matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_tail_depth.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/syntax/path_tail.ma".
  16 include "delayed_updating/syntax/path_depth.ma".
  17 include "delayed_updating/syntax/path_height.ma".
  18 (*
  19 include "ground/arith/nat_plus.ma".
  20 include "ground/arith/nat_pred_succ.ma".
  21 *)
  22 (* TAIL FOR PATH ************************************************************)
  23
  24 (* Constructions with depth and height **************************************)
  25
  26 axiom depth_lablels_L (n):
  27       n = ♭(𝗟∗∗n).
  28
  29 axiom height_lablels_L (n):
  30       (𝟎) = ♯(𝗟∗∗n).
  31
  32 lemma tail_depth (p) (n):
  33       n + ♯(↳[n]p) = ♭↳[n]p.
  34 #p elim p -p // #l #p #IH
  35 #n @(nat_ind_succ … n) -n // #n #_
  36 cases l [ #m ]
  37 [ <tail_d_sn <height_d_sn <depth_d_sn //
  38 | <tail_m_sn <height_m_sn <depth_m_sn //
  39 | <tail_L_sn <height_L_sn <depth_L_sn
  40   <nplus_succ_sn //
  41 | <tail_A_sn <height_A_sn <depth_A_sn //
  42 | <tail_S_sn <height_S_sn <depth_S_sn //
  43 ]
  44 qed.