src/HOL/HOLCF/Universal.thy
author wenzelm
Sat Nov 04 15:24:40 2017 +0100 (20 months ago)
changeset 67003 49850a679c2c
parent 66453 cc19f7ca2ed6
child 67399 eab6ce8368fa
permissions -rw-r--r--
more robust sorted_entries;
     1 (*  Title:      HOL/HOLCF/Universal.thy
     2     Author:     Brian Huffman
     3 *)
     4 
     5 section \<open>A universal bifinite domain\<close>
     6 
     7 theory Universal
     8 imports Bifinite Completion "HOL-Library.Nat_Bijection"
     9 begin
    10 
    11 no_notation binomial  (infixl "choose" 65)
    12 
    13 subsection \<open>Basis for universal domain\<close>
    14 
    15 subsubsection \<open>Basis datatype\<close>
    16 
    17 type_synonym ubasis = nat
    18 
    19 definition
    20   node :: "nat \<Rightarrow> ubasis \<Rightarrow> ubasis set \<Rightarrow> ubasis"
    21 where
    22   "node i a S = Suc (prod_encode (i, prod_encode (a, set_encode S)))"
    23 
    24 lemma node_not_0 [simp]: "node i a S \<noteq> 0"
    25 unfolding node_def by simp
    26 
    27 lemma node_gt_0 [simp]: "0 < node i a S"
    28 unfolding node_def by simp
    29 
    30 lemma node_inject [simp]:
    31   "\<lbrakk>finite S; finite T\<rbrakk>
    32     \<Longrightarrow> node i a S = node j b T \<longleftrightarrow> i = j \<and> a = b \<and> S = T"
    33 unfolding node_def by (simp add: prod_encode_eq set_encode_eq)
    34 
    35 lemma node_gt0: "i < node i a S"
    36 unfolding node_def less_Suc_eq_le
    37 by (rule le_prod_encode_1)
    38 
    39 lemma node_gt1: "a < node i a S"
    40 unfolding node_def less_Suc_eq_le
    41 by (rule order_trans [OF le_prod_encode_1 le_prod_encode_2])
    42 
    43 lemma nat_less_power2: "n < 2^n"
    44 by (induct n) simp_all
    45 
    46 lemma node_gt2: "\<lbrakk>finite S; b \<in> S\<rbrakk> \<Longrightarrow> b < node i a S"
    47 unfolding node_def less_Suc_eq_le set_encode_def
    48 apply (rule order_trans [OF _ le_prod_encode_2])
    49 apply (rule order_trans [OF _ le_prod_encode_2])
    50 apply (rule order_trans [where y="sum (op ^ 2) {b}"])
    51 apply (simp add: nat_less_power2 [THEN order_less_imp_le])
    52 apply (erule sum_mono2, simp, simp)
    53 done
    54 
    55 lemma eq_prod_encode_pairI:
    56   "\<lbrakk>fst (prod_decode x) = a; snd (prod_decode x) = b\<rbrakk> \<Longrightarrow> x = prod_encode (a, b)"
    57 by (erule subst, erule subst, simp)
    58 
    59 lemma node_cases:
    60   assumes 1: "x = 0 \<Longrightarrow> P"
    61   assumes 2: "\<And>i a S. \<lbrakk>finite S; x = node i a S\<rbrakk> \<Longrightarrow> P"
    62   shows "P"
    63  apply (cases x)
    64   apply (erule 1)
    65  apply (rule 2)
    66   apply (rule finite_set_decode)
    67  apply (simp add: node_def)
    68  apply (rule eq_prod_encode_pairI [OF refl])
    69  apply (rule eq_prod_encode_pairI [OF refl refl])
    70 done
    71 
    72 lemma node_induct:
    73   assumes 1: "P 0"
    74   assumes 2: "\<And>i a S. \<lbrakk>P a; finite S; \<forall>b\<in>S. P b\<rbrakk> \<Longrightarrow> P (node i a S)"
    75   shows "P x"
    76  apply (induct x rule: nat_less_induct)
    77  apply (case_tac n rule: node_cases)
    78   apply (simp add: 1)
    79  apply (simp add: 2 node_gt1 node_gt2)
    80 done
    81 
    82 subsubsection \<open>Basis ordering\<close>
    83 
    84 inductive
    85   ubasis_le :: "nat \<Rightarrow> nat \<Rightarrow> bool"
    86 where
    87   ubasis_le_refl: "ubasis_le a a"
    88 | ubasis_le_trans:
    89     "\<lbrakk>ubasis_le a b; ubasis_le b c\<rbrakk> \<Longrightarrow> ubasis_le a c"
    90 | ubasis_le_lower:
    91     "finite S \<Longrightarrow> ubasis_le a (node i a S)"
    92 | ubasis_le_upper:
    93     "\<lbrakk>finite S; b \<in> S; ubasis_le a b\<rbrakk> \<Longrightarrow> ubasis_le (node i a S) b"
    94 
    95 lemma ubasis_le_minimal: "ubasis_le 0 x"
    96 apply (induct x rule: node_induct)
    97 apply (rule ubasis_le_refl)
    98 apply (erule ubasis_le_trans)
    99 apply (erule ubasis_le_lower)
   100 done
   101 
   102 interpretation udom: preorder ubasis_le
   103 apply standard
   104 apply (rule ubasis_le_refl)
   105 apply (erule (1) ubasis_le_trans)
   106 done
   107 
   108 subsubsection \<open>Generic take function\<close>
   109 
   110 function
   111   ubasis_until :: "(ubasis \<Rightarrow> bool) \<Rightarrow> ubasis \<Rightarrow> ubasis"
   112 where
   113   "ubasis_until P 0 = 0"
   114 | "finite S \<Longrightarrow> ubasis_until P (node i a S) =
   115     (if P (node i a S) then node i a S else ubasis_until P a)"
   116    apply clarify
   117    apply (rule_tac x=b in node_cases)
   118     apply simp
   119    apply simp
   120    apply fast
   121   apply simp
   122  apply simp
   123 done
   124 
   125 termination ubasis_until
   126 apply (relation "measure snd")
   127 apply (rule wf_measure)
   128 apply (simp add: node_gt1)
   129 done
   130 
   131 lemma ubasis_until: "P 0 \<Longrightarrow> P (ubasis_until P x)"
   132 by (induct x rule: node_induct) simp_all
   133 
   134 lemma ubasis_until': "0 < ubasis_until P x \<Longrightarrow> P (ubasis_until P x)"
   135 by (induct x rule: node_induct) auto
   136 
   137 lemma ubasis_until_same: "P x \<Longrightarrow> ubasis_until P x = x"
   138 by (induct x rule: node_induct) simp_all
   139 
   140 lemma ubasis_until_idem:
   141   "P 0 \<Longrightarrow> ubasis_until P (ubasis_until P x) = ubasis_until P x"
   142 by (rule ubasis_until_same [OF ubasis_until])
   143 
   144 lemma ubasis_until_0:
   145   "\<forall>x. x \<noteq> 0 \<longrightarrow> \<not> P x \<Longrightarrow> ubasis_until P x = 0"
   146 by (induct x rule: node_induct) simp_all
   147 
   148 lemma ubasis_until_less: "ubasis_le (ubasis_until P x) x"
   149 apply (induct x rule: node_induct)
   150 apply (simp add: ubasis_le_refl)
   151 apply (simp add: ubasis_le_refl)
   152 apply (rule impI)
   153 apply (erule ubasis_le_trans)
   154 apply (erule ubasis_le_lower)
   155 done
   156 
   157 lemma ubasis_until_chain:
   158   assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
   159   shows "ubasis_le (ubasis_until P x) (ubasis_until Q x)"
   160 apply (induct x rule: node_induct)
   161 apply (simp add: ubasis_le_refl)
   162 apply (simp add: ubasis_le_refl)
   163 apply (simp add: PQ)
   164 apply clarify
   165 apply (rule ubasis_le_trans)
   166 apply (rule ubasis_until_less)
   167 apply (erule ubasis_le_lower)
   168 done
   169 
   170 lemma ubasis_until_mono:
   171   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"
   172   shows "ubasis_le a b \<Longrightarrow> ubasis_le (ubasis_until P a) (ubasis_until P b)"
   173 proof (induct set: ubasis_le)
   174   case (ubasis_le_refl a) show ?case by (rule ubasis_le.ubasis_le_refl)
   175 next
   176   case (ubasis_le_trans a b c) thus ?case by - (rule ubasis_le.ubasis_le_trans)
   177 next
   178   case (ubasis_le_lower S a i) thus ?case
   179     apply (clarsimp simp add: ubasis_le_refl)
   180     apply (rule ubasis_le_trans [OF ubasis_until_less])
   181     apply (erule ubasis_le.ubasis_le_lower)
   182     done
   183 next
   184   case (ubasis_le_upper S b a i) thus ?case
   185     apply clarsimp
   186     apply (subst ubasis_until_same)
   187      apply (erule (3) assms)
   188     apply (erule (2) ubasis_le.ubasis_le_upper)
   189     done
   190 qed
   191 
   192 lemma finite_range_ubasis_until:
   193   "finite {x. P x} \<Longrightarrow> finite (range (ubasis_until P))"
   194 apply (rule finite_subset [where B="insert 0 {x. P x}"])
   195 apply (clarsimp simp add: ubasis_until')
   196 apply simp
   197 done
   198 
   199 
   200 subsection \<open>Defining the universal domain by ideal completion\<close>
   201 
   202 typedef udom = "{S. udom.ideal S}"
   203 by (rule udom.ex_ideal)
   204 
   205 instantiation udom :: below
   206 begin
   207 
   208 definition
   209   "x \<sqsubseteq> y \<longleftrightarrow> Rep_udom x \<subseteq> Rep_udom y"
   210 
   211 instance ..
   212 end
   213 
   214 instance udom :: po
   215 using type_definition_udom below_udom_def
   216 by (rule udom.typedef_ideal_po)
   217 
   218 instance udom :: cpo
   219 using type_definition_udom below_udom_def
   220 by (rule udom.typedef_ideal_cpo)
   221 
   222 definition
   223   udom_principal :: "nat \<Rightarrow> udom" where
   224   "udom_principal t = Abs_udom {u. ubasis_le u t}"
   225 
   226 lemma ubasis_countable: "\<exists>f::ubasis \<Rightarrow> nat. inj f"
   227 by (rule exI, rule inj_on_id)
   228 
   229 interpretation udom:
   230   ideal_completion ubasis_le udom_principal Rep_udom
   231 using type_definition_udom below_udom_def
   232 using udom_principal_def ubasis_countable
   233 by (rule udom.typedef_ideal_completion)
   234 
   235 text \<open>Universal domain is pointed\<close>
   236 
   237 lemma udom_minimal: "udom_principal 0 \<sqsubseteq> x"
   238 apply (induct x rule: udom.principal_induct)
   239 apply (simp, simp add: ubasis_le_minimal)
   240 done
   241 
   242 instance udom :: pcpo
   243 by intro_classes (fast intro: udom_minimal)
   244 
   245 lemma inst_udom_pcpo: "\<bottom> = udom_principal 0"
   246 by (rule udom_minimal [THEN bottomI, symmetric])
   247 
   248 
   249 subsection \<open>Compact bases of domains\<close>
   250 
   251 typedef 'a compact_basis = "{x::'a::pcpo. compact x}"
   252 by auto
   253 
   254 lemma Rep_compact_basis' [simp]: "compact (Rep_compact_basis a)"
   255 by (rule Rep_compact_basis [unfolded mem_Collect_eq])
   256 
   257 lemma Abs_compact_basis_inverse' [simp]:
   258    "compact x \<Longrightarrow> Rep_compact_basis (Abs_compact_basis x) = x"
   259 by (rule Abs_compact_basis_inverse [unfolded mem_Collect_eq])
   260 
   261 instantiation compact_basis :: (pcpo) below
   262 begin
   263 
   264 definition
   265   compact_le_def:
   266     "(op \<sqsubseteq>) \<equiv> (\<lambda>x y. Rep_compact_basis x \<sqsubseteq> Rep_compact_basis y)"
   267 
   268 instance ..
   269 end
   270 
   271 instance compact_basis :: (pcpo) po
   272 using type_definition_compact_basis compact_le_def
   273 by (rule typedef_po)
   274 
   275 definition
   276   approximants :: "'a \<Rightarrow> 'a compact_basis set" where
   277   "approximants = (\<lambda>x. {a. Rep_compact_basis a \<sqsubseteq> x})"
   278 
   279 definition
   280   compact_bot :: "'a::pcpo compact_basis" where
   281   "compact_bot = Abs_compact_basis \<bottom>"
   282 
   283 lemma Rep_compact_bot [simp]: "Rep_compact_basis compact_bot = \<bottom>"
   284 unfolding compact_bot_def by simp
   285 
   286 lemma compact_bot_minimal [simp]: "compact_bot \<sqsubseteq> a"
   287 unfolding compact_le_def Rep_compact_bot by simp
   288 
   289 
   290 subsection \<open>Universality of \emph{udom}\<close>
   291 
   292 text \<open>We use a locale to parameterize the construction over a chain
   293 of approx functions on the type to be embedded.\<close>
   294 
   295 locale bifinite_approx_chain =
   296   approx_chain approx for approx :: "nat \<Rightarrow> 'a::bifinite \<rightarrow> 'a"
   297 begin
   298 
   299 subsubsection \<open>Choosing a maximal element from a finite set\<close>
   300 
   301 lemma finite_has_maximal:
   302   fixes A :: "'a compact_basis set"
   303   shows "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y"
   304 proof (induct rule: finite_ne_induct)
   305   case (singleton x)
   306     show ?case by simp
   307 next
   308   case (insert a A)
   309   from \<open>\<exists>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y\<close>
   310   obtain x where x: "x \<in> A"
   311            and x_eq: "\<And>y. \<lbrakk>y \<in> A; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> x = y" by fast
   312   show ?case
   313   proof (intro bexI ballI impI)
   314     fix y
   315     assume "y \<in> insert a A" and "(if x \<sqsubseteq> a then a else x) \<sqsubseteq> y"
   316     thus "(if x \<sqsubseteq> a then a else x) = y"
   317       apply auto
   318       apply (frule (1) below_trans)
   319       apply (frule (1) x_eq)
   320       apply (rule below_antisym, assumption)
   321       apply simp
   322       apply (erule (1) x_eq)
   323       done
   324   next
   325     show "(if x \<sqsubseteq> a then a else x) \<in> insert a A"
   326       by (simp add: x)
   327   qed
   328 qed
   329 
   330 definition
   331   choose :: "'a compact_basis set \<Rightarrow> 'a compact_basis"
   332 where
   333   "choose A = (SOME x. x \<in> {x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y})"
   334 
   335 lemma choose_lemma:
   336   "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> choose A \<in> {x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y}"
   337 unfolding choose_def
   338 apply (rule someI_ex)
   339 apply (frule (1) finite_has_maximal, fast)
   340 done
   341 
   342 lemma maximal_choose:
   343   "\<lbrakk>finite A; y \<in> A; choose A \<sqsubseteq> y\<rbrakk> \<Longrightarrow> choose A = y"
   344 apply (cases "A = {}", simp)
   345 apply (frule (1) choose_lemma, simp)
   346 done
   347 
   348 lemma choose_in: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> choose A \<in> A"
   349 by (frule (1) choose_lemma, simp)
   350 
   351 function
   352   choose_pos :: "'a compact_basis set \<Rightarrow> 'a compact_basis \<Rightarrow> nat"
   353 where
   354   "choose_pos A x =
   355     (if finite A \<and> x \<in> A \<and> x \<noteq> choose A
   356       then Suc (choose_pos (A - {choose A}) x) else 0)"
   357 by auto
   358 
   359 termination choose_pos
   360 apply (relation "measure (card \<circ> fst)", simp)
   361 apply clarsimp
   362 apply (rule card_Diff1_less)
   363 apply assumption
   364 apply (erule choose_in)
   365 apply clarsimp
   366 done
   367 
   368 declare choose_pos.simps [simp del]
   369 
   370 lemma choose_pos_choose: "finite A \<Longrightarrow> choose_pos A (choose A) = 0"
   371 by (simp add: choose_pos.simps)
   372 
   373 lemma inj_on_choose_pos [OF refl]:
   374   "\<lbrakk>card A = n; finite A\<rbrakk> \<Longrightarrow> inj_on (choose_pos A) A"
   375  apply (induct n arbitrary: A)
   376   apply simp
   377  apply (case_tac "A = {}", simp)
   378  apply (frule (1) choose_in)
   379  apply (rule inj_onI)
   380  apply (drule_tac x="A - {choose A}" in meta_spec, simp)
   381  apply (simp add: choose_pos.simps)
   382  apply (simp split: if_split_asm)
   383  apply (erule (1) inj_onD, simp, simp)
   384 done
   385 
   386 lemma choose_pos_bounded [OF refl]:
   387   "\<lbrakk>card A = n; finite A; x \<in> A\<rbrakk> \<Longrightarrow> choose_pos A x < n"
   388 apply (induct n arbitrary: A)
   389 apply simp
   390  apply (case_tac "A = {}", simp)
   391  apply (frule (1) choose_in)
   392 apply (subst choose_pos.simps)
   393 apply simp
   394 done
   395 
   396 lemma choose_pos_lessD:
   397   "\<lbrakk>choose_pos A x < choose_pos A y; finite A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x \<notsqsubseteq> y"
   398  apply (induct A x arbitrary: y rule: choose_pos.induct)
   399  apply simp
   400  apply (case_tac "x = choose A")
   401   apply simp
   402   apply (rule notI)
   403   apply (frule (2) maximal_choose)
   404   apply simp
   405  apply (case_tac "y = choose A")
   406   apply (simp add: choose_pos_choose)
   407  apply (drule_tac x=y in meta_spec)
   408  apply simp
   409  apply (erule meta_mp)
   410  apply (simp add: choose_pos.simps)
   411 done
   412 
   413 subsubsection \<open>Compact basis take function\<close>
   414 
   415 primrec
   416   cb_take :: "nat \<Rightarrow> 'a compact_basis \<Rightarrow> 'a compact_basis" where
   417   "cb_take 0 = (\<lambda>x. compact_bot)"
   418 | "cb_take (Suc n) = (\<lambda>a. Abs_compact_basis (approx n\<cdot>(Rep_compact_basis a)))"
   419 
   420 declare cb_take.simps [simp del]
   421 
   422 lemma cb_take_zero [simp]: "cb_take 0 a = compact_bot"
   423 by (simp only: cb_take.simps)
   424 
   425 lemma Rep_cb_take:
   426   "Rep_compact_basis (cb_take (Suc n) a) = approx n\<cdot>(Rep_compact_basis a)"
   427 by (simp add: cb_take.simps(2))
   428 
   429 lemmas approx_Rep_compact_basis = Rep_cb_take [symmetric]
   430 
   431 lemma cb_take_covers: "\<exists>n. cb_take n x = x"
   432 apply (subgoal_tac "\<exists>n. cb_take (Suc n) x = x", fast)
   433 apply (simp add: Rep_compact_basis_inject [symmetric])
   434 apply (simp add: Rep_cb_take)
   435 apply (rule compact_eq_approx)
   436 apply (rule Rep_compact_basis')
   437 done
   438 
   439 lemma cb_take_less: "cb_take n x \<sqsubseteq> x"
   440 unfolding compact_le_def
   441 by (cases n, simp, simp add: Rep_cb_take approx_below)
   442 
   443 lemma cb_take_idem: "cb_take n (cb_take n x) = cb_take n x"
   444 unfolding Rep_compact_basis_inject [symmetric]
   445 by (cases n, simp, simp add: Rep_cb_take approx_idem)
   446 
   447 lemma cb_take_mono: "x \<sqsubseteq> y \<Longrightarrow> cb_take n x \<sqsubseteq> cb_take n y"
   448 unfolding compact_le_def
   449 by (cases n, simp, simp add: Rep_cb_take monofun_cfun_arg)
   450 
   451 lemma cb_take_chain_le: "m \<le> n \<Longrightarrow> cb_take m x \<sqsubseteq> cb_take n x"
   452 unfolding compact_le_def
   453 apply (cases m, simp, cases n, simp)
   454 apply (simp add: Rep_cb_take, rule chain_mono, simp, simp)
   455 done
   456 
   457 lemma finite_range_cb_take: "finite (range (cb_take n))"
   458 apply (cases n)
   459 apply (subgoal_tac "range (cb_take 0) = {compact_bot}", simp, force)
   460 apply (rule finite_imageD [where f="Rep_compact_basis"])
   461 apply (rule finite_subset [where B="range (\<lambda>x. approx (n - 1)\<cdot>x)"])
   462 apply (clarsimp simp add: Rep_cb_take)
   463 apply (rule finite_range_approx)
   464 apply (rule inj_onI, simp add: Rep_compact_basis_inject)
   465 done
   466 
   467 subsubsection \<open>Rank of basis elements\<close>
   468 
   469 definition
   470   rank :: "'a compact_basis \<Rightarrow> nat"
   471 where
   472   "rank x = (LEAST n. cb_take n x = x)"
   473 
   474 lemma compact_approx_rank: "cb_take (rank x) x = x"
   475 unfolding rank_def
   476 apply (rule LeastI_ex)
   477 apply (rule cb_take_covers)
   478 done
   479 
   480 lemma rank_leD: "rank x \<le> n \<Longrightarrow> cb_take n x = x"
   481 apply (rule below_antisym [OF cb_take_less])
   482 apply (subst compact_approx_rank [symmetric])
   483 apply (erule cb_take_chain_le)
   484 done
   485 
   486 lemma rank_leI: "cb_take n x = x \<Longrightarrow> rank x \<le> n"
   487 unfolding rank_def by (rule Least_le)
   488 
   489 lemma rank_le_iff: "rank x \<le> n \<longleftrightarrow> cb_take n x = x"
   490 by (rule iffI [OF rank_leD rank_leI])
   491 
   492 lemma rank_compact_bot [simp]: "rank compact_bot = 0"
   493 using rank_leI [of 0 compact_bot] by simp
   494 
   495 lemma rank_eq_0_iff [simp]: "rank x = 0 \<longleftrightarrow> x = compact_bot"
   496 using rank_le_iff [of x 0] by auto
   497 
   498 definition
   499   rank_le :: "'a compact_basis \<Rightarrow> 'a compact_basis set"
   500 where
   501   "rank_le x = {y. rank y \<le> rank x}"
   502 
   503 definition
   504   rank_lt :: "'a compact_basis \<Rightarrow> 'a compact_basis set"
   505 where
   506   "rank_lt x = {y. rank y < rank x}"
   507 
   508 definition
   509   rank_eq :: "'a compact_basis \<Rightarrow> 'a compact_basis set"
   510 where
   511   "rank_eq x = {y. rank y = rank x}"
   512 
   513 lemma rank_eq_cong: "rank x = rank y \<Longrightarrow> rank_eq x = rank_eq y"
   514 unfolding rank_eq_def by simp
   515 
   516 lemma rank_lt_cong: "rank x = rank y \<Longrightarrow> rank_lt x = rank_lt y"
   517 unfolding rank_lt_def by simp
   518 
   519 lemma rank_eq_subset: "rank_eq x \<subseteq> rank_le x"
   520 unfolding rank_eq_def rank_le_def by auto
   521 
   522 lemma rank_lt_subset: "rank_lt x \<subseteq> rank_le x"
   523 unfolding rank_lt_def rank_le_def by auto
   524 
   525 lemma finite_rank_le: "finite (rank_le x)"
   526 unfolding rank_le_def
   527 apply (rule finite_subset [where B="range (cb_take (rank x))"])
   528 apply clarify
   529 apply (rule range_eqI)
   530 apply (erule rank_leD [symmetric])
   531 apply (rule finite_range_cb_take)
   532 done
   533 
   534 lemma finite_rank_eq: "finite (rank_eq x)"
   535 by (rule finite_subset [OF rank_eq_subset finite_rank_le])
   536 
   537 lemma finite_rank_lt: "finite (rank_lt x)"
   538 by (rule finite_subset [OF rank_lt_subset finite_rank_le])
   539 
   540 lemma rank_lt_Int_rank_eq: "rank_lt x \<inter> rank_eq x = {}"
   541 unfolding rank_lt_def rank_eq_def rank_le_def by auto
   542 
   543 lemma rank_lt_Un_rank_eq: "rank_lt x \<union> rank_eq x = rank_le x"
   544 unfolding rank_lt_def rank_eq_def rank_le_def by auto
   545 
   546 subsubsection \<open>Sequencing basis elements\<close>
   547 
   548 definition
   549   place :: "'a compact_basis \<Rightarrow> nat"
   550 where
   551   "place x = card (rank_lt x) + choose_pos (rank_eq x) x"
   552 
   553 lemma place_bounded: "place x < card (rank_le x)"
   554 unfolding place_def
   555  apply (rule ord_less_eq_trans)
   556   apply (rule add_strict_left_mono)
   557   apply (rule choose_pos_bounded)
   558    apply (rule finite_rank_eq)
   559   apply (simp add: rank_eq_def)
   560  apply (subst card_Un_disjoint [symmetric])
   561     apply (rule finite_rank_lt)
   562    apply (rule finite_rank_eq)
   563   apply (rule rank_lt_Int_rank_eq)
   564  apply (simp add: rank_lt_Un_rank_eq)
   565 done
   566 
   567 lemma place_ge: "card (rank_lt x) \<le> place x"
   568 unfolding place_def by simp
   569 
   570 lemma place_rank_mono:
   571   fixes x y :: "'a compact_basis"
   572   shows "rank x < rank y \<Longrightarrow> place x < place y"
   573 apply (rule less_le_trans [OF place_bounded])
   574 apply (rule order_trans [OF _ place_ge])
   575 apply (rule card_mono)
   576 apply (rule finite_rank_lt)
   577 apply (simp add: rank_le_def rank_lt_def subset_eq)
   578 done
   579 
   580 lemma place_eqD: "place x = place y \<Longrightarrow> x = y"
   581  apply (rule linorder_cases [where x="rank x" and y="rank y"])
   582    apply (drule place_rank_mono, simp)
   583   apply (simp add: place_def)
   584   apply (rule inj_on_choose_pos [where A="rank_eq x", THEN inj_onD])
   585      apply (rule finite_rank_eq)
   586     apply (simp cong: rank_lt_cong rank_eq_cong)
   587    apply (simp add: rank_eq_def)
   588   apply (simp add: rank_eq_def)
   589  apply (drule place_rank_mono, simp)
   590 done
   591 
   592 lemma inj_place: "inj place"
   593 by (rule inj_onI, erule place_eqD)
   594 
   595 subsubsection \<open>Embedding and projection on basis elements\<close>
   596 
   597 definition
   598   sub :: "'a compact_basis \<Rightarrow> 'a compact_basis"
   599 where
   600   "sub x = (case rank x of 0 \<Rightarrow> compact_bot | Suc k \<Rightarrow> cb_take k x)"
   601 
   602 lemma rank_sub_less: "x \<noteq> compact_bot \<Longrightarrow> rank (sub x) < rank x"
   603 unfolding sub_def
   604 apply (cases "rank x", simp)
   605 apply (simp add: less_Suc_eq_le)
   606 apply (rule rank_leI)
   607 apply (rule cb_take_idem)
   608 done
   609 
   610 lemma place_sub_less: "x \<noteq> compact_bot \<Longrightarrow> place (sub x) < place x"
   611 apply (rule place_rank_mono)
   612 apply (erule rank_sub_less)
   613 done
   614 
   615 lemma sub_below: "sub x \<sqsubseteq> x"
   616 unfolding sub_def by (cases "rank x", simp_all add: cb_take_less)
   617 
   618 lemma rank_less_imp_below_sub: "\<lbrakk>x \<sqsubseteq> y; rank x < rank y\<rbrakk> \<Longrightarrow> x \<sqsubseteq> sub y"
   619 unfolding sub_def
   620 apply (cases "rank y", simp)
   621 apply (simp add: less_Suc_eq_le)
   622 apply (subgoal_tac "cb_take nat x \<sqsubseteq> cb_take nat y")
   623 apply (simp add: rank_leD)
   624 apply (erule cb_take_mono)
   625 done
   626 
   627 function
   628   basis_emb :: "'a compact_basis \<Rightarrow> ubasis"
   629 where
   630   "basis_emb x = (if x = compact_bot then 0 else
   631     node (place x) (basis_emb (sub x))
   632       (basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y}))"
   633 by auto
   634 
   635 termination basis_emb
   636 apply (relation "measure place", simp)
   637 apply (simp add: place_sub_less)
   638 apply simp
   639 done
   640 
   641 declare basis_emb.simps [simp del]
   642 
   643 lemma basis_emb_compact_bot [simp]: "basis_emb compact_bot = 0"
   644 by (simp add: basis_emb.simps)
   645 
   646 lemma fin1: "finite {y. place y < place x \<and> x \<sqsubseteq> y}"
   647 apply (subst Collect_conj_eq)
   648 apply (rule finite_Int)
   649 apply (rule disjI1)
   650 apply (subgoal_tac "finite (place -` {n. n < place x})", simp)
   651 apply (rule finite_vimageI [OF _ inj_place])
   652 apply (simp add: lessThan_def [symmetric])
   653 done
   654 
   655 lemma fin2: "finite (basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y})"
   656 by (rule finite_imageI [OF fin1])
   657 
   658 lemma rank_place_mono:
   659   "\<lbrakk>place x < place y; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> rank x < rank y"
   660 apply (rule linorder_cases, assumption)
   661 apply (simp add: place_def cong: rank_lt_cong rank_eq_cong)
   662 apply (drule choose_pos_lessD)
   663 apply (rule finite_rank_eq)
   664 apply (simp add: rank_eq_def)
   665 apply (simp add: rank_eq_def)
   666 apply simp
   667 apply (drule place_rank_mono, simp)
   668 done
   669 
   670 lemma basis_emb_mono:
   671   "x \<sqsubseteq> y \<Longrightarrow> ubasis_le (basis_emb x) (basis_emb y)"
   672 proof (induct "max (place x) (place y)" arbitrary: x y rule: less_induct)
   673   case less
   674   show ?case proof (rule linorder_cases)
   675     assume "place x < place y"
   676     then have "rank x < rank y"
   677       using \<open>x \<sqsubseteq> y\<close> by (rule rank_place_mono)
   678     with \<open>place x < place y\<close> show ?case
   679       apply (case_tac "y = compact_bot", simp)
   680       apply (simp add: basis_emb.simps [of y])
   681       apply (rule ubasis_le_trans [OF _ ubasis_le_lower [OF fin2]])
   682       apply (rule less)
   683        apply (simp add: less_max_iff_disj)
   684        apply (erule place_sub_less)
   685       apply (erule rank_less_imp_below_sub [OF \<open>x \<sqsubseteq> y\<close>])
   686       done
   687   next
   688     assume "place x = place y"
   689     hence "x = y" by (rule place_eqD)
   690     thus ?case by (simp add: ubasis_le_refl)
   691   next
   692     assume "place x > place y"
   693     with \<open>x \<sqsubseteq> y\<close> show ?case
   694       apply (case_tac "x = compact_bot", simp add: ubasis_le_minimal)
   695       apply (simp add: basis_emb.simps [of x])
   696       apply (rule ubasis_le_upper [OF fin2], simp)
   697       apply (rule less)
   698        apply (simp add: less_max_iff_disj)
   699        apply (erule place_sub_less)
   700       apply (erule rev_below_trans)
   701       apply (rule sub_below)
   702       done
   703   qed
   704 qed
   705 
   706 lemma inj_basis_emb: "inj basis_emb"
   707  apply (rule inj_onI)
   708  apply (case_tac "x = compact_bot")
   709   apply (case_tac [!] "y = compact_bot")
   710     apply simp
   711    apply (simp add: basis_emb.simps)
   712   apply (simp add: basis_emb.simps)
   713  apply (simp add: basis_emb.simps)
   714  apply (simp add: fin2 inj_eq [OF inj_place])
   715 done
   716 
   717 definition
   718   basis_prj :: "ubasis \<Rightarrow> 'a compact_basis"
   719 where
   720   "basis_prj x = inv basis_emb
   721     (ubasis_until (\<lambda>x. x \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> ubasis)) x)"
   722 
   723 lemma basis_prj_basis_emb: "\<And>x. basis_prj (basis_emb x) = x"
   724 unfolding basis_prj_def
   725  apply (subst ubasis_until_same)
   726   apply (rule rangeI)
   727  apply (rule inv_f_f)
   728  apply (rule inj_basis_emb)
   729 done
   730 
   731 lemma basis_prj_node:
   732   "\<lbrakk>finite S; node i a S \<notin> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)\<rbrakk>
   733     \<Longrightarrow> basis_prj (node i a S) = (basis_prj a :: 'a compact_basis)"
   734 unfolding basis_prj_def by simp
   735 
   736 lemma basis_prj_0: "basis_prj 0 = compact_bot"
   737 apply (subst basis_emb_compact_bot [symmetric])
   738 apply (rule basis_prj_basis_emb)
   739 done
   740 
   741 lemma node_eq_basis_emb_iff:
   742   "finite S \<Longrightarrow> node i a S = basis_emb x \<longleftrightarrow>
   743     x \<noteq> compact_bot \<and> i = place x \<and> a = basis_emb (sub x) \<and>
   744         S = basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y}"
   745 apply (cases "x = compact_bot", simp)
   746 apply (simp add: basis_emb.simps [of x])
   747 apply (simp add: fin2)
   748 done
   749 
   750 lemma basis_prj_mono: "ubasis_le a b \<Longrightarrow> basis_prj a \<sqsubseteq> basis_prj b"
   751 proof (induct a b rule: ubasis_le.induct)
   752   case (ubasis_le_refl a) show ?case by (rule below_refl)
   753 next
   754   case (ubasis_le_trans a b c) thus ?case by - (rule below_trans)
   755 next
   756   case (ubasis_le_lower S a i) thus ?case
   757     apply (cases "node i a S \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)")
   758      apply (erule rangeE, rename_tac x)
   759      apply (simp add: basis_prj_basis_emb)
   760      apply (simp add: node_eq_basis_emb_iff)
   761      apply (simp add: basis_prj_basis_emb)
   762      apply (rule sub_below)
   763     apply (simp add: basis_prj_node)
   764     done
   765 next
   766   case (ubasis_le_upper S b a i) thus ?case
   767     apply (cases "node i a S \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)")
   768      apply (erule rangeE, rename_tac x)
   769      apply (simp add: basis_prj_basis_emb)
   770      apply (clarsimp simp add: node_eq_basis_emb_iff)
   771      apply (simp add: basis_prj_basis_emb)
   772     apply (simp add: basis_prj_node)
   773     done
   774 qed
   775 
   776 lemma basis_emb_prj_less: "ubasis_le (basis_emb (basis_prj x)) x"
   777 unfolding basis_prj_def
   778  apply (subst f_inv_into_f [where f=basis_emb])
   779   apply (rule ubasis_until)
   780   apply (rule range_eqI [where x=compact_bot])
   781   apply simp
   782  apply (rule ubasis_until_less)
   783 done
   784 
   785 lemma ideal_completion:
   786   "ideal_completion below Rep_compact_basis (approximants :: 'a \<Rightarrow> _)"
   787 proof
   788   fix w :: "'a"
   789   show "below.ideal (approximants w)"
   790   proof (rule below.idealI)
   791     have "Abs_compact_basis (approx 0\<cdot>w) \<in> approximants w"
   792       by (simp add: approximants_def approx_below)
   793     thus "\<exists>x. x \<in> approximants w" ..
   794   next
   795     fix x y :: "'a compact_basis"
   796     assume x: "x \<in> approximants w" and y: "y \<in> approximants w"
   797     obtain i where i: "approx i\<cdot>(Rep_compact_basis x) = Rep_compact_basis x"
   798       using compact_eq_approx Rep_compact_basis' by fast
   799     obtain j where j: "approx j\<cdot>(Rep_compact_basis y) = Rep_compact_basis y"
   800       using compact_eq_approx Rep_compact_basis' by fast
   801     let ?z = "Abs_compact_basis (approx (max i j)\<cdot>w)"
   802     have "?z \<in> approximants w"
   803       by (simp add: approximants_def approx_below)
   804     moreover from x y have "x \<sqsubseteq> ?z \<and> y \<sqsubseteq> ?z"
   805       by (simp add: approximants_def compact_le_def)
   806          (metis i j monofun_cfun chain_mono chain_approx max.cobounded1 max.cobounded2)
   807     ultimately show "\<exists>z \<in> approximants w. x \<sqsubseteq> z \<and> y \<sqsubseteq> z" ..
   808   next
   809     fix x y :: "'a compact_basis"
   810     assume "x \<sqsubseteq> y" "y \<in> approximants w" thus "x \<in> approximants w"
   811       unfolding approximants_def compact_le_def
   812       by (auto elim: below_trans)
   813   qed
   814 next
   815   fix Y :: "nat \<Rightarrow> 'a"
   816   assume "chain Y"
   817   thus "approximants (\<Squnion>i. Y i) = (\<Union>i. approximants (Y i))"
   818     unfolding approximants_def
   819     by (auto simp add: compact_below_lub_iff)
   820 next
   821   fix a :: "'a compact_basis"
   822   show "approximants (Rep_compact_basis a) = {b. b \<sqsubseteq> a}"
   823     unfolding approximants_def compact_le_def ..
   824 next
   825   fix x y :: "'a"
   826   assume "approximants x \<subseteq> approximants y"
   827   hence "\<forall>z. compact z \<longrightarrow> z \<sqsubseteq> x \<longrightarrow> z \<sqsubseteq> y"
   828     by (simp add: approximants_def subset_eq)
   829        (metis Abs_compact_basis_inverse')
   830   hence "(\<Squnion>i. approx i\<cdot>x) \<sqsubseteq> y"
   831     by (simp add: lub_below approx_below)
   832   thus "x \<sqsubseteq> y"
   833     by (simp add: lub_distribs)
   834 next
   835   show "\<exists>f::'a compact_basis \<Rightarrow> nat. inj f"
   836     by (rule exI, rule inj_place)
   837 qed
   838 
   839 end
   840 
   841 interpretation compact_basis:
   842   ideal_completion below Rep_compact_basis
   843     "approximants :: 'a::bifinite \<Rightarrow> 'a compact_basis set"
   844 proof -
   845   obtain a :: "nat \<Rightarrow> 'a \<rightarrow> 'a" where "approx_chain a"
   846     using bifinite ..
   847   hence "bifinite_approx_chain a"
   848     unfolding bifinite_approx_chain_def .
   849   thus "ideal_completion below Rep_compact_basis (approximants :: 'a \<Rightarrow> _)"
   850     by (rule bifinite_approx_chain.ideal_completion)
   851 qed
   852 
   853 subsubsection \<open>EP-pair from any bifinite domain into \emph{udom}\<close>
   854 
   855 context bifinite_approx_chain begin
   856 
   857 definition
   858   udom_emb :: "'a \<rightarrow> udom"
   859 where
   860   "udom_emb = compact_basis.extension (\<lambda>x. udom_principal (basis_emb x))"
   861 
   862 definition
   863   udom_prj :: "udom \<rightarrow> 'a"
   864 where
   865   "udom_prj = udom.extension (\<lambda>x. Rep_compact_basis (basis_prj x))"
   866 
   867 lemma udom_emb_principal:
   868   "udom_emb\<cdot>(Rep_compact_basis x) = udom_principal (basis_emb x)"
   869 unfolding udom_emb_def
   870 apply (rule compact_basis.extension_principal)
   871 apply (rule udom.principal_mono)
   872 apply (erule basis_emb_mono)
   873 done
   874 
   875 lemma udom_prj_principal:
   876   "udom_prj\<cdot>(udom_principal x) = Rep_compact_basis (basis_prj x)"
   877 unfolding udom_prj_def
   878 apply (rule udom.extension_principal)
   879 apply (rule compact_basis.principal_mono)
   880 apply (erule basis_prj_mono)
   881 done
   882 
   883 lemma ep_pair_udom: "ep_pair udom_emb udom_prj"
   884  apply standard
   885   apply (rule compact_basis.principal_induct, simp)
   886   apply (simp add: udom_emb_principal udom_prj_principal)
   887   apply (simp add: basis_prj_basis_emb)
   888  apply (rule udom.principal_induct, simp)
   889  apply (simp add: udom_emb_principal udom_prj_principal)
   890  apply (rule basis_emb_prj_less)
   891 done
   892 
   893 end
   894 
   895 abbreviation "udom_emb \<equiv> bifinite_approx_chain.udom_emb"
   896 abbreviation "udom_prj \<equiv> bifinite_approx_chain.udom_prj"
   897 
   898 lemmas ep_pair_udom =
   899   bifinite_approx_chain.ep_pair_udom [unfolded bifinite_approx_chain_def]
   900 
   901 subsection \<open>Chain of approx functions for type \emph{udom}\<close>
   902 
   903 definition
   904   udom_approx :: "nat \<Rightarrow> udom \<rightarrow> udom"
   905 where
   906   "udom_approx i =
   907     udom.extension (\<lambda>x. udom_principal (ubasis_until (\<lambda>y. y \<le> i) x))"
   908 
   909 lemma udom_approx_mono:
   910   "ubasis_le a b \<Longrightarrow>
   911     udom_principal (ubasis_until (\<lambda>y. y \<le> i) a) \<sqsubseteq>
   912     udom_principal (ubasis_until (\<lambda>y. y \<le> i) b)"
   913 apply (rule udom.principal_mono)
   914 apply (rule ubasis_until_mono)
   915 apply (frule (2) order_less_le_trans [OF node_gt2])
   916 apply (erule order_less_imp_le)
   917 apply assumption
   918 done
   919 
   920 lemma adm_mem_finite: "\<lbrakk>cont f; finite S\<rbrakk> \<Longrightarrow> adm (\<lambda>x. f x \<in> S)"
   921 by (erule adm_subst, induct set: finite, simp_all)
   922 
   923 lemma udom_approx_principal:
   924   "udom_approx i\<cdot>(udom_principal x) =
   925     udom_principal (ubasis_until (\<lambda>y. y \<le> i) x)"
   926 unfolding udom_approx_def
   927 apply (rule udom.extension_principal)
   928 apply (erule udom_approx_mono)
   929 done
   930 
   931 lemma finite_deflation_udom_approx: "finite_deflation (udom_approx i)"
   932 proof
   933   fix x show "udom_approx i\<cdot>(udom_approx i\<cdot>x) = udom_approx i\<cdot>x"
   934     by (induct x rule: udom.principal_induct, simp)
   935        (simp add: udom_approx_principal ubasis_until_idem)
   936 next
   937   fix x show "udom_approx i\<cdot>x \<sqsubseteq> x"
   938     by (induct x rule: udom.principal_induct, simp)
   939        (simp add: udom_approx_principal ubasis_until_less)
   940 next
   941   have *: "finite (range (\<lambda>x. udom_principal (ubasis_until (\<lambda>y. y \<le> i) x)))"
   942     apply (subst range_composition [where f=udom_principal])
   943     apply (simp add: finite_range_ubasis_until)
   944     done
   945   show "finite {x. udom_approx i\<cdot>x = x}"
   946     apply (rule finite_range_imp_finite_fixes)
   947     apply (rule rev_finite_subset [OF *])
   948     apply (clarsimp, rename_tac x)
   949     apply (induct_tac x rule: udom.principal_induct)
   950     apply (simp add: adm_mem_finite *)
   951     apply (simp add: udom_approx_principal)
   952     done
   953 qed
   954 
   955 interpretation udom_approx: finite_deflation "udom_approx i"
   956 by (rule finite_deflation_udom_approx)
   957 
   958 lemma chain_udom_approx [simp]: "chain (\<lambda>i. udom_approx i)"
   959 unfolding udom_approx_def
   960 apply (rule chainI)
   961 apply (rule udom.extension_mono)
   962 apply (erule udom_approx_mono)
   963 apply (erule udom_approx_mono)
   964 apply (rule udom.principal_mono)
   965 apply (rule ubasis_until_chain, simp)
   966 done
   967 
   968 lemma lub_udom_approx [simp]: "(\<Squnion>i. udom_approx i) = ID"
   969 apply (rule cfun_eqI, simp add: contlub_cfun_fun)
   970 apply (rule below_antisym)
   971 apply (rule lub_below)
   972 apply (simp)
   973 apply (rule udom_approx.below)
   974 apply (rule_tac x=x in udom.principal_induct)
   975 apply (simp add: lub_distribs)
   976 apply (rule_tac i=a in below_lub)
   977 apply simp
   978 apply (simp add: udom_approx_principal)
   979 apply (simp add: ubasis_until_same ubasis_le_refl)
   980 done
   981 
   982 lemma udom_approx [simp]: "approx_chain udom_approx"
   983 proof
   984   show "chain (\<lambda>i. udom_approx i)"
   985     by (rule chain_udom_approx)
   986   show "(\<Squnion>i. udom_approx i) = ID"
   987     by (rule lub_udom_approx)
   988 qed
   989 
   990 instance udom :: bifinite
   991   by standard (fast intro: udom_approx)
   992 
   993 hide_const (open) node
   994 
   995 notation binomial  (infixl "choose" 65)
   996 
   997 end