matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_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.ma".
  16 include "delayed_updating/notation/functions/flat_1.ma".
  17 include "ground/arith/nat_plus.ma".
  18
  19 (* DEPTH FOR PATH ***********************************************************)
  20
  21 rec definition depth (p) on p: nat ≝
  22 match p with
  23 [ list_empty     ⇒ 𝟎
  24 | list_lcons l q ⇒
  25   match l with
  26   [ label_d k    ⇒ depth q
  27   | label_d2 k d ⇒ depth q
  28   | label_m      ⇒ depth q
  29   | label_L      ⇒ ↑(depth q)
  30   | label_A      ⇒ depth q
  31   | label_S      ⇒ depth q
  32   ]
  33 ].
  34
  35 interpretation
  36   "depth (path)"
  37   'Flat p = (depth p).
  38
  39 (* Basic constructions ******************************************************)
  40
  41 lemma depth_empty: 𝟎 = ♭𝐞.
  42 // qed.
  43
  44 lemma depth_d_dx (p) (k):
  45       ♭p = ♭(p◖𝗱k).
  46 // qed.
  47
  48 lemma depth_d2_dx (p) (k) (d):
  49       ♭p = ♭(p◖𝗱❨k,d❩).
  50 // qed.
  51
  52 lemma depth_m_dx (p):
  53       ♭p = ♭(p◖𝗺).
  54 // qed.
  55
  56 lemma depth_L_dx (p):
  57       ↑♭p = ♭(p◖𝗟).
  58 // qed.
  59
  60 lemma depth_A_dx (p):
  61       ♭p = ♭(p◖𝗔).
  62 // qed.
  63
  64 lemma depth_S_dx (p):
  65       ♭p = ♭(p◖𝗦).
  66 // qed.
  67
  68 (* Main constructions *******************************************************)
  69
  70 theorem depth_append (p) (q):
  71         (♭p)+(♭q) = ♭(p●q).
  72 #p #q elim q -q //
  73 * [ #k | #k #d ] #q #IH <list_append_lcons_sn
  74 [ <depth_d_dx <depth_d_dx //
  75 | <depth_d2_dx <depth_d2_dx //
  76 | <depth_m_dx <depth_m_dx //
  77 | <depth_L_dx <depth_L_dx //
  78 | <depth_A_dx <depth_A_dx //
  79 | <depth_S_dx <depth_S_dx //
  80 ]
  81 qed.
  82
  83 (* Constructions with path_lcons ********************************************)
  84
  85 lemma depth_d_sn (p) (k):
  86       ♭p = ♭(𝗱k◗p).
  87 // qed.
  88
  89 lemma depth_d2_sn (p) (k) (d):
  90       ♭p = ♭(𝗱❨k,d❩◗p).
  91 // qed.
  92
  93 lemma depth_m_sn (p):
  94       ♭p = ♭(𝗺◗p).
  95 // qed.
  96
  97 lemma depth_L_sn (p):
  98       ↑♭p = ♭(𝗟◗p).
  99 // qed.
 100
 101 lemma depth_A_sn (p):
 102       ♭p = ♭(𝗔◗p).
 103 // qed.
 104
 105 lemma depth_S_sn (p):
 106       ♭p = ♭(𝗦◗p).
 107 // qed.