milestone update in ground, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / ynat.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/ynat.ma b/matita/matita/contribs/lambdadelta/ground/arith/ynat.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/pnat.ma".
+include "ground/notation/functions/zero_0.ma".
+include "ground/notation/functions/infinity_0.ma".
+
+(* NON-NEGATIVE INTEGERS WITH INFINITY **************************************)
+
+(*** ynat *)
+inductive ynat: Type[0] ≝
+| yzero: ynat
+| yinj : pnat → ynat
+| yinf : ynat
+.
+
+coercion yinj.
+
+interpretation
+  "zero (non-negative integers with infinity)"
+  'Zero = yzero.
+
+interpretation
+  "infinity (non-negative integers with infinity)"
+  'Infinity = yinf.
+
+(* Inversion lemmas *********************************************************)
+
+(* Note: destruct *)
+(*** yinj_inj *)
+lemma eq_inv_yinj_bi (y1) (y2): yinj y1 = yinj y2 → y1 = y2.
+#x #y #H destruct //
+qed-.
+
+(* Basic properties *********************************************************)
+
+(*** eq_ynat_dec *)
+lemma eq_ynat_dec (y1,y2:ynat): Decidable (y1 = y2).
+* [| #p1 |] *
+[1,9: /2 width=1 by or_introl/ |2,5,8: #p2 ]
+[2: elim (eq_pnat_dec p1 p2)
+    /4 width=1 by eq_inv_yinj_bi, or_intror, or_introl/
+|*: @or_intror #H destruct
+]
+qed-.