matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_height.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 "ground/arith/nat_plus.ma".
  16 include "delayed_updating/syntax/path.ma".
  17 include "delayed_updating/notation/functions/sharp_1.ma".
  18
  19 (* HEIGHT FOR PATH **********************************************************)
  20
  21 rec definition height (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 ⇒ height q + k
  27   | label_m   ⇒ height q
  28   | label_L   ⇒ height q
  29   | label_A   ⇒ height q
  30   | label_S   ⇒ height q
  31   ]
  32 ].
  33
  34 interpretation
  35   "height (path)"
  36   'Sharp p = (height p).
  37
  38 (* Basic constructions ******************************************************)
  39
  40 lemma height_empty: 𝟎 = ♯𝐞.
  41 // qed.
  42
  43 lemma height_d_dx (p) (k:pnat):
  44       (♯p)+k = ♯(p◖𝗱k).
  45 // qed.
  46
  47 lemma height_m_dx (p):
  48       (♯p) = ♯(p◖𝗺).
  49 // qed.
  50
  51 lemma height_L_dx (p):
  52       (♯p) = ♯(p◖𝗟).
  53 // qed.
  54
  55 lemma height_A_dx (p):
  56       (♯p) = ♯(p◖𝗔).
  57 // qed.
  58
  59 lemma height_S_dx (p):
  60       (♯p) = ♯(p◖𝗦).
  61 // qed.
  62
  63 (* Main constructions *******************************************************)
  64
  65 theorem height_append (p) (q):
  66         (♯p+♯q) = ♯(p●q).
  67 #p #q elim q -q //
  68 * [ #k ] #q #IH <list_append_lcons_sn
  69 [ <height_d_dx <height_d_dx //
  70 | <height_m_dx <height_m_dx //
  71 | <height_L_dx <height_L_dx //
  72 | <height_A_dx <height_A_dx //
  73 | <height_S_dx <height_S_dx //
  74 ]
  75 qed.
  76
  77 (* Constructions with path_lcons ********************************************)
  78
  79 lemma height_d_sn (p) (k:pnat):
  80       k+♯p = ♯(𝗱k◗p).
  81 // qed.
  82
  83 lemma height_m_sn (p):
  84       ♯p = ♯(𝗺◗p).
  85 // qed.
  86
  87 lemma height_L_sn (p):
  88       ♯p = ♯(𝗟◗p).
  89 // qed.
  90
  91 lemma height_A_sn (p):
  92       ♯p = ♯(𝗔◗p).
  93 // qed.
  94
  95 lemma height_S_sn (p):
  96       ♯p = ♯(𝗦◗p).
  97 // qed.