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