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
  18 include "delayed_updating/unwind1/unwind_fsubst.ma".
  19 include "delayed_updating/unwind1/unwind_constructors.ma".
  20 include "delayed_updating/unwind1/unwind_preterm_eq.ma".
  21 include "delayed_updating/unwind1/unwind_structure_depth.ma".
  22 include "delayed_updating/unwind1/unwind_depth.ma".
  23
  24 include "delayed_updating/substitution/fsubst_eq.ma".
  25 include "delayed_updating/substitution/lift_prototerm_eq.ma".
  26 include "delayed_updating/syntax/prototerm_proper_constructors.ma".
  27 include "delayed_updating/syntax/path_structure_depth.ma".
  28 include "ground/relocation/tr_uni_compose.ma".
  29 include "ground/relocation/tr_pap_pushs.ma".
  30
  31 (* DELAYED FOCUSED REDUCTION ************************************************)
  32
  33 (* COMMENT
  34 axiom pippo (b) (q) (n):
  35       ↑❘q❘ = (↑[q]𝐢)＠⧣❨n❩ →
  36       ↑❘q❘+❘b❘= (↑[b●𝗟◗q]𝐢)＠⧣❨n+❘b❘❩.
  37
  38 lemma unwind_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 <unwind_rmap_pap_d_dx #H0
  45     @(stream_eq_trans … (unwind_rmap_tls_d_dx …))
  46     @(stream_eq_trans … (IH …)) -IH //
  47   | /2 width=1 by/
  48   | <depth_L_dx <unwind_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 unwind_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 [| @unwind_rmap_decompose | skip ]
  67 <tr_compose_tls <Hp
  68
  69 @(stream_eq_canc_dx) … (unwind_rmap_decompose …))
  70
  71 *)
  72 lemma dfr_unwind_id_bi (p) (q) (t1) (t2): t1 ϵ 𝐓 →
  73       t1 ➡𝐝𝐟[p,q] t2 → ▼[𝐢]t1 ➡𝐟[⊗p,⊗q] ▼[𝐢]t2.
  74 #p #q #t1 #t2 #H0t1
  75 * #b #n * #Hb #Hn #Ht1 #Ht2
  76 @(ex1_2_intro … (⊗b) (↑♭⊗q)) @and4_intro
  77 [ //
  78 | (*//*)
  79 | lapply (in_comp_unwind_bi (𝐢) … Ht1) -Ht1 -H0t1 -Hb -Ht2
  80   <unwind_path_d_empty_dx <depth_structure //
  81 | lapply (unwind_term_eq_repl_dx (𝐢) … Ht2) -Ht2 #Ht2
  82   @(subset_eq_trans … Ht2) -t2
  83   @(subset_eq_trans … (unwind_fsubst …))
  84   [ (*<unwind_rmap_append <unwind_rmap_A_sn <unwind_rmap_append <unwind_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 … (unwind_iref …))
  89
  90     elim Hb -Hb #Hb #H0 <H0 -H0 <nrplus_zero_dx <nplus_zero_dx <nsucc_unfold
  91     >Hn
  92     @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
  93     [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
  94     <Hn <Hn
  95 (*
  96     @(subset_eq_trans … (lift_term_eq_repl_dx …))
  97     [ @(unwind_term_eq_repl_sn … (tls_succ_unwind q …)) | skip ]
  98 *)
  99 (*
 100
 101     @subset_eq_trans
 102     [2: @unwind_term_eq_repl_dx
 103     @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
 104
 105     @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
 106     [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
 107
 108     @(subset_eq_trans … (unwind_term_after …))
 109     @(subset_eq_canc_dx … (unwind_term_after …))
 110     @unwind_term_eq_repl_sn -t1
 111     @(stream_eq_trans … (tr_compose_uni_dx …))
 112     lapply (Hn (𝐢)) -Hn >tr_id_unfold #Hn
 113     lapply (pippo … b … Hn) -Hn #Hn
 114     @tr_compose_eq_repl
 115     [ <unwind_rmap_pap_le //
 116       <Hn <nrplus_inj_sn //
 117     |
 118     ]
 119 *)
 120   | //
 121   | /2 width=2 by ex_intro/
 122   | //
 123   ]
 124 ]
 125
 126 (*
 127 Hn : ↑❘q❘ = ↑[p●𝗔◗b●𝗟◗q]𝐢＠⧣❨n❩
 128 ---------------------------
 129 ↑[𝐮❨↑❘q❘+❘b❘❩] ↑[↑[p]𝐢] t ⇔ ↑[𝐮❨↑[p●𝗔◗b●𝗟◗q]𝐢＠⧣❨n+❘b❘❩❩] t
 130 *)
 131 (*
 132 (↑[𝐮❨↑❘q❘❩]▼[⇂*[↑❘q❘]▼[p●𝗟◗q]𝐢](t1⋔(p◖𝗦))⇔▼[𝐮❨↑❘q❘❩∘▼[p●𝗔◗b●𝗟◗q]𝐢](t1⋔(p◖𝗦))
 133 *)