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 "ground/arith/nat_plus.ma".
  17 include "ground/notation/functions/verticalbars_1.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 _ ⇒ 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   'VerticalBars p = (depth p).
  37
  38 (* Basic constructions ******************************************************)
  39
  40 lemma depth_empty: 𝟎 = ❘𝐞❘.
  41 // qed.
  42
  43 lemma depth_d_sn (q) (n): ❘q❘ = ❘𝗱n◗q❘.
  44 // qed.
  45
  46 lemma depth_m_sn (q): ❘q❘ = ❘𝗺◗q❘.
  47 // qed.
  48
  49 lemma depth_L_sn (q): ↑❘q❘ = ❘𝗟◗q❘.
  50 // qed.
  51
  52 lemma depth_A_sn (q): ❘q❘ = ❘𝗔◗q❘.
  53 // qed.
  54
  55 lemma depth_S_sn (q): ❘q❘ = ❘𝗦◗q❘.
  56 // qed.
  57
  58 (* Advanced constructions with nplus ****************************************)
  59
  60 lemma depth_plus (p1) (p2):
  61       ❘p2❘+❘p1❘ = ❘p1●p2❘.
  62 #p1 elim p1 -p1 //
  63 * [ #n ] #p1 #IH #p2 <list_append_lcons_sn
  64 [ <depth_d_sn <depth_d_sn //
  65 | <depth_m_sn <depth_m_sn //
  66 | <depth_L_sn <depth_L_sn //
  67 | <depth_A_sn <depth_A_sn //
  68 | <depth_S_sn <depth_S_sn //
  69 ]
  70 qed.