1(* Title: HOL/HOLCF/Universal.thy 2 Author: Brian Huffman 3*) 4 5section \<open>A universal bifinite domain\<close> 6 7theory Universal 8imports Bifinite Completion "HOL-Library.Nat_Bijection" 9begin 10 11no_notation binomial (infixl "choose" 65) 12 13subsection \<open>Basis for universal domain\<close> 14 15subsubsection \<open>Basis datatype\<close> 16 17type_synonym ubasis = nat 18 19definition 20 node :: "nat \<Rightarrow> ubasis \<Rightarrow> ubasis set \<Rightarrow> ubasis" 21where 22 "node i a S = Suc (prod_encode (i, prod_encode (a, set_encode S)))" 23 24lemma node_not_0 [simp]: "node i a S \<noteq> 0" 25unfolding node_def by simp 26 27lemma node_gt_0 [simp]: "0 < node i a S" 28unfolding node_def by simp 29 30lemma 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" 33unfolding node_def by (simp add: prod_encode_eq set_encode_eq) 34 35lemma node_gt0: "i < node i a S" 36unfolding node_def less_Suc_eq_le 37by (rule le_prod_encode_1) 38 39lemma node_gt1: "a < node i a S" 40unfolding node_def less_Suc_eq_le 41by (rule order_trans [OF le_prod_encode_1 le_prod_encode_2]) 42 43lemma nat_less_power2: "n < 2^n" 44by (induct n) simp_all 45 46lemma node_gt2: "\<lbrakk>finite S; b \<in> S\<rbrakk> \<Longrightarrow> b < node i a S" 47unfolding node_def less_Suc_eq_le set_encode_def 48apply (rule order_trans [OF _ le_prod_encode_2]) 49apply (rule order_trans [OF _ le_prod_encode_2]) 50apply (rule order_trans [where y="sum ((^) 2) {b}"]) 51apply (simp add: nat_less_power2 [THEN order_less_imp_le]) 52apply (erule sum_mono2, simp, simp) 53done 54 55lemma eq_prod_encode_pairI: 56 "\<lbrakk>fst (prod_decode x) = a; snd (prod_decode x) = b\<rbrakk> \<Longrightarrow> x = prod_encode (a, b)" 57by (erule subst, erule subst, simp) 58 59lemma 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]) 70done 71 72lemma 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) 80done 81 82subsubsection \<open>Basis ordering\<close> 83 84inductive 85 ubasis_le :: "nat \<Rightarrow> nat \<Rightarrow> bool" 86where 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 95lemma ubasis_le_minimal: "ubasis_le 0 x" 96apply (induct x rule: node_induct) 97apply (rule ubasis_le_refl) 98apply (erule ubasis_le_trans) 99apply (erule ubasis_le_lower) 100done 101 102interpretation udom: preorder ubasis_le 103apply standard 104apply (rule ubasis_le_refl) 105apply (erule (1) ubasis_le_trans) 106done 107 108subsubsection \<open>Generic take function\<close> 109 110function 111 ubasis_until :: "(ubasis \<Rightarrow> bool) \<Rightarrow> ubasis \<Rightarrow> ubasis" 112where 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 123done 124 125termination ubasis_until 126apply (relation "measure snd") 127apply (rule wf_measure) 128apply (simp add: node_gt1) 129done 130 131lemma ubasis_until: "P 0 \<Longrightarrow> P (ubasis_until P x)" 132by (induct x rule: node_induct) simp_all 133 134lemma ubasis_until': "0 < ubasis_until P x \<Longrightarrow> P (ubasis_until P x)" 135by (induct x rule: node_induct) auto 136 137lemma ubasis_until_same: "P x \<Longrightarrow> ubasis_until P x = x" 138by (induct x rule: node_induct) simp_all 139 140lemma ubasis_until_idem: 141 "P 0 \<Longrightarrow> ubasis_until P (ubasis_until P x) = ubasis_until P x" 142by (rule ubasis_until_same [OF ubasis_until]) 143 144lemma ubasis_until_0: 145 "\<forall>x. x \<noteq> 0 \<longrightarrow> \<not> P x \<Longrightarrow> ubasis_until P x = 0" 146by (induct x rule: node_induct) simp_all 147 148lemma ubasis_until_less: "ubasis_le (ubasis_until P x) x" 149apply (induct x rule: node_induct) 150apply (simp add: ubasis_le_refl) 151apply (simp add: ubasis_le_refl) 152apply (rule impI) 153apply (erule ubasis_le_trans) 154apply (erule ubasis_le_lower) 155done 156 157lemma 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)" 160apply (induct x rule: node_induct) 161apply (simp add: ubasis_le_refl) 162apply (simp add: ubasis_le_refl) 163apply (simp add: PQ) 164apply clarify 165apply (rule ubasis_le_trans) 166apply (rule ubasis_until_less) 167apply (erule ubasis_le_lower) 168done 169 170lemma 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)" 173proof (induct set: ubasis_le) 174 case (ubasis_le_refl a) show ?case by (rule ubasis_le.ubasis_le_refl) 175next 176 case (ubasis_le_trans a b c) thus ?case by - (rule ubasis_le.ubasis_le_trans) 177next 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 183next 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 190qed 191 192lemma finite_range_ubasis_until: 193 "finite {x. P x} \<Longrightarrow> finite (range (ubasis_until P))" 194apply (rule finite_subset [where B="insert 0 {x. P x}"]) 195apply (clarsimp simp add: ubasis_until') 196apply simp 197done 198 199 200subsection \<open>Defining the universal domain by ideal completion\<close> 201 202typedef udom = "{S. udom.ideal S}" 203by (rule udom.ex_ideal) 204 205instantiation udom :: below 206begin 207 208definition 209 "x \<sqsubseteq> y \<longleftrightarrow> Rep_udom x \<subseteq> Rep_udom y" 210 211instance .. 212end 213 214instance udom :: po 215using type_definition_udom below_udom_def 216by (rule udom.typedef_ideal_po) 217 218instance udom :: cpo 219using type_definition_udom below_udom_def 220by (rule udom.typedef_ideal_cpo) 221 222definition 223 udom_principal :: "nat \<Rightarrow> udom" where 224 "udom_principal t = Abs_udom {u. ubasis_le u t}" 225 226lemma ubasis_countable: "\<exists>f::ubasis \<Rightarrow> nat. inj f" 227by (rule exI, rule inj_on_id) 228 229interpretation udom: 230 ideal_completion ubasis_le udom_principal Rep_udom 231using type_definition_udom below_udom_def 232using udom_principal_def ubasis_countable 233by (rule udom.typedef_ideal_completion) 234 235text \<open>Universal domain is pointed\<close> 236 237lemma udom_minimal: "udom_principal 0 \<sqsubseteq> x" 238apply (induct x rule: udom.principal_induct) 239apply (simp, simp add: ubasis_le_minimal) 240done 241 242instance udom :: pcpo 243by intro_classes (fast intro: udom_minimal) 244 245lemma inst_udom_pcpo: "\<bottom> = udom_principal 0" 246by (rule udom_minimal [THEN bottomI, symmetric]) 247 248 249subsection \<open>Compact bases of domains\<close> 250 251typedef 'a compact_basis = "{x::'a::pcpo. compact x}" 252by auto 253 254lemma Rep_compact_basis' [simp]: "compact (Rep_compact_basis a)" 255by (rule Rep_compact_basis [unfolded mem_Collect_eq]) 256 257lemma Abs_compact_basis_inverse' [simp]: 258 "compact x \<Longrightarrow> Rep_compact_basis (Abs_compact_basis x) = x" 259by (rule Abs_compact_basis_inverse [unfolded mem_Collect_eq]) 260 261instantiation compact_basis :: (pcpo) below 262begin 263 264definition 265 compact_le_def: 266 "(\<sqsubseteq>) \<equiv> (\<lambda>x y. Rep_compact_basis x \<sqsubseteq> Rep_compact_basis y)" 267 268instance .. 269end 270 271instance compact_basis :: (pcpo) po 272using type_definition_compact_basis compact_le_def 273by (rule typedef_po) 274 275definition 276 approximants :: "'a \<Rightarrow> 'a compact_basis set" where 277 "approximants = (\<lambda>x. {a. Rep_compact_basis a \<sqsubseteq> x})" 278 279definition 280 compact_bot :: "'a::pcpo compact_basis" where 281 "compact_bot = Abs_compact_basis \<bottom>" 282 283lemma Rep_compact_bot [simp]: "Rep_compact_basis compact_bot = \<bottom>" 284unfolding compact_bot_def by simp 285 286lemma compact_bot_minimal [simp]: "compact_bot \<sqsubseteq> a" 287unfolding compact_le_def Rep_compact_bot by simp 288 289 290subsection \<open>Universality of \emph{udom}\<close> 291 292text \<open>We use a locale to parameterize the construction over a chain 293of approx functions on the type to be embedded.\<close> 294 295locale bifinite_approx_chain = 296 approx_chain approx for approx :: "nat \<Rightarrow> 'a::bifinite \<rightarrow> 'a" 297begin 298 299subsubsection \<open>Choosing a maximal element from a finite set\<close> 300 301lemma 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" 304proof (induct rule: finite_ne_induct) 305 case (singleton x) 306 show ?case by simp 307next 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 328qed 329 330definition 331 choose :: "'a compact_basis set \<Rightarrow> 'a compact_basis" 332where 333 "choose A = (SOME x. x \<in> {x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y})" 334 335lemma 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}" 337unfolding choose_def 338apply (rule someI_ex) 339apply (frule (1) finite_has_maximal, fast) 340done 341 342lemma maximal_choose: 343 "\<lbrakk>finite A; y \<in> A; choose A \<sqsubseteq> y\<rbrakk> \<Longrightarrow> choose A = y" 344apply (cases "A = {}", simp) 345apply (frule (1) choose_lemma, simp) 346done 347 348lemma choose_in: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> choose A \<in> A" 349by (frule (1) choose_lemma, simp) 350 351function 352 choose_pos :: "'a compact_basis set \<Rightarrow> 'a compact_basis \<Rightarrow> nat" 353where 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)" 357by auto 358 359termination choose_pos 360apply (relation "measure (card \<circ> fst)", simp) 361apply clarsimp 362apply (rule card_Diff1_less) 363apply assumption 364apply (erule choose_in) 365apply clarsimp 366done 367 368declare choose_pos.simps [simp del] 369 370lemma choose_pos_choose: "finite A \<Longrightarrow> choose_pos A (choose A) = 0" 371by (simp add: choose_pos.simps) 372 373lemma 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) 384done 385 386lemma choose_pos_bounded [OF refl]: 387 "\<lbrakk>card A = n; finite A; x \<in> A\<rbrakk> \<Longrightarrow> choose_pos A x < n" 388apply (induct n arbitrary: A) 389apply simp 390 apply (case_tac "A = {}", simp) 391 apply (frule (1) choose_in) 392apply (subst choose_pos.simps) 393apply simp 394done 395 396lemma 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) 411done 412 413subsubsection \<open>Compact basis take function\<close> 414 415primrec 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 420declare cb_take.simps [simp del] 421 422lemma cb_take_zero [simp]: "cb_take 0 a = compact_bot" 423by (simp only: cb_take.simps) 424 425lemma Rep_cb_take: 426 "Rep_compact_basis (cb_take (Suc n) a) = approx n\<cdot>(Rep_compact_basis a)" 427by (simp add: cb_take.simps(2)) 428 429lemmas approx_Rep_compact_basis = Rep_cb_take [symmetric] 430 431lemma cb_take_covers: "\<exists>n. cb_take n x = x" 432apply (subgoal_tac "\<exists>n. cb_take (Suc n) x = x", fast) 433apply (simp add: Rep_compact_basis_inject [symmetric]) 434apply (simp add: Rep_cb_take) 435apply (rule compact_eq_approx) 436apply (rule Rep_compact_basis') 437done 438 439lemma cb_take_less: "cb_take n x \<sqsubseteq> x" 440unfolding compact_le_def 441by (cases n, simp, simp add: Rep_cb_take approx_below) 442 443lemma cb_take_idem: "cb_take n (cb_take n x) = cb_take n x" 444unfolding Rep_compact_basis_inject [symmetric] 445by (cases n, simp, simp add: Rep_cb_take approx_idem) 446 447lemma cb_take_mono: "x \<sqsubseteq> y \<Longrightarrow> cb_take n x \<sqsubseteq> cb_take n y" 448unfolding compact_le_def 449by (cases n, simp, simp add: Rep_cb_take monofun_cfun_arg) 450 451lemma cb_take_chain_le: "m \<le> n \<Longrightarrow> cb_take m x \<sqsubseteq> cb_take n x" 452unfolding compact_le_def 453apply (cases m, simp, cases n, simp) 454apply (simp add: Rep_cb_take, rule chain_mono, simp, simp) 455done 456 457lemma finite_range_cb_take: "finite (range (cb_take n))" 458apply (cases n) 459apply (subgoal_tac "range (cb_take 0) = {compact_bot}", simp, force) 460apply (rule finite_imageD [where f="Rep_compact_basis"]) 461apply (rule finite_subset [where B="range (\<lambda>x. approx (n - 1)\<cdot>x)"]) 462apply (clarsimp simp add: Rep_cb_take) 463apply (rule finite_range_approx) 464apply (rule inj_onI, simp add: Rep_compact_basis_inject) 465done 466 467subsubsection \<open>Rank of basis elements\<close> 468 469definition 470 rank :: "'a compact_basis \<Rightarrow> nat" 471where 472 "rank x = (LEAST n. cb_take n x = x)" 473 474lemma compact_approx_rank: "cb_take (rank x) x = x" 475unfolding rank_def 476apply (rule LeastI_ex) 477apply (rule cb_take_covers) 478done 479 480lemma rank_leD: "rank x \<le> n \<Longrightarrow> cb_take n x = x" 481apply (rule below_antisym [OF cb_take_less]) 482apply (subst compact_approx_rank [symmetric]) 483apply (erule cb_take_chain_le) 484done 485 486lemma rank_leI: "cb_take n x = x \<Longrightarrow> rank x \<le> n" 487unfolding rank_def by (rule Least_le) 488 489lemma rank_le_iff: "rank x \<le> n \<longleftrightarrow> cb_take n x = x" 490by (rule iffI [OF rank_leD rank_leI]) 491 492lemma rank_compact_bot [simp]: "rank compact_bot = 0" 493using rank_leI [of 0 compact_bot] by simp 494 495lemma rank_eq_0_iff [simp]: "rank x = 0 \<longleftrightarrow> x = compact_bot" 496using rank_le_iff [of x 0] by auto 497 498definition 499 rank_le :: "'a compact_basis \<Rightarrow> 'a compact_basis set" 500where 501 "rank_le x = {y. rank y \<le> rank x}" 502 503definition 504 rank_lt :: "'a compact_basis \<Rightarrow> 'a compact_basis set" 505where 506 "rank_lt x = {y. rank y < rank x}" 507 508definition 509 rank_eq :: "'a compact_basis \<Rightarrow> 'a compact_basis set" 510where 511 "rank_eq x = {y. rank y = rank x}" 512 513lemma rank_eq_cong: "rank x = rank y \<Longrightarrow> rank_eq x = rank_eq y" 514unfolding rank_eq_def by simp 515 516lemma rank_lt_cong: "rank x = rank y \<Longrightarrow> rank_lt x = rank_lt y" 517unfolding rank_lt_def by simp 518 519lemma rank_eq_subset: "rank_eq x \<subseteq> rank_le x" 520unfolding rank_eq_def rank_le_def by auto 521 522lemma rank_lt_subset: "rank_lt x \<subseteq> rank_le x" 523unfolding rank_lt_def rank_le_def by auto 524 525lemma finite_rank_le: "finite (rank_le x)" 526unfolding rank_le_def 527apply (rule finite_subset [where B="range (cb_take (rank x))"]) 528apply clarify 529apply (rule range_eqI) 530apply (erule rank_leD [symmetric]) 531apply (rule finite_range_cb_take) 532done 533 534lemma finite_rank_eq: "finite (rank_eq x)" 535by (rule finite_subset [OF rank_eq_subset finite_rank_le]) 536 537lemma finite_rank_lt: "finite (rank_lt x)" 538by (rule finite_subset [OF rank_lt_subset finite_rank_le]) 539 540lemma rank_lt_Int_rank_eq: "rank_lt x \<inter> rank_eq x = {}" 541unfolding rank_lt_def rank_eq_def rank_le_def by auto 542 543lemma rank_lt_Un_rank_eq: "rank_lt x \<union> rank_eq x = rank_le x" 544unfolding rank_lt_def rank_eq_def rank_le_def by auto 545 546subsubsection \<open>Sequencing basis elements\<close> 547 548definition 549 place :: "'a compact_basis \<Rightarrow> nat" 550where 551 "place x = card (rank_lt x) + choose_pos (rank_eq x) x" 552 553lemma place_bounded: "place x < card (rank_le x)" 554unfolding 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) 565done 566 567lemma place_ge: "card (rank_lt x) \<le> place x" 568unfolding place_def by simp 569 570lemma place_rank_mono: 571 fixes x y :: "'a compact_basis" 572 shows "rank x < rank y \<Longrightarrow> place x < place y" 573apply (rule less_le_trans [OF place_bounded]) 574apply (rule order_trans [OF _ place_ge]) 575apply (rule card_mono) 576apply (rule finite_rank_lt) 577apply (simp add: rank_le_def rank_lt_def subset_eq) 578done 579 580lemma 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) 590done 591 592lemma inj_place: "inj place" 593by (rule inj_onI, erule place_eqD) 594 595subsubsection \<open>Embedding and projection on basis elements\<close> 596 597definition 598 sub :: "'a compact_basis \<Rightarrow> 'a compact_basis" 599where 600 "sub x = (case rank x of 0 \<Rightarrow> compact_bot | Suc k \<Rightarrow> cb_take k x)" 601 602lemma rank_sub_less: "x \<noteq> compact_bot \<Longrightarrow> rank (sub x) < rank x" 603unfolding sub_def 604apply (cases "rank x", simp) 605apply (simp add: less_Suc_eq_le) 606apply (rule rank_leI) 607apply (rule cb_take_idem) 608done 609 610lemma place_sub_less: "x \<noteq> compact_bot \<Longrightarrow> place (sub x) < place x" 611apply (rule place_rank_mono) 612apply (erule rank_sub_less) 613done 614 615lemma sub_below: "sub x \<sqsubseteq> x" 616unfolding sub_def by (cases "rank x", simp_all add: cb_take_less) 617 618lemma rank_less_imp_below_sub: "\<lbrakk>x \<sqsubseteq> y; rank x < rank y\<rbrakk> \<Longrightarrow> x \<sqsubseteq> sub y" 619unfolding sub_def 620apply (cases "rank y", simp) 621apply (simp add: less_Suc_eq_le) 622apply (subgoal_tac "cb_take nat x \<sqsubseteq> cb_take nat y") 623apply (simp add: rank_leD) 624apply (erule cb_take_mono) 625done 626 627function 628 basis_emb :: "'a compact_basis \<Rightarrow> ubasis" 629where 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}))" 633by auto 634 635termination basis_emb 636apply (relation "measure place", simp) 637apply (simp add: place_sub_less) 638apply simp 639done 640 641declare basis_emb.simps [simp del] 642 643lemma basis_emb_compact_bot [simp]: "basis_emb compact_bot = 0" 644by (simp add: basis_emb.simps) 645 646lemma fin1: "finite {y. place y < place x \<and> x \<sqsubseteq> y}" 647apply (subst Collect_conj_eq) 648apply (rule finite_Int) 649apply (rule disjI1) 650apply (subgoal_tac "finite (place -` {n. n < place x})", simp) 651apply (rule finite_vimageI [OF _ inj_place]) 652apply (simp add: lessThan_def [symmetric]) 653done 654 655lemma fin2: "finite (basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y})" 656by (rule finite_imageI [OF fin1]) 657 658lemma rank_place_mono: 659 "\<lbrakk>place x < place y; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> rank x < rank y" 660apply (rule linorder_cases, assumption) 661apply (simp add: place_def cong: rank_lt_cong rank_eq_cong) 662apply (drule choose_pos_lessD) 663apply (rule finite_rank_eq) 664apply (simp add: rank_eq_def) 665apply (simp add: rank_eq_def) 666apply simp 667apply (drule place_rank_mono, simp) 668done 669 670lemma basis_emb_mono: 671 "x \<sqsubseteq> y \<Longrightarrow> ubasis_le (basis_emb x) (basis_emb y)" 672proof (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 704qed 705 706lemma 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]) 715done 716 717definition 718 basis_prj :: "ubasis \<Rightarrow> 'a compact_basis" 719where 720 "basis_prj x = inv basis_emb 721 (ubasis_until (\<lambda>x. x \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> ubasis)) x)" 722 723lemma basis_prj_basis_emb: "\<And>x. basis_prj (basis_emb x) = x" 724unfolding 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) 729done 730 731lemma 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)" 734unfolding basis_prj_def by simp 735 736lemma basis_prj_0: "basis_prj 0 = compact_bot" 737apply (subst basis_emb_compact_bot [symmetric]) 738apply (rule basis_prj_basis_emb) 739done 740 741lemma 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}" 745apply (cases "x = compact_bot", simp) 746apply (simp add: basis_emb.simps [of x]) 747apply (simp add: fin2) 748done 749 750lemma basis_prj_mono: "ubasis_le a b \<Longrightarrow> basis_prj a \<sqsubseteq> basis_prj b" 751proof (induct a b rule: ubasis_le.induct) 752 case (ubasis_le_refl a) show ?case by (rule below_refl) 753next 754 case (ubasis_le_trans a b c) thus ?case by - (rule below_trans) 755next 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 765next 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 774qed 775 776lemma basis_emb_prj_less: "ubasis_le (basis_emb (basis_prj x)) x" 777unfolding 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) 783done 784 785lemma ideal_completion: 786 "ideal_completion below Rep_compact_basis (approximants :: 'a \<Rightarrow> _)" 787proof 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 814next 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) 820next 821 fix a :: "'a compact_basis" 822 show "approximants (Rep_compact_basis a) = {b. b \<sqsubseteq> a}" 823 unfolding approximants_def compact_le_def .. 824next 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) 834next 835 show "\<exists>f::'a compact_basis \<Rightarrow> nat. inj f" 836 by (rule exI, rule inj_place) 837qed 838 839end 840 841interpretation compact_basis: 842 ideal_completion below Rep_compact_basis 843 "approximants :: 'a::bifinite \<Rightarrow> 'a compact_basis set" 844proof - 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) 851qed 852 853subsubsection \<open>EP-pair from any bifinite domain into \emph{udom}\<close> 854 855context bifinite_approx_chain begin 856 857definition 858 udom_emb :: "'a \<rightarrow> udom" 859where 860 "udom_emb = compact_basis.extension (\<lambda>x. udom_principal (basis_emb x))" 861 862definition 863 udom_prj :: "udom \<rightarrow> 'a" 864where 865 "udom_prj = udom.extension (\<lambda>x. Rep_compact_basis (basis_prj x))" 866 867lemma udom_emb_principal: 868 "udom_emb\<cdot>(Rep_compact_basis x) = udom_principal (basis_emb x)" 869unfolding udom_emb_def 870apply (rule compact_basis.extension_principal) 871apply (rule udom.principal_mono) 872apply (erule basis_emb_mono) 873done 874 875lemma udom_prj_principal: 876 "udom_prj\<cdot>(udom_principal x) = Rep_compact_basis (basis_prj x)" 877unfolding udom_prj_def 878apply (rule udom.extension_principal) 879apply (rule compact_basis.principal_mono) 880apply (erule basis_prj_mono) 881done 882 883lemma 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) 891done 892 893end 894 895abbreviation "udom_emb \<equiv> bifinite_approx_chain.udom_emb" 896abbreviation "udom_prj \<equiv> bifinite_approx_chain.udom_prj" 897 898lemmas ep_pair_udom = 899 bifinite_approx_chain.ep_pair_udom [unfolded bifinite_approx_chain_def] 900 901subsection \<open>Chain of approx functions for type \emph{udom}\<close> 902 903definition 904 udom_approx :: "nat \<Rightarrow> udom \<rightarrow> udom" 905where 906 "udom_approx i = 907 udom.extension (\<lambda>x. udom_principal (ubasis_until (\<lambda>y. y \<le> i) x))" 908 909lemma 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)" 913apply (rule udom.principal_mono) 914apply (rule ubasis_until_mono) 915apply (frule (2) order_less_le_trans [OF node_gt2]) 916apply (erule order_less_imp_le) 917apply assumption 918done 919 920lemma adm_mem_finite: "\<lbrakk>cont f; finite S\<rbrakk> \<Longrightarrow> adm (\<lambda>x. f x \<in> S)" 921by (erule adm_subst, induct set: finite, simp_all) 922 923lemma udom_approx_principal: 924 "udom_approx i\<cdot>(udom_principal x) = 925 udom_principal (ubasis_until (\<lambda>y. y \<le> i) x)" 926unfolding udom_approx_def 927apply (rule udom.extension_principal) 928apply (erule udom_approx_mono) 929done 930 931lemma finite_deflation_udom_approx: "finite_deflation (udom_approx i)" 932proof 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) 936next 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) 940next 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 953qed 954 955interpretation udom_approx: finite_deflation "udom_approx i" 956by (rule finite_deflation_udom_approx) 957 958lemma chain_udom_approx [simp]: "chain (\<lambda>i. udom_approx i)" 959unfolding udom_approx_def 960apply (rule chainI) 961apply (rule udom.extension_mono) 962apply (erule udom_approx_mono) 963apply (erule udom_approx_mono) 964apply (rule udom.principal_mono) 965apply (rule ubasis_until_chain, simp) 966done 967 968lemma lub_udom_approx [simp]: "(\<Squnion>i. udom_approx i) = ID" 969apply (rule cfun_eqI, simp add: contlub_cfun_fun) 970apply (rule below_antisym) 971apply (rule lub_below) 972apply (simp) 973apply (rule udom_approx.below) 974apply (rule_tac x=x in udom.principal_induct) 975apply (simp add: lub_distribs) 976apply (rule_tac i=a in below_lub) 977apply simp 978apply (simp add: udom_approx_principal) 979apply (simp add: ubasis_until_same ubasis_le_refl) 980done 981 982lemma udom_approx [simp]: "approx_chain udom_approx" 983proof 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) 988qed 989 990instance udom :: bifinite 991 by standard (fast intro: udom_approx) 992 993hide_const (open) node 994 995notation binomial (infixl "choose" 65) 996 997end 998