matita/matita/contribs/lambdadelta/ground/arith/nat_pred_succ.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/nat_succ.ma".
  16 include "ground/arith/nat_pred.ma".
  17
  18 (* PREDECESSOR FOR NON-NEGATIVE INTEGERS ************************************)
  19
  20 (* Constructions with npsucc ************************************************)
  21
  22 lemma pnpred_succ (n): n = pnpred (npsucc n).
  23 * //
  24 qed.
  25
  26 lemma npsucc_pred (p): p = npsucc (pnpred p).
  27 * //
  28 qed.
  29
  30 (* Constructions with nsucc and psucc ***************************************)
  31
  32 lemma pnpred_psucc (p): pnpred (psucc p) = nsucc (pnpred p).
  33 * // qed.
  34
  35 (* Constructions with nsucc *************************************************)
  36
  37 lemma nsucc_pnpred (p):
  38       ninj p = ↑(pnpred p).
  39 // qed.
  40
  41 (*** pred_Sn pred_S *)
  42 lemma npred_succ (n): n = ↓↑n.
  43 * //
  44 qed.
  45
  46 (* Basic inversions *********************************************************)
  47
  48 lemma eq_inv_pnpred_bi:
  49       injective … pnpred.
  50 #p1 #p2 #Hp
  51 >(npsucc_pred p1) >(npsucc_pred p2)
  52 <Hp -Hp @refl
  53 qed-.
  54
  55 (* Inversions with nsucc ****************************************************)
  56
  57 (*** nat_split *)
  58 lemma nat_split_zero_pos (n): ∨∨ 𝟎 = n | n = ↑↓n.
  59 #n @(nat_ind_succ … n) -n
  60 /2 width=1 by or_introl, or_intror/
  61 qed-.