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:                                           *)
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   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_m   ⇒ depth q
  28   | label_L   ⇒ ↑(depth q)
  29   | label_A   ⇒ depth q
  30   | label_S   ⇒ depth q
  31   ]
  32 ].
  33
  34 interpretation
  35   "depth (path)"
  36   'Flat p = (depth p).
  37
  38 (* Basic constructions ******************************************************)
  39
  40 lemma depth_empty: 𝟎 = ♭𝐞.
  41 // qed.
  42
  43 lemma depth_d_dx (p) (k):
  44       ♭p = ♭(p◖𝗱k).
  45 // qed.
  46
  47 lemma depth_m_dx (p):
  48       ♭p = ♭(p◖𝗺).
  49 // qed.
  50
  51 lemma depth_L_dx (p):
  52       ↑♭p = ♭(p◖𝗟).
  53 // qed.
  54
  55 lemma depth_A_dx (p):
  56       ♭p = ♭(p◖𝗔).
  57 // qed.
  58
  59 lemma depth_S_dx (p):
  60       ♭p = ♭(p◖𝗦).
  61 // qed.
  62
  63 (* Main constructions *******************************************************)
  64
  65 theorem depth_append (p) (q):
  66         (♭p)+(♭q) = ♭(p●q).
  67 #p #q elim q -q //
  68 * [ #k ] #q #IH <list_append_lcons_sn
  69 [ <depth_d_dx <depth_d_dx //
  70 | <depth_m_dx <depth_m_dx //
  71 | <depth_L_dx <depth_L_dx //
  72 | <depth_A_dx <depth_A_dx //
  73 | <depth_S_dx <depth_S_dx //
  74 ]
  75 qed.
  76
  77 (* Constructions with path_lcons ********************************************)
  78
  79 lemma depth_d_sn (p) (k):
  80       ♭p = ♭(𝗱k◗p).
  81 // qed.
  82
  83 lemma depth_m_sn (p):
  84       ♭p = ♭(𝗺◗p).
  85 // qed.
  86
  87 lemma depth_L_sn (p):
  88       ↑♭p = ♭(𝗟◗p).
  89 // qed.
  90
  91 lemma depth_A_sn (p):
  92       ♭p = ♭(𝗔◗p).
  93 // qed.
  94
  95 lemma depth_S_sn (p):
  96       ♭p = ♭(𝗦◗p).
  97 // qed.