propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / pnat_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/notation/functions/downarrow_1.ma".
+include "ground/arith/pnat_split.ma".
+
+(* PREDECESSOR FOR POSITIVE INTEGERS ****************************************)
+
+definition ppred (p): pnat ≝ psplit … (𝟏) (λp.p) p.
+
+interpretation
+  "predecessor (positive integers)"
+  'DownArrow p = (ppred p).
+
+(* Basic constructions ******************************************************)
+
+lemma ppred_unit: 𝟏 = ↓𝟏.
+// qed.
+
+lemma ppred_succ (p): p = ↓↑p.
+// qed.
+
+(* Basic inversions *********************************************************)
+
+lemma ppred_inv_refl (p): p = ↓p → 𝟏 = p.
+#p elim p -p //
+#p #IH #H /2 width=1 by/
+qed-.
+
+lemma pnat_split_unit_pos (p): ∨∨ 𝟏 = p | p = ↑↓p.
+#p elim p -p
+/2 width=1 by or_introl, or_intror/
+qed-.