matita/matita/contribs/lambdadelta/ground/relocation/gr_nat_basic.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/relocation/gr_basic.ma".
  16 include "ground/relocation/gr_nat_uni.ma".
  17
  18 (* NON-NEGATIVE APPLICATION FOR GENERIC RELOCATION MAPS *****************************)
  19
  20 (* Properties with gr_basic **********************************************)
  21
  22 lemma gr_nat_basic_lt (m) (n) (l):
  23       l < m → @↑❪l, 𝐛❨m,n❩❫ ≘ l.
  24 #m @(nat_ind_succ … m) -m
  25 [ #n #i #H elim (nlt_inv_zero_dx … H)
  26 | #m #IH #n #l @(nat_ind_succ … l) -l
  27   [ #_ /2 width=2 by refl, gr_pat_refl/
  28   | #l #_ #H
  29     lapply (nlt_inv_succ_bi … H) -H #Hlm
  30     /3 width=7 by refl, gr_pat_push/
  31   ]
  32 ]
  33 qed.
  34
  35 lemma gr_nat_basic_ge (m) (n) (l):
  36       m ≤ l → @↑❪l, 𝐛❨m,n❩❫ ≘ l+n.
  37 #m @(nat_ind_succ … m) -m //
  38 #m #IH #n #l #H
  39 elim (nle_inv_succ_sn … H) -H #Hml #H >H -H
  40 /3 width=7 by gr_nat_push/
  41 qed.
  42
  43 (* Inversion lemmas with gr_basic ****************************************)
  44
  45 lemma gr_nat_basic_inv_lt (m) (n) (l) (k):
  46       l < m → @↑❪l, 𝐛❨m,n❩❫ ≘ k → l = k.
  47 /3 width=4 by gr_nat_basic_lt, gr_nat_mono/ qed-.
  48
  49 lemma gr_nat_basic_inv_ge (m) (n) (l) (k):
  50       m ≤ l → @↑❪l, 𝐛❨m,n❩❫ ≘ k → l+n = k.
  51 /3 width=4 by gr_nat_basic_ge, gr_nat_mono/ qed-.