/
-| >yminus_Y_inj #H destruct
-]
-qed.
-
-lemma monotonic_yle_minus_dx: ∀x,y. x ≤ y → ∀z. x - z ≤ y - z.
-#x #y #Hxy * //
-#z elim z -z /3 width=1 by yle_pred/
-qed.
-
-(* Properties on strict order ***********************************************)
-
-lemma monotonic_ylt_minus_dx: ∀x,y:ynat. x < y → ∀z:nat. z ≤ x → x - z < y - z.
-#x #y * -x -y
-/4 width=1 by ylt_inj, yle_inv_inj, monotonic_lt_minus_l/
-qed.