matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.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/dfr.ma".
  16 include "delayed_updating/reduction/ifr.ma".
  17 include "delayed_updating/substitution/fsubst_lift.ma".
  18 include "delayed_updating/substitution/fsubst_eq.ma".
  19 include "delayed_updating/substitution/lift_constructors.ma".
  20 include "delayed_updating/substitution/lift_preterm_eq.ma".
  21 include "delayed_updating/substitution/lift_structure_depth.ma".
  22 include "delayed_updating/substitution/lift_depth.ma".
  23 include "delayed_updating/syntax/prototerm_proper_constructors.ma".
  24 include "delayed_updating/syntax/path_structure_depth.ma".
  25 include "ground/relocation/tr_uni_compose.ma".
  26 include "ground/relocation/tr_pap_pushs.ma".
  27
  28 (* DELAYED FOCUSED REDUCTION ************************************************)
  29
  30 lemma tr_uni_eq_repl (n1) (n2):
  31       n1 = n2 → 𝐮❨n1❩ ≗ 𝐮❨n2❩.
  32 // qed.
  33
  34 axiom pippo (b) (q) (n):
  35       ↑❘q❘ = (↑[q]𝐢)@❨n❩ →
  36       ↑❘q❘+❘b❘= (↑[b●𝗟◗q]𝐢)@❨n+❘b❘❩.
  37
  38 lemma lift_rmap_tls_eq_id (p) (n):
  39       ❘p❘ = ↑[p]𝐢@❨n❩ →
  40       (𝐢) ≗ ⇂*[n]↑[p]𝐢.
  41 #p @(list_ind_rcons … p) -p
  42 [ #n <depth_empty #H destruct
  43 | #p * [ #m ] #IH #n
  44   [ <depth_d_dx <lift_rmap_pap_d_dx #H0
  45     @(stream_eq_trans … (lift_rmap_tls_d_dx …))
  46     @(stream_eq_trans … (IH …)) -IH //
  47   | /2 width=1 by/
  48   | <depth_L_dx <lift_rmap_L_dx
  49     cases n -n [| #n ] #H0
  50     [
  51     |
  52     ]
  53   | /2 width=1 by/
  54   | /2 width=1 by/
  55   ]
  56 ]
  57
  58
  59 (*  (↑❘q❘+❘b❘=↑[b●𝗟◗q]𝐢@❨n+❘b❘❩ *)
  60 (* [↑[p]𝐢@❨n❩]⫯*[❘p❘]f∘⇂*[n]↑[p]𝐢) *)
  61 lemma lift_rmap_tls_eq (f) (p) (n):
  62       ❘p❘ = ↑[p]𝐢@❨n❩ →
  63       f ≗ ⇂*[n]↑[p]f.
  64 #f #p #n #Hp
  65 @(stream_eq_canc_dx … (stream_tls_eq_repl …))
  66 [| @lift_rmap_decompose | skip ]
  67 <tr_compose_tls <Hp
  68
  69 @(stream_eq_canc_dx) … (lift_rmap_decompose …))
  70
  71 lemma dfr_lift_bi (f) (p) (q) (t1) (t2): t1 ϵ 𝐓 →
  72       t1 ➡𝐝𝐟[p,q] t2 → ↑[f]t1 ➡𝐟[⊗p,⊗q] ↑[f]t2.
  73 #f #p #q #t1 #t2 #H0t1
  74 * #b #n * #Hb #Hn  #Ht1 #Ht2
  75 @(ex1_2_intro … (⊗b) (↑❘⊗q❘)) @and4_intro
  76 [ //
  77 | #g <lift_rmap_structure <depth_structure
  78   >tr_pushs_swap <tr_pap_pushs_le //
  79 | lapply (in_comp_lift_bi f … Ht1) -Ht1 -H0t1 -Hb -Ht2
  80   <lift_path_d_empty_dx //
  81 | lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
  82   @(subset_eq_trans … Ht2) -t2
  83   @(subset_eq_trans … (lift_fsubst …))
  84   [ <lift_rmap_append <lift_rmap_A_sn (* <lift_rmap_append <lift_rmap_L_sn *)
  85     <structure_append <structure_A_sn <structure_append <structure_L_sn
  86     <depth_append <depth_L_sn <depth_structure <depth_structure
  87     @fsubst_eq_repl [ // ]
  88     @(subset_eq_trans … (lift_iref …))
  89     @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
  90     [ @lift_grafted_S /2 width=2 by ex_intro/ | skip ]
  91     @(subset_eq_trans … (lift_term_after …))
  92     @(subset_eq_canc_dx … (lift_term_after …))
  93     @lift_term_eq_repl_sn -t1
  94     @(stream_eq_trans … (tr_compose_uni_dx …))
  95     lapply (Hn (𝐢)) -Hn >tr_id_unfold #Hn
  96     lapply (pippo … b … Hn) -Hn #Hn
  97     @tr_compose_eq_repl
  98     [ <lift_rmap_pap_le //
  99       <Hn <nrplus_inj_sn //
 100     |
 101     ]
 102   | //
 103   | /2 width=2 by ex_intro/
 104   | //
 105   ]
 106 ]