propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / ynat_lminus.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/nat_minus.ma".
+include "ground/arith/ynat_pred.ma".
+
+(* LEFT SUBTRACTION FOR NON-NEGATIVE INTEGERS WITH INFINITY *****************)
+
+(*** yminus_sn *)
+definition ylminus (x) (n): ynat ≝
+           ypred^n x.
+
+interpretation
+  "left minus (non-negative integers with infinity)"
+  'minus x n = (ylminus x n).
+
+(* Basic constructions ******************************************************)
+
+(*** yminus_O2 *)
+lemma ylminus_zero_dx (x:ynat): x = x - 𝟎 .
+// qed.
+
+(*** yminus_pred1 *)
+lemma yminus_pred_sn (x) (n): ↓(x-n) = ↓x - n.
+#x #n @(niter_appl … ypred)
+qed.
+
+(*** yminus_succ2 yminus_S2 *)
+lemma ylminus_succ_dx (x:ynat) (n): ↓(x-n) = x - ↑n.
+#x #n @(niter_succ … ypred)
+qed.
+
+(*** yminus_SO2 *)
+lemma ylminus_one_dx (x): ↓x = x - (𝟏).
+// qed.
+
+(*** yminus_Y_inj *)
+lemma ylminus_inf_sn (n): ∞ = ∞ - n.
+#n @(nat_ind_succ … n) -n //
+qed.
+
+(* Constructions with nminus ************************************************)
+
+(*** yminus_inj *)
+lemma ylminus_inj_sn (m) (n): yinj_nat (m - n) = yinj_nat m - n.
+#m #n
+@(niter_compose ???? yinj_nat)
+@ypred_inj
+qed.
+
+(* Advanced constructions ***************************************************)
+
+(* yminus_O1 *)
+lemma ylminus_zero_sn (n): 𝟎 = 𝟎 - n.
+// qed.
+
+(*** yminus_refl *)
+lemma ylminus_refl (n): 𝟎 = yinj_nat n - n.
+// qed.
+
+(*** yminus_minus_comm *)
+lemma ylminus_minus_comm (x) (n) (o):
+      x - n - o = x - o - n.
+#x @(ynat_split_nat_inf … x) -x //
+qed.