matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_lift.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/reduction/ifr.ma".
  16
  17 include "delayed_updating/substitution/fsubst_lift.ma".
  18 include "delayed_updating/substitution/fsubst_eq.ma".
  19 include "delayed_updating/substitution/lift_prototerm_after.ma".
  20 include "delayed_updating/substitution/lift_path_head.ma".
  21 include "delayed_updating/substitution/lift_rmap_head.ma".
  22
  23 include "ground/relocation/tr_uni_compose.ma".
  24 include "ground/relocation/tr_compose_eq.ma".
  25
  26 (* IMMEDIATE FOCUSED REDUCTION **********************************************)
  27
  28 (* Constructions with lift **************************************************)
  29
  30 theorem ifr_lift_bi (f) (p) (q) (t1) (t2):
  31         t1 ➡𝐢𝐟[p,q] t2 → ↑[f]t1 ➡𝐢𝐟[↑[f]p,↑[↑[p◖𝗔◖𝗟]f]q] ↑[f]t2.
  32 #f #p #q #t1 #t2
  33 * #k * #H1k #Ht1 #Ht2
  34 @(ex_intro … ((↑[p●𝗔◗𝗟◗q]f)＠⧣❨k❩)) @and3_intro
  35 [ -Ht1 -Ht2
  36   <lift_rmap_L_dx >lift_path_L_sn
  37   <(lift_path_head_closed … H1k) in ⊢ (??%?); -H1k //
  38 | lapply (in_comp_lift_path_term f … Ht1) -Ht2 -Ht1 -H1k
  39   <lift_path_d_dx #Ht1 //
  40 | lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2 -Ht1
  41   @(subset_eq_trans … Ht2) -t2
  42   @(subset_eq_trans … (lift_term_fsubst …))
  43   @fsubst_eq_repl [ // | <lift_path_append // ]
  44   @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
  45   [ @lift_term_grafted_S | skip ]
  46   @(subset_eq_trans … (lift_term_after …))
  47   @(subset_eq_canc_dx … (lift_term_after …))
  48   @lift_term_eq_repl_sn
  49 (* Note: crux of the proof begins *)
  50   @(stream_eq_trans … (tr_compose_uni_dx …))
  51   @tr_compose_eq_repl //
  52   >list_append_rcons_sn in H1k; #H1k >lift_rmap_A_dx
  53   /2 width=1 by tls_lift_rmap_closed/
  54 (* Note: crux of the proof ends *)
  55 ]
  56 qed.