matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_inner.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 "delayed_updating/notation/functions/class_i_0.ma".
  17 include "ground/lib/subset.ma".
  18
  19 (* INNER CONDITION FOR PATH *************************************************)
  20
  21 definition pic: predicate path ≝
  22            λp. ∀q,k. q◖𝗱k = p → ⊥
  23 .
  24
  25 interpretation
  26   "inner condition (path)"
  27   'ClassI = (pic).
  28
  29 (* Basic constructions ******************************************************)
  30
  31 lemma pic_empty:
  32       (𝐞) ϵ 𝐈.
  33 #q #k #H0 destruct
  34 qed.
  35
  36 lemma pic_m_dx (p):
  37       p◖𝗺 ϵ 𝐈.
  38 #p #q #k #H0 destruct
  39 qed.
  40
  41 lemma pic_L_dx (p):
  42       p◖𝗟 ϵ 𝐈.
  43 #p #q #k #H0 destruct
  44 qed.
  45
  46 lemma pic_A_dx (p):
  47       p◖𝗔 ϵ 𝐈.
  48 #p #q #k #H0 destruct
  49 qed.
  50
  51 lemma pic_S_dx (p):
  52       p◖𝗦 ϵ 𝐈.
  53 #p #q #k #H0 destruct
  54 qed.
  55
  56 (* Basic inversions ********************************************************)
  57
  58 lemma pic_inv_d_dx (p) (k):
  59       p◖𝗱k ϵ 𝐈 → ⊥.
  60 #p #k #H0 @H0 -H0 //
  61 qed-.
  62
  63 (* Constructions with path_lcons ********************************************)
  64
  65 lemma pic_m_sn (p):
  66       p ϵ 𝐈 → 𝗺◗p ϵ 𝐈.
  67 * [| * [ #k ] #p #Hp <list_cons_shift ] //
  68 [ #_ <list_cons_comm //
  69 | elim (pic_inv_d_dx … Hp)
  70 ]
  71 qed.
  72
  73 lemma pic_L_sn (p):
  74       p ϵ 𝐈 → 𝗟◗p ϵ 𝐈.
  75 * [| * [ #k ] #p #Hp <list_cons_shift ] //
  76 [ #_ <list_cons_comm //
  77 | elim (pic_inv_d_dx … Hp)
  78 ]
  79 qed.
  80
  81 lemma pic_A_sn (p):
  82       p ϵ 𝐈 → 𝗔◗p ϵ 𝐈.
  83 * [| * [ #k ] #p #Hp <list_cons_shift ] //
  84 [ #_ <list_cons_comm //
  85 | elim (pic_inv_d_dx … Hp)
  86 ]
  87 qed.
  88
  89 lemma pic_S_sn (p):
  90       p ϵ 𝐈 → 𝗦◗p ϵ 𝐈.
  91 * [| * [ #k ] #p #Hp <list_cons_shift ] //
  92 [ #_ <list_cons_comm //
  93 | elim (pic_inv_d_dx … Hp)
  94 ]
  95 qed.