matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind.etc

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/relocation/tr_compose_pap.ma".
  16 include "ground/relocation/tr_uni_pap.ma".
  17 include "delayed_updating/syntax/path.ma".
  18 include "delayed_updating/notation/functions/black_downtriangle_4.ma".
  19 include "delayed_updating/notation/functions/black_downtriangle_2.ma".
  20
  21 (* UNWIND FOR PATH **********************************************************)
  22
  23 definition unwind_continuation (A:Type[0]) ≝
  24 tr_map → path → A.
  25
  26 rec definition unwind_gen (A:Type[0]) (k:unwind_continuation A) (f) (p) on p ≝
  27 match p with
  28 [ list_empty     ⇒ k f (𝐞)
  29 | list_lcons l q ⇒
  30   match l with
  31   [ label_d n ⇒
  32     match q with
  33     [ list_empty     ⇒ unwind_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (f∘𝐮❨n❩) q
  34     | list_lcons _ _ ⇒ unwind_gen (A) k (f∘𝐮❨n❩) q
  35     ]
  36   | label_m   ⇒ unwind_gen (A) k f q
  37   | label_L   ⇒ unwind_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
  38   | label_A   ⇒ unwind_gen (A) (λg,p. k g (𝗔◗p)) f q
  39   | label_S   ⇒ unwind_gen (A) (λg,p. k g (𝗦◗p)) f q
  40   ]
  41 ].
  42
  43 interpretation
  44   "unwind (gneric)"
  45   'BlackDownTriangle A k f p = (unwind_gen A k f p).
  46
  47 definition proj_path: unwind_continuation … ≝
  48            λf,p.p.
  49
  50 definition proj_rmap: unwind_continuation … ≝
  51            λf,p.f.
  52
  53 interpretation
  54   "unwind (path)"
  55   'BlackDownTriangle f p = (unwind_gen ? proj_path f p).
  56
  57 interpretation
  58   "unwind (relocation map)"
  59   'BlackDownTriangle p f = (unwind_gen ? proj_rmap f p).
  60
  61 (* Basic constructions ******************************************************)
  62
  63 lemma unwind_empty (A) (k) (f):
  64       k f (𝐞) = ▼{A}❨k, f, 𝐞❩.
  65 // qed.
  66
  67 lemma unwind_d_empty_sn (A) (k) (n) (f):
  68       ▼❨(λg,p. k g (𝗱(f@❨n❩)◗p)), f∘𝐮❨ninj n❩, 𝐞❩ = ▼{A}❨k, f,
  69 𝗱n◗𝐞❩.
  70 // qed.
  71
  72 lemma unwind_d_lcons_sn (A) (k) (p) (l) (n) (f):
  73       ▼❨k, f∘𝐮❨ninj n❩, l◗p❩ = ▼{A}❨k, f, 𝗱n◗l◗p❩.
  74 // qed.
  75
  76 lemma unwind_m_sn (A) (k) (p) (f):
  77       ▼❨k, f, p❩ = ▼{A}❨k, f, 𝗺◗p❩.
  78 // qed.
  79
  80 lemma unwind_L_sn (A) (k) (p) (f):
  81       ▼❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ▼{A}❨k, f, 𝗟◗p❩.
  82 // qed.
  83
  84 lemma unwind_A_sn (A) (k) (p) (f):
  85       ▼❨(λg,p. k g (𝗔◗p)), f, p❩ = ▼{A}❨k, f, 𝗔◗p❩.
  86 // qed.
  87
  88 lemma unwind_S_sn (A) (k) (p) (f):
  89       ▼❨(λg,p. k g (𝗦◗p)), f, p❩ = ▼{A}❨k, f, 𝗦◗p❩.
  90 // qed.
  91
  92 (* Basic constructions with proj_path ***************************************)
  93
  94 lemma unwind_path_empty (f):
  95       (𝐞) = ▼[f]𝐞.
  96 // qed.
  97
  98 lemma unwind_path_d_empty_sn (f) (n):
  99       𝗱(f@❨n❩)◗𝐞 = ▼[f](𝗱n◗𝐞).
 100 // qed.
 101
 102 lemma unwind_path_d_lcons_sn (f) (p) (l) (n):
 103       ▼[f∘𝐮❨ninj n❩](l◗p) = ▼[f](𝗱n◗l◗p).
 104 // qed.
 105
 106 lemma unwind_path_m_sn (f) (p):
 107       ▼[f]p = ▼[f](𝗺◗p).
 108 // qed.
 109
 110 (* Basic constructions with proj_rmap ***************************************)
 111
 112 lemma unwind_rmap_empty (f):
 113       f = ▼[𝐞]f.
 114 // qed.
 115
 116 lemma unwind_rmap_d_sn (f) (p) (n):
 117       ▼[p](f∘𝐮❨ninj n❩) = ▼[𝗱n◗p]f.
 118 #f * // qed.
 119
 120 lemma unwind_rmap_m_sn (f) (p):
 121       ▼[p]f = ▼[𝗺◗p]f.
 122 // qed.
 123
 124 lemma unwind_rmap_L_sn (f) (p):
 125       ▼[p](⫯f) = ▼[𝗟◗p]f.
 126 // qed.
 127
 128 lemma unwind_rmap_A_sn (f) (p):
 129       ▼[p]f = ▼[𝗔◗p]f.
 130 // qed.
 131
 132 lemma unwind_rmap_S_sn (f) (p):
 133       ▼[p]f = ▼[𝗦◗p]f.
 134 // qed.
 135
 136 (* Advanced constructions with proj_rmap and path_append ********************)
 137
 138 lemma unwind_rmap_append (p2) (p1) (f):
 139       ▼[p2]▼[p1]f = ▼[p1●p2]f.
 140 #p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
 141 [ <unwind_rmap_m_sn <unwind_rmap_m_sn //
 142 | <unwind_rmap_A_sn <unwind_rmap_A_sn //
 143 | <unwind_rmap_S_sn <unwind_rmap_S_sn //
 144 ]
 145 qed.
 146
 147 (* Advanced constructions with proj_rmap and path_rcons *********************)
 148
 149 lemma unwind_rmap_d_dx (f) (p) (n):
 150       (▼[p]f)∘𝐮❨ninj n❩ = ▼[p◖𝗱n]f.
 151 // qed.
 152
 153 lemma unwind_rmap_m_dx (f) (p):
 154       ▼[p]f = ▼[p◖𝗺]f.
 155 // qed.
 156
 157 lemma unwind_rmap_L_dx (f) (p):
 158       (⫯▼[p]f) = ▼[p◖𝗟]f.
 159 // qed.
 160
 161 lemma unwind_rmap_A_dx (f) (p):
 162       ▼[p]f = ▼[p◖𝗔]f.
 163 // qed.
 164
 165 lemma unwind_rmap_S_dx (f) (p):
 166 ▼[p]f = ▼[p◖𝗦]f.
 167 // qed.
 168
 169 lemma unwind_rmap_pap_d_dx (f) (p) (n) (m):
 170       ▼[p]f@❨m+n❩ = ▼[p◖𝗱n]f@❨m❩.
 171 #f #p #n #m
 172 <unwind_rmap_d_dx <tr_compose_pap <tr_uni_pap //
 173 qed.
 174
 175 (* Advanced eliminations with path ******************************************)
 176
 177 lemma path_ind_unwind (Q:predicate …):
 178       Q (𝐞) →
 179       (∀n. Q (𝐞) → Q (𝗱n◗𝐞)) →
 180       (∀n,l,p. Q (l◗p) → Q (𝗱n◗l◗p)) →
 181       (∀p. Q p → Q (𝗺◗p)) →
 182       (∀p. Q p → Q (𝗟◗p)) →
 183       (∀p. Q p → Q (𝗔◗p)) →
 184       (∀p. Q p → Q (𝗦◗p)) →
 185       ∀p. Q p.
 186 #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #p
 187 elim p -p [| * [ #n * ] ]
 188 /2 width=1 by/
 189 qed-.