matita/matita/contribs/lambdadelta/delayed_updating/relocation/tr_minus_pap.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 "delayed_updating/relocation/tr_minus_pn.ma".
  16 include "ground/relocation/tr_pap_pn.ma".
  17 include "ground/relocation/tr_pap_lt.ma".
  18 include "ground/arith/nat_rplus_succ.ma".
  19 include "ground/arith/pnat_le.ma".
  20
  21 (* RIGHT SUBTRACTION FOR TOTAL RELOCATION MAPS ******************************)
  22
  23 (* Constructions with tr_pap ************************************************)
  24
  25 lemma tr_pap_minus_le (n) (p) (f):
  26       f＠⧣❨p❩ ≤ p + n →
  27       p = (f-n)＠⧣❨p❩.
  28 #n @(nat_ind_succ … n) -n [| #n #IHn ]
  29 [ #p #f #H1f
  30   lapply (tr_pap_increasing f p) #H2f
  31   >(ple_antisym … H2f H1f) in ⊢ (??%?); -H1f -H2f //
  32 | #p elim p -p [| #p #IHp ]
  33   #f elim (tr_map_split f) * #g #H0 destruct
  34   [ //
  35   |2,4:
  36     <tr_pap_next <nrplus_succ_dx #Hf
  37     lapply (ple_inv_succ_bi … Hf) -Hf #Hf
  38     <tr_minus_next_succ /2 width=1 by/
  39   | <tr_minus_push_succ <tr_pap_push <tr_pap_push <nrplus_succ_sn #Hf
  40     lapply (ple_inv_succ_bi … Hf) -Hf #Hf
  41     <IHp -IHp //
  42   ]
  43 ]
  44 qed-.