matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
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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/black_righttriangle_2.ma".
  17 include "delayed_updating/notation/functions/black_righttriangle_1.ma".
  18 include "ground/relocation/tr_uni.ma".
  19 include "ground/relocation/tr_compose.ma".
  20
  21 (* UNWIND MAP FOR PATH ******************************************************)
  22
  23 rec definition unwind2_rmap (f) (p) on p: tr_map ≝
  24 match p with
  25 [ list_empty     ⇒ f
  26 | list_lcons l q ⇒
  27   match l with
  28   [ label_d n ⇒ (unwind2_rmap f q)∘𝐮❨n❩
  29   | label_m   ⇒ unwind2_rmap f q
  30   | label_L   ⇒ ⫯(unwind2_rmap f q)
  31   | label_A   ⇒ unwind2_rmap f q
  32   | label_S   ⇒ unwind2_rmap f q
  33   ]
  34 ].
  35
  36 interpretation
  37   "tailed unwind map (reversed path)"
  38   'BlackRightTriangle f p = (unwind2_rmap f p).
  39
  40 interpretation
  41   "unwind map (reversed path)"
  42   'BlackRightTriangle p = (unwind2_rmap tr_id p).
  43
  44 (* Basic constructions ******************************************************)
  45
  46 lemma unwind2_rmap_empty (f):
  47       f = ▶[f]𝐞.
  48 // qed.
  49
  50 lemma unwind2_rmap_d_sn (f) (p) (n:pnat):
  51       (▶[f]p∘𝐮❨n❩) = ▶[f](𝗱n◗p).
  52 // qed.
  53
  54 lemma unwind2_rmap_m_sn (f) (p):
  55       ▶[f]p = ▶[f](𝗺◗p).
  56 // qed.
  57
  58 lemma unwind2_rmap_L_sn (f) (p):
  59       (⫯▶[f]p) = ▶[f](𝗟◗p).
  60 // qed.
  61
  62 lemma unwind2_rmap_A_sn (f) (p):
  63       ▶[f]p = ▶[f](𝗔◗p).
  64 // qed.
  65
  66 lemma unwind2_rmap_S_sn (f) (p):
  67       ▶[f]p = ▶[f](𝗦◗p).
  68 // qed.
  69
  70 (* Constructions with list_append *******************************************)
  71
  72 lemma unwind2_rmap_append (f) (p1) (p2):
  73       ▶[▶[f]p2]p1 = ▶[f](p1●p2).
  74 #f #p1 elim p1 -p1 //
  75 * [ #n ] #p1 #IH #p2 //
  76 [ <unwind2_rmap_m_sn <unwind2_rmap_m_sn //
  77 | <unwind2_rmap_L_sn <unwind2_rmap_L_sn //
  78 | <unwind2_rmap_A_sn <unwind2_rmap_A_sn //
  79 | <unwind2_rmap_S_sn <unwind2_rmap_S_sn //
  80 ]
  81 qed.
  82
  83 (* Constructions with list_rcons ********************************************)
  84
  85 lemma unwind2_rmap_d_dx (f) (p) (n:pnat):
  86       ▶[f∘𝐮❨n❩]p = ▶[f](p◖𝗱n).
  87 // qed.
  88
  89 lemma unwind2_rmap_m_dx (f) (p):
  90       ▶[f]p = ▶[f](p◖𝗺).
  91 // qed.
  92
  93 lemma unwind2_rmap_L_dx (f) (p):
  94       ▶[⫯f]p = ▶[f](p◖𝗟).
  95 // qed.
  96
  97 lemma unwind2_rmap_A_dx (f) (p):
  98       ▶[f]p = ▶[f](p◖𝗔).
  99 // qed.
 100
 101 lemma unwind2_rmap_S_dx (f) (p):
 102       ▶[f]p = ▶[f](p◖𝗦).
 103 // qed.