src/HOLCF/Completion.thy
author huffman
Fri May 08 16:19:51 2009 -0700 (2009-05-08)
changeset 31076 99fe356cbbc2
parent 30729 461ee3e49ad3
child 39967 1c6dce3ef477
permissions -rw-r--r--
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
     1 (*  Title:      HOLCF/Completion.thy
     2     Author:     Brian Huffman
     3 *)
     4 
     5 header {* Defining bifinite domains by ideal completion *}
     6 
     7 theory Completion
     8 imports Bifinite
     9 begin
    10 
    11 subsection {* Ideals over a preorder *}
    12 
    13 locale 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"
    17 begin
    18 
    19 definition
    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 
    24 lemma 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"
    29 unfolding ideal_def using prems by fast
    30 
    31 lemma idealD1:
    32   "ideal A \<Longrightarrow> \<exists>x. x \<in> A"
    33 unfolding ideal_def by fast
    34 
    35 lemma idealD2:
    36   "\<lbrakk>ideal A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. x \<preceq> z \<and> y \<preceq> z"
    37 unfolding ideal_def by fast
    38 
    39 lemma idealD3:
    40   "\<lbrakk>ideal A; x \<preceq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A"
    41 unfolding ideal_def by fast
    42 
    43 lemma ideal_directed_finite:
    44   assumes A: "ideal A"
    45   shows "\<lbrakk>finite U; U \<subseteq> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. \<forall>x\<in>U. x \<preceq> z"
    46 apply (induct U set: finite)
    47 apply (simp add: idealD1 [OF A])
    48 apply (simp, clarify, rename_tac y)
    49 apply (drule (1) idealD2 [OF A])
    50 apply (clarify, erule_tac x=z in rev_bexI)
    51 apply (fast intro: r_trans)
    52 done
    53 
    54 lemma ideal_principal: "ideal {x. x \<preceq> z}"
    55 apply (rule idealI)
    56 apply (rule_tac x=z in exI)
    57 apply (fast intro: r_refl)
    58 apply (rule_tac x=z in bexI, fast)
    59 apply (fast intro: r_refl)
    60 apply (fast intro: r_trans)
    61 done
    62 
    63 lemma ex_ideal: "\<exists>A. ideal A"
    64 by (rule exI, rule ideal_principal)
    65 
    66 lemma directed_image_ideal:
    67   assumes A: "ideal A"
    68   assumes f: "\<And>x y. x \<preceq> y \<Longrightarrow> f x \<sqsubseteq> f y"
    69   shows "directed (f ` A)"
    70 apply (rule directedI)
    71 apply (cut_tac idealD1 [OF A], fast)
    72 apply (clarify, rename_tac a b)
    73 apply (drule (1) idealD2 [OF A])
    74 apply (clarify, rename_tac c)
    75 apply (rule_tac x="f c" in rev_bexI)
    76 apply (erule imageI)
    77 apply (simp add: f)
    78 done
    79 
    80 lemma lub_image_principal:
    81   assumes f: "\<And>x y. x \<preceq> y \<Longrightarrow> f x \<sqsubseteq> f y"
    82   shows "(\<Squnion>x\<in>{x. x \<preceq> y}. f x) = f y"
    83 apply (rule thelubI)
    84 apply (rule is_lub_maximal)
    85 apply (rule ub_imageI)
    86 apply (simp add: f)
    87 apply (rule imageI)
    88 apply (simp add: r_refl)
    89 done
    90 
    91 text {* The set of ideals is a cpo *}
    92 
    93 lemma ideal_UN:
    94   fixes A :: "nat \<Rightarrow> 'a set"
    95   assumes ideal_A: "\<And>i. ideal (A i)"
    96   assumes chain_A: "\<And>i j. i \<le> j \<Longrightarrow> A i \<subseteq> A j"
    97   shows "ideal (\<Union>i. A i)"
    98  apply (rule idealI)
    99    apply (cut_tac idealD1 [OF ideal_A], fast)
   100   apply (clarify, rename_tac i j)
   101   apply (drule subsetD [OF chain_A [OF le_maxI1]])
   102   apply (drule subsetD [OF chain_A [OF le_maxI2]])
   103   apply (drule (1) idealD2 [OF ideal_A])
   104   apply blast
   105  apply clarify
   106  apply (drule (1) idealD3 [OF ideal_A])
   107  apply fast
   108 done
   109 
   110 lemma typedef_ideal_po:
   111   fixes Abs :: "'a set \<Rightarrow> 'b::below"
   112   assumes type: "type_definition Rep Abs {S. ideal S}"
   113   assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"
   114   shows "OFCLASS('b, po_class)"
   115  apply (intro_classes, unfold below)
   116    apply (rule subset_refl)
   117   apply (erule (1) subset_trans)
   118  apply (rule type_definition.Rep_inject [OF type, THEN iffD1])
   119  apply (erule (1) subset_antisym)
   120 done
   121 
   122 lemma
   123   fixes Abs :: "'a set \<Rightarrow> 'b::po"
   124   assumes type: "type_definition Rep Abs {S. ideal S}"
   125   assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"
   126   assumes S: "chain S"
   127   shows typedef_ideal_lub: "range S <<| Abs (\<Union>i. Rep (S i))"
   128     and typedef_ideal_rep_contlub: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))"
   129 proof -
   130   have 1: "ideal (\<Union>i. Rep (S i))"
   131     apply (rule ideal_UN)
   132      apply (rule type_definition.Rep [OF type, unfolded mem_Collect_eq])
   133     apply (subst below [symmetric])
   134     apply (erule chain_mono [OF S])
   135     done
   136   hence 2: "Rep (Abs (\<Union>i. Rep (S i))) = (\<Union>i. Rep (S i))"
   137     by (simp add: type_definition.Abs_inverse [OF type])
   138   show 3: "range S <<| Abs (\<Union>i. Rep (S i))"
   139     apply (rule is_lubI)
   140      apply (rule is_ubI)
   141      apply (simp add: below 2, fast)
   142     apply (simp add: below 2 is_ub_def, fast)
   143     done
   144   hence 4: "(\<Squnion>i. S i) = Abs (\<Union>i. Rep (S i))"
   145     by (rule thelubI)
   146   show 5: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))"
   147     by (simp add: 4 2)
   148 qed
   149 
   150 lemma typedef_ideal_cpo:
   151   fixes Abs :: "'a set \<Rightarrow> 'b::po"
   152   assumes type: "type_definition Rep Abs {S. ideal S}"
   153   assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"
   154   shows "OFCLASS('b, cpo_class)"
   155 by (default, rule exI, erule typedef_ideal_lub [OF type below])
   156 
   157 end
   158 
   159 interpretation below: preorder "below :: 'a::po \<Rightarrow> 'a \<Rightarrow> bool"
   160 apply unfold_locales
   161 apply (rule below_refl)
   162 apply (erule (1) below_trans)
   163 done
   164 
   165 subsection {* Lemmas about least upper bounds *}
   166 
   167 lemma finite_directed_contains_lub:
   168   "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> \<exists>u\<in>S. S <<| u"
   169 apply (drule (1) directed_finiteD, rule subset_refl)
   170 apply (erule bexE)
   171 apply (rule rev_bexI, assumption)
   172 apply (erule (1) is_lub_maximal)
   173 done
   174 
   175 lemma lub_finite_directed_in_self:
   176   "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> lub S \<in> S"
   177 apply (drule (1) finite_directed_contains_lub, clarify)
   178 apply (drule thelubI, simp)
   179 done
   180 
   181 lemma finite_directed_has_lub: "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> \<exists>u. S <<| u"
   182 by (drule (1) finite_directed_contains_lub, fast)
   183 
   184 lemma is_ub_thelub0: "\<lbrakk>\<exists>u. S <<| u; x \<in> S\<rbrakk> \<Longrightarrow> x \<sqsubseteq> lub S"
   185 apply (erule exE, drule lubI)
   186 apply (drule is_lubD1)
   187 apply (erule (1) is_ubD)
   188 done
   189 
   190 lemma is_lub_thelub0: "\<lbrakk>\<exists>u. S <<| u; S <| x\<rbrakk> \<Longrightarrow> lub S \<sqsubseteq> x"
   191 by (erule exE, drule lubI, erule is_lub_lub)
   192 
   193 subsection {* Locale for ideal completion *}
   194 
   195 locale basis_take = preorder +
   196   fixes take :: "nat \<Rightarrow> 'a::type \<Rightarrow> 'a"
   197   assumes take_less: "take n a \<preceq> a"
   198   assumes take_take: "take n (take n a) = take n a"
   199   assumes take_mono: "a \<preceq> b \<Longrightarrow> take n a \<preceq> take n b"
   200   assumes take_chain: "take n a \<preceq> take (Suc n) a"
   201   assumes finite_range_take: "finite (range (take n))"
   202   assumes take_covers: "\<exists>n. take n a = a"
   203 begin
   204 
   205 lemma take_chain_less: "m < n \<Longrightarrow> take m a \<preceq> take n a"
   206 by (erule less_Suc_induct, rule take_chain, erule (1) r_trans)
   207 
   208 lemma take_chain_le: "m \<le> n \<Longrightarrow> take m a \<preceq> take n a"
   209 by (cases "m = n", simp add: r_refl, simp add: take_chain_less)
   210 
   211 end
   212 
   213 locale ideal_completion = basis_take +
   214   fixes principal :: "'a::type \<Rightarrow> 'b::cpo"
   215   fixes rep :: "'b::cpo \<Rightarrow> 'a::type set"
   216   assumes ideal_rep: "\<And>x. preorder.ideal r (rep x)"
   217   assumes rep_contlub: "\<And>Y. chain Y \<Longrightarrow> rep (\<Squnion>i. Y i) = (\<Union>i. rep (Y i))"
   218   assumes rep_principal: "\<And>a. rep (principal a) = {b. b \<preceq> a}"
   219   assumes subset_repD: "\<And>x y. rep x \<subseteq> rep y \<Longrightarrow> x \<sqsubseteq> y"
   220 begin
   221 
   222 lemma finite_take_rep: "finite (take n ` rep x)"
   223 by (rule finite_subset [OF image_mono [OF subset_UNIV] finite_range_take])
   224 
   225 lemma rep_mono: "x \<sqsubseteq> y \<Longrightarrow> rep x \<subseteq> rep y"
   226 apply (frule bin_chain)
   227 apply (drule rep_contlub)
   228 apply (simp only: thelubI [OF lub_bin_chain])
   229 apply (rule subsetI, rule UN_I [where a=0], simp_all)
   230 done
   231 
   232 lemma below_def: "x \<sqsubseteq> y \<longleftrightarrow> rep x \<subseteq> rep y"
   233 by (rule iffI [OF rep_mono subset_repD])
   234 
   235 lemma rep_eq: "rep x = {a. principal a \<sqsubseteq> x}"
   236 unfolding below_def rep_principal
   237 apply safe
   238 apply (erule (1) idealD3 [OF ideal_rep])
   239 apply (erule subsetD, simp add: r_refl)
   240 done
   241 
   242 lemma mem_rep_iff_principal_below: "a \<in> rep x \<longleftrightarrow> principal a \<sqsubseteq> x"
   243 by (simp add: rep_eq)
   244 
   245 lemma principal_below_iff_mem_rep: "principal a \<sqsubseteq> x \<longleftrightarrow> a \<in> rep x"
   246 by (simp add: rep_eq)
   247 
   248 lemma principal_below_iff [simp]: "principal a \<sqsubseteq> principal b \<longleftrightarrow> a \<preceq> b"
   249 by (simp add: principal_below_iff_mem_rep rep_principal)
   250 
   251 lemma principal_eq_iff: "principal a = principal b \<longleftrightarrow> a \<preceq> b \<and> b \<preceq> a"
   252 unfolding po_eq_conv [where 'a='b] principal_below_iff ..
   253 
   254 lemma repD: "a \<in> rep x \<Longrightarrow> principal a \<sqsubseteq> x"
   255 by (simp add: rep_eq)
   256 
   257 lemma principal_mono: "a \<preceq> b \<Longrightarrow> principal a \<sqsubseteq> principal b"
   258 by (simp only: principal_below_iff)
   259 
   260 lemma belowI: "(\<And>a. principal a \<sqsubseteq> x \<Longrightarrow> principal a \<sqsubseteq> u) \<Longrightarrow> x \<sqsubseteq> u"
   261 unfolding principal_below_iff_mem_rep
   262 by (simp add: below_def subset_eq)
   263 
   264 lemma lub_principal_rep: "principal ` rep x <<| x"
   265 apply (rule is_lubI)
   266 apply (rule ub_imageI)
   267 apply (erule repD)
   268 apply (subst below_def)
   269 apply (rule subsetI)
   270 apply (drule (1) ub_imageD)
   271 apply (simp add: rep_eq)
   272 done
   273 
   274 subsection {* Defining functions in terms of basis elements *}
   275 
   276 definition
   277   basis_fun :: "('a::type \<Rightarrow> 'c::cpo) \<Rightarrow> 'b \<rightarrow> 'c" where
   278   "basis_fun = (\<lambda>f. (\<Lambda> x. lub (f ` rep x)))"
   279 
   280 lemma basis_fun_lemma0:
   281   fixes f :: "'a::type \<Rightarrow> 'c::cpo"
   282   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
   283   shows "\<exists>u. f ` take i ` rep x <<| u"
   284 apply (rule finite_directed_has_lub)
   285 apply (rule finite_imageI)
   286 apply (rule finite_take_rep)
   287 apply (subst image_image)
   288 apply (rule directed_image_ideal)
   289 apply (rule ideal_rep)
   290 apply (rule f_mono)
   291 apply (erule take_mono)
   292 done
   293 
   294 lemma basis_fun_lemma1:
   295   fixes f :: "'a::type \<Rightarrow> 'c::cpo"
   296   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
   297   shows "chain (\<lambda>i. lub (f ` take i ` rep x))"
   298  apply (rule chainI)
   299  apply (rule is_lub_thelub0)
   300   apply (rule basis_fun_lemma0, erule f_mono)
   301  apply (rule is_ubI, clarsimp, rename_tac a)
   302  apply (rule below_trans [OF f_mono [OF take_chain]])
   303  apply (rule is_ub_thelub0)
   304   apply (rule basis_fun_lemma0, erule f_mono)
   305  apply simp
   306 done
   307 
   308 lemma basis_fun_lemma2:
   309   fixes f :: "'a::type \<Rightarrow> 'c::cpo"
   310   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
   311   shows "f ` rep x <<| (\<Squnion>i. lub (f ` take i ` rep x))"
   312  apply (rule is_lubI)
   313  apply (rule ub_imageI, rename_tac a)
   314   apply (cut_tac a=a in take_covers, erule exE, rename_tac i)
   315   apply (erule subst)
   316   apply (rule rev_below_trans)
   317    apply (rule_tac x=i in is_ub_thelub)
   318    apply (rule basis_fun_lemma1, erule f_mono)
   319   apply (rule is_ub_thelub0)
   320    apply (rule basis_fun_lemma0, erule f_mono)
   321   apply simp
   322  apply (rule is_lub_thelub [OF _ ub_rangeI])
   323   apply (rule basis_fun_lemma1, erule f_mono)
   324  apply (rule is_lub_thelub0)
   325   apply (rule basis_fun_lemma0, erule f_mono)
   326  apply (rule is_ubI, clarsimp, rename_tac a)
   327  apply (rule below_trans [OF f_mono [OF take_less]])
   328  apply (erule (1) ub_imageD)
   329 done
   330 
   331 lemma basis_fun_lemma:
   332   fixes f :: "'a::type \<Rightarrow> 'c::cpo"
   333   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
   334   shows "\<exists>u. f ` rep x <<| u"
   335 by (rule exI, rule basis_fun_lemma2, erule f_mono)
   336 
   337 lemma basis_fun_beta:
   338   fixes f :: "'a::type \<Rightarrow> 'c::cpo"
   339   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
   340   shows "basis_fun f\<cdot>x = lub (f ` rep x)"
   341 unfolding basis_fun_def
   342 proof (rule beta_cfun)
   343   have lub: "\<And>x. \<exists>u. f ` rep x <<| u"
   344     using f_mono by (rule basis_fun_lemma)
   345   show cont: "cont (\<lambda>x. lub (f ` rep x))"
   346     apply (rule contI2)
   347      apply (rule monofunI)
   348      apply (rule is_lub_thelub0 [OF lub ub_imageI])
   349      apply (rule is_ub_thelub0 [OF lub imageI])
   350      apply (erule (1) subsetD [OF rep_mono])
   351     apply (rule is_lub_thelub0 [OF lub ub_imageI])
   352     apply (simp add: rep_contlub, clarify)
   353     apply (erule rev_below_trans [OF is_ub_thelub])
   354     apply (erule is_ub_thelub0 [OF lub imageI])
   355     done
   356 qed
   357 
   358 lemma basis_fun_principal:
   359   fixes f :: "'a::type \<Rightarrow> 'c::cpo"
   360   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
   361   shows "basis_fun f\<cdot>(principal a) = f a"
   362 apply (subst basis_fun_beta, erule f_mono)
   363 apply (subst rep_principal)
   364 apply (rule lub_image_principal, erule f_mono)
   365 done
   366 
   367 lemma basis_fun_mono:
   368   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
   369   assumes g_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> g a \<sqsubseteq> g b"
   370   assumes below: "\<And>a. f a \<sqsubseteq> g a"
   371   shows "basis_fun f \<sqsubseteq> basis_fun g"
   372  apply (rule below_cfun_ext)
   373  apply (simp only: basis_fun_beta f_mono g_mono)
   374  apply (rule is_lub_thelub0)
   375   apply (rule basis_fun_lemma, erule f_mono)
   376  apply (rule ub_imageI, rename_tac a)
   377  apply (rule below_trans [OF below])
   378  apply (rule is_ub_thelub0)
   379   apply (rule basis_fun_lemma, erule g_mono)
   380  apply (erule imageI)
   381 done
   382 
   383 lemma compact_principal [simp]: "compact (principal a)"
   384 by (rule compactI2, simp add: principal_below_iff_mem_rep rep_contlub)
   385 
   386 subsection {* Bifiniteness of ideal completions *}
   387 
   388 definition
   389   completion_approx :: "nat \<Rightarrow> 'b \<rightarrow> 'b" where
   390   "completion_approx = (\<lambda>i. basis_fun (\<lambda>a. principal (take i a)))"
   391 
   392 lemma completion_approx_beta:
   393   "completion_approx i\<cdot>x = (\<Squnion>a\<in>rep x. principal (take i a))"
   394 unfolding completion_approx_def
   395 by (simp add: basis_fun_beta principal_mono take_mono)
   396 
   397 lemma completion_approx_principal:
   398   "completion_approx i\<cdot>(principal a) = principal (take i a)"
   399 unfolding completion_approx_def
   400 by (simp add: basis_fun_principal principal_mono take_mono)
   401 
   402 lemma chain_completion_approx: "chain completion_approx"
   403 unfolding completion_approx_def
   404 apply (rule chainI)
   405 apply (rule basis_fun_mono)
   406 apply (erule principal_mono [OF take_mono])
   407 apply (erule principal_mono [OF take_mono])
   408 apply (rule principal_mono [OF take_chain])
   409 done
   410 
   411 lemma lub_completion_approx: "(\<Squnion>i. completion_approx i\<cdot>x) = x"
   412 unfolding completion_approx_beta
   413  apply (subst image_image [where f=principal, symmetric])
   414  apply (rule unique_lub [OF _ lub_principal_rep])
   415  apply (rule basis_fun_lemma2, erule principal_mono)
   416 done
   417 
   418 lemma completion_approx_eq_principal:
   419   "\<exists>a\<in>rep x. completion_approx i\<cdot>x = principal (take i a)"
   420 unfolding completion_approx_beta
   421  apply (subst image_image [where f=principal, symmetric])
   422  apply (subgoal_tac "finite (principal ` take i ` rep x)")
   423   apply (subgoal_tac "directed (principal ` take i ` rep x)")
   424    apply (drule (1) lub_finite_directed_in_self, fast)
   425   apply (subst image_image)
   426   apply (rule directed_image_ideal)
   427    apply (rule ideal_rep)
   428   apply (erule principal_mono [OF take_mono])
   429  apply (rule finite_imageI)
   430  apply (rule finite_take_rep)
   431 done
   432 
   433 lemma completion_approx_idem:
   434   "completion_approx i\<cdot>(completion_approx i\<cdot>x) = completion_approx i\<cdot>x"
   435 using completion_approx_eq_principal [where i=i and x=x]
   436 by (auto simp add: completion_approx_principal take_take)
   437 
   438 lemma finite_fixes_completion_approx:
   439   "finite {x. completion_approx i\<cdot>x = x}" (is "finite ?S")
   440 apply (subgoal_tac "?S \<subseteq> principal ` range (take i)")
   441 apply (erule finite_subset)
   442 apply (rule finite_imageI)
   443 apply (rule finite_range_take)
   444 apply (clarify, erule subst)
   445 apply (cut_tac x=x and i=i in completion_approx_eq_principal)
   446 apply fast
   447 done
   448 
   449 lemma principal_induct:
   450   assumes adm: "adm P"
   451   assumes P: "\<And>a. P (principal a)"
   452   shows "P x"
   453  apply (subgoal_tac "P (\<Squnion>i. completion_approx i\<cdot>x)")
   454  apply (simp add: lub_completion_approx)
   455  apply (rule admD [OF adm])
   456   apply (simp add: chain_completion_approx)
   457  apply (cut_tac x=x and i=i in completion_approx_eq_principal)
   458  apply (clarify, simp add: P)
   459 done
   460 
   461 lemma principal_induct2:
   462   "\<lbrakk>\<And>y. adm (\<lambda>x. P x y); \<And>x. adm (\<lambda>y. P x y);
   463     \<And>a b. P (principal a) (principal b)\<rbrakk> \<Longrightarrow> P x y"
   464 apply (rule_tac x=y in spec)
   465 apply (rule_tac x=x in principal_induct, simp)
   466 apply (rule allI, rename_tac y)
   467 apply (rule_tac x=y in principal_induct, simp)
   468 apply simp
   469 done
   470 
   471 lemma compact_imp_principal: "compact x \<Longrightarrow> \<exists>a. x = principal a"
   472 apply (drule adm_compact_neq [OF _ cont_id])
   473 apply (drule admD2 [where Y="\<lambda>n. completion_approx n\<cdot>x"])
   474 apply (simp add: chain_completion_approx)
   475 apply (simp add: lub_completion_approx)
   476 apply (erule exE, erule ssubst)
   477 apply (cut_tac i=i and x=x in completion_approx_eq_principal)
   478 apply (clarify, erule exI)
   479 done
   480 
   481 end
   482 
   483 end