update in lambdadelta

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

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_pred.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_pred.ma
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+++ b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_pred.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_2A/notation/functions/predecessor_1.ma".
+include "ground_2A/lib/arith.ma".
+include "ground_2A/ynat/ynat.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* the predecessor function *)
+definition ypred: ynat → ynat ≝ λm. match m with
+[ yinj m ⇒ pred m
+| Y      ⇒ Y
+].
+
+interpretation "ynat predecessor" 'Predecessor m = (ypred m).
+
+(* Properties ***************************************************************)
+
+lemma ypred_inj_rew: ∀m:nat. ⫰m = pred m.
+// qed.
+
+(* Inversion lemmas *********************************************************)
+
+lemma ypred_inv_refl: ∀m. ⫰m = m → m = 0 ∨ m = ∞.
+* // #m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
+/4 width=1 by pred_inv_refl, or_introl, eq_f/
+qed-.