propagating the arithmetics library, partial commit

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/pnat_le_pred.ma b/matita/matita/contribs/lambdadelta/ground/arith/pnat_le_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/arith/pnat_pred.ma".
+include "ground/arith/pnat_le.ma".
+
+(* ORDER FOR POSITIVE INTEGERS **********************************************)
+
+(* Destructions with ppred **************************************************)
+
+lemma ple_inv_pred_sn (p) (q): ↓p ≤ q → p ≤ ↑q.
+#p #q elim p -p
+/2 width=1 by ple_succ_bi/
+qed-.
+
+(* Constructions with ppred *************************************************)
+
+lemma ple_succ_pred_dx_refl (p): p ≤ ↑↓p.
+#p @ple_inv_pred_sn // qed.
+
+lemma ple_pred_sn_refl (p): ↓p ≤ p.
+#p elim p -p //
+qed.
+
+lemma ple_pred_bi (p) (q): p ≤ q → ↓p ≤ ↓q.
+#p #q #H elim H -q //
+/2 width=3 by ple_trans/
+qed.
+
+lemma ple_pred_sn (p) (q): p ≤ ↑q → ↓p ≤ q.
+#p #q elim p -p //
+/2 width=1 by ple_pred_bi/
+qed-.
+
+(* Inversions with ppred ****************************************************)
+
+lemma ple_inv_succ_sn (p) (q):
+      ↑p ≤ q → ∧∧ p ≤ ↓q & q = ↑↓q.
+#p #q * -q
+[ /2 width=3 by ple_refl, conj/
+| #q #Hq /3 width=1 by ple_des_succ_sn, conj/
+]
+qed-.