matita/matita/contribs/lambdadelta/delayed_updating/etc/head/lift_path_head.etc

   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                  *)
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  13 (**************************************************************************)
  14
  15 include "delayed_updating/substitution/lift_path.ma".
  16 include "delayed_updating/syntax/path_head.ma".
  17 include "ground/relocation/xap.ma".
  18
  19 (* LIFT FOR PATH ************************************************************)
  20
  21 (* Constructions with head for path *****************************************)
  22
  23 lemma lift_path_head_closed (f) (p) (q) (n):
  24       q = ↳[n]q →
  25       ↳[↑[p●q]f＠❨n❩]↑[↑[p]f]q = ↑[↑[p]f]q.
  26 #f #p #q elim q -q
  27 [ #n #H0
  28   <(eq_inv_path_empty_head … H0) -H0
  29   <path_head_zero //
  30 | #l #q #IH #n @(nat_ind_succ …n) -n [| #n #_ ]
  31   [ <path_head_zero #H0 destruct
  32   | cases l [ #k ]
  33     [ <path_head_d_dx #H0
  34       elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
  35       <lift_rmap_d_dx <lift_path_d_dx
  36       <tr_xap_succ_nap <path_head_d_dx
  37       <(IH … H0) in ⊢ (???%); -IH -H0 <tr_xap_plus //
  38     | <path_head_m_dx #H0
  39       elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
  40       <lift_rmap_m_dx <lift_path_m_dx
  41       <tr_xap_succ_nap <path_head_m_dx
  42       <(IH … H0) in ⊢ (???%); -IH -H0 //
  43     | <path_head_L_dx #H0
  44       elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
  45       <lift_rmap_L_dx <lift_path_L_dx
  46       <tr_xap_succ_nap <path_head_L_dx
  47       <(IH … H0) in ⊢ (???%); -IH -H0 //
  48     | <path_head_A_dx #H0
  49       elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
  50       <lift_rmap_A_dx <lift_path_A_dx
  51       <tr_xap_succ_nap <path_head_A_dx
  52       <(IH … H0) in ⊢ (???%); -IH -H0 //
  53     | <path_head_S_dx #H0
  54       elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
  55       <lift_rmap_S_dx <lift_path_S_dx
  56       <tr_xap_succ_nap <path_head_S_dx
  57       <(IH … H0) in ⊢ (???%); -IH -H0 //
  58     ]
  59   ]
  60 ]
  61 qed.