matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_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_head.ma".
  16 include "delayed_updating/syntax/path_height_labels.ma".
  17 include "delayed_updating/syntax/path_depth_labels.ma".
  18
  19 (* HEAD FOR PATH ************************************************************)
  20
  21 (* Constructions with depth and height **************************************)
  22
  23 lemma path_head_depth (p) (n):
  24       n + ♯(↳[n]p) = ♭↳[n]p.
  25 #p elim p -p // #l #p #IH
  26 #n @(nat_ind_succ … n) -n // #n #_
  27 cases l [ #m ]
  28 [ <path_head_d_sn <height_d_sn <depth_d_sn //
  29 | <path_head_m_sn <height_m_sn <depth_m_sn //
  30 | <path_head_L_sn <height_L_sn <depth_L_sn
  31   <nplus_succ_sn //
  32 | <path_head_A_sn <height_A_sn <depth_A_sn //
  33 | <path_head_S_sn <height_S_sn <depth_S_sn //
  34 ]
  35 qed-. (**) (* gets in the way with auto *)