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/nat_minus_pminus.ma".
  20 include "ground/arith/pnat_le.ma".
  21
  22 (* RIGHT SUBTRACTION FOR TOTAL RELOCATION MAPS ******************************)
  23
  24 (* Constructions with tr_pap ************************************************)
  25
  26 lemma tr_pap_minus_le (n) (p) (f):
  27       f＠⧣❨p❩ ≤ p + n →
  28       p = (f-n)＠⧣❨p❩.
  29 #n @(nat_ind_succ … n) -n [| #n #IHn ]
  30 [ #p #f #H1f
  31   lapply (tr_pap_increasing f p) #H2f
  32   >(ple_antisym … H2f H1f) in ⊢ (??%?); -H1f -H2f //
  33 | #p elim p -p [| #p #IHp ]
  34   #f elim (tr_map_split f) * #g #H0 destruct
  35   [ //
  36   |2,4:
  37     <tr_pap_next <nrplus_succ_dx #Hf
  38     lapply (ple_inv_succ_bi … Hf) -Hf #Hf
  39     <tr_minus_next_succ /2 width=1 by/
  40   | <tr_minus_push_succ <tr_pap_push <tr_pap_push <nrplus_succ_sn #Hf
  41     lapply (ple_inv_succ_bi … Hf) -Hf #Hf
  42     <IHp -IHp //
  43   ]
  44 ]
  45 qed-.
  46
  47 lemma tr_pap_minus_ge (n:nat) (p:pnat) (f):
  48       p + n ≤ f＠⧣❨p❩ →
  49       f＠⧣❨p❩-n = (f-n)＠⧣❨p❩.
  50 #n @(nat_ind_succ … n) -n [| #n #IHn ]
  51 [ #p #f #_ //
  52 | #p elim p -p [| #p #IHp ]
  53   #f elim (tr_map_split f) * #g #H0 destruct
  54   [ <tr_cons_pap_unit <nrplus_unit_sn #H0
  55     elim (ple_inv_succ_unit … H0)
  56   |2,4:
  57     <tr_pap_next <nrplus_succ_dx #Hf
  58     lapply (ple_inv_succ_bi … Hf) -Hf #Hf
  59     <tr_minus_next_succ >nsucc_inj /2 width=1 by/
  60   | <tr_minus_push_succ <tr_pap_push <tr_pap_push <nrplus_succ_sn #Hf
  61     lapply (ple_inv_succ_bi … Hf) -Hf #Hf
  62     >nsucc_inj >nsucc_inj <IHp -IHp //
  63     <nminus_inj_bi
  64     [ <nminus_inj_bi
  65       [ <nsucc_inj @eq_f
  66 (*
  67     <nminus_succ_sn //
  68 *)