1(* Title: HOL/HOLCF/Completion.thy 2 Author: Brian Huffman 3*) 4 5section \<open>Defining algebraic domains by ideal completion\<close> 6 7theory Completion 8imports Cfun 9begin 10 11subsection \<open>Ideals over a preorder\<close> 12 13locale preorder = 14 fixes r :: "'a::type \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<preceq>" 50) 15 assumes r_refl: "x \<preceq> x" 16 assumes r_trans: "\<lbrakk>x \<preceq> y; y \<preceq> z\<rbrakk> \<Longrightarrow> x \<preceq> z" 17begin 18 19definition 20 ideal :: "'a set \<Rightarrow> bool" where 21 "ideal A = ((\<exists>x. x \<in> A) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. \<exists>z\<in>A. x \<preceq> z \<and> y \<preceq> z) \<and> 22 (\<forall>x y. x \<preceq> y \<longrightarrow> y \<in> A \<longrightarrow> x \<in> A))" 23 24lemma idealI: 25 assumes "\<exists>x. x \<in> A" 26 assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. x \<preceq> z \<and> y \<preceq> z" 27 assumes "\<And>x y. \<lbrakk>x \<preceq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A" 28 shows "ideal A" 29unfolding ideal_def using assms by fast 30 31lemma idealD1: 32 "ideal A \<Longrightarrow> \<exists>x. x \<in> A" 33unfolding ideal_def by fast 34 35lemma idealD2: 36 "\<lbrakk>ideal A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. x \<preceq> z \<and> y \<preceq> z" 37unfolding ideal_def by fast 38 39lemma idealD3: 40 "\<lbrakk>ideal A; x \<preceq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A" 41unfolding ideal_def by fast 42 43lemma ideal_principal: "ideal {x. x \<preceq> z}" 44 apply (rule idealI) 45 apply (rule exI [where x = z]) 46 apply (fast intro: r_refl) 47 apply (rule bexI [where x = z], fast) 48 apply (fast intro: r_refl) 49 apply (fast intro: r_trans) 50 done 51 52lemma ex_ideal: "\<exists>A. A \<in> {A. ideal A}" 53by (fast intro: ideal_principal) 54 55text \<open>The set of ideals is a cpo\<close> 56 57lemma ideal_UN: 58 fixes A :: "nat \<Rightarrow> 'a set" 59 assumes ideal_A: "\<And>i. ideal (A i)" 60 assumes chain_A: "\<And>i j. i \<le> j \<Longrightarrow> A i \<subseteq> A j" 61 shows "ideal (\<Union>i. A i)" 62 apply (rule idealI) 63 using idealD1 [OF ideal_A] apply fast 64 apply (clarify) 65 subgoal for i j 66 apply (drule subsetD [OF chain_A [OF max.cobounded1]]) 67 apply (drule subsetD [OF chain_A [OF max.cobounded2]]) 68 apply (drule (1) idealD2 [OF ideal_A]) 69 apply blast 70 done 71 apply clarify 72 apply (drule (1) idealD3 [OF ideal_A]) 73 apply fast 74 done 75 76lemma typedef_ideal_po: 77 fixes Abs :: "'a set \<Rightarrow> 'b::below" 78 assumes type: "type_definition Rep Abs {S. ideal S}" 79 assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y" 80 shows "OFCLASS('b, po_class)" 81 apply (intro_classes, unfold below) 82 apply (rule subset_refl) 83 apply (erule (1) subset_trans) 84 apply (rule type_definition.Rep_inject [OF type, THEN iffD1]) 85 apply (erule (1) subset_antisym) 86done 87 88lemma 89 fixes Abs :: "'a set \<Rightarrow> 'b::po" 90 assumes type: "type_definition Rep Abs {S. ideal S}" 91 assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y" 92 assumes S: "chain S" 93 shows typedef_ideal_lub: "range S <<| Abs (\<Union>i. Rep (S i))" 94 and typedef_ideal_rep_lub: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))" 95proof - 96 have 1: "ideal (\<Union>i. Rep (S i))" 97 apply (rule ideal_UN) 98 apply (rule type_definition.Rep [OF type, unfolded mem_Collect_eq]) 99 apply (subst below [symmetric]) 100 apply (erule chain_mono [OF S]) 101 done 102 hence 2: "Rep (Abs (\<Union>i. Rep (S i))) = (\<Union>i. Rep (S i))" 103 by (simp add: type_definition.Abs_inverse [OF type]) 104 show 3: "range S <<| Abs (\<Union>i. Rep (S i))" 105 apply (rule is_lubI) 106 apply (rule is_ubI) 107 apply (simp add: below 2, fast) 108 apply (simp add: below 2 is_ub_def, fast) 109 done 110 hence 4: "(\<Squnion>i. S i) = Abs (\<Union>i. Rep (S i))" 111 by (rule lub_eqI) 112 show 5: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))" 113 by (simp add: 4 2) 114qed 115 116lemma typedef_ideal_cpo: 117 fixes Abs :: "'a set \<Rightarrow> 'b::po" 118 assumes type: "type_definition Rep Abs {S. ideal S}" 119 assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y" 120 shows "OFCLASS('b, cpo_class)" 121 by standard (rule exI, erule typedef_ideal_lub [OF type below]) 122 123end 124 125interpretation below: preorder "below :: 'a::po \<Rightarrow> 'a \<Rightarrow> bool" 126apply unfold_locales 127apply (rule below_refl) 128apply (erule (1) below_trans) 129done 130 131subsection \<open>Lemmas about least upper bounds\<close> 132 133lemma is_ub_thelub_ex: "\<lbrakk>\<exists>u. S <<| u; x \<in> S\<rbrakk> \<Longrightarrow> x \<sqsubseteq> lub S" 134apply (erule exE, drule is_lub_lub) 135apply (drule is_lubD1) 136apply (erule (1) is_ubD) 137done 138 139lemma is_lub_thelub_ex: "\<lbrakk>\<exists>u. S <<| u; S <| x\<rbrakk> \<Longrightarrow> lub S \<sqsubseteq> x" 140by (erule exE, drule is_lub_lub, erule is_lubD2) 141 142 143subsection \<open>Locale for ideal completion\<close> 144 145hide_const (open) Filter.principal 146 147locale ideal_completion = preorder + 148 fixes principal :: "'a::type \<Rightarrow> 'b::cpo" 149 fixes rep :: "'b::cpo \<Rightarrow> 'a::type set" 150 assumes ideal_rep: "\<And>x. ideal (rep x)" 151 assumes rep_lub: "\<And>Y. chain Y \<Longrightarrow> rep (\<Squnion>i. Y i) = (\<Union>i. rep (Y i))" 152 assumes rep_principal: "\<And>a. rep (principal a) = {b. b \<preceq> a}" 153 assumes belowI: "\<And>x y. rep x \<subseteq> rep y \<Longrightarrow> x \<sqsubseteq> y" 154 assumes countable: "\<exists>f::'a \<Rightarrow> nat. inj f" 155begin 156 157lemma rep_mono: "x \<sqsubseteq> y \<Longrightarrow> rep x \<subseteq> rep y" 158apply (frule bin_chain) 159apply (drule rep_lub) 160apply (simp only: lub_eqI [OF is_lub_bin_chain]) 161apply (rule subsetI, rule UN_I [where a=0], simp_all) 162done 163 164lemma below_def: "x \<sqsubseteq> y \<longleftrightarrow> rep x \<subseteq> rep y" 165by (rule iffI [OF rep_mono belowI]) 166 167lemma principal_below_iff_mem_rep: "principal a \<sqsubseteq> x \<longleftrightarrow> a \<in> rep x" 168unfolding below_def rep_principal 169by (auto intro: r_refl elim: idealD3 [OF ideal_rep]) 170 171lemma principal_below_iff [simp]: "principal a \<sqsubseteq> principal b \<longleftrightarrow> a \<preceq> b" 172by (simp add: principal_below_iff_mem_rep rep_principal) 173 174lemma principal_eq_iff: "principal a = principal b \<longleftrightarrow> a \<preceq> b \<and> b \<preceq> a" 175unfolding po_eq_conv [where 'a='b] principal_below_iff .. 176 177lemma eq_iff: "x = y \<longleftrightarrow> rep x = rep y" 178unfolding po_eq_conv below_def by auto 179 180lemma principal_mono: "a \<preceq> b \<Longrightarrow> principal a \<sqsubseteq> principal b" 181by (simp only: principal_below_iff) 182 183lemma ch2ch_principal [simp]: 184 "\<forall>i. Y i \<preceq> Y (Suc i) \<Longrightarrow> chain (\<lambda>i. principal (Y i))" 185by (simp add: chainI principal_mono) 186 187subsubsection \<open>Principal ideals approximate all elements\<close> 188 189lemma compact_principal [simp]: "compact (principal a)" 190by (rule compactI2, simp add: principal_below_iff_mem_rep rep_lub) 191 192text \<open>Construct a chain whose lub is the same as a given ideal\<close> 193 194lemma obtain_principal_chain: 195 obtains Y where "\<forall>i. Y i \<preceq> Y (Suc i)" and "x = (\<Squnion>i. principal (Y i))" 196proof - 197 obtain count :: "'a \<Rightarrow> nat" where inj: "inj count" 198 using countable .. 199 define enum where "enum i = (THE a. count a = i)" for i 200 have enum_count [simp]: "\<And>x. enum (count x) = x" 201 unfolding enum_def by (simp add: inj_eq [OF inj]) 202 define a where "a = (LEAST i. enum i \<in> rep x)" 203 define b where "b i = (LEAST j. enum j \<in> rep x \<and> \<not> enum j \<preceq> enum i)" for i 204 define c where "c i j = (LEAST k. enum k \<in> rep x \<and> enum i \<preceq> enum k \<and> enum j \<preceq> enum k)" for i j 205 define P where "P i \<longleftrightarrow> (\<exists>j. enum j \<in> rep x \<and> \<not> enum j \<preceq> enum i)" for i 206 define X where "X = rec_nat a (\<lambda>n i. if P i then c i (b i) else i)" 207 have X_0: "X 0 = a" unfolding X_def by simp 208 have X_Suc: "\<And>n. X (Suc n) = (if P (X n) then c (X n) (b (X n)) else X n)" 209 unfolding X_def by simp 210 have a_mem: "enum a \<in> rep x" 211 unfolding a_def 212 apply (rule LeastI_ex) 213 apply (insert ideal_rep [of x]) 214 apply (drule idealD1) 215 apply (clarify) 216 subgoal for a by (rule exI [where x="count a"]) simp 217 done 218 have b: "\<And>i. P i \<Longrightarrow> enum i \<in> rep x 219 \<Longrightarrow> enum (b i) \<in> rep x \<and> \<not> enum (b i) \<preceq> enum i" 220 unfolding P_def b_def by (erule LeastI2_ex, simp) 221 have c: "\<And>i j. enum i \<in> rep x \<Longrightarrow> enum j \<in> rep x 222 \<Longrightarrow> enum (c i j) \<in> rep x \<and> enum i \<preceq> enum (c i j) \<and> enum j \<preceq> enum (c i j)" 223 unfolding c_def 224 apply (drule (1) idealD2 [OF ideal_rep], clarify) 225 subgoal for \<dots> z by (rule LeastI2 [where a="count z"], simp, simp) 226 done 227 have X_mem: "enum (X n) \<in> rep x" for n 228 proof (induct n) 229 case 0 230 then show ?case by (simp add: X_0 a_mem) 231 next 232 case (Suc n) 233 with b c show ?case by (auto simp: X_Suc) 234 qed 235 have X_chain: "\<And>n. enum (X n) \<preceq> enum (X (Suc n))" 236 apply (clarsimp simp add: X_Suc r_refl) 237 apply (simp add: b c X_mem) 238 done 239 have less_b: "\<And>n i. n < b i \<Longrightarrow> enum n \<in> rep x \<Longrightarrow> enum n \<preceq> enum i" 240 unfolding b_def by (drule not_less_Least, simp) 241 have X_covers: "\<forall>k\<le>n. enum k \<in> rep x \<longrightarrow> enum k \<preceq> enum (X n)" for n 242 proof (induct n) 243 case 0 244 then show ?case 245 apply (clarsimp simp add: X_0 a_def) 246 apply (drule Least_le [where k=0], simp add: r_refl) 247 done 248 next 249 case (Suc n) 250 then show ?case 251 apply clarsimp 252 apply (erule le_SucE) 253 apply (rule r_trans [OF _ X_chain], simp) 254 apply (cases "P (X n)", simp add: X_Suc) 255 apply (rule linorder_cases [where x="b (X n)" and y="Suc n"]) 256 apply (simp only: less_Suc_eq_le) 257 apply (drule spec, drule (1) mp, simp add: b X_mem) 258 apply (simp add: c X_mem) 259 apply (drule (1) less_b) 260 apply (erule r_trans) 261 apply (simp add: b c X_mem) 262 apply (simp add: X_Suc) 263 apply (simp add: P_def) 264 done 265 qed 266 have 1: "\<forall>i. enum (X i) \<preceq> enum (X (Suc i))" 267 by (simp add: X_chain) 268 have "x = (\<Squnion>n. principal (enum (X n)))" 269 apply (simp add: eq_iff rep_lub 1 rep_principal) 270 apply auto 271 subgoal for a 272 apply (subgoal_tac "\<exists>i. a = enum i", erule exE) 273 apply (rule_tac x=i in exI, simp add: X_covers) 274 apply (rule_tac x="count a" in exI, simp) 275 done 276 subgoal 277 apply (erule idealD3 [OF ideal_rep]) 278 apply (rule X_mem) 279 done 280 done 281 with 1 show ?thesis .. 282qed 283 284lemma principal_induct: 285 assumes adm: "adm P" 286 assumes P: "\<And>a. P (principal a)" 287 shows "P x" 288apply (rule obtain_principal_chain [of x]) 289apply (simp add: admD [OF adm] P) 290done 291 292lemma compact_imp_principal: "compact x \<Longrightarrow> \<exists>a. x = principal a" 293apply (rule obtain_principal_chain [of x]) 294apply (drule adm_compact_neq [OF _ cont_id]) 295apply (subgoal_tac "chain (\<lambda>i. principal (Y i))") 296apply (drule (2) admD2, fast, simp) 297done 298 299subsection \<open>Defining functions in terms of basis elements\<close> 300 301definition 302 extension :: "('a::type \<Rightarrow> 'c::cpo) \<Rightarrow> 'b \<rightarrow> 'c" where 303 "extension = (\<lambda>f. (\<Lambda> x. lub (f ` rep x)))" 304 305lemma extension_lemma: 306 fixes f :: "'a::type \<Rightarrow> 'c::cpo" 307 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" 308 shows "\<exists>u. f ` rep x <<| u" 309proof - 310 obtain Y where Y: "\<forall>i. Y i \<preceq> Y (Suc i)" 311 and x: "x = (\<Squnion>i. principal (Y i))" 312 by (rule obtain_principal_chain [of x]) 313 have chain: "chain (\<lambda>i. f (Y i))" 314 by (rule chainI, simp add: f_mono Y) 315 have rep_x: "rep x = (\<Union>n. {a. a \<preceq> Y n})" 316 by (simp add: x rep_lub Y rep_principal) 317 have "f ` rep x <<| (\<Squnion>n. f (Y n))" 318 apply (rule is_lubI) 319 apply (rule ub_imageI) 320 subgoal for a 321 apply (clarsimp simp add: rep_x) 322 apply (drule f_mono) 323 apply (erule below_lub [OF chain]) 324 done 325 apply (rule lub_below [OF chain]) 326 subgoal for \<dots> n 327 apply (drule ub_imageD [where x="Y n"]) 328 apply (simp add: rep_x, fast intro: r_refl) 329 apply assumption 330 done 331 done 332 then show ?thesis .. 333qed 334 335lemma extension_beta: 336 fixes f :: "'a::type \<Rightarrow> 'c::cpo" 337 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" 338 shows "extension f\<cdot>x = lub (f ` rep x)" 339unfolding extension_def 340proof (rule beta_cfun) 341 have lub: "\<And>x. \<exists>u. f ` rep x <<| u" 342 using f_mono by (rule extension_lemma) 343 show cont: "cont (\<lambda>x. lub (f ` rep x))" 344 apply (rule contI2) 345 apply (rule monofunI) 346 apply (rule is_lub_thelub_ex [OF lub ub_imageI]) 347 apply (rule is_ub_thelub_ex [OF lub imageI]) 348 apply (erule (1) subsetD [OF rep_mono]) 349 apply (rule is_lub_thelub_ex [OF lub ub_imageI]) 350 apply (simp add: rep_lub, clarify) 351 apply (erule rev_below_trans [OF is_ub_thelub]) 352 apply (erule is_ub_thelub_ex [OF lub imageI]) 353 done 354qed 355 356lemma extension_principal: 357 fixes f :: "'a::type \<Rightarrow> 'c::cpo" 358 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" 359 shows "extension f\<cdot>(principal a) = f a" 360apply (subst extension_beta, erule f_mono) 361apply (subst rep_principal) 362apply (rule lub_eqI) 363apply (rule is_lub_maximal) 364apply (rule ub_imageI) 365apply (simp add: f_mono) 366apply (rule imageI) 367apply (simp add: r_refl) 368done 369 370lemma extension_mono: 371 assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b" 372 assumes g_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> g a \<sqsubseteq> g b" 373 assumes below: "\<And>a. f a \<sqsubseteq> g a" 374 shows "extension f \<sqsubseteq> extension g" 375 apply (rule cfun_belowI) 376 apply (simp only: extension_beta f_mono g_mono) 377 apply (rule is_lub_thelub_ex) 378 apply (rule extension_lemma, erule f_mono) 379 apply (rule ub_imageI) 380 subgoal for x a 381 apply (rule below_trans [OF below]) 382 apply (rule is_ub_thelub_ex) 383 apply (rule extension_lemma, erule g_mono) 384 apply (erule imageI) 385 done 386 done 387 388lemma cont_extension: 389 assumes f_mono: "\<And>a b x. a \<preceq> b \<Longrightarrow> f x a \<sqsubseteq> f x b" 390 assumes f_cont: "\<And>a. cont (\<lambda>x. f x a)" 391 shows "cont (\<lambda>x. extension (\<lambda>a. f x a))" 392 apply (rule contI2) 393 apply (rule monofunI) 394 apply (rule extension_mono, erule f_mono, erule f_mono) 395 apply (erule cont2monofunE [OF f_cont]) 396 apply (rule cfun_belowI) 397 apply (rule principal_induct, simp) 398 apply (simp only: contlub_cfun_fun) 399 apply (simp only: extension_principal f_mono) 400 apply (simp add: cont2contlubE [OF f_cont]) 401done 402 403end 404 405lemma (in preorder) typedef_ideal_completion: 406 fixes Abs :: "'a set \<Rightarrow> 'b::cpo" 407 assumes type: "type_definition Rep Abs {S. ideal S}" 408 assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y" 409 assumes principal: "\<And>a. principal a = Abs {b. b \<preceq> a}" 410 assumes countable: "\<exists>f::'a \<Rightarrow> nat. inj f" 411 shows "ideal_completion r principal Rep" 412proof 413 interpret type_definition Rep Abs "{S. ideal S}" by fact 414 fix a b :: 'a and x y :: 'b and Y :: "nat \<Rightarrow> 'b" 415 show "ideal (Rep x)" 416 using Rep [of x] by simp 417 show "chain Y \<Longrightarrow> Rep (\<Squnion>i. Y i) = (\<Union>i. Rep (Y i))" 418 using type below by (rule typedef_ideal_rep_lub) 419 show "Rep (principal a) = {b. b \<preceq> a}" 420 by (simp add: principal Abs_inverse ideal_principal) 421 show "Rep x \<subseteq> Rep y \<Longrightarrow> x \<sqsubseteq> y" 422 by (simp only: below) 423 show "\<exists>f::'a \<Rightarrow> nat. inj f" 424 by (rule countable) 425qed 426 427end 428