src/HOLCF/Completion.thy
 author huffman Tue Oct 12 06:20:05 2010 -0700 (2010-10-12) changeset 40006 116e94f9543b parent 40002 c5b5f7a3a3b1 child 40500 ee9c8d36318e permissions -rw-r--r--
remove unneeded lemmas from Fun_Cpo.thy
```     1 (*  Title:      HOLCF/Completion.thy
```
```     2     Author:     Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Defining algebraic domains by ideal completion *}
```
```     6
```
```     7 theory Completion
```
```     8 imports Cfun
```
```     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_principal: "ideal {x. x \<preceq> z}"
```
```    44 apply (rule idealI)
```
```    45 apply (rule_tac x=z in exI)
```
```    46 apply (fast intro: r_refl)
```
```    47 apply (rule_tac x=z in bexI, fast)
```
```    48 apply (fast intro: r_refl)
```
```    49 apply (fast intro: r_trans)
```
```    50 done
```
```    51
```
```    52 lemma ex_ideal: "\<exists>A. ideal A"
```
```    53 by (rule exI, rule ideal_principal)
```
```    54
```
```    55 lemma lub_image_principal:
```
```    56   assumes f: "\<And>x y. x \<preceq> y \<Longrightarrow> f x \<sqsubseteq> f y"
```
```    57   shows "(\<Squnion>x\<in>{x. x \<preceq> y}. f x) = f y"
```
```    58 apply (rule thelubI)
```
```    59 apply (rule is_lub_maximal)
```
```    60 apply (rule ub_imageI)
```
```    61 apply (simp add: f)
```
```    62 apply (rule imageI)
```
```    63 apply (simp add: r_refl)
```
```    64 done
```
```    65
```
```    66 text {* The set of ideals is a cpo *}
```
```    67
```
```    68 lemma ideal_UN:
```
```    69   fixes A :: "nat \<Rightarrow> 'a set"
```
```    70   assumes ideal_A: "\<And>i. ideal (A i)"
```
```    71   assumes chain_A: "\<And>i j. i \<le> j \<Longrightarrow> A i \<subseteq> A j"
```
```    72   shows "ideal (\<Union>i. A i)"
```
```    73  apply (rule idealI)
```
```    74    apply (cut_tac idealD1 [OF ideal_A], fast)
```
```    75   apply (clarify, rename_tac i j)
```
```    76   apply (drule subsetD [OF chain_A [OF le_maxI1]])
```
```    77   apply (drule subsetD [OF chain_A [OF le_maxI2]])
```
```    78   apply (drule (1) idealD2 [OF ideal_A])
```
```    79   apply blast
```
```    80  apply clarify
```
```    81  apply (drule (1) idealD3 [OF ideal_A])
```
```    82  apply fast
```
```    83 done
```
```    84
```
```    85 lemma typedef_ideal_po:
```
```    86   fixes Abs :: "'a set \<Rightarrow> 'b::below"
```
```    87   assumes type: "type_definition Rep Abs {S. ideal S}"
```
```    88   assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"
```
```    89   shows "OFCLASS('b, po_class)"
```
```    90  apply (intro_classes, unfold below)
```
```    91    apply (rule subset_refl)
```
```    92   apply (erule (1) subset_trans)
```
```    93  apply (rule type_definition.Rep_inject [OF type, THEN iffD1])
```
```    94  apply (erule (1) subset_antisym)
```
```    95 done
```
```    96
```
```    97 lemma
```
```    98   fixes Abs :: "'a set \<Rightarrow> 'b::po"
```
```    99   assumes type: "type_definition Rep Abs {S. ideal S}"
```
```   100   assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"
```
```   101   assumes S: "chain S"
```
```   102   shows typedef_ideal_lub: "range S <<| Abs (\<Union>i. Rep (S i))"
```
```   103     and typedef_ideal_rep_contlub: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))"
```
```   104 proof -
```
```   105   have 1: "ideal (\<Union>i. Rep (S i))"
```
```   106     apply (rule ideal_UN)
```
```   107      apply (rule type_definition.Rep [OF type, unfolded mem_Collect_eq])
```
```   108     apply (subst below [symmetric])
```
```   109     apply (erule chain_mono [OF S])
```
```   110     done
```
```   111   hence 2: "Rep (Abs (\<Union>i. Rep (S i))) = (\<Union>i. Rep (S i))"
```
```   112     by (simp add: type_definition.Abs_inverse [OF type])
```
```   113   show 3: "range S <<| Abs (\<Union>i. Rep (S i))"
```
```   114     apply (rule is_lubI)
```
```   115      apply (rule is_ubI)
```
```   116      apply (simp add: below 2, fast)
```
```   117     apply (simp add: below 2 is_ub_def, fast)
```
```   118     done
```
```   119   hence 4: "(\<Squnion>i. S i) = Abs (\<Union>i. Rep (S i))"
```
```   120     by (rule thelubI)
```
```   121   show 5: "Rep (\<Squnion>i. S i) = (\<Union>i. Rep (S i))"
```
```   122     by (simp add: 4 2)
```
```   123 qed
```
```   124
```
```   125 lemma typedef_ideal_cpo:
```
```   126   fixes Abs :: "'a set \<Rightarrow> 'b::po"
```
```   127   assumes type: "type_definition Rep Abs {S. ideal S}"
```
```   128   assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"
```
```   129   shows "OFCLASS('b, cpo_class)"
```
```   130 by (default, rule exI, erule typedef_ideal_lub [OF type below])
```
```   131
```
```   132 end
```
```   133
```
```   134 interpretation below: preorder "below :: 'a::po \<Rightarrow> 'a \<Rightarrow> bool"
```
```   135 apply unfold_locales
```
```   136 apply (rule below_refl)
```
```   137 apply (erule (1) below_trans)
```
```   138 done
```
```   139
```
```   140 subsection {* Lemmas about least upper bounds *}
```
```   141
```
```   142 lemma is_ub_thelub_ex: "\<lbrakk>\<exists>u. S <<| u; x \<in> S\<rbrakk> \<Longrightarrow> x \<sqsubseteq> lub S"
```
```   143 apply (erule exE, drule lubI)
```
```   144 apply (drule is_lubD1)
```
```   145 apply (erule (1) is_ubD)
```
```   146 done
```
```   147
```
```   148 lemma is_lub_thelub_ex: "\<lbrakk>\<exists>u. S <<| u; S <| x\<rbrakk> \<Longrightarrow> lub S \<sqsubseteq> x"
```
```   149 by (erule exE, drule lubI, erule is_lub_lub)
```
```   150
```
```   151 subsection {* Locale for ideal completion *}
```
```   152
```
```   153 locale ideal_completion = preorder +
```
```   154   fixes principal :: "'a::type \<Rightarrow> 'b::cpo"
```
```   155   fixes rep :: "'b::cpo \<Rightarrow> 'a::type set"
```
```   156   assumes ideal_rep: "\<And>x. ideal (rep x)"
```
```   157   assumes rep_contlub: "\<And>Y. chain Y \<Longrightarrow> rep (\<Squnion>i. Y i) = (\<Union>i. rep (Y i))"
```
```   158   assumes rep_principal: "\<And>a. rep (principal a) = {b. b \<preceq> a}"
```
```   159   assumes subset_repD: "\<And>x y. rep x \<subseteq> rep y \<Longrightarrow> x \<sqsubseteq> y"
```
```   160   assumes countable: "\<exists>f::'a \<Rightarrow> nat. inj f"
```
```   161 begin
```
```   162
```
```   163 lemma rep_mono: "x \<sqsubseteq> y \<Longrightarrow> rep x \<subseteq> rep y"
```
```   164 apply (frule bin_chain)
```
```   165 apply (drule rep_contlub)
```
```   166 apply (simp only: thelubI [OF lub_bin_chain])
```
```   167 apply (rule subsetI, rule UN_I [where a=0], simp_all)
```
```   168 done
```
```   169
```
```   170 lemma below_def: "x \<sqsubseteq> y \<longleftrightarrow> rep x \<subseteq> rep y"
```
```   171 by (rule iffI [OF rep_mono subset_repD])
```
```   172
```
```   173 lemma rep_eq: "rep x = {a. principal a \<sqsubseteq> x}"
```
```   174 unfolding below_def rep_principal
```
```   175 apply safe
```
```   176 apply (erule (1) idealD3 [OF ideal_rep])
```
```   177 apply (erule subsetD, simp add: r_refl)
```
```   178 done
```
```   179
```
```   180 lemma mem_rep_iff_principal_below: "a \<in> rep x \<longleftrightarrow> principal a \<sqsubseteq> x"
```
```   181 by (simp add: rep_eq)
```
```   182
```
```   183 lemma principal_below_iff_mem_rep: "principal a \<sqsubseteq> x \<longleftrightarrow> a \<in> rep x"
```
```   184 by (simp add: rep_eq)
```
```   185
```
```   186 lemma principal_below_iff [simp]: "principal a \<sqsubseteq> principal b \<longleftrightarrow> a \<preceq> b"
```
```   187 by (simp add: principal_below_iff_mem_rep rep_principal)
```
```   188
```
```   189 lemma principal_eq_iff: "principal a = principal b \<longleftrightarrow> a \<preceq> b \<and> b \<preceq> a"
```
```   190 unfolding po_eq_conv [where 'a='b] principal_below_iff ..
```
```   191
```
```   192 lemma eq_iff: "x = y \<longleftrightarrow> rep x = rep y"
```
```   193 unfolding po_eq_conv below_def by auto
```
```   194
```
```   195 lemma repD: "a \<in> rep x \<Longrightarrow> principal a \<sqsubseteq> x"
```
```   196 by (simp add: rep_eq)
```
```   197
```
```   198 lemma principal_mono: "a \<preceq> b \<Longrightarrow> principal a \<sqsubseteq> principal b"
```
```   199 by (simp only: principal_below_iff)
```
```   200
```
```   201 lemma ch2ch_principal [simp]:
```
```   202   "\<forall>i. Y i \<preceq> Y (Suc i) \<Longrightarrow> chain (\<lambda>i. principal (Y i))"
```
```   203 by (simp add: chainI principal_mono)
```
```   204
```
```   205 lemma lub_principal_rep: "principal ` rep x <<| x"
```
```   206 apply (rule is_lubI)
```
```   207 apply (rule ub_imageI)
```
```   208 apply (erule repD)
```
```   209 apply (subst below_def)
```
```   210 apply (rule subsetI)
```
```   211 apply (drule (1) ub_imageD)
```
```   212 apply (simp add: rep_eq)
```
```   213 done
```
```   214
```
```   215 subsubsection {* Principal ideals approximate all elements *}
```
```   216
```
```   217 lemma compact_principal [simp]: "compact (principal a)"
```
```   218 by (rule compactI2, simp add: principal_below_iff_mem_rep rep_contlub)
```
```   219
```
```   220 text {* Construct a chain whose lub is the same as a given ideal *}
```
```   221
```
```   222 lemma obtain_principal_chain:
```
```   223   obtains Y where "\<forall>i. Y i \<preceq> Y (Suc i)" and "x = (\<Squnion>i. principal (Y i))"
```
```   224 proof -
```
```   225   obtain count :: "'a \<Rightarrow> nat" where inj: "inj count"
```
```   226     using countable ..
```
```   227   def enum \<equiv> "\<lambda>i. THE a. count a = i"
```
```   228   have enum_count [simp]: "\<And>x. enum (count x) = x"
```
```   229     unfolding enum_def by (simp add: inj_eq [OF inj])
```
```   230   def a \<equiv> "LEAST i. enum i \<in> rep x"
```
```   231   def b \<equiv> "\<lambda>i. LEAST j. enum j \<in> rep x \<and> \<not> enum j \<preceq> enum i"
```
```   232   def c \<equiv> "\<lambda>i j. LEAST k. enum k \<in> rep x \<and> enum i \<preceq> enum k \<and> enum j \<preceq> enum k"
```
```   233   def P \<equiv> "\<lambda>i. \<exists>j. enum j \<in> rep x \<and> \<not> enum j \<preceq> enum i"
```
```   234   def X \<equiv> "nat_rec a (\<lambda>n i. if P i then c i (b i) else i)"
```
```   235   have X_0: "X 0 = a" unfolding X_def by simp
```
```   236   have X_Suc: "\<And>n. X (Suc n) = (if P (X n) then c (X n) (b (X n)) else X n)"
```
```   237     unfolding X_def by simp
```
```   238   have a_mem: "enum a \<in> rep x"
```
```   239     unfolding a_def
```
```   240     apply (rule LeastI_ex)
```
```   241     apply (cut_tac ideal_rep [of x])
```
```   242     apply (drule idealD1)
```
```   243     apply (clarify, rename_tac a)
```
```   244     apply (rule_tac x="count a" in exI, simp)
```
```   245     done
```
```   246   have b: "\<And>i. P i \<Longrightarrow> enum i \<in> rep x
```
```   247     \<Longrightarrow> enum (b i) \<in> rep x \<and> \<not> enum (b i) \<preceq> enum i"
```
```   248     unfolding P_def b_def by (erule LeastI2_ex, simp)
```
```   249   have c: "\<And>i j. enum i \<in> rep x \<Longrightarrow> enum j \<in> rep x
```
```   250     \<Longrightarrow> enum (c i j) \<in> rep x \<and> enum i \<preceq> enum (c i j) \<and> enum j \<preceq> enum (c i j)"
```
```   251     unfolding c_def
```
```   252     apply (drule (1) idealD2 [OF ideal_rep], clarify)
```
```   253     apply (rule_tac a="count z" in LeastI2, simp, simp)
```
```   254     done
```
```   255   have X_mem: "\<And>n. enum (X n) \<in> rep x"
```
```   256     apply (induct_tac n)
```
```   257     apply (simp add: X_0 a_mem)
```
```   258     apply (clarsimp simp add: X_Suc, rename_tac n)
```
```   259     apply (simp add: b c)
```
```   260     done
```
```   261   have X_chain: "\<And>n. enum (X n) \<preceq> enum (X (Suc n))"
```
```   262     apply (clarsimp simp add: X_Suc r_refl)
```
```   263     apply (simp add: b c X_mem)
```
```   264     done
```
```   265   have less_b: "\<And>n i. n < b i \<Longrightarrow> enum n \<in> rep x \<Longrightarrow> enum n \<preceq> enum i"
```
```   266     unfolding b_def by (drule not_less_Least, simp)
```
```   267   have X_covers: "\<And>n. \<forall>k\<le>n. enum k \<in> rep x \<longrightarrow> enum k \<preceq> enum (X n)"
```
```   268     apply (induct_tac n)
```
```   269     apply (clarsimp simp add: X_0 a_def)
```
```   270     apply (drule_tac k=0 in Least_le, simp add: r_refl)
```
```   271     apply (clarsimp, rename_tac n k)
```
```   272     apply (erule le_SucE)
```
```   273     apply (rule r_trans [OF _ X_chain], simp)
```
```   274     apply (case_tac "P (X n)", simp add: X_Suc)
```
```   275     apply (rule_tac x="b (X n)" and y="Suc n" in linorder_cases)
```
```   276     apply (simp only: less_Suc_eq_le)
```
```   277     apply (drule spec, drule (1) mp, simp add: b X_mem)
```
```   278     apply (simp add: c X_mem)
```
```   279     apply (drule (1) less_b)
```
```   280     apply (erule r_trans)
```
```   281     apply (simp add: b c X_mem)
```
```   282     apply (simp add: X_Suc)
```
```   283     apply (simp add: P_def)
```
```   284     done
```
```   285   have 1: "\<forall>i. enum (X i) \<preceq> enum (X (Suc i))"
```
```   286     by (simp add: X_chain)
```
```   287   have 2: "x = (\<Squnion>n. principal (enum (X n)))"
```
```   288     apply (simp add: eq_iff rep_contlub 1 rep_principal)
```
```   289     apply (auto, rename_tac a)
```
```   290     apply (subgoal_tac "\<exists>i. a = enum i", erule exE)
```
```   291     apply (rule_tac x=i in exI, simp add: X_covers)
```
```   292     apply (rule_tac x="count a" in exI, simp)
```
```   293     apply (erule idealD3 [OF ideal_rep])
```
```   294     apply (rule X_mem)
```
```   295     done
```
```   296   from 1 2 show ?thesis ..
```
```   297 qed
```
```   298
```
```   299 lemma principal_induct:
```
```   300   assumes adm: "adm P"
```
```   301   assumes P: "\<And>a. P (principal a)"
```
```   302   shows "P x"
```
```   303 apply (rule obtain_principal_chain [of x])
```
```   304 apply (simp add: admD [OF adm] P)
```
```   305 done
```
```   306
```
```   307 lemma principal_induct2:
```
```   308   "\<lbrakk>\<And>y. adm (\<lambda>x. P x y); \<And>x. adm (\<lambda>y. P x y);
```
```   309     \<And>a b. P (principal a) (principal b)\<rbrakk> \<Longrightarrow> P x y"
```
```   310 apply (rule_tac x=y in spec)
```
```   311 apply (rule_tac x=x in principal_induct, simp)
```
```   312 apply (rule allI, rename_tac y)
```
```   313 apply (rule_tac x=y in principal_induct, simp)
```
```   314 apply simp
```
```   315 done
```
```   316
```
```   317 lemma compact_imp_principal: "compact x \<Longrightarrow> \<exists>a. x = principal a"
```
```   318 apply (rule obtain_principal_chain [of x])
```
```   319 apply (drule adm_compact_neq [OF _ cont_id])
```
```   320 apply (subgoal_tac "chain (\<lambda>i. principal (Y i))")
```
```   321 apply (drule (2) admD2, fast, simp)
```
```   322 done
```
```   323
```
```   324 lemma obtain_compact_chain:
```
```   325   obtains Y :: "nat \<Rightarrow> 'b"
```
```   326   where "chain Y" and "\<forall>i. compact (Y i)" and "x = (\<Squnion>i. Y i)"
```
```   327 apply (rule obtain_principal_chain [of x])
```
```   328 apply (rule_tac Y="\<lambda>i. principal (Y i)" in that, simp_all)
```
```   329 done
```
```   330
```
```   331 subsection {* Defining functions in terms of basis elements *}
```
```   332
```
```   333 definition
```
```   334   basis_fun :: "('a::type \<Rightarrow> 'c::cpo) \<Rightarrow> 'b \<rightarrow> 'c" where
```
```   335   "basis_fun = (\<lambda>f. (\<Lambda> x. lub (f ` rep x)))"
```
```   336
```
```   337 lemma basis_fun_lemma:
```
```   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 "\<exists>u. f ` rep x <<| u"
```
```   341 proof -
```
```   342   obtain Y where Y: "\<forall>i. Y i \<preceq> Y (Suc i)"
```
```   343   and x: "x = (\<Squnion>i. principal (Y i))"
```
```   344     by (rule obtain_principal_chain [of x])
```
```   345   have chain: "chain (\<lambda>i. f (Y i))"
```
```   346     by (rule chainI, simp add: f_mono Y)
```
```   347   have rep_x: "rep x = (\<Union>n. {a. a \<preceq> Y n})"
```
```   348     by (simp add: x rep_contlub Y rep_principal)
```
```   349   have "f ` rep x <<| (\<Squnion>n. f (Y n))"
```
```   350     apply (rule is_lubI)
```
```   351     apply (rule ub_imageI, rename_tac a)
```
```   352     apply (clarsimp simp add: rep_x)
```
```   353     apply (drule f_mono)
```
```   354     apply (erule below_trans)
```
```   355     apply (rule is_ub_thelub [OF chain])
```
```   356     apply (rule is_lub_thelub [OF chain])
```
```   357     apply (rule ub_rangeI)
```
```   358     apply (drule_tac x="Y i" in ub_imageD)
```
```   359     apply (simp add: rep_x, fast intro: r_refl)
```
```   360     apply assumption
```
```   361     done
```
```   362   thus ?thesis ..
```
```   363 qed
```
```   364
```
```   365 lemma basis_fun_beta:
```
```   366   fixes f :: "'a::type \<Rightarrow> 'c::cpo"
```
```   367   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
```
```   368   shows "basis_fun f\<cdot>x = lub (f ` rep x)"
```
```   369 unfolding basis_fun_def
```
```   370 proof (rule beta_cfun)
```
```   371   have lub: "\<And>x. \<exists>u. f ` rep x <<| u"
```
```   372     using f_mono by (rule basis_fun_lemma)
```
```   373   show cont: "cont (\<lambda>x. lub (f ` rep x))"
```
```   374     apply (rule contI2)
```
```   375      apply (rule monofunI)
```
```   376      apply (rule is_lub_thelub_ex [OF lub ub_imageI])
```
```   377      apply (rule is_ub_thelub_ex [OF lub imageI])
```
```   378      apply (erule (1) subsetD [OF rep_mono])
```
```   379     apply (rule is_lub_thelub_ex [OF lub ub_imageI])
```
```   380     apply (simp add: rep_contlub, clarify)
```
```   381     apply (erule rev_below_trans [OF is_ub_thelub])
```
```   382     apply (erule is_ub_thelub_ex [OF lub imageI])
```
```   383     done
```
```   384 qed
```
```   385
```
```   386 lemma basis_fun_principal:
```
```   387   fixes f :: "'a::type \<Rightarrow> 'c::cpo"
```
```   388   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
```
```   389   shows "basis_fun f\<cdot>(principal a) = f a"
```
```   390 apply (subst basis_fun_beta, erule f_mono)
```
```   391 apply (subst rep_principal)
```
```   392 apply (rule lub_image_principal, erule f_mono)
```
```   393 done
```
```   394
```
```   395 lemma basis_fun_mono:
```
```   396   assumes f_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> f a \<sqsubseteq> f b"
```
```   397   assumes g_mono: "\<And>a b. a \<preceq> b \<Longrightarrow> g a \<sqsubseteq> g b"
```
```   398   assumes below: "\<And>a. f a \<sqsubseteq> g a"
```
```   399   shows "basis_fun f \<sqsubseteq> basis_fun g"
```
```   400  apply (rule cfun_belowI)
```
```   401  apply (simp only: basis_fun_beta f_mono g_mono)
```
```   402  apply (rule is_lub_thelub_ex)
```
```   403   apply (rule basis_fun_lemma, erule f_mono)
```
```   404  apply (rule ub_imageI, rename_tac a)
```
```   405  apply (rule below_trans [OF below])
```
```   406  apply (rule is_ub_thelub_ex)
```
```   407   apply (rule basis_fun_lemma, erule g_mono)
```
```   408  apply (erule imageI)
```
```   409 done
```
```   410
```
```   411 end
```
```   412
```
```   413 lemma (in preorder) typedef_ideal_completion:
```
```   414   fixes Abs :: "'a set \<Rightarrow> 'b::cpo"
```
```   415   assumes type: "type_definition Rep Abs {S. ideal S}"
```
```   416   assumes below: "\<And>x y. x \<sqsubseteq> y \<longleftrightarrow> Rep x \<subseteq> Rep y"
```
```   417   assumes principal: "\<And>a. principal a = Abs {b. b \<preceq> a}"
```
```   418   assumes countable: "\<exists>f::'a \<Rightarrow> nat. inj f"
```
```   419   shows "ideal_completion r principal Rep"
```
```   420 proof
```
```   421   interpret type_definition Rep Abs "{S. ideal S}" by fact
```
```   422   fix a b :: 'a and x y :: 'b and Y :: "nat \<Rightarrow> 'b"
```
```   423   show "ideal (Rep x)"
```
```   424     using Rep [of x] by simp
```
```   425   show "chain Y \<Longrightarrow> Rep (\<Squnion>i. Y i) = (\<Union>i. Rep (Y i))"
```
```   426     using type below by (rule typedef_ideal_rep_contlub)
```
```   427   show "Rep (principal a) = {b. b \<preceq> a}"
```
```   428     by (simp add: principal Abs_inverse ideal_principal)
```
```   429   show "Rep x \<subseteq> Rep y \<Longrightarrow> x \<sqsubseteq> y"
```
```   430     by (simp only: below)
```
```   431   show "\<exists>f::'a \<Rightarrow> nat. inj f"
```
```   432     by (rule countable)
```
```   433 qed
```
```   434
```
```   435 end
```