matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_update.etc

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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  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/unwind1/unwind_eq.ma".
  16 include "delayed_updating/syntax/path_height.ma".
  17 include "delayed_updating/syntax/path_depth.ma".
  18 include "ground/lib/stream_eq_eq.ma".
  19
  20 (* UNWIND FOR PATH *********************************************************)
  21 (* COMMMENT
  22 (* Constructions with update ***********************************************)
  23
  24 lemma unwind_rmap_pap_le (f1) (f2) (p) (m:pnat) (l:nat):
  25       ninj (m+⧣p+l) = ❘p❘ →
  26       (▼[p]f1)＠⧣❨m❩ = (▼[p]f2)＠⧣❨m❩.
  27 #f1 #f2 #p @(list_ind_rcons … p) -p
  28 [ #m #l <depth_empty #H0 destruct
  29 | #p * [ #n ] #IH #m #l
  30   [ <update_d_dx <depth_d_dx <unwind_rmap_pap_d_dx <unwind_rmap_pap_d_dx
  31     <nplus_comm <nrplus_inj_sn <nrplus_inj_dx <nrplus_pplus_assoc
  32     #H0 <(IH … l) -IH //
  33   | /2 width=2 by/
  34   | <update_L_dx <depth_L_dx <unwind_rmap_L_dx <unwind_rmap_L_dx
  35     cases m -m // #m
  36     <nrplus_succ_sn <nrplus_succ_sn >nsucc_inj #H0
  37     <tr_pap_push <tr_pap_push
  38     <(IH … l) -IH //
  39   | /2 width=2 by/
  40   | /2 width=2 by/
  41   ]
  42 ]
  43 qed.
  44
  45 lemma unwind_rmap_pap_gt (f) (p) (m):
  46       f＠⧣❨m+⧣p❩+❘p❘ = (▼[p]f)＠⧣❨m+❘p❘❩.
  47 #f #p @(list_ind_rcons … p) -p [ // ]
  48 #p * [ #n ] #IH #m
  49 [ <update_d_dx <depth_d_dx
  50   <nplus_comm <nrplus_inj_sn <nrplus_inj_dx <nrplus_pplus_assoc
  51   <unwind_rmap_pap_d_dx >IH -IH //
  52 | //
  53 | <update_L_dx <depth_L_dx
  54   <nrplus_succ_dx <nrplus_succ_dx <unwind_rmap_L_dx <tr_pap_push //
  55 | //
  56 | //
  57 ]
  58 qed.
  59
  60 lemma unwind_rmap_tls_gt (f) (p) (m:pnat):
  61       ⇂*[ninj (m+⧣p)]f ≗ ⇂*[ninj (m+❘p❘)]▼[p]f.
  62 #f #p @(list_ind_rcons … p) -p [ // ]
  63 #p * [ #n ] #IH #m
  64 [ <update_d_dx <depth_d_dx
  65   <nplus_comm <nrplus_inj_sn <nrplus_inj_dx <nrplus_pplus_assoc
  66   @(stream_eq_trans … (unwind_rmap_tls_d_dx …))
  67   @(stream_eq_canc_dx … (IH …)) -IH //
  68 | //
  69 | <update_L_dx <depth_L_dx
  70   <nrplus_succ_dx >nsucc_inj //
  71 | //
  72 | //
  73 ]
  74 qed.
  75
  76 lemma unwind_rmap_tls_eq (f) (p):
  77       ⇂*[⧣p]f ≗ ⇂*[▼❘p❘]▼[p]⫯f.
  78 (*
  79 #f #p @(list_ind_rcons … p) -p //
  80 #p * [ #n ] #IH //
  81 <update_d_dx <depth_d_dx <unwind_rmap_d_dx
  82 @(stream_eq_trans … (tr_tls_compose_uni_dx …))
  83 *)
  84 *)