matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/dfr_ifr.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                  *)
  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/substitution/lift_prototerm_eq.ma".
  24 include "delayed_updating/syntax/prototerm_proper_constructors.ma".
  25 include "delayed_updating/syntax/path_structure_depth.ma".
  26 include "ground/relocation/tr_uni_compose.ma".
  27 include "ground/relocation/tr_pap_pushs.ma".
  28
  29 (* DELAYED FOCUSED REDUCTION ************************************************)
  30
  31 (* COMMENT
  32 axiom pippo (b) (q) (n):
  33       ↑❘q❘ = (↑[q]𝐢)@❨n❩ →
  34       ↑❘q❘+❘b❘= (↑[b●𝗟◗q]𝐢)@❨n+❘b❘❩.
  35
  36 lemma unwind_rmap_tls_eq_id (p) (n):
  37       ❘p❘ = ↑[p]𝐢@❨n❩ →
  38       (𝐢) ≗ ⇂*[n]↑[p]𝐢.
  39 #p @(list_ind_rcons … p) -p
  40 [ #n <depth_empty #H destruct
  41 | #p * [ #m ] #IH #n
  42   [ <depth_d_dx <unwind_rmap_pap_d_dx #H0
  43     @(stream_eq_trans … (unwind_rmap_tls_d_dx …))
  44     @(stream_eq_trans … (IH …)) -IH //
  45   | /2 width=1 by/
  46   | <depth_L_dx <unwind_rmap_L_dx
  47     cases n -n [| #n ] #H0
  48     [
  49     |
  50     ]
  51   | /2 width=1 by/
  52   | /2 width=1 by/
  53   ]
  54 ]
  55
  56
  57 (*  (↑❘q❘+❘b❘=↑[b●𝗟◗q]𝐢@❨n+❘b❘❩ *)
  58 (* [↑[p]𝐢@❨n❩]⫯*[❘p❘]f∘⇂*[n]↑[p]𝐢) *)
  59 lemma unwind_rmap_tls_eq (f) (p) (n):
  60       ❘p❘ = ↑[p]𝐢@❨n❩ →
  61       f ≗ ⇂*[n]↑[p]f.
  62 #f #p #n #Hp
  63 @(stream_eq_canc_dx … (stream_tls_eq_repl …))
  64 [| @unwind_rmap_decompose | skip ]
  65 <tr_compose_tls <Hp
  66
  67 @(stream_eq_canc_dx) … (unwind_rmap_decompose …))
  68
  69 *)
  70 lemma dfr_unwind_id_bi (p) (q) (t1) (t2): t1 ϵ 𝐓 →
  71       t1 ➡𝐝𝐟[p,q] t2 → ▼[𝐢]t1 ➡𝐟[⊗p,⊗q] ▼[𝐢]t2.
  72 #p #q #t1 #t2 #H0t1
  73 * #b #n * #Hb #Hn #Ht1 #Ht2
  74 @(ex1_2_intro … (⊗b) (↑❘⊗q❘)) @and4_intro
  75 [ //
  76 | //
  77 | lapply (in_comp_unwind_bi (𝐢) … Ht1) -Ht1 -H0t1 -Hb -Ht2
  78   <unwind_path_d_empty_dx <depth_structure //
  79 | lapply (unwind_term_eq_repl_dx (𝐢) … Ht2) -Ht2 #Ht2
  80   @(subset_eq_trans … Ht2) -t2
  81   @(subset_eq_trans … (unwind_fsubst …))
  82   [ (*<unwind_rmap_append <unwind_rmap_A_sn <unwind_rmap_append <unwind_rmap_L_sn *)
  83     <structure_append <structure_A_sn <structure_append <structure_L_sn
  84     <depth_append <depth_L_sn <depth_structure <depth_structure
  85     @fsubst_eq_repl [ // ]
  86     @(subset_eq_trans … (unwind_iref …))
  87
  88     elim Hb -Hb #Hb #H0 <H0 -H0 <nrplus_zero_dx <nplus_zero_dx <Hn
  89     @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
  90     [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
  91
  92 (*
  93     @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
  94     [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
  95
  96     @(subset_eq_trans … (unwind_term_after …))
  97     @(subset_eq_canc_dx … (unwind_term_after …))
  98     @unwind_term_eq_repl_sn -t1
  99     @(stream_eq_trans … (tr_compose_uni_dx …))
 100     lapply (Hn (𝐢)) -Hn >tr_id_unfold #Hn
 101     lapply (pippo … b … Hn) -Hn #Hn
 102     @tr_compose_eq_repl
 103     [ <unwind_rmap_pap_le //
 104       <Hn <nrplus_inj_sn //
 105     |
 106     ]
 107 *)
 108   | //
 109   | /2 width=2 by ex_intro/
 110   | //
 111   ]
 112 ]
 113
 114 (*
 115 Hn : ↑❘q❘ = ↑[p●𝗔◗b●𝗟◗q]𝐢@❨n❩
 116 ---------------------------
 117 ↑[𝐮❨↑❘q❘+❘b❘❩] ↑[↑[p]𝐢] t ⇔ ↑[𝐮❨↑[p●𝗔◗b●𝗟◗q]𝐢@❨n+❘b❘❩❩] t
 118 *)