arithmetics for λδ

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/pnat.ma b/matita/matita/contribs/lambdadelta/ground/arith/pnat.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/notation/functions/one_0.ma".
+include "ground/notation/functions/uparrow_1.ma".
+include "ground/lib/relations.ma".
+
+(* POSITIVE INTEGERS ********************************************************)
+
+inductive pnat: Type[0] ≝
+| punit: pnat
+| psucc: pnat → pnat
+.
+
+interpretation
+  "unit (positive integers)"
+  'One = (punit).
+
+interpretation
+  "successor (positive integers)"
+  'UpArrow p = (psucc p).
+
+(* Basic inversions *********************************************************)
+
+(* Note: destruct *)
+lemma eq_inv_psucc_bi: injective … psucc.
+#p #q #H destruct //
+qed.
+
+(* Basic constructions ******************************************************)
+
+lemma eq_pnat_dec (p1,p2:pnat): Decidable (p1 = p2).
+#p1 elim p1 -p1 [| #p1 #IH ] * [2,4: #p2 ]
+[1,4: @or_intror #H destruct
+| elim (IH p2) -IH #H destruct
+  /4 width=1 by eq_inv_psucc_bi, or_intror, or_introl/
+| /2 width=1 by or_introl/
+]
+qed-.