matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_structure.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/substitution/lift_eq.ma".
  16 include "delayed_updating/syntax/path_structure.ma".
  17 include "delayed_updating/syntax/path_proper.ma".
  18 include "ground/xoa/ex_4_2.ma".
  19 include "ground/xoa/ex_3_2.ma".
  20
  21 (* LIFT FOR PATH ***********************************************************)
  22
  23 (* Basic constructions with structure **************************************)
  24
  25 lemma structure_lift (p) (f):
  26       ⊗p = ⊗↑[f]p.
  27 #p @(path_ind_lift … p) -p // #p #IH #f
  28 <lift_path_L_sn //
  29 qed.
  30
  31 lemma lift_structure (p) (f):
  32       ⊗p = ↑[f]⊗p.
  33 #p @(path_ind_lift … p) -p //
  34 qed.
  35
  36 (* Destructions with structure **********************************************)
  37
  38 lemma lift_des_structure (q) (p) (f):
  39       ⊗q = ↑[f]p → ⊗q = ⊗p.
  40 // qed-.
  41
  42 (* Constructions with proper condition for path *****************************)
  43
  44 lemma lift_append_proper_dx (p2) (p1) (f): Ꝕp2 →
  45       (⊗p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
  46 #p2 #p1 @(path_ind_lift … p1) -p1 //
  47 [ #n | #n #l #p1 |*: #p1 ] #IH #f #Hp2
  48 [ elim (ppc_inv_lcons … Hp2) -Hp2 #l #q #H destruct //
  49 | <lift_path_d_lcons_sn <IH //
  50 | <lift_path_L_sn <IH //
  51 | <lift_path_A_sn <IH //
  52 | <lift_path_S_sn <IH //
  53 ]
  54 qed-.
  55
  56 (* Advanced constructions with structure ************************************)
  57
  58 lemma lift_d_empty_dx (n) (p) (f):
  59       (⊗p)◖𝗱((↑[p]f)@❨n❩) = ↑[f](p◖𝗱n).
  60 #n #p #f <lift_append_proper_dx //
  61 qed.
  62
  63 lemma lift_L_dx (p) (f):
  64       (⊗p)◖𝗟 = ↑[f](p◖𝗟).
  65 #p #f <lift_append_proper_dx //
  66 qed.
  67
  68 lemma lift_A_dx (p) (f):
  69       (⊗p)◖𝗔 = ↑[f](p◖𝗔).
  70 #p #f <lift_append_proper_dx //
  71 qed.
  72
  73 lemma lift_S_dx (p) (f):
  74       (⊗p)◖𝗦 = ↑[f](p◖𝗦).
  75 #p #f <lift_append_proper_dx //
  76 qed.
  77
  78 lemma lift_root (f) (p):
  79       ∃∃r. 𝐞 = ⊗r & ⊗p●r = ↑[f]p.
  80 #f #p @(list_ind_rcons … p) -p
  81 [ /2 width=3 by ex2_intro/
  82 | #p * [ #n ] /2 width=3 by ex2_intro/
  83 ]
  84 qed-.
  85
  86 (* Advanced inversions with proj_path ***************************************)
  87
  88 lemma lift_path_inv_d_sn (k) (q) (p) (f):
  89       (𝗱k◗q) = ↑[f]p →
  90       ∃∃r,h. 𝐞 = ⊗r & (↑[r]f)@❨h❩ = k & 𝐞 = q & r◖𝗱h = p.
  91 #k #q #p @(path_ind_lift … p) -p
  92 [| #n | #n #l #p |*: #p ] [|*: #IH ] #f
  93 [ <lift_path_empty #H destruct
  94 | <lift_path_d_empty_sn #H destruct -IH
  95   /2 width=5 by ex4_2_intro/
  96 | <lift_path_d_lcons_sn #H
  97   elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
  98   /2 width=5 by ex4_2_intro/
  99 | <lift_path_L_sn #H destruct
 100 | <lift_path_A_sn #H destruct
 101 | <lift_path_S_sn #H destruct
 102 ]
 103 qed-.
 104
 105 lemma lift_path_inv_L_sn (q) (p) (f):
 106       (𝗟◗q) = ↑[f]p →
 107       ∃∃r1,r2. 𝐞 = ⊗r1 & q = ↑[⫯↑[r1]f]r2 & r1●𝗟◗r2 = p.
 108 #q #p @(path_ind_lift … p) -p
 109 [| #n | #n #l #p |*: #p ] [|*: #IH ] #f
 110 [ <lift_path_empty #H destruct
 111 | <lift_path_d_empty_sn #H destruct
 112 | <lift_path_d_lcons_sn #H
 113   elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
 114   /2 width=5 by ex3_2_intro/
 115 | <lift_path_L_sn #H destruct -IH
 116   /2 width=5 by ex3_2_intro/
 117 | <lift_path_A_sn #H destruct
 118 | <lift_path_S_sn #H destruct
 119 ]
 120 qed-.
 121
 122 lemma lift_path_inv_A_sn (q) (p) (f):
 123       (𝗔◗q) = ↑[f]p →
 124       ∃∃r1,r2. 𝐞 = ⊗r1 & q = ↑[↑[r1]f]r2 & r1●𝗔◗r2 = p.
 125 #q #p @(path_ind_lift … p) -p
 126 [| #n | #n #l #p |*: #p ] [|*: #IH ] #f
 127 [ <lift_path_empty #H destruct
 128 | <lift_path_d_empty_sn #H destruct
 129 | <lift_path_d_lcons_sn #H
 130   elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
 131   /2 width=5 by ex3_2_intro/
 132 | <lift_path_L_sn #H destruct
 133 | <lift_path_A_sn #H destruct -IH
 134   /2 width=5 by ex3_2_intro/
 135 | <lift_path_S_sn #H destruct
 136 ]
 137 qed-.
 138
 139 lemma lift_path_inv_S_sn (q) (p) (f):
 140       (𝗦◗q) = ↑[f]p →
 141       ∃∃r1,r2. 𝐞 = ⊗r1 & q = ↑[↑[r1]f]r2 & r1●𝗦◗r2 = p.
 142 #q #p @(path_ind_lift … p) -p
 143 [| #n | #n #l #p |*: #p ] [|*: #IH ] #f
 144 [ <lift_path_empty #H destruct
 145 | <lift_path_d_empty_sn #H destruct
 146 | <lift_path_d_lcons_sn #H
 147   elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
 148   /2 width=5 by ex3_2_intro/
 149 | <lift_path_L_sn #H destruct
 150 | <lift_path_A_sn #H destruct
 151 | <lift_path_S_sn #H destruct -IH
 152   /2 width=5 by ex3_2_intro/
 153 ]
 154 qed-.
 155
 156 (* Inversions with proper condition for path ********************************)
 157
 158 lemma lift_inv_append_proper_dx (q2) (q1) (p) (f): Ꝕq2 →
 159       q1●q2 = ↑[f]p →
 160       ∃∃p1,p2. ⊗p1 = q1 & ↑[↑[p1]f]p2 = q2 & p1●p2 = p.
 161 #q2 #q1 elim q1 -q1
 162 [ #p #f #Hq2 <list_append_empty_sn #H destruct
 163   /2 width=5 by ex3_2_intro/
 164 | * [ #n1 ] #q1 #IH #p #f #Hq2 <list_append_lcons_sn #H
 165   [ elim (lift_path_inv_d_sn … H) -H #r1 #m1 #_ #_ #H0 #_ -IH
 166     elim (eq_inv_list_empty_append … H0) -H0 #_ #H0 destruct
 167     elim Hq2 -Hq2 //
 168   | elim (lift_path_inv_L_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
 169     elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
 170     @(ex3_2_intro … (r1●𝗟◗p1)) //
 171     <structure_append <Hr1 -Hr1 //
 172   | elim (lift_path_inv_A_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
 173     elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
 174     @(ex3_2_intro … (r1●𝗔◗p1)) //
 175     <structure_append <Hr1 -Hr1 //
 176   | elim (lift_path_inv_S_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
 177     elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
 178     @(ex3_2_intro … (r1●𝗦◗p1)) //
 179     <structure_append <Hr1 -Hr1 //
 180   ]
 181 ]
 182 qed-.