matita/matita/contribs/lambdadelta/ground/arith/pnat_le_pred.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/arith/pnat_pred.ma".
  16 include "ground/arith/pnat_le.ma".
  17
  18 (* ORDER FOR POSITIVE INTEGERS **********************************************)
  19
  20 (* Destructions with ppred **************************************************)
  21
  22 lemma ple_inv_pred_sn (p) (q): ↓p ≤ q → p ≤ ↑q.
  23 #p #q elim p -p
  24 /2 width=1 by ple_succ_bi/
  25 qed-.
  26
  27 (* Constructions with ppred *************************************************)
  28
  29 lemma ple_succ_pred_dx_refl (p): p ≤ ↑↓p.
  30 #p @ple_inv_pred_sn // qed.
  31
  32 lemma ple_pred_sn_refl (p): ↓p ≤ p.
  33 #p elim p -p //
  34 qed.
  35
  36 lemma ple_pred_bi (p) (q): p ≤ q → ↓p ≤ ↓q.
  37 #p #q #H elim H -q //
  38 /2 width=3 by ple_trans/
  39 qed.
  40
  41 lemma ple_pred_sn (p) (q): p ≤ ↑q → ↓p ≤ q.
  42 #p #q elim p -p //
  43 /2 width=1 by ple_pred_bi/
  44 qed-.
  45
  46 (* Inversions with ppred ****************************************************)
  47
  48 lemma ple_inv_succ_sn (p) (q):
  49       ↑p ≤ q → ∧∧ p ≤ ↓q & q = ↑↓q.
  50 #p #q * -q
  51 [ /2 width=3 by ple_refl, conj/
  52 | #q #Hq /3 width=1 by ple_des_succ_sn, conj/
  53 ]
  54 qed-.