matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.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/substitution/lift.ma".
  16 include "ground/relocation/tr_pap_pap.ma".
  17 include "ground/relocation/tr_pap_eq.ma".
  18 include "ground/relocation/tr_pn_eq.ma".
  19 include "ground/lib/stream_tls_eq.ma".
  20
  21 (* LIFT FOR PATH ************************************************************)
  22
  23 definition lift_exteq (A): relation2 (lift_continuation A) (lift_continuation A) ≝
  24            λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
  25
  26 interpretation
  27   "extensional equivalence (lift continuation)"
  28   'RingEq A k1 k2 = (lift_exteq A k1 k2).
  29
  30 (* Constructions with lift_exteq ********************************************)
  31
  32 lemma lift_eq_repl (A) (p) (k1) (k2):
  33       k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ↑❨k1, f1, p❩ = ↑❨k2, f2, p❩).
  34 #A #p elim p -p [| * [ #n ] #q #IH ]
  35 #k1 #k2 #Hk #f1 #f2 #Hf
  36 [ <lift_empty <lift_empty /2 width=1 by/
  37 | <lift_d_sn <lift_d_sn <(tr_pap_eq_repl … Hf)
  38   /3 width=3 by stream_tls_eq_repl, compose_repl_fwd_sn/
  39 | /3 width=1 by/
  40 | /3 width=1 by tr_push_eq_repl/
  41 | /3 width=1 by/
  42 | /3 width=1 by/
  43 ]
  44 qed-.
  45
  46 (* Advanced constructions ***************************************************)
  47
  48 lemma lift_lcons_alt (A) (k) (f) (p) (l): k ≗ k →
  49       ↑❨λg,p2. k g (l◗p2), f, p❩ = ↑{A}❨λg,p2. k g ((l◗𝐞)●p2), f, p❩.
  50 #A #k #f #p #l #Hk
  51 @lift_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
  52 qed.
  53
  54 lemma lift_append_rcons_sn (A) (k) (f) (p1) (p) (l): k ≗ k →
  55       ↑❨λg,p2. k g (p1●l◗p2), f, p❩ = ↑{A}❨λg,p2. k g (p1◖l●p2), f, p❩.
  56 #A #k #f #p1 #p #l #Hk
  57 @lift_eq_repl // #g1 #g2 #p2 #Hg
  58 <list_append_rcons_sn @Hk -Hk // (**) (* auto fail *)
  59 qed.
  60
  61 (* Advanced constructions with proj_path ************************************)
  62
  63 lemma proj_path_proper:
  64       proj_path ≗ proj_path.
  65 // qed.
  66
  67 lemma lift_path_eq_repl (p):
  68       stream_eq_repl … (λf1,f2. ↑[f1]p = ↑[f2]p).
  69 /2 width=1 by lift_eq_repl/ qed.
  70
  71 lemma lift_path_append_sn (p) (f) (q):
  72       q●↑[f]p = ↑❨(λg,p. proj_path g (q●p)), f, p❩.
  73 #p elim p -p // * [ #n ] #p #IH #f #q
  74 [ <lift_d_sn <lift_d_sn
  75 | <lift_m_sn <lift_m_sn
  76 | <lift_L_sn <lift_L_sn
  77 | <lift_A_sn <lift_A_sn
  78 | <lift_S_sn <lift_S_sn
  79 ]
  80 >lift_lcons_alt // >lift_append_rcons_sn //
  81 <IH <IH -IH <list_append_rcons_sn //
  82 qed.
  83
  84 lemma lift_path_lcons (f) (p) (l):
  85       l◗↑[f]p = ↑❨(λg,p. proj_path g (l◗p)), f, p❩.
  86 #f #p #l
  87 >lift_lcons_alt <lift_path_append_sn //
  88 qed.
  89
  90 lemma lift_path_d_sn (f) (p) (n):
  91       (𝗱(f＠⧣❨n❩)◗↑[⇂*[n]f]p) = ↑[f](𝗱n◗p).
  92 // qed.
  93
  94 lemma lift_path_m_sn (f) (p):
  95       (𝗺◗↑[f]p) = ↑[f](𝗺◗p).
  96 // qed.
  97
  98 lemma lift_path_L_sn (f) (p):
  99       (𝗟◗↑[⫯f]p) = ↑[f](𝗟◗p).
 100 // qed.
 101
 102 lemma lift_path_A_sn (f) (p):
 103       (𝗔◗↑[f]p) = ↑[f](𝗔◗p).
 104 // qed.
 105
 106 lemma lift_path_S_sn (f) (p):
 107       (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
 108 // qed.
 109
 110 lemma lift_path_append (p2) (p1) (f):
 111       (↑[f]p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
 112 #p2 #p1 elim p1 -p1 //
 113 * [ #n1 ] #p1 #IH #f
 114 [ <lift_path_d_sn <lift_path_d_sn <IH //
 115 | <lift_path_m_sn <lift_path_m_sn <IH //
 116 | <lift_path_L_sn <lift_path_L_sn <IH //
 117 | <lift_path_A_sn <lift_path_A_sn <IH //
 118 | <lift_path_S_sn <lift_path_S_sn <IH //
 119 ]
 120 qed.
 121
 122 lemma lift_path_d_dx (f) (p) (n):
 123       (↑[f]p)◖𝗱((↑[p]f)＠⧣❨n❩) = ↑[f](p◖𝗱n).
 124 #f #p #n <lift_path_append //
 125 qed.
 126
 127 lemma lift_path_m_dx (f) (p):
 128       (↑[f]p)◖𝗺 = ↑[f](p◖𝗺).
 129 #f #p <lift_path_append //
 130 qed.
 131
 132 lemma lift_path_L_dx (f) (p):
 133       (↑[f]p)◖𝗟 = ↑[f](p◖𝗟).
 134 #f #p <lift_path_append //
 135 qed.
 136
 137 lemma lift_path_A_dx (f) (p):
 138       (↑[f]p)◖𝗔 = ↑[f](p◖𝗔).
 139 #f #p <lift_path_append //
 140 qed.
 141
 142 lemma lift_path_S_dx (f) (p):
 143       (↑[f]p)◖𝗦 = ↑[f](p◖𝗦).
 144 #f #p <lift_path_append //
 145 qed.
 146
 147 (* Advanced inversions ******************************************************)
 148
 149 lemma lift_path_inv_empty (f) (p):
 150       (𝐞) = ↑[f]p → 𝐞 = p.
 151 #f * // * [ #n ] #p
 152 [ <lift_path_d_sn
 153 | <lift_path_m_sn
 154 | <lift_path_L_sn
 155 | <lift_path_A_sn
 156 | <lift_path_S_sn
 157 ] #H destruct
 158 qed-.
 159
 160 lemma lift_path_inv_d_sn (f) (p) (q) (k):
 161       (𝗱k◗q) = ↑[f]p →
 162       ∃∃r,h. k = f＠⧣❨h❩ & q = ↑[⇂*[h]f]r & 𝗱h◗r = p.
 163 #f * [| * [ #n ] #p ] #q #k
 164 [ <lift_path_empty
 165 | <lift_path_d_sn
 166 | <lift_path_m_sn
 167 | <lift_path_L_sn
 168 | <lift_path_A_sn
 169 | <lift_path_S_sn
 170 ] #H destruct
 171 /2 width=5 by ex3_2_intro/
 172 qed-.
 173
 174 lemma lift_path_inv_m_sn (f) (p) (q):
 175       (𝗺◗q) = ↑[f]p →
 176       ∃∃r. q = ↑[f]r & 𝗺◗r = p.
 177 #f * [| * [ #n ] #p ] #q
 178 [ <lift_path_empty
 179 | <lift_path_d_sn
 180 | <lift_path_m_sn
 181 | <lift_path_L_sn
 182 | <lift_path_A_sn
 183 | <lift_path_S_sn
 184 ] #H destruct
 185 /2 width=3 by ex2_intro/
 186 qed-.
 187
 188 lemma lift_path_inv_L_sn (f) (p) (q):
 189       (𝗟◗q) = ↑[f]p →
 190       ∃∃r. q = ↑[⫯f]r & 𝗟◗r = p.
 191 #f * [| * [ #n ] #p ] #q
 192 [ <lift_path_empty
 193 | <lift_path_d_sn
 194 | <lift_path_m_sn
 195 | <lift_path_L_sn
 196 | <lift_path_A_sn
 197 | <lift_path_S_sn
 198 ] #H destruct
 199 /2 width=3 by ex2_intro/
 200 qed-.
 201
 202 lemma lift_path_inv_A_sn (f) (p) (q):
 203       (𝗔◗q) = ↑[f]p →
 204       ∃∃r. q = ↑[f]r & 𝗔◗r = p.
 205 #f * [| * [ #n ] #p ] #q
 206 [ <lift_path_empty
 207 | <lift_path_d_sn
 208 | <lift_path_m_sn
 209 | <lift_path_L_sn
 210 | <lift_path_A_sn
 211 | <lift_path_S_sn
 212 ] #H destruct
 213 /2 width=3 by ex2_intro/
 214 qed-.
 215
 216 lemma lift_path_inv_S_sn (f) (p) (q):
 217       (𝗦◗q) = ↑[f]p →
 218       ∃∃r. q = ↑[f]r & 𝗦◗r = p.
 219 #f * [| * [ #n ] #p ] #q
 220 [ <lift_path_empty
 221 | <lift_path_d_sn
 222 | <lift_path_m_sn
 223 | <lift_path_L_sn
 224 | <lift_path_A_sn
 225 | <lift_path_S_sn
 226 ] #H destruct
 227 /2 width=3 by ex2_intro/
 228 qed-.
 229
 230 lemma lift_path_inv_append_dx (q2) (q1) (p) (f):
 231       q1●q2 = ↑[f]p →
 232       ∃∃p1,p2. q1 = ↑[f]p1 & q2 = ↑[↑[p1]f]p2 & p1●p2 = p.
 233 #q2 #q1 elim q1 -q1
 234 [| * [ #n1 ] #q1 #IH ] #p #f
 235 [ <list_append_empty_sn #H0 destruct
 236   /2 width=5 by ex3_2_intro/
 237 | <list_append_lcons_sn #H0
 238   elim (lift_path_inv_d_sn … H0) -H0 #r1 #m1 #_ #_ #H0 #_ -IH
 239     elim (eq_inv_list_empty_append … H0) -H0 #_ #H0 destruct
 240     elim Hq2 -Hq2 //
 241   | elim (lift_path_inv_m_sn … H)
 242   | elim (lift_path_inv_L_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
 243     elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
 244     @(ex3_2_intro … (r1●𝗟◗p1)) //
 245     <structure_append <Hr1 -Hr1 //
 246   | elim (lift_path_inv_A_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
 247     elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
 248     @(ex3_2_intro … (r1●𝗔◗p1)) //
 249     <structure_append <Hr1 -Hr1 //
 250   | elim (lift_path_inv_S_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
 251     elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
 252     @(ex3_2_intro … (r1●𝗦◗p1)) //
 253     <structure_append <Hr1 -Hr1 //
 254   ]
 255 ]
 256 qed-.
 257
 258 (* Main inversions **********************************************************)
 259
 260 theorem lift_path_inj (q:path) (p) (f):
 261         ↑[f]q = ↑[f]p → q = p.
 262 #q elim q -q [| * [ #k ] #q #IH ] #p #f
 263 [ <lift_path_empty #H0
 264   <(lift_path_inv_empty … H0) -H0 //
 265 | <lift_path_d_sn #H0
 266   elim (lift_path_inv_d_sn … H0) -H0 #r #h #H0
 267   <(tr_pap_inj ????? H0) -h [1,3: // ] #Hr #H0 destruct
 268 | <lift_path_m_sn #H0
 269   elim (lift_path_inv_m_sn … H0) -H0 #r #Hr #H0 destruct
 270 | <lift_path_L_sn #H0
 271   elim (lift_path_inv_L_sn … H0) -H0 #r #Hr #H0 destruct
 272 | <lift_path_A_sn #H0
 273   elim (lift_path_inv_A_sn … H0) -H0 #r #Hr #H0 destruct
 274 | <lift_path_S_sn #H0
 275   elim (lift_path_inv_S_sn … H0) -H0 #r #Hr #H0 destruct
 276 ]
 277 <(IH … Hr) -r -IH //
 278 qed-.
 279
 280 (* COMMENT
 281
 282 (* Advanced constructions with proj_rmap and stream_tls *********************)
 283
 284 lemma lift_rmap_tls_d_dx (f) (p) (m) (n):
 285       ⇂*[m+n]↑[p]f ≗ ⇂*[m]↑[p◖𝗱n]f.
 286 #f #p #m #n
 287 <lift_rmap_d_dx >nrplus_inj_dx
 288 /2 width=1 by tr_tls_compose_uni_dx/
 289 qed.
 290
 291 *)