src/HOLCF/Universal.thy
author huffman
Wed Nov 10 17:56:08 2010 -0800 (2010-11-10)
changeset 40502 8e92772bc0e8
parent 40500 ee9c8d36318e
permissions -rw-r--r--
move map functions to new theory file Map_Functions; add theory file Plain_HOLCF
     1 (*  Title:      HOLCF/Universal.thy
     2     Author:     Brian Huffman
     3 *)
     4 
     5 header {* A universal bifinite domain *}
     6 
     7 theory Universal
     8 imports Completion Deflation Nat_Bijection
     9 begin
    10 
    11 subsection {* Basis for universal domain *}
    12 
    13 subsubsection {* Basis datatype *}
    14 
    15 types ubasis = nat
    16 
    17 definition
    18   node :: "nat \<Rightarrow> ubasis \<Rightarrow> ubasis set \<Rightarrow> ubasis"
    19 where
    20   "node i a S = Suc (prod_encode (i, prod_encode (a, set_encode S)))"
    21 
    22 lemma node_not_0 [simp]: "node i a S \<noteq> 0"
    23 unfolding node_def by simp
    24 
    25 lemma node_gt_0 [simp]: "0 < node i a S"
    26 unfolding node_def by simp
    27 
    28 lemma node_inject [simp]:
    29   "\<lbrakk>finite S; finite T\<rbrakk>
    30     \<Longrightarrow> node i a S = node j b T \<longleftrightarrow> i = j \<and> a = b \<and> S = T"
    31 unfolding node_def by (simp add: prod_encode_eq set_encode_eq)
    32 
    33 lemma node_gt0: "i < node i a S"
    34 unfolding node_def less_Suc_eq_le
    35 by (rule le_prod_encode_1)
    36 
    37 lemma node_gt1: "a < node i a S"
    38 unfolding node_def less_Suc_eq_le
    39 by (rule order_trans [OF le_prod_encode_1 le_prod_encode_2])
    40 
    41 lemma nat_less_power2: "n < 2^n"
    42 by (induct n) simp_all
    43 
    44 lemma node_gt2: "\<lbrakk>finite S; b \<in> S\<rbrakk> \<Longrightarrow> b < node i a S"
    45 unfolding node_def less_Suc_eq_le set_encode_def
    46 apply (rule order_trans [OF _ le_prod_encode_2])
    47 apply (rule order_trans [OF _ le_prod_encode_2])
    48 apply (rule order_trans [where y="setsum (op ^ 2) {b}"])
    49 apply (simp add: nat_less_power2 [THEN order_less_imp_le])
    50 apply (erule setsum_mono2, simp, simp)
    51 done
    52 
    53 lemma eq_prod_encode_pairI:
    54   "\<lbrakk>fst (prod_decode x) = a; snd (prod_decode x) = b\<rbrakk> \<Longrightarrow> x = prod_encode (a, b)"
    55 by (erule subst, erule subst, simp)
    56 
    57 lemma node_cases:
    58   assumes 1: "x = 0 \<Longrightarrow> P"
    59   assumes 2: "\<And>i a S. \<lbrakk>finite S; x = node i a S\<rbrakk> \<Longrightarrow> P"
    60   shows "P"
    61  apply (cases x)
    62   apply (erule 1)
    63  apply (rule 2)
    64   apply (rule finite_set_decode)
    65  apply (simp add: node_def)
    66  apply (rule eq_prod_encode_pairI [OF refl])
    67  apply (rule eq_prod_encode_pairI [OF refl refl])
    68 done
    69 
    70 lemma node_induct:
    71   assumes 1: "P 0"
    72   assumes 2: "\<And>i a S. \<lbrakk>P a; finite S; \<forall>b\<in>S. P b\<rbrakk> \<Longrightarrow> P (node i a S)"
    73   shows "P x"
    74  apply (induct x rule: nat_less_induct)
    75  apply (case_tac n rule: node_cases)
    76   apply (simp add: 1)
    77  apply (simp add: 2 node_gt1 node_gt2)
    78 done
    79 
    80 subsubsection {* Basis ordering *}
    81 
    82 inductive
    83   ubasis_le :: "nat \<Rightarrow> nat \<Rightarrow> bool"
    84 where
    85   ubasis_le_refl: "ubasis_le a a"
    86 | ubasis_le_trans:
    87     "\<lbrakk>ubasis_le a b; ubasis_le b c\<rbrakk> \<Longrightarrow> ubasis_le a c"
    88 | ubasis_le_lower:
    89     "finite S \<Longrightarrow> ubasis_le a (node i a S)"
    90 | ubasis_le_upper:
    91     "\<lbrakk>finite S; b \<in> S; ubasis_le a b\<rbrakk> \<Longrightarrow> ubasis_le (node i a S) b"
    92 
    93 lemma ubasis_le_minimal: "ubasis_le 0 x"
    94 apply (induct x rule: node_induct)
    95 apply (rule ubasis_le_refl)
    96 apply (erule ubasis_le_trans)
    97 apply (erule ubasis_le_lower)
    98 done
    99 
   100 interpretation udom: preorder ubasis_le
   101 apply default
   102 apply (rule ubasis_le_refl)
   103 apply (erule (1) ubasis_le_trans)
   104 done
   105 
   106 subsubsection {* Generic take function *}
   107 
   108 function
   109   ubasis_until :: "(ubasis \<Rightarrow> bool) \<Rightarrow> ubasis \<Rightarrow> ubasis"
   110 where
   111   "ubasis_until P 0 = 0"
   112 | "finite S \<Longrightarrow> ubasis_until P (node i a S) =
   113     (if P (node i a S) then node i a S else ubasis_until P a)"
   114     apply clarify
   115     apply (rule_tac x=b in node_cases)
   116      apply simp
   117     apply simp
   118     apply fast
   119    apply simp
   120   apply simp
   121  apply simp
   122 done
   123 
   124 termination ubasis_until
   125 apply (relation "measure snd")
   126 apply (rule wf_measure)
   127 apply (simp add: node_gt1)
   128 done
   129 
   130 lemma ubasis_until: "P 0 \<Longrightarrow> P (ubasis_until P x)"
   131 by (induct x rule: node_induct) simp_all
   132 
   133 lemma ubasis_until': "0 < ubasis_until P x \<Longrightarrow> P (ubasis_until P x)"
   134 by (induct x rule: node_induct) auto
   135 
   136 lemma ubasis_until_same: "P x \<Longrightarrow> ubasis_until P x = x"
   137 by (induct x rule: node_induct) simp_all
   138 
   139 lemma ubasis_until_idem:
   140   "P 0 \<Longrightarrow> ubasis_until P (ubasis_until P x) = ubasis_until P x"
   141 by (rule ubasis_until_same [OF ubasis_until])
   142 
   143 lemma ubasis_until_0:
   144   "\<forall>x. x \<noteq> 0 \<longrightarrow> \<not> P x \<Longrightarrow> ubasis_until P x = 0"
   145 by (induct x rule: node_induct) simp_all
   146 
   147 lemma ubasis_until_less: "ubasis_le (ubasis_until P x) x"
   148 apply (induct x rule: node_induct)
   149 apply (simp add: ubasis_le_refl)
   150 apply (simp add: ubasis_le_refl)
   151 apply (rule impI)
   152 apply (erule ubasis_le_trans)
   153 apply (erule ubasis_le_lower)
   154 done
   155 
   156 lemma ubasis_until_chain:
   157   assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
   158   shows "ubasis_le (ubasis_until P x) (ubasis_until Q x)"
   159 apply (induct x rule: node_induct)
   160 apply (simp add: ubasis_le_refl)
   161 apply (simp add: ubasis_le_refl)
   162 apply (simp add: PQ)
   163 apply clarify
   164 apply (rule ubasis_le_trans)
   165 apply (rule ubasis_until_less)
   166 apply (erule ubasis_le_lower)
   167 done
   168 
   169 lemma ubasis_until_mono:
   170   assumes "\<And>i a S b. \<lbrakk>finite S; P (node i a S); b \<in> S; ubasis_le a b\<rbrakk> \<Longrightarrow> P b"
   171   shows "ubasis_le a b \<Longrightarrow> ubasis_le (ubasis_until P a) (ubasis_until P b)"
   172 proof (induct set: ubasis_le)
   173   case (ubasis_le_refl a) show ?case by (rule ubasis_le.ubasis_le_refl)
   174 next
   175   case (ubasis_le_trans a b c) thus ?case by - (rule ubasis_le.ubasis_le_trans)
   176 next
   177   case (ubasis_le_lower S a i) thus ?case
   178     apply (clarsimp simp add: ubasis_le_refl)
   179     apply (rule ubasis_le_trans [OF ubasis_until_less])
   180     apply (erule ubasis_le.ubasis_le_lower)
   181     done
   182 next
   183   case (ubasis_le_upper S b a i) thus ?case
   184     apply clarsimp
   185     apply (subst ubasis_until_same)
   186      apply (erule (3) prems)
   187     apply (erule (2) ubasis_le.ubasis_le_upper)
   188     done
   189 qed
   190 
   191 lemma finite_range_ubasis_until:
   192   "finite {x. P x} \<Longrightarrow> finite (range (ubasis_until P))"
   193 apply (rule finite_subset [where B="insert 0 {x. P x}"])
   194 apply (clarsimp simp add: ubasis_until')
   195 apply simp
   196 done
   197 
   198 
   199 subsection {* Defining the universal domain by ideal completion *}
   200 
   201 typedef (open) udom = "{S. udom.ideal S}"
   202 by (fast intro: udom.ideal_principal)
   203 
   204 instantiation udom :: below
   205 begin
   206 
   207 definition
   208   "x \<sqsubseteq> y \<longleftrightarrow> Rep_udom x \<subseteq> Rep_udom y"
   209 
   210 instance ..
   211 end
   212 
   213 instance udom :: po
   214 using type_definition_udom below_udom_def
   215 by (rule udom.typedef_ideal_po)
   216 
   217 instance udom :: cpo
   218 using type_definition_udom below_udom_def
   219 by (rule udom.typedef_ideal_cpo)
   220 
   221 definition
   222   udom_principal :: "nat \<Rightarrow> udom" where
   223   "udom_principal t = Abs_udom {u. ubasis_le u t}"
   224 
   225 lemma ubasis_countable: "\<exists>f::ubasis \<Rightarrow> nat. inj f"
   226 by (rule exI, rule inj_on_id)
   227 
   228 interpretation udom:
   229   ideal_completion ubasis_le udom_principal Rep_udom
   230 using type_definition_udom below_udom_def
   231 using udom_principal_def ubasis_countable
   232 by (rule udom.typedef_ideal_completion)
   233 
   234 text {* Universal domain is pointed *}
   235 
   236 lemma udom_minimal: "udom_principal 0 \<sqsubseteq> x"
   237 apply (induct x rule: udom.principal_induct)
   238 apply (simp, simp add: ubasis_le_minimal)
   239 done
   240 
   241 instance udom :: pcpo
   242 by intro_classes (fast intro: udom_minimal)
   243 
   244 lemma inst_udom_pcpo: "\<bottom> = udom_principal 0"
   245 by (rule udom_minimal [THEN UU_I, symmetric])
   246 
   247 
   248 subsection {* Compact bases of domains *}
   249 
   250 typedef (open) 'a compact_basis = "{x::'a::pcpo. compact x}"
   251 by auto
   252 
   253 lemma compact_Rep_compact_basis: "compact (Rep_compact_basis a)"
   254 by (rule Rep_compact_basis [unfolded mem_Collect_eq])
   255 
   256 instantiation compact_basis :: (pcpo) below
   257 begin
   258 
   259 definition
   260   compact_le_def:
   261     "(op \<sqsubseteq>) \<equiv> (\<lambda>x y. Rep_compact_basis x \<sqsubseteq> Rep_compact_basis y)"
   262 
   263 instance ..
   264 end
   265 
   266 instance compact_basis :: (pcpo) po
   267 using type_definition_compact_basis compact_le_def
   268 by (rule typedef_po)
   269 
   270 definition
   271   approximants :: "'a \<Rightarrow> 'a compact_basis set" where
   272   "approximants = (\<lambda>x. {a. Rep_compact_basis a \<sqsubseteq> x})"
   273 
   274 definition
   275   compact_bot :: "'a::pcpo compact_basis" where
   276   "compact_bot = Abs_compact_basis \<bottom>"
   277 
   278 lemma Rep_compact_bot [simp]: "Rep_compact_basis compact_bot = \<bottom>"
   279 unfolding compact_bot_def by (simp add: Abs_compact_basis_inverse)
   280 
   281 lemma compact_bot_minimal [simp]: "compact_bot \<sqsubseteq> a"
   282 unfolding compact_le_def Rep_compact_bot by simp
   283 
   284 
   285 subsection {* Universality of \emph{udom} *}
   286 
   287 text {* We use a locale to parameterize the construction over a chain
   288 of approx functions on the type to be embedded. *}
   289 
   290 locale approx_chain =
   291   fixes approx :: "nat \<Rightarrow> 'a::pcpo \<rightarrow> 'a"
   292   assumes chain_approx [simp]: "chain (\<lambda>i. approx i)"
   293   assumes lub_approx [simp]: "(\<Squnion>i. approx i) = ID"
   294   assumes finite_deflation_approx: "\<And>i. finite_deflation (approx i)"
   295 begin
   296 
   297 subsubsection {* Choosing a maximal element from a finite set *}
   298 
   299 lemma finite_has_maximal:
   300   fixes A :: "'a compact_basis set"
   301   shows "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y"
   302 proof (induct rule: finite_ne_induct)
   303   case (singleton x)
   304     show ?case by simp
   305 next
   306   case (insert a A)
   307   from `\<exists>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y`
   308   obtain x where x: "x \<in> A"
   309            and x_eq: "\<And>y. \<lbrakk>y \<in> A; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> x = y" by fast
   310   show ?case
   311   proof (intro bexI ballI impI)
   312     fix y
   313     assume "y \<in> insert a A" and "(if x \<sqsubseteq> a then a else x) \<sqsubseteq> y"
   314     thus "(if x \<sqsubseteq> a then a else x) = y"
   315       apply auto
   316       apply (frule (1) below_trans)
   317       apply (frule (1) x_eq)
   318       apply (rule below_antisym, assumption)
   319       apply simp
   320       apply (erule (1) x_eq)
   321       done
   322   next
   323     show "(if x \<sqsubseteq> a then a else x) \<in> insert a A"
   324       by (simp add: x)
   325   qed
   326 qed
   327 
   328 definition
   329   choose :: "'a compact_basis set \<Rightarrow> 'a compact_basis"
   330 where
   331   "choose A = (SOME x. x \<in> {x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y})"
   332 
   333 lemma choose_lemma:
   334   "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> choose A \<in> {x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y}"
   335 unfolding choose_def
   336 apply (rule someI_ex)
   337 apply (frule (1) finite_has_maximal, fast)
   338 done
   339 
   340 lemma maximal_choose:
   341   "\<lbrakk>finite A; y \<in> A; choose A \<sqsubseteq> y\<rbrakk> \<Longrightarrow> choose A = y"
   342 apply (cases "A = {}", simp)
   343 apply (frule (1) choose_lemma, simp)
   344 done
   345 
   346 lemma choose_in: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> choose A \<in> A"
   347 by (frule (1) choose_lemma, simp)
   348 
   349 function
   350   choose_pos :: "'a compact_basis set \<Rightarrow> 'a compact_basis \<Rightarrow> nat"
   351 where
   352   "choose_pos A x =
   353     (if finite A \<and> x \<in> A \<and> x \<noteq> choose A
   354       then Suc (choose_pos (A - {choose A}) x) else 0)"
   355 by auto
   356 
   357 termination choose_pos
   358 apply (relation "measure (card \<circ> fst)", simp)
   359 apply clarsimp
   360 apply (rule card_Diff1_less)
   361 apply assumption
   362 apply (erule choose_in)
   363 apply clarsimp
   364 done
   365 
   366 declare choose_pos.simps [simp del]
   367 
   368 lemma choose_pos_choose: "finite A \<Longrightarrow> choose_pos A (choose A) = 0"
   369 by (simp add: choose_pos.simps)
   370 
   371 lemma inj_on_choose_pos [OF refl]:
   372   "\<lbrakk>card A = n; finite A\<rbrakk> \<Longrightarrow> inj_on (choose_pos A) A"
   373  apply (induct n arbitrary: A)
   374   apply simp
   375  apply (case_tac "A = {}", simp)
   376  apply (frule (1) choose_in)
   377  apply (rule inj_onI)
   378  apply (drule_tac x="A - {choose A}" in meta_spec, simp)
   379  apply (simp add: choose_pos.simps)
   380  apply (simp split: split_if_asm)
   381  apply (erule (1) inj_onD, simp, simp)
   382 done
   383 
   384 lemma choose_pos_bounded [OF refl]:
   385   "\<lbrakk>card A = n; finite A; x \<in> A\<rbrakk> \<Longrightarrow> choose_pos A x < n"
   386 apply (induct n arbitrary: A)
   387 apply simp
   388  apply (case_tac "A = {}", simp)
   389  apply (frule (1) choose_in)
   390 apply (subst choose_pos.simps)
   391 apply simp
   392 done
   393 
   394 lemma choose_pos_lessD:
   395   "\<lbrakk>choose_pos A x < choose_pos A y; finite A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<not> x \<sqsubseteq> y"
   396  apply (induct A x arbitrary: y rule: choose_pos.induct)
   397  apply simp
   398  apply (case_tac "x = choose A")
   399   apply simp
   400   apply (rule notI)
   401   apply (frule (2) maximal_choose)
   402   apply simp
   403  apply (case_tac "y = choose A")
   404   apply (simp add: choose_pos_choose)
   405  apply (drule_tac x=y in meta_spec)
   406  apply simp
   407  apply (erule meta_mp)
   408  apply (simp add: choose_pos.simps)
   409 done
   410 
   411 subsubsection {* Properties of approx function *}
   412 
   413 lemma deflation_approx: "deflation (approx i)"
   414 using finite_deflation_approx by (rule finite_deflation_imp_deflation)
   415 
   416 lemma approx_idem: "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
   417 using deflation_approx by (rule deflation.idem)
   418 
   419 lemma approx_below: "approx i\<cdot>x \<sqsubseteq> x"
   420 using deflation_approx by (rule deflation.below)
   421 
   422 lemma finite_range_approx: "finite (range (\<lambda>x. approx i\<cdot>x))"
   423 apply (rule finite_deflation.finite_range)
   424 apply (rule finite_deflation_approx)
   425 done
   426 
   427 lemma compact_approx: "compact (approx n\<cdot>x)"
   428 apply (rule finite_deflation.compact)
   429 apply (rule finite_deflation_approx)
   430 done
   431 
   432 lemma compact_eq_approx: "compact x \<Longrightarrow> \<exists>i. approx i\<cdot>x = x"
   433 by (rule admD2, simp_all)
   434 
   435 subsubsection {* Compact basis take function *}
   436 
   437 primrec
   438   cb_take :: "nat \<Rightarrow> 'a compact_basis \<Rightarrow> 'a compact_basis" where
   439   "cb_take 0 = (\<lambda>x. compact_bot)"
   440 | "cb_take (Suc n) = (\<lambda>a. Abs_compact_basis (approx n\<cdot>(Rep_compact_basis a)))"
   441 
   442 declare cb_take.simps [simp del]
   443 
   444 lemma cb_take_zero [simp]: "cb_take 0 a = compact_bot"
   445 by (simp only: cb_take.simps)
   446 
   447 lemma Rep_cb_take:
   448   "Rep_compact_basis (cb_take (Suc n) a) = approx n\<cdot>(Rep_compact_basis a)"
   449 by (simp add: Abs_compact_basis_inverse cb_take.simps(2) compact_approx)
   450 
   451 lemmas approx_Rep_compact_basis = Rep_cb_take [symmetric]
   452 
   453 lemma cb_take_covers: "\<exists>n. cb_take n x = x"
   454 apply (subgoal_tac "\<exists>n. cb_take (Suc n) x = x", fast)
   455 apply (simp add: Rep_compact_basis_inject [symmetric])
   456 apply (simp add: Rep_cb_take)
   457 apply (rule compact_eq_approx)
   458 apply (rule compact_Rep_compact_basis)
   459 done
   460 
   461 lemma cb_take_less: "cb_take n x \<sqsubseteq> x"
   462 unfolding compact_le_def
   463 by (cases n, simp, simp add: Rep_cb_take approx_below)
   464 
   465 lemma cb_take_idem: "cb_take n (cb_take n x) = cb_take n x"
   466 unfolding Rep_compact_basis_inject [symmetric]
   467 by (cases n, simp, simp add: Rep_cb_take approx_idem)
   468 
   469 lemma cb_take_mono: "x \<sqsubseteq> y \<Longrightarrow> cb_take n x \<sqsubseteq> cb_take n y"
   470 unfolding compact_le_def
   471 by (cases n, simp, simp add: Rep_cb_take monofun_cfun_arg)
   472 
   473 lemma cb_take_chain_le: "m \<le> n \<Longrightarrow> cb_take m x \<sqsubseteq> cb_take n x"
   474 unfolding compact_le_def
   475 apply (cases m, simp, cases n, simp)
   476 apply (simp add: Rep_cb_take, rule chain_mono, simp, simp)
   477 done
   478 
   479 lemma finite_range_cb_take: "finite (range (cb_take n))"
   480 apply (cases n)
   481 apply (subgoal_tac "range (cb_take 0) = {compact_bot}", simp, force)
   482 apply (rule finite_imageD [where f="Rep_compact_basis"])
   483 apply (rule finite_subset [where B="range (\<lambda>x. approx (n - 1)\<cdot>x)"])
   484 apply (clarsimp simp add: Rep_cb_take)
   485 apply (rule finite_range_approx)
   486 apply (rule inj_onI, simp add: Rep_compact_basis_inject)
   487 done
   488 
   489 subsubsection {* Rank of basis elements *}
   490 
   491 definition
   492   rank :: "'a compact_basis \<Rightarrow> nat"
   493 where
   494   "rank x = (LEAST n. cb_take n x = x)"
   495 
   496 lemma compact_approx_rank: "cb_take (rank x) x = x"
   497 unfolding rank_def
   498 apply (rule LeastI_ex)
   499 apply (rule cb_take_covers)
   500 done
   501 
   502 lemma rank_leD: "rank x \<le> n \<Longrightarrow> cb_take n x = x"
   503 apply (rule below_antisym [OF cb_take_less])
   504 apply (subst compact_approx_rank [symmetric])
   505 apply (erule cb_take_chain_le)
   506 done
   507 
   508 lemma rank_leI: "cb_take n x = x \<Longrightarrow> rank x \<le> n"
   509 unfolding rank_def by (rule Least_le)
   510 
   511 lemma rank_le_iff: "rank x \<le> n \<longleftrightarrow> cb_take n x = x"
   512 by (rule iffI [OF rank_leD rank_leI])
   513 
   514 lemma rank_compact_bot [simp]: "rank compact_bot = 0"
   515 using rank_leI [of 0 compact_bot] by simp
   516 
   517 lemma rank_eq_0_iff [simp]: "rank x = 0 \<longleftrightarrow> x = compact_bot"
   518 using rank_le_iff [of x 0] by auto
   519 
   520 definition
   521   rank_le :: "'a compact_basis \<Rightarrow> 'a compact_basis set"
   522 where
   523   "rank_le x = {y. rank y \<le> rank x}"
   524 
   525 definition
   526   rank_lt :: "'a compact_basis \<Rightarrow> 'a compact_basis set"
   527 where
   528   "rank_lt x = {y. rank y < rank x}"
   529 
   530 definition
   531   rank_eq :: "'a compact_basis \<Rightarrow> 'a compact_basis set"
   532 where
   533   "rank_eq x = {y. rank y = rank x}"
   534 
   535 lemma rank_eq_cong: "rank x = rank y \<Longrightarrow> rank_eq x = rank_eq y"
   536 unfolding rank_eq_def by simp
   537 
   538 lemma rank_lt_cong: "rank x = rank y \<Longrightarrow> rank_lt x = rank_lt y"
   539 unfolding rank_lt_def by simp
   540 
   541 lemma rank_eq_subset: "rank_eq x \<subseteq> rank_le x"
   542 unfolding rank_eq_def rank_le_def by auto
   543 
   544 lemma rank_lt_subset: "rank_lt x \<subseteq> rank_le x"
   545 unfolding rank_lt_def rank_le_def by auto
   546 
   547 lemma finite_rank_le: "finite (rank_le x)"
   548 unfolding rank_le_def
   549 apply (rule finite_subset [where B="range (cb_take (rank x))"])
   550 apply clarify
   551 apply (rule range_eqI)
   552 apply (erule rank_leD [symmetric])
   553 apply (rule finite_range_cb_take)
   554 done
   555 
   556 lemma finite_rank_eq: "finite (rank_eq x)"
   557 by (rule finite_subset [OF rank_eq_subset finite_rank_le])
   558 
   559 lemma finite_rank_lt: "finite (rank_lt x)"
   560 by (rule finite_subset [OF rank_lt_subset finite_rank_le])
   561 
   562 lemma rank_lt_Int_rank_eq: "rank_lt x \<inter> rank_eq x = {}"
   563 unfolding rank_lt_def rank_eq_def rank_le_def by auto
   564 
   565 lemma rank_lt_Un_rank_eq: "rank_lt x \<union> rank_eq x = rank_le x"
   566 unfolding rank_lt_def rank_eq_def rank_le_def by auto
   567 
   568 subsubsection {* Sequencing basis elements *}
   569 
   570 definition
   571   place :: "'a compact_basis \<Rightarrow> nat"
   572 where
   573   "place x = card (rank_lt x) + choose_pos (rank_eq x) x"
   574 
   575 lemma place_bounded: "place x < card (rank_le x)"
   576 unfolding place_def
   577  apply (rule ord_less_eq_trans)
   578   apply (rule add_strict_left_mono)
   579   apply (rule choose_pos_bounded)
   580    apply (rule finite_rank_eq)
   581   apply (simp add: rank_eq_def)
   582  apply (subst card_Un_disjoint [symmetric])
   583     apply (rule finite_rank_lt)
   584    apply (rule finite_rank_eq)
   585   apply (rule rank_lt_Int_rank_eq)
   586  apply (simp add: rank_lt_Un_rank_eq)
   587 done
   588 
   589 lemma place_ge: "card (rank_lt x) \<le> place x"
   590 unfolding place_def by simp
   591 
   592 lemma place_rank_mono:
   593   fixes x y :: "'a compact_basis"
   594   shows "rank x < rank y \<Longrightarrow> place x < place y"
   595 apply (rule less_le_trans [OF place_bounded])
   596 apply (rule order_trans [OF _ place_ge])
   597 apply (rule card_mono)
   598 apply (rule finite_rank_lt)
   599 apply (simp add: rank_le_def rank_lt_def subset_eq)
   600 done
   601 
   602 lemma place_eqD: "place x = place y \<Longrightarrow> x = y"
   603  apply (rule linorder_cases [where x="rank x" and y="rank y"])
   604    apply (drule place_rank_mono, simp)
   605   apply (simp add: place_def)
   606   apply (rule inj_on_choose_pos [where A="rank_eq x", THEN inj_onD])
   607      apply (rule finite_rank_eq)
   608     apply (simp cong: rank_lt_cong rank_eq_cong)
   609    apply (simp add: rank_eq_def)
   610   apply (simp add: rank_eq_def)
   611  apply (drule place_rank_mono, simp)
   612 done
   613 
   614 lemma inj_place: "inj place"
   615 by (rule inj_onI, erule place_eqD)
   616 
   617 subsubsection {* Embedding and projection on basis elements *}
   618 
   619 definition
   620   sub :: "'a compact_basis \<Rightarrow> 'a compact_basis"
   621 where
   622   "sub x = (case rank x of 0 \<Rightarrow> compact_bot | Suc k \<Rightarrow> cb_take k x)"
   623 
   624 lemma rank_sub_less: "x \<noteq> compact_bot \<Longrightarrow> rank (sub x) < rank x"
   625 unfolding sub_def
   626 apply (cases "rank x", simp)
   627 apply (simp add: less_Suc_eq_le)
   628 apply (rule rank_leI)
   629 apply (rule cb_take_idem)
   630 done
   631 
   632 lemma place_sub_less: "x \<noteq> compact_bot \<Longrightarrow> place (sub x) < place x"
   633 apply (rule place_rank_mono)
   634 apply (erule rank_sub_less)
   635 done
   636 
   637 lemma sub_below: "sub x \<sqsubseteq> x"
   638 unfolding sub_def by (cases "rank x", simp_all add: cb_take_less)
   639 
   640 lemma rank_less_imp_below_sub: "\<lbrakk>x \<sqsubseteq> y; rank x < rank y\<rbrakk> \<Longrightarrow> x \<sqsubseteq> sub y"
   641 unfolding sub_def
   642 apply (cases "rank y", simp)
   643 apply (simp add: less_Suc_eq_le)
   644 apply (subgoal_tac "cb_take nat x \<sqsubseteq> cb_take nat y")
   645 apply (simp add: rank_leD)
   646 apply (erule cb_take_mono)
   647 done
   648 
   649 function
   650   basis_emb :: "'a compact_basis \<Rightarrow> ubasis"
   651 where
   652   "basis_emb x = (if x = compact_bot then 0 else
   653     node (place x) (basis_emb (sub x))
   654       (basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y}))"
   655 by auto
   656 
   657 termination basis_emb
   658 apply (relation "measure place", simp)
   659 apply (simp add: place_sub_less)
   660 apply simp
   661 done
   662 
   663 declare basis_emb.simps [simp del]
   664 
   665 lemma basis_emb_compact_bot [simp]: "basis_emb compact_bot = 0"
   666 by (simp add: basis_emb.simps)
   667 
   668 lemma fin1: "finite {y. place y < place x \<and> x \<sqsubseteq> y}"
   669 apply (subst Collect_conj_eq)
   670 apply (rule finite_Int)
   671 apply (rule disjI1)
   672 apply (subgoal_tac "finite (place -` {n. n < place x})", simp)
   673 apply (rule finite_vimageI [OF _ inj_place])
   674 apply (simp add: lessThan_def [symmetric])
   675 done
   676 
   677 lemma fin2: "finite (basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y})"
   678 by (rule finite_imageI [OF fin1])
   679 
   680 lemma rank_place_mono:
   681   "\<lbrakk>place x < place y; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> rank x < rank y"
   682 apply (rule linorder_cases, assumption)
   683 apply (simp add: place_def cong: rank_lt_cong rank_eq_cong)
   684 apply (drule choose_pos_lessD)
   685 apply (rule finite_rank_eq)
   686 apply (simp add: rank_eq_def)
   687 apply (simp add: rank_eq_def)
   688 apply simp
   689 apply (drule place_rank_mono, simp)
   690 done
   691 
   692 lemma basis_emb_mono:
   693   "x \<sqsubseteq> y \<Longrightarrow> ubasis_le (basis_emb x) (basis_emb y)"
   694 proof (induct "max (place x) (place y)" arbitrary: x y rule: less_induct)
   695   case less
   696   show ?case proof (rule linorder_cases)
   697     assume "place x < place y"
   698     then have "rank x < rank y"
   699       using `x \<sqsubseteq> y` by (rule rank_place_mono)
   700     with `place x < place y` show ?case
   701       apply (case_tac "y = compact_bot", simp)
   702       apply (simp add: basis_emb.simps [of y])
   703       apply (rule ubasis_le_trans [OF _ ubasis_le_lower [OF fin2]])
   704       apply (rule less)
   705        apply (simp add: less_max_iff_disj)
   706        apply (erule place_sub_less)
   707       apply (erule rank_less_imp_below_sub [OF `x \<sqsubseteq> y`])
   708       done
   709   next
   710     assume "place x = place y"
   711     hence "x = y" by (rule place_eqD)
   712     thus ?case by (simp add: ubasis_le_refl)
   713   next
   714     assume "place x > place y"
   715     with `x \<sqsubseteq> y` show ?case
   716       apply (case_tac "x = compact_bot", simp add: ubasis_le_minimal)
   717       apply (simp add: basis_emb.simps [of x])
   718       apply (rule ubasis_le_upper [OF fin2], simp)
   719       apply (rule less)
   720        apply (simp add: less_max_iff_disj)
   721        apply (erule place_sub_less)
   722       apply (erule rev_below_trans)
   723       apply (rule sub_below)
   724       done
   725   qed
   726 qed
   727 
   728 lemma inj_basis_emb: "inj basis_emb"
   729  apply (rule inj_onI)
   730  apply (case_tac "x = compact_bot")
   731   apply (case_tac [!] "y = compact_bot")
   732     apply simp
   733    apply (simp add: basis_emb.simps)
   734   apply (simp add: basis_emb.simps)
   735  apply (simp add: basis_emb.simps)
   736  apply (simp add: fin2 inj_eq [OF inj_place])
   737 done
   738 
   739 definition
   740   basis_prj :: "ubasis \<Rightarrow> 'a compact_basis"
   741 where
   742   "basis_prj x = inv basis_emb
   743     (ubasis_until (\<lambda>x. x \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> ubasis)) x)"
   744 
   745 lemma basis_prj_basis_emb: "\<And>x. basis_prj (basis_emb x) = x"
   746 unfolding basis_prj_def
   747  apply (subst ubasis_until_same)
   748   apply (rule rangeI)
   749  apply (rule inv_f_f)
   750  apply (rule inj_basis_emb)
   751 done
   752 
   753 lemma basis_prj_node:
   754   "\<lbrakk>finite S; node i a S \<notin> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)\<rbrakk>
   755     \<Longrightarrow> basis_prj (node i a S) = (basis_prj a :: 'a compact_basis)"
   756 unfolding basis_prj_def by simp
   757 
   758 lemma basis_prj_0: "basis_prj 0 = compact_bot"
   759 apply (subst basis_emb_compact_bot [symmetric])
   760 apply (rule basis_prj_basis_emb)
   761 done
   762 
   763 lemma node_eq_basis_emb_iff:
   764   "finite S \<Longrightarrow> node i a S = basis_emb x \<longleftrightarrow>
   765     x \<noteq> compact_bot \<and> i = place x \<and> a = basis_emb (sub x) \<and>
   766         S = basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y}"
   767 apply (cases "x = compact_bot", simp)
   768 apply (simp add: basis_emb.simps [of x])
   769 apply (simp add: fin2)
   770 done
   771 
   772 lemma basis_prj_mono: "ubasis_le a b \<Longrightarrow> basis_prj a \<sqsubseteq> basis_prj b"
   773 proof (induct a b rule: ubasis_le.induct)
   774   case (ubasis_le_refl a) show ?case by (rule below_refl)
   775 next
   776   case (ubasis_le_trans a b c) thus ?case by - (rule below_trans)
   777 next
   778   case (ubasis_le_lower S a i) thus ?case
   779     apply (cases "node i a S \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)")
   780      apply (erule rangeE, rename_tac x)
   781      apply (simp add: basis_prj_basis_emb)
   782      apply (simp add: node_eq_basis_emb_iff)
   783      apply (simp add: basis_prj_basis_emb)
   784      apply (rule sub_below)
   785     apply (simp add: basis_prj_node)
   786     done
   787 next
   788   case (ubasis_le_upper S b a i) thus ?case
   789     apply (cases "node i a S \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)")
   790      apply (erule rangeE, rename_tac x)
   791      apply (simp add: basis_prj_basis_emb)
   792      apply (clarsimp simp add: node_eq_basis_emb_iff)
   793      apply (simp add: basis_prj_basis_emb)
   794     apply (simp add: basis_prj_node)
   795     done
   796 qed
   797 
   798 lemma basis_emb_prj_less: "ubasis_le (basis_emb (basis_prj x)) x"
   799 unfolding basis_prj_def
   800  apply (subst f_inv_into_f [where f=basis_emb])
   801   apply (rule ubasis_until)
   802   apply (rule range_eqI [where x=compact_bot])
   803   apply simp
   804  apply (rule ubasis_until_less)
   805 done
   806 
   807 end
   808 
   809 sublocale approx_chain \<subseteq> compact_basis!:
   810   ideal_completion below Rep_compact_basis
   811     "approximants :: 'a \<Rightarrow> 'a compact_basis set"
   812 proof
   813   fix w :: "'a"
   814   show "below.ideal (approximants w)"
   815   proof (rule below.idealI)
   816     show "\<exists>x. x \<in> approximants w"
   817       unfolding approximants_def
   818       apply (rule_tac x="Abs_compact_basis (approx 0\<cdot>w)" in exI)
   819       apply (simp add: Abs_compact_basis_inverse approx_below compact_approx)
   820       done
   821   next
   822     fix x y :: "'a compact_basis"
   823     assume "x \<in> approximants w" "y \<in> approximants w"
   824     thus "\<exists>z \<in> approximants w. x \<sqsubseteq> z \<and> y \<sqsubseteq> z"
   825       unfolding approximants_def
   826       apply simp
   827       apply (cut_tac a=x in compact_Rep_compact_basis)
   828       apply (cut_tac a=y in compact_Rep_compact_basis)
   829       apply (drule compact_eq_approx)
   830       apply (drule compact_eq_approx)
   831       apply (clarify, rename_tac i j)
   832       apply (rule_tac x="Abs_compact_basis (approx (max i j)\<cdot>w)" in exI)
   833       apply (simp add: compact_le_def)
   834       apply (simp add: Abs_compact_basis_inverse approx_below compact_approx)
   835       apply (erule subst, erule subst)
   836       apply (simp add: monofun_cfun chain_mono [OF chain_approx])
   837       done
   838   next
   839     fix x y :: "'a compact_basis"
   840     assume "x \<sqsubseteq> y" "y \<in> approximants w" thus "x \<in> approximants w"
   841       unfolding approximants_def
   842       apply simp
   843       apply (simp add: compact_le_def)
   844       apply (erule (1) below_trans)
   845       done
   846   qed
   847 next
   848   fix Y :: "nat \<Rightarrow> 'a"
   849   assume Y: "chain Y"
   850   show "approximants (\<Squnion>i. Y i) = (\<Union>i. approximants (Y i))"
   851     unfolding approximants_def
   852     apply safe
   853     apply (simp add: compactD2 [OF compact_Rep_compact_basis Y])
   854     apply (erule below_lub [OF Y])
   855     done
   856 next
   857   fix a :: "'a compact_basis"
   858   show "approximants (Rep_compact_basis a) = {b. b \<sqsubseteq> a}"
   859     unfolding approximants_def compact_le_def ..
   860 next
   861   fix x y :: "'a"
   862   assume "approximants x \<subseteq> approximants y" thus "x \<sqsubseteq> y"
   863     apply (subgoal_tac "(\<Squnion>i. approx i\<cdot>x) \<sqsubseteq> y")
   864     apply (simp add: lub_distribs)
   865     apply (rule admD, simp, simp)
   866     apply (drule_tac c="Abs_compact_basis (approx i\<cdot>x)" in subsetD)
   867     apply (simp add: approximants_def Abs_compact_basis_inverse
   868                      approx_below compact_approx)
   869     apply (simp add: approximants_def Abs_compact_basis_inverse compact_approx)
   870     done
   871 next
   872   show "\<exists>f::'a compact_basis \<Rightarrow> nat. inj f"
   873     by (rule exI, rule inj_place)
   874 qed
   875 
   876 subsubsection {* EP-pair from any bifinite domain into \emph{udom} *}
   877 
   878 context approx_chain begin
   879 
   880 definition
   881   udom_emb :: "'a \<rightarrow> udom"
   882 where
   883   "udom_emb = compact_basis.basis_fun (\<lambda>x. udom_principal (basis_emb x))"
   884 
   885 definition
   886   udom_prj :: "udom \<rightarrow> 'a"
   887 where
   888   "udom_prj = udom.basis_fun (\<lambda>x. Rep_compact_basis (basis_prj x))"
   889 
   890 lemma udom_emb_principal:
   891   "udom_emb\<cdot>(Rep_compact_basis x) = udom_principal (basis_emb x)"
   892 unfolding udom_emb_def
   893 apply (rule compact_basis.basis_fun_principal)
   894 apply (rule udom.principal_mono)
   895 apply (erule basis_emb_mono)
   896 done
   897 
   898 lemma udom_prj_principal:
   899   "udom_prj\<cdot>(udom_principal x) = Rep_compact_basis (basis_prj x)"
   900 unfolding udom_prj_def
   901 apply (rule udom.basis_fun_principal)
   902 apply (rule compact_basis.principal_mono)
   903 apply (erule basis_prj_mono)
   904 done
   905 
   906 lemma ep_pair_udom: "ep_pair udom_emb udom_prj"
   907  apply default
   908   apply (rule compact_basis.principal_induct, simp)
   909   apply (simp add: udom_emb_principal udom_prj_principal)
   910   apply (simp add: basis_prj_basis_emb)
   911  apply (rule udom.principal_induct, simp)
   912  apply (simp add: udom_emb_principal udom_prj_principal)
   913  apply (rule basis_emb_prj_less)
   914 done
   915 
   916 end
   917 
   918 abbreviation "udom_emb \<equiv> approx_chain.udom_emb"
   919 abbreviation "udom_prj \<equiv> approx_chain.udom_prj"
   920 
   921 lemmas ep_pair_udom = approx_chain.ep_pair_udom
   922 
   923 subsection {* Chain of approx functions for type \emph{udom} *}
   924 
   925 definition
   926   udom_approx :: "nat \<Rightarrow> udom \<rightarrow> udom"
   927 where
   928   "udom_approx i =
   929     udom.basis_fun (\<lambda>x. udom_principal (ubasis_until (\<lambda>y. y \<le> i) x))"
   930 
   931 lemma udom_approx_mono:
   932   "ubasis_le a b \<Longrightarrow>
   933     udom_principal (ubasis_until (\<lambda>y. y \<le> i) a) \<sqsubseteq>
   934     udom_principal (ubasis_until (\<lambda>y. y \<le> i) b)"
   935 apply (rule udom.principal_mono)
   936 apply (rule ubasis_until_mono)
   937 apply (frule (2) order_less_le_trans [OF node_gt2])
   938 apply (erule order_less_imp_le)
   939 apply assumption
   940 done
   941 
   942 lemma adm_mem_finite: "\<lbrakk>cont f; finite S\<rbrakk> \<Longrightarrow> adm (\<lambda>x. f x \<in> S)"
   943 by (erule adm_subst, induct set: finite, simp_all)
   944 
   945 lemma udom_approx_principal:
   946   "udom_approx i\<cdot>(udom_principal x) =
   947     udom_principal (ubasis_until (\<lambda>y. y \<le> i) x)"
   948 unfolding udom_approx_def
   949 apply (rule udom.basis_fun_principal)
   950 apply (erule udom_approx_mono)
   951 done
   952 
   953 lemma finite_deflation_udom_approx: "finite_deflation (udom_approx i)"
   954 proof
   955   fix x show "udom_approx i\<cdot>(udom_approx i\<cdot>x) = udom_approx i\<cdot>x"
   956     by (induct x rule: udom.principal_induct, simp)
   957        (simp add: udom_approx_principal ubasis_until_idem)
   958 next
   959   fix x show "udom_approx i\<cdot>x \<sqsubseteq> x"
   960     by (induct x rule: udom.principal_induct, simp)
   961        (simp add: udom_approx_principal ubasis_until_less)
   962 next
   963   have *: "finite (range (\<lambda>x. udom_principal (ubasis_until (\<lambda>y. y \<le> i) x)))"
   964     apply (subst range_composition [where f=udom_principal])
   965     apply (simp add: finite_range_ubasis_until)
   966     done
   967   show "finite {x. udom_approx i\<cdot>x = x}"
   968     apply (rule finite_range_imp_finite_fixes)
   969     apply (rule rev_finite_subset [OF *])
   970     apply (clarsimp, rename_tac x)
   971     apply (induct_tac x rule: udom.principal_induct)
   972     apply (simp add: adm_mem_finite *)
   973     apply (simp add: udom_approx_principal)
   974     done
   975 qed
   976 
   977 interpretation udom_approx: finite_deflation "udom_approx i"
   978 by (rule finite_deflation_udom_approx)
   979 
   980 lemma chain_udom_approx [simp]: "chain (\<lambda>i. udom_approx i)"
   981 unfolding udom_approx_def
   982 apply (rule chainI)
   983 apply (rule udom.basis_fun_mono)
   984 apply (erule udom_approx_mono)
   985 apply (erule udom_approx_mono)
   986 apply (rule udom.principal_mono)
   987 apply (rule ubasis_until_chain, simp)
   988 done
   989 
   990 lemma lub_udom_approx [simp]: "(\<Squnion>i. udom_approx i) = ID"
   991 apply (rule cfun_eqI, simp add: contlub_cfun_fun)
   992 apply (rule below_antisym)
   993 apply (rule lub_below)
   994 apply (simp)
   995 apply (rule udom_approx.below)
   996 apply (rule_tac x=x in udom.principal_induct)
   997 apply (simp add: lub_distribs)
   998 apply (rule_tac i=a in below_lub)
   999 apply simp
  1000 apply (simp add: udom_approx_principal)
  1001 apply (simp add: ubasis_until_same ubasis_le_refl)
  1002 done
  1003  
  1004 lemma udom_approx: "approx_chain udom_approx"
  1005 proof
  1006   show "chain (\<lambda>i. udom_approx i)"
  1007     by (rule chain_udom_approx)
  1008   show "(\<Squnion>i. udom_approx i) = ID"
  1009     by (rule lub_udom_approx)
  1010 qed
  1011 
  1012 hide_const (open) node
  1013 
  1014 end