matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label.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/notation/functions/uparrow_2.ma".
  16 include "delayed_updating/syntax/label.ma".
  17 include "ground/relocation/tr_pap_pap.ma".
  18
  19 (* PRELIFT FOR LABEL ********************************************************)
  20
  21 definition prelift_label (f) (l): label ≝
  22 match l with
  23 [ label_d k ⇒ 𝗱(f＠⧣❨k❩)
  24 | label_m   ⇒ 𝗺
  25 | label_L   ⇒ 𝗟
  26 | label_A   ⇒ 𝗔
  27 | label_S   ⇒ 𝗦
  28 ].
  29
  30 interpretation
  31   "prelift (label)"
  32   'UpArrow f l = (prelift_label f l).
  33
  34 (* Basic constructions ******************************************************)
  35
  36 lemma prelift_label_d (f) (k):
  37       (𝗱(f＠⧣❨k❩)) = ↑[f]𝗱k.
  38 // qed.
  39
  40 lemma prelift_label_m (f):
  41       (𝗺) = ↑[f]𝗺.
  42 // qed.
  43
  44 lemma prelift_label_L (f):
  45       (𝗟) = ↑[f]𝗟.
  46 // qed.
  47
  48 lemma prelift_label_A (f):
  49       (𝗔) = ↑[f]𝗔.
  50 // qed.
  51
  52 lemma prelift_label_S (f):
  53       (𝗦) = ↑[f]𝗦.
  54 // qed.
  55
  56 (* Basic inversions *********************************************************)
  57
  58 lemma prelift_label_inv_d_sn (f) (l) (k1):
  59       (𝗱k1) = ↑[f]l →
  60       ∃∃k2. k1 = f＠⧣❨k2❩ & 𝗱k2 = l.
  61 #f * [ #k ] #k1
  62 [ <prelift_label_d
  63 | <prelift_label_m
  64 | <prelift_label_L
  65 | <prelift_label_A
  66 | <prelift_label_S
  67 ] #H0 destruct
  68 /2 width=3 by ex2_intro/
  69 qed-.
  70
  71 lemma prelift_label_inv_m_sn (f) (l):
  72       (𝗺) = ↑[f]l → 𝗺 = l.
  73 #f * [ #k ]
  74 [ <prelift_label_d
  75 | <prelift_label_m
  76 | <prelift_label_L
  77 | <prelift_label_A
  78 | <prelift_label_S
  79 ] #H0 destruct //
  80 qed-.
  81
  82 lemma prelift_label_inv_L_sn (f) (l):
  83       (𝗟) = ↑[f]l → 𝗟 = l.
  84 #f * [ #k ]
  85 [ <prelift_label_d
  86 | <prelift_label_m
  87 | <prelift_label_L
  88 | <prelift_label_A
  89 | <prelift_label_S
  90 ] #H0 destruct //
  91 qed-.
  92
  93 lemma prelift_label_inv_A_sn (f) (l):
  94       (𝗔) = ↑[f]l → 𝗔 = l.
  95 #f * [ #k ]
  96 [ <prelift_label_d
  97 | <prelift_label_m
  98 | <prelift_label_L
  99 | <prelift_label_A
 100 | <prelift_label_S
 101 ] #H0 destruct //
 102 qed-.
 103
 104 lemma prelift_label_inv_S_sn (f) (l):
 105       (𝗦) = ↑[f]l → 𝗦 = l.
 106 #f * [ #k ]
 107 [ <prelift_label_d
 108 | <prelift_label_m
 109 | <prelift_label_L
 110 | <prelift_label_A
 111 | <prelift_label_S
 112 ] #H0 destruct //
 113 qed-.
 114
 115 (* Main inversions **********************************************************)
 116
 117 theorem prelift_label_inj (f) (l1) (l2):
 118         ↑[f]l1 = ↑[f]l2 → l1 = l2.
 119 #f * [ #k1 ] #l2 #Hl
 120 [ elim (prelift_label_inv_d_sn … Hl) -Hl #k2 #Hk #H0 destruct
 121   >(tr_pap_inj ???? Hk) -Hk //
 122 | <(prelift_label_inv_m_sn … Hl) -l2 //
 123 | <(prelift_label_inv_L_sn … Hl) -l2 //
 124 | <(prelift_label_inv_A_sn … Hl) -l2 //
 125 | <(prelift_label_inv_S_sn … Hl) -l2 //
 126 ]
 127 qed-.