matita/matita/contribs/lambdadelta/delayed_updating/etc/lift/lift_eq.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/substitution/lift.ma".
  16 (*
  17 include "ground/relocation/tr_uni_compose.ma".
  18 include "ground/relocation/tr_compose_compose.ma".
  19 include "ground/relocation/tr_compose_eq.ma".
  20 *)
  21 include "ground/relocation/tr_pap_eq.ma".
  22 include "ground/relocation/tr_pn_eq.ma".
  23 include "ground/lib/stream_tls_eq.ma".
  24
  25 (* LIFT FOR PATH ***********************************************************)
  26
  27 definition lift_exteq (A): relation2 (lift_continuation A) (lift_continuation A) ≝
  28            λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
  29
  30 interpretation
  31   "extensional equivalence (lift continuation)"
  32   'RingEq A k1 k2 = (lift_exteq A k1 k2).
  33
  34 (* Constructions with lift_exteq ********************************************)
  35
  36 lemma lift_eq_repl (A) (p) (k1) (k2):
  37       k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ↑❨k1, f1, p❩ = ↑❨k2, f2, p❩).
  38 #A #p elim p -p [| * [ #n ] #q #IH ]
  39 #k1 #k2 #Hk #f1 #f2 #Hf
  40 [ <lift_empty <lift_empty /2 width=1 by/
  41 | <lift_d_sn <lift_d_sn <(tr_pap_eq_repl … Hf)
  42   /3 width=3 by stream_tls_eq_repl, compose_repl_fwd_sn/
  43 | /3 width=1 by/
  44 | /3 width=1 by tr_push_eq_repl/
  45 | /3 width=1 by/
  46 | /3 width=1 by/
  47 ]
  48 qed-.
  49
  50 (* Advanced constructions ***************************************************)
  51
  52 lemma lift_lcons_alt (A) (k) (f) (p) (l): k ≗ k →
  53       ↑❨λg,p2. k g (l◗p2), f, p❩ = ↑{A}❨λg,p2. k g ((l◗𝐞)●p2), f, p❩.
  54 #A #k #f #p #l #Hk
  55 @lift_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
  56 qed.
  57
  58 lemma lift_append_rcons_sn (A) (k) (f) (p1) (p) (l): k ≗ k →
  59       ↑❨λg,p2. k g (p1●l◗p2), f, p❩ = ↑{A}❨λg,p2. k g (p1◖l●p2), f, p❩.
  60 #A #k #f #p1 #p #l #Hk
  61 @lift_eq_repl // #g1 #g2 #p2 #Hg
  62 <list_append_rcons_sn @Hk -Hk // (**) (* auto fail *)
  63 qed.
  64
  65 (* Advanced constructions with proj_path ************************************)
  66
  67 lemma proj_path_proper:
  68       proj_path ≗ proj_path.
  69 // qed.
  70
  71 lemma lift_path_eq_repl (p):
  72       stream_eq_repl … (λf1,f2. ↑[f1]p = ↑[f2]p).
  73 /2 width=1 by lift_eq_repl/ qed.
  74
  75 lemma lift_path_append_sn (p) (f) (q):
  76       q●↑[f]p = ↑❨(λg,p. proj_path g (q●p)), f, p❩.
  77 #p elim p -p // * [ #n ] #p #IH #f #q
  78 [ <lift_d_sn <lift_d_sn
  79 | <lift_m_sn <lift_m_sn
  80 | <lift_L_sn <lift_L_sn
  81 | <lift_A_sn <lift_A_sn
  82 | <lift_S_sn <lift_S_sn
  83 ]
  84 >lift_lcons_alt // >lift_append_rcons_sn //
  85 <IH <IH -IH <list_append_rcons_sn //
  86 qed.
  87
  88 lemma lift_path_lcons (f) (p) (l):
  89       l◗↑[f]p = ↑❨(λg,p. proj_path g (l◗p)), f, p❩.
  90 #f #p #l
  91 >lift_lcons_alt <lift_path_append_sn //
  92 qed.
  93
  94 lemma lift_path_d_sn (f) (p) (n):
  95       (𝗱(f@❨n❩)◗↑[⇂*[n]f]p) = ↑[f](𝗱n◗p).
  96 // qed.
  97
  98 lemma lift_path_m_sn (f) (p):
  99       (𝗺◗↑[f]p) = ↑[f](𝗺◗p).
 100 // qed.
 101
 102 lemma lift_path_L_sn (f) (p):
 103       (𝗟◗↑[⫯f]p) = ↑[f](𝗟◗p).
 104 // qed.
 105
 106 lemma lift_path_A_sn (f) (p):
 107       (𝗔◗↑[f]p) = ↑[f](𝗔◗p).
 108 // qed.
 109
 110 lemma lift_path_S_sn (f) (p):
 111       (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
 112 // qed.
 113 (* COMMENT
 114
 115 (* Advanced constructions with proj_rmap and stream_tls *********************)
 116
 117 lemma lift_rmap_tls_d_dx (f) (p) (m) (n):
 118       ⇂*[m+n]↑[p]f ≗ ⇂*[m]↑[p◖𝗱n]f.
 119 #f #p #m #n
 120 <lift_rmap_d_dx >nrplus_inj_dx
 121 /2 width=1 by tr_tls_compose_uni_dx/
 122 qed.
 123 *)