matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_update.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/hash_1.ma".
  18
  19 (* UPDATE COUNT FOR PATH ****************************************************)
  20
  21 rec definition update (p) on p: nat ≝
  22 match p with
  23 [ list_empty     ⇒ 𝟎
  24 | list_lcons l q ⇒
  25   match l with
  26   [ label_d n ⇒ n + update q
  27   | label_m   ⇒ update q
  28   | label_L   ⇒ update q
  29   | label_A   ⇒ update q
  30   | label_S   ⇒ update q
  31   ]
  32 ].
  33
  34 interpretation
  35   "update count (path)"
  36   'Hash p = (update p).
  37
  38 (* Basic constructions ******************************************************)
  39
  40 lemma update_empty: 𝟎 = ⧣𝐞.
  41 // qed.
  42
  43 lemma update_d_sn (q) (n): ninj n+⧣q = ⧣(𝗱n◗q).
  44 // qed.
  45
  46 lemma update_m_sn (q): ⧣q = ⧣(𝗺◗q).
  47 // qed.
  48
  49 lemma update_L_sn (q): ⧣q = ⧣(𝗟◗q).
  50 // qed.
  51
  52 lemma update_A_sn (q): ⧣q = ⧣(𝗔◗q).
  53 // qed.
  54
  55 lemma update_S_sn (q): ⧣q = ⧣(𝗦◗q).
  56 // qed.
  57
  58 (* Main constructions *******************************************************)
  59
  60 theorem update_append (p1) (p2):
  61         (⧣p2+⧣p1) = ⧣(p1●p2).
  62 #p1 elim p1 -p1 //
  63 * [ #n ] #p1 #IH #p2 <list_append_lcons_sn
  64 [ <update_d_sn <update_d_sn //
  65 | <update_m_sn <update_m_sn //
  66 | <update_L_sn <update_L_sn //
  67 | <update_A_sn <update_A_sn //
  68 | <update_S_sn <update_S_sn //
  69 ]
  70 qed.
  71
  72 (* Constructions with list_rcons ********************************************)
  73
  74 lemma update_d_dx (p) (n):
  75       (⧣p)+(ninj n) = ⧣(p◖𝗱n).
  76 // qed.
  77
  78 lemma update_m_dx (p):
  79       (⧣p) = ⧣(p◖𝗺).
  80 // qed.
  81
  82 lemma update_L_dx (p):
  83       (⧣p) = ⧣(p◖𝗟).
  84 // qed.
  85
  86 lemma update_A_dx (p):
  87       (⧣p) = ⧣(p◖𝗔).
  88 // qed.
  89
  90 lemma update_S_dx (p):
  91       (⧣p) = ⧣(p◖𝗦).
  92 // qed.