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/unwind1/unwind_fsubst.ma".
  18 include "delayed_updating/unwind1/unwind_constructors.ma".
  19 include "delayed_updating/unwind1/unwind_preterm_eq.ma".
  20 include "delayed_updating/unwind1/unwind_structure_depth.ma".
  21 include "delayed_updating/unwind1/unwind_depth.ma".
  22 include "delayed_updating/substitution/fsubst_eq.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 (* COMMENT
  31 axiom pippo (b) (q) (n):
  32       ↑❘q❘ = (↑[q]𝐢)@❨n❩ →
  33       ↑❘q❘+❘b❘= (↑[b●𝗟◗q]𝐢)@❨n+❘b❘❩.
  34
  35 lemma unwind_rmap_tls_eq_id (p) (n):
  36       ❘p❘ = ↑[p]𝐢@❨n❩ →
  37       (𝐢) ≗ ⇂*[n]↑[p]𝐢.
  38 #p @(list_ind_rcons … p) -p
  39 [ #n <depth_empty #H destruct
  40 | #p * [ #m ] #IH #n
  41   [ <depth_d_dx <unwind_rmap_pap_d_dx #H0
  42     @(stream_eq_trans … (unwind_rmap_tls_d_dx …))
  43     @(stream_eq_trans … (IH …)) -IH //
  44   | /2 width=1 by/
  45   | <depth_L_dx <unwind_rmap_L_dx
  46     cases n -n [| #n ] #H0
  47     [
  48     |
  49     ]
  50   | /2 width=1 by/
  51   | /2 width=1 by/
  52   ]
  53 ]
  54
  55
  56 (*  (↑❘q❘+❘b❘=↑[b●𝗟◗q]𝐢@❨n+❘b❘❩ *)
  57 (* [↑[p]𝐢@❨n❩]⫯*[❘p❘]f∘⇂*[n]↑[p]𝐢) *)
  58 lemma unwind_rmap_tls_eq (f) (p) (n):
  59       ❘p❘ = ↑[p]𝐢@❨n❩ →
  60       f ≗ ⇂*[n]↑[p]f.
  61 #f #p #n #Hp
  62 @(stream_eq_canc_dx … (stream_tls_eq_repl …))
  63 [| @unwind_rmap_decompose | skip ]
  64 <tr_compose_tls <Hp
  65
  66 @(stream_eq_canc_dx) … (unwind_rmap_decompose …))
  67
  68 *)
  69 lemma dfr_unwind_id_bi (p) (q) (t1) (t2): t1 ϵ 𝐓 →
  70       t1 ➡𝐝𝐟[p,q] t2 → ▼[𝐢]t1 ➡𝐟[⊗p,⊗q] ▼[𝐢]t2.
  71 #p #q #t1 #t2 #H0t1
  72 * #b #n * #Hb #Hn #Ht1 #Ht2
  73 @(ex1_2_intro … (⊗b) (↑❘⊗q❘)) @and4_intro
  74 [ //
  75 | //
  76 | lapply (in_comp_unwind_bi (𝐢) … Ht1) -Ht1 -H0t1 -Hb -Ht2
  77   <unwind_path_d_empty_dx <depth_structure //
  78 | lapply (unwind_term_eq_repl_dx (𝐢) … Ht2) -Ht2 #Ht2
  79   @(subset_eq_trans … Ht2) -t2
  80   @(subset_eq_trans … (unwind_fsubst …))
  81   [ (*<unwind_rmap_append <unwind_rmap_A_sn <unwind_rmap_append <unwind_rmap_L_sn *)
  82     <structure_append <structure_A_sn <structure_append <structure_L_sn
  83     <depth_append <depth_L_sn <depth_structure <depth_structure
  84     @fsubst_eq_repl [ // ]
  85     @(subset_eq_trans … (unwind_iref …))
  86 (*
  87     @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
  88     [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
  89
  90     @(subset_eq_trans … (unwind_term_after …))
  91     @(subset_eq_canc_dx … (unwind_term_after …))
  92     @unwind_term_eq_repl_sn -t1
  93     @(stream_eq_trans … (tr_compose_uni_dx …))
  94     lapply (Hn (𝐢)) -Hn >tr_id_unfold #Hn
  95     lapply (pippo … b … Hn) -Hn #Hn
  96     @tr_compose_eq_repl
  97     [ <unwind_rmap_pap_le //
  98       <Hn <nrplus_inj_sn //
  99     |
 100     ]
 101 *)
 102   | //
 103   | /2 width=2 by ex_intro/
 104   | //
 105   ]
 106 ]
 107
 108 (*
 109 Hn : ↑❘q❘ = ↑[p●𝗔◗b●𝗟◗q]𝐢@❨n❩
 110 ---------------------------
 111 ↑[𝐮❨↑❘q❘+❘b❘❩] ↑[↑[p]𝐢] t ⇔ ↑[𝐮❨↑[p●𝗔◗b●𝗟◗q]𝐢@❨n+❘b❘❩❩] t
 112 *)