src/HOLCF/Universal.thy
 author haftmann Fri Jun 19 21:08:07 2009 +0200 (2009-06-19) changeset 31726 ffd2dc631d88 parent 31076 99fe356cbbc2 child 32997 e760950ba6c5 permissions -rw-r--r--
merged
```     1 (*  Title:      HOLCF/Universal.thy
```
```     2     Author:     Brian Huffman
```
```     3 *)
```
```     4
```
```     5 theory Universal
```
```     6 imports CompactBasis NatIso
```
```     7 begin
```
```     8
```
```     9 subsection {* Basis datatype *}
```
```    10
```
```    11 types ubasis = nat
```
```    12
```
```    13 definition
```
```    14   node :: "nat \<Rightarrow> ubasis \<Rightarrow> ubasis set \<Rightarrow> ubasis"
```
```    15 where
```
```    16   "node i a S = Suc (prod2nat (i, prod2nat (a, set2nat S)))"
```
```    17
```
```    18 lemma node_not_0 [simp]: "node i a S \<noteq> 0"
```
```    19 unfolding node_def by simp
```
```    20
```
```    21 lemma node_gt_0 [simp]: "0 < node i a S"
```
```    22 unfolding node_def by simp
```
```    23
```
```    24 lemma node_inject [simp]:
```
```    25   "\<lbrakk>finite S; finite T\<rbrakk>
```
```    26     \<Longrightarrow> node i a S = node j b T \<longleftrightarrow> i = j \<and> a = b \<and> S = T"
```
```    27 unfolding node_def by simp
```
```    28
```
```    29 lemma node_gt0: "i < node i a S"
```
```    30 unfolding node_def less_Suc_eq_le
```
```    31 by (rule le_prod2nat_1)
```
```    32
```
```    33 lemma node_gt1: "a < node i a S"
```
```    34 unfolding node_def less_Suc_eq_le
```
```    35 by (rule order_trans [OF le_prod2nat_1 le_prod2nat_2])
```
```    36
```
```    37 lemma nat_less_power2: "n < 2^n"
```
```    38 by (induct n) simp_all
```
```    39
```
```    40 lemma node_gt2: "\<lbrakk>finite S; b \<in> S\<rbrakk> \<Longrightarrow> b < node i a S"
```
```    41 unfolding node_def less_Suc_eq_le set2nat_def
```
```    42 apply (rule order_trans [OF _ le_prod2nat_2])
```
```    43 apply (rule order_trans [OF _ le_prod2nat_2])
```
```    44 apply (rule order_trans [where y="setsum (op ^ 2) {b}"])
```
```    45 apply (simp add: nat_less_power2 [THEN order_less_imp_le])
```
```    46 apply (erule setsum_mono2, simp, simp)
```
```    47 done
```
```    48
```
```    49 lemma eq_prod2nat_pairI:
```
```    50   "\<lbrakk>fst (nat2prod x) = a; snd (nat2prod x) = b\<rbrakk> \<Longrightarrow> x = prod2nat (a, b)"
```
```    51 by (erule subst, erule subst, simp)
```
```    52
```
```    53 lemma node_cases:
```
```    54   assumes 1: "x = 0 \<Longrightarrow> P"
```
```    55   assumes 2: "\<And>i a S. \<lbrakk>finite S; x = node i a S\<rbrakk> \<Longrightarrow> P"
```
```    56   shows "P"
```
```    57  apply (cases x)
```
```    58   apply (erule 1)
```
```    59  apply (rule 2)
```
```    60   apply (rule finite_nat2set)
```
```    61  apply (simp add: node_def)
```
```    62  apply (rule eq_prod2nat_pairI [OF refl])
```
```    63  apply (rule eq_prod2nat_pairI [OF refl refl])
```
```    64 done
```
```    65
```
```    66 lemma node_induct:
```
```    67   assumes 1: "P 0"
```
```    68   assumes 2: "\<And>i a S. \<lbrakk>P a; finite S; \<forall>b\<in>S. P b\<rbrakk> \<Longrightarrow> P (node i a S)"
```
```    69   shows "P x"
```
```    70  apply (induct x rule: nat_less_induct)
```
```    71  apply (case_tac n rule: node_cases)
```
```    72   apply (simp add: 1)
```
```    73  apply (simp add: 2 node_gt1 node_gt2)
```
```    74 done
```
```    75
```
```    76 subsection {* Basis ordering *}
```
```    77
```
```    78 inductive
```
```    79   ubasis_le :: "nat \<Rightarrow> nat \<Rightarrow> bool"
```
```    80 where
```
```    81   ubasis_le_refl: "ubasis_le a a"
```
```    82 | ubasis_le_trans:
```
```    83     "\<lbrakk>ubasis_le a b; ubasis_le b c\<rbrakk> \<Longrightarrow> ubasis_le a c"
```
```    84 | ubasis_le_lower:
```
```    85     "finite S \<Longrightarrow> ubasis_le a (node i a S)"
```
```    86 | ubasis_le_upper:
```
```    87     "\<lbrakk>finite S; b \<in> S; ubasis_le a b\<rbrakk> \<Longrightarrow> ubasis_le (node i a S) b"
```
```    88
```
```    89 lemma ubasis_le_minimal: "ubasis_le 0 x"
```
```    90 apply (induct x rule: node_induct)
```
```    91 apply (rule ubasis_le_refl)
```
```    92 apply (erule ubasis_le_trans)
```
```    93 apply (erule ubasis_le_lower)
```
```    94 done
```
```    95
```
```    96 subsubsection {* Generic take function *}
```
```    97
```
```    98 function
```
```    99   ubasis_until :: "(ubasis \<Rightarrow> bool) \<Rightarrow> ubasis \<Rightarrow> ubasis"
```
```   100 where
```
```   101   "ubasis_until P 0 = 0"
```
```   102 | "finite S \<Longrightarrow> ubasis_until P (node i a S) =
```
```   103     (if P (node i a S) then node i a S else ubasis_until P a)"
```
```   104     apply clarify
```
```   105     apply (rule_tac x=b in node_cases)
```
```   106      apply simp
```
```   107     apply simp
```
```   108     apply fast
```
```   109    apply simp
```
```   110   apply simp
```
```   111  apply simp
```
```   112 done
```
```   113
```
```   114 termination ubasis_until
```
```   115 apply (relation "measure snd")
```
```   116 apply (rule wf_measure)
```
```   117 apply (simp add: node_gt1)
```
```   118 done
```
```   119
```
```   120 lemma ubasis_until: "P 0 \<Longrightarrow> P (ubasis_until P x)"
```
```   121 by (induct x rule: node_induct) simp_all
```
```   122
```
```   123 lemma ubasis_until': "0 < ubasis_until P x \<Longrightarrow> P (ubasis_until P x)"
```
```   124 by (induct x rule: node_induct) auto
```
```   125
```
```   126 lemma ubasis_until_same: "P x \<Longrightarrow> ubasis_until P x = x"
```
```   127 by (induct x rule: node_induct) simp_all
```
```   128
```
```   129 lemma ubasis_until_idem:
```
```   130   "P 0 \<Longrightarrow> ubasis_until P (ubasis_until P x) = ubasis_until P x"
```
```   131 by (rule ubasis_until_same [OF ubasis_until])
```
```   132
```
```   133 lemma ubasis_until_0:
```
```   134   "\<forall>x. x \<noteq> 0 \<longrightarrow> \<not> P x \<Longrightarrow> ubasis_until P x = 0"
```
```   135 by (induct x rule: node_induct) simp_all
```
```   136
```
```   137 lemma ubasis_until_less: "ubasis_le (ubasis_until P x) x"
```
```   138 apply (induct x rule: node_induct)
```
```   139 apply (simp add: ubasis_le_refl)
```
```   140 apply (simp add: ubasis_le_refl)
```
```   141 apply (rule impI)
```
```   142 apply (erule ubasis_le_trans)
```
```   143 apply (erule ubasis_le_lower)
```
```   144 done
```
```   145
```
```   146 lemma ubasis_until_chain:
```
```   147   assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
```
```   148   shows "ubasis_le (ubasis_until P x) (ubasis_until Q x)"
```
```   149 apply (induct x rule: node_induct)
```
```   150 apply (simp add: ubasis_le_refl)
```
```   151 apply (simp add: ubasis_le_refl)
```
```   152 apply (simp add: PQ)
```
```   153 apply clarify
```
```   154 apply (rule ubasis_le_trans)
```
```   155 apply (rule ubasis_until_less)
```
```   156 apply (erule ubasis_le_lower)
```
```   157 done
```
```   158
```
```   159 lemma ubasis_until_mono:
```
```   160   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"
```
```   161   shows "ubasis_le a b \<Longrightarrow> ubasis_le (ubasis_until P a) (ubasis_until P b)"
```
```   162 proof (induct set: ubasis_le)
```
```   163   case (ubasis_le_refl a) show ?case by (rule ubasis_le.ubasis_le_refl)
```
```   164 next
```
```   165   case (ubasis_le_trans a b c) thus ?case by - (rule ubasis_le.ubasis_le_trans)
```
```   166 next
```
```   167   case (ubasis_le_lower S a i) thus ?case
```
```   168     apply (clarsimp simp add: ubasis_le_refl)
```
```   169     apply (rule ubasis_le_trans [OF ubasis_until_less])
```
```   170     apply (erule ubasis_le.ubasis_le_lower)
```
```   171     done
```
```   172 next
```
```   173   case (ubasis_le_upper S b a i) thus ?case
```
```   174     apply clarsimp
```
```   175     apply (subst ubasis_until_same)
```
```   176      apply (erule (3) prems)
```
```   177     apply (erule (2) ubasis_le.ubasis_le_upper)
```
```   178     done
```
```   179 qed
```
```   180
```
```   181 lemma finite_range_ubasis_until:
```
```   182   "finite {x. P x} \<Longrightarrow> finite (range (ubasis_until P))"
```
```   183 apply (rule finite_subset [where B="insert 0 {x. P x}"])
```
```   184 apply (clarsimp simp add: ubasis_until')
```
```   185 apply simp
```
```   186 done
```
```   187
```
```   188 subsubsection {* Take function for @{typ ubasis} *}
```
```   189
```
```   190 definition
```
```   191   ubasis_take :: "nat \<Rightarrow> ubasis \<Rightarrow> ubasis"
```
```   192 where
```
```   193   "ubasis_take n = ubasis_until (\<lambda>x. x \<le> n)"
```
```   194
```
```   195 lemma ubasis_take_le: "ubasis_take n x \<le> n"
```
```   196 unfolding ubasis_take_def by (rule ubasis_until, rule le0)
```
```   197
```
```   198 lemma ubasis_take_same: "x \<le> n \<Longrightarrow> ubasis_take n x = x"
```
```   199 unfolding ubasis_take_def by (rule ubasis_until_same)
```
```   200
```
```   201 lemma ubasis_take_idem: "ubasis_take n (ubasis_take n x) = ubasis_take n x"
```
```   202 by (rule ubasis_take_same [OF ubasis_take_le])
```
```   203
```
```   204 lemma ubasis_take_0 [simp]: "ubasis_take 0 x = 0"
```
```   205 unfolding ubasis_take_def by (simp add: ubasis_until_0)
```
```   206
```
```   207 lemma ubasis_take_less: "ubasis_le (ubasis_take n x) x"
```
```   208 unfolding ubasis_take_def by (rule ubasis_until_less)
```
```   209
```
```   210 lemma ubasis_take_chain: "ubasis_le (ubasis_take n x) (ubasis_take (Suc n) x)"
```
```   211 unfolding ubasis_take_def by (rule ubasis_until_chain) simp
```
```   212
```
```   213 lemma ubasis_take_mono:
```
```   214   assumes "ubasis_le x y"
```
```   215   shows "ubasis_le (ubasis_take n x) (ubasis_take n y)"
```
```   216 unfolding ubasis_take_def
```
```   217  apply (rule ubasis_until_mono [OF _ prems])
```
```   218  apply (frule (2) order_less_le_trans [OF node_gt2])
```
```   219  apply (erule order_less_imp_le)
```
```   220 done
```
```   221
```
```   222 lemma finite_range_ubasis_take: "finite (range (ubasis_take n))"
```
```   223 apply (rule finite_subset [where B="{..n}"])
```
```   224 apply (simp add: subset_eq ubasis_take_le)
```
```   225 apply simp
```
```   226 done
```
```   227
```
```   228 lemma ubasis_take_covers: "\<exists>n. ubasis_take n x = x"
```
```   229 apply (rule exI [where x=x])
```
```   230 apply (simp add: ubasis_take_same)
```
```   231 done
```
```   232
```
```   233 interpretation udom: preorder ubasis_le
```
```   234 apply default
```
```   235 apply (rule ubasis_le_refl)
```
```   236 apply (erule (1) ubasis_le_trans)
```
```   237 done
```
```   238
```
```   239 interpretation udom: basis_take ubasis_le ubasis_take
```
```   240 apply default
```
```   241 apply (rule ubasis_take_less)
```
```   242 apply (rule ubasis_take_idem)
```
```   243 apply (erule ubasis_take_mono)
```
```   244 apply (rule ubasis_take_chain)
```
```   245 apply (rule finite_range_ubasis_take)
```
```   246 apply (rule ubasis_take_covers)
```
```   247 done
```
```   248
```
```   249 subsection {* Defining the universal domain by ideal completion *}
```
```   250
```
```   251 typedef (open) udom = "{S. udom.ideal S}"
```
```   252 by (fast intro: udom.ideal_principal)
```
```   253
```
```   254 instantiation udom :: below
```
```   255 begin
```
```   256
```
```   257 definition
```
```   258   "x \<sqsubseteq> y \<longleftrightarrow> Rep_udom x \<subseteq> Rep_udom y"
```
```   259
```
```   260 instance ..
```
```   261 end
```
```   262
```
```   263 instance udom :: po
```
```   264 by (rule udom.typedef_ideal_po
```
```   265     [OF type_definition_udom below_udom_def])
```
```   266
```
```   267 instance udom :: cpo
```
```   268 by (rule udom.typedef_ideal_cpo
```
```   269     [OF type_definition_udom below_udom_def])
```
```   270
```
```   271 lemma Rep_udom_lub:
```
```   272   "chain Y \<Longrightarrow> Rep_udom (\<Squnion>i. Y i) = (\<Union>i. Rep_udom (Y i))"
```
```   273 by (rule udom.typedef_ideal_rep_contlub
```
```   274     [OF type_definition_udom below_udom_def])
```
```   275
```
```   276 lemma ideal_Rep_udom: "udom.ideal (Rep_udom xs)"
```
```   277 by (rule Rep_udom [unfolded mem_Collect_eq])
```
```   278
```
```   279 definition
```
```   280   udom_principal :: "nat \<Rightarrow> udom" where
```
```   281   "udom_principal t = Abs_udom {u. ubasis_le u t}"
```
```   282
```
```   283 lemma Rep_udom_principal:
```
```   284   "Rep_udom (udom_principal t) = {u. ubasis_le u t}"
```
```   285 unfolding udom_principal_def
```
```   286 by (simp add: Abs_udom_inverse udom.ideal_principal)
```
```   287
```
```   288 interpretation udom:
```
```   289   ideal_completion ubasis_le ubasis_take udom_principal Rep_udom
```
```   290 apply unfold_locales
```
```   291 apply (rule ideal_Rep_udom)
```
```   292 apply (erule Rep_udom_lub)
```
```   293 apply (rule Rep_udom_principal)
```
```   294 apply (simp only: below_udom_def)
```
```   295 done
```
```   296
```
```   297 text {* Universal domain is pointed *}
```
```   298
```
```   299 lemma udom_minimal: "udom_principal 0 \<sqsubseteq> x"
```
```   300 apply (induct x rule: udom.principal_induct)
```
```   301 apply (simp, simp add: ubasis_le_minimal)
```
```   302 done
```
```   303
```
```   304 instance udom :: pcpo
```
```   305 by intro_classes (fast intro: udom_minimal)
```
```   306
```
```   307 lemma inst_udom_pcpo: "\<bottom> = udom_principal 0"
```
```   308 by (rule udom_minimal [THEN UU_I, symmetric])
```
```   309
```
```   310 text {* Universal domain is bifinite *}
```
```   311
```
```   312 instantiation udom :: bifinite
```
```   313 begin
```
```   314
```
```   315 definition
```
```   316   approx_udom_def: "approx = udom.completion_approx"
```
```   317
```
```   318 instance
```
```   319 apply (intro_classes, unfold approx_udom_def)
```
```   320 apply (rule udom.chain_completion_approx)
```
```   321 apply (rule udom.lub_completion_approx)
```
```   322 apply (rule udom.completion_approx_idem)
```
```   323 apply (rule udom.finite_fixes_completion_approx)
```
```   324 done
```
```   325
```
```   326 end
```
```   327
```
```   328 lemma approx_udom_principal [simp]:
```
```   329   "approx n\<cdot>(udom_principal x) = udom_principal (ubasis_take n x)"
```
```   330 unfolding approx_udom_def
```
```   331 by (rule udom.completion_approx_principal)
```
```   332
```
```   333 lemma approx_eq_udom_principal:
```
```   334   "\<exists>a\<in>Rep_udom x. approx n\<cdot>x = udom_principal (ubasis_take n a)"
```
```   335 unfolding approx_udom_def
```
```   336 by (rule udom.completion_approx_eq_principal)
```
```   337
```
```   338
```
```   339 subsection {* Universality of @{typ udom} *}
```
```   340
```
```   341 defaultsort bifinite
```
```   342
```
```   343 subsubsection {* Choosing a maximal element from a finite set *}
```
```   344
```
```   345 lemma finite_has_maximal:
```
```   346   fixes A :: "'a::po set"
```
```   347   shows "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y"
```
```   348 proof (induct rule: finite_ne_induct)
```
```   349   case (singleton x)
```
```   350     show ?case by simp
```
```   351 next
```
```   352   case (insert a A)
```
```   353   from `\<exists>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y`
```
```   354   obtain x where x: "x \<in> A"
```
```   355            and x_eq: "\<And>y. \<lbrakk>y \<in> A; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> x = y" by fast
```
```   356   show ?case
```
```   357   proof (intro bexI ballI impI)
```
```   358     fix y
```
```   359     assume "y \<in> insert a A" and "(if x \<sqsubseteq> a then a else x) \<sqsubseteq> y"
```
```   360     thus "(if x \<sqsubseteq> a then a else x) = y"
```
```   361       apply auto
```
```   362       apply (frule (1) below_trans)
```
```   363       apply (frule (1) x_eq)
```
```   364       apply (rule below_antisym, assumption)
```
```   365       apply simp
```
```   366       apply (erule (1) x_eq)
```
```   367       done
```
```   368   next
```
```   369     show "(if x \<sqsubseteq> a then a else x) \<in> insert a A"
```
```   370       by (simp add: x)
```
```   371   qed
```
```   372 qed
```
```   373
```
```   374 definition
```
```   375   choose :: "'a compact_basis set \<Rightarrow> 'a compact_basis"
```
```   376 where
```
```   377   "choose A = (SOME x. x \<in> {x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y})"
```
```   378
```
```   379 lemma choose_lemma:
```
```   380   "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> choose A \<in> {x\<in>A. \<forall>y\<in>A. x \<sqsubseteq> y \<longrightarrow> x = y}"
```
```   381 unfolding choose_def
```
```   382 apply (rule someI_ex)
```
```   383 apply (frule (1) finite_has_maximal, fast)
```
```   384 done
```
```   385
```
```   386 lemma maximal_choose:
```
```   387   "\<lbrakk>finite A; y \<in> A; choose A \<sqsubseteq> y\<rbrakk> \<Longrightarrow> choose A = y"
```
```   388 apply (cases "A = {}", simp)
```
```   389 apply (frule (1) choose_lemma, simp)
```
```   390 done
```
```   391
```
```   392 lemma choose_in: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> choose A \<in> A"
```
```   393 by (frule (1) choose_lemma, simp)
```
```   394
```
```   395 function
```
```   396   choose_pos :: "'a compact_basis set \<Rightarrow> 'a compact_basis \<Rightarrow> nat"
```
```   397 where
```
```   398   "choose_pos A x =
```
```   399     (if finite A \<and> x \<in> A \<and> x \<noteq> choose A
```
```   400       then Suc (choose_pos (A - {choose A}) x) else 0)"
```
```   401 by auto
```
```   402
```
```   403 termination choose_pos
```
```   404 apply (relation "measure (card \<circ> fst)", simp)
```
```   405 apply clarsimp
```
```   406 apply (rule card_Diff1_less)
```
```   407 apply assumption
```
```   408 apply (erule choose_in)
```
```   409 apply clarsimp
```
```   410 done
```
```   411
```
```   412 declare choose_pos.simps [simp del]
```
```   413
```
```   414 lemma choose_pos_choose: "finite A \<Longrightarrow> choose_pos A (choose A) = 0"
```
```   415 by (simp add: choose_pos.simps)
```
```   416
```
```   417 lemma inj_on_choose_pos [OF refl]:
```
```   418   "\<lbrakk>card A = n; finite A\<rbrakk> \<Longrightarrow> inj_on (choose_pos A) A"
```
```   419  apply (induct n arbitrary: A)
```
```   420   apply simp
```
```   421  apply (case_tac "A = {}", simp)
```
```   422  apply (frule (1) choose_in)
```
```   423  apply (rule inj_onI)
```
```   424  apply (drule_tac x="A - {choose A}" in meta_spec, simp)
```
```   425  apply (simp add: choose_pos.simps)
```
```   426  apply (simp split: split_if_asm)
```
```   427  apply (erule (1) inj_onD, simp, simp)
```
```   428 done
```
```   429
```
```   430 lemma choose_pos_bounded [OF refl]:
```
```   431   "\<lbrakk>card A = n; finite A; x \<in> A\<rbrakk> \<Longrightarrow> choose_pos A x < n"
```
```   432 apply (induct n arbitrary: A)
```
```   433 apply simp
```
```   434  apply (case_tac "A = {}", simp)
```
```   435  apply (frule (1) choose_in)
```
```   436 apply (subst choose_pos.simps)
```
```   437 apply simp
```
```   438 done
```
```   439
```
```   440 lemma choose_pos_lessD:
```
```   441   "\<lbrakk>choose_pos A x < choose_pos A y; finite A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<not> x \<sqsubseteq> y"
```
```   442  apply (induct A x arbitrary: y rule: choose_pos.induct)
```
```   443  apply simp
```
```   444  apply (case_tac "x = choose A")
```
```   445   apply simp
```
```   446   apply (rule notI)
```
```   447   apply (frule (2) maximal_choose)
```
```   448   apply simp
```
```   449  apply (case_tac "y = choose A")
```
```   450   apply (simp add: choose_pos_choose)
```
```   451  apply (drule_tac x=y in meta_spec)
```
```   452  apply simp
```
```   453  apply (erule meta_mp)
```
```   454  apply (simp add: choose_pos.simps)
```
```   455 done
```
```   456
```
```   457 subsubsection {* Rank of basis elements *}
```
```   458
```
```   459 primrec
```
```   460   cb_take :: "nat \<Rightarrow> 'a compact_basis \<Rightarrow> 'a compact_basis"
```
```   461 where
```
```   462   "cb_take 0 = (\<lambda>x. compact_bot)"
```
```   463 | "cb_take (Suc n) = compact_take n"
```
```   464
```
```   465 lemma cb_take_covers: "\<exists>n. cb_take n x = x"
```
```   466 apply (rule exE [OF compact_basis.take_covers [where a=x]])
```
```   467 apply (rename_tac n, rule_tac x="Suc n" in exI, simp)
```
```   468 done
```
```   469
```
```   470 lemma cb_take_less: "cb_take n x \<sqsubseteq> x"
```
```   471 by (cases n, simp, simp add: compact_basis.take_less)
```
```   472
```
```   473 lemma cb_take_idem: "cb_take n (cb_take n x) = cb_take n x"
```
```   474 by (cases n, simp, simp add: compact_basis.take_take)
```
```   475
```
```   476 lemma cb_take_mono: "x \<sqsubseteq> y \<Longrightarrow> cb_take n x \<sqsubseteq> cb_take n y"
```
```   477 by (cases n, simp, simp add: compact_basis.take_mono)
```
```   478
```
```   479 lemma cb_take_chain_le: "m \<le> n \<Longrightarrow> cb_take m x \<sqsubseteq> cb_take n x"
```
```   480 apply (cases m, simp)
```
```   481 apply (cases n, simp)
```
```   482 apply (simp add: compact_basis.take_chain_le)
```
```   483 done
```
```   484
```
```   485 lemma range_const: "range (\<lambda>x. c) = {c}"
```
```   486 by auto
```
```   487
```
```   488 lemma finite_range_cb_take: "finite (range (cb_take n))"
```
```   489 apply (cases n)
```
```   490 apply (simp add: range_const)
```
```   491 apply (simp add: compact_basis.finite_range_take)
```
```   492 done
```
```   493
```
```   494 definition
```
```   495   rank :: "'a compact_basis \<Rightarrow> nat"
```
```   496 where
```
```   497   "rank x = (LEAST n. cb_take n x = x)"
```
```   498
```
```   499 lemma compact_approx_rank: "cb_take (rank x) x = x"
```
```   500 unfolding rank_def
```
```   501 apply (rule LeastI_ex)
```
```   502 apply (rule cb_take_covers)
```
```   503 done
```
```   504
```
```   505 lemma rank_leD: "rank x \<le> n \<Longrightarrow> cb_take n x = x"
```
```   506 apply (rule below_antisym [OF cb_take_less])
```
```   507 apply (subst compact_approx_rank [symmetric])
```
```   508 apply (erule cb_take_chain_le)
```
```   509 done
```
```   510
```
```   511 lemma rank_leI: "cb_take n x = x \<Longrightarrow> rank x \<le> n"
```
```   512 unfolding rank_def by (rule Least_le)
```
```   513
```
```   514 lemma rank_le_iff: "rank x \<le> n \<longleftrightarrow> cb_take n x = x"
```
```   515 by (rule iffI [OF rank_leD rank_leI])
```
```   516
```
```   517 lemma rank_compact_bot [simp]: "rank compact_bot = 0"
```
```   518 using rank_leI [of 0 compact_bot] by simp
```
```   519
```
```   520 lemma rank_eq_0_iff [simp]: "rank x = 0 \<longleftrightarrow> x = compact_bot"
```
```   521 using rank_le_iff [of x 0] by auto
```
```   522
```
```   523 definition
```
```   524   rank_le :: "'a compact_basis \<Rightarrow> 'a compact_basis set"
```
```   525 where
```
```   526   "rank_le x = {y. rank y \<le> rank x}"
```
```   527
```
```   528 definition
```
```   529   rank_lt :: "'a compact_basis \<Rightarrow> 'a compact_basis set"
```
```   530 where
```
```   531   "rank_lt x = {y. rank y < rank x}"
```
```   532
```
```   533 definition
```
```   534   rank_eq :: "'a compact_basis \<Rightarrow> 'a compact_basis set"
```
```   535 where
```
```   536   "rank_eq x = {y. rank y = rank x}"
```
```   537
```
```   538 lemma rank_eq_cong: "rank x = rank y \<Longrightarrow> rank_eq x = rank_eq y"
```
```   539 unfolding rank_eq_def by simp
```
```   540
```
```   541 lemma rank_lt_cong: "rank x = rank y \<Longrightarrow> rank_lt x = rank_lt y"
```
```   542 unfolding rank_lt_def by simp
```
```   543
```
```   544 lemma rank_eq_subset: "rank_eq x \<subseteq> rank_le x"
```
```   545 unfolding rank_eq_def rank_le_def by auto
```
```   546
```
```   547 lemma rank_lt_subset: "rank_lt x \<subseteq> rank_le x"
```
```   548 unfolding rank_lt_def rank_le_def by auto
```
```   549
```
```   550 lemma finite_rank_le: "finite (rank_le x)"
```
```   551 unfolding rank_le_def
```
```   552 apply (rule finite_subset [where B="range (cb_take (rank x))"])
```
```   553 apply clarify
```
```   554 apply (rule range_eqI)
```
```   555 apply (erule rank_leD [symmetric])
```
```   556 apply (rule finite_range_cb_take)
```
```   557 done
```
```   558
```
```   559 lemma finite_rank_eq: "finite (rank_eq x)"
```
```   560 by (rule finite_subset [OF rank_eq_subset finite_rank_le])
```
```   561
```
```   562 lemma finite_rank_lt: "finite (rank_lt x)"
```
```   563 by (rule finite_subset [OF rank_lt_subset finite_rank_le])
```
```   564
```
```   565 lemma rank_lt_Int_rank_eq: "rank_lt x \<inter> rank_eq x = {}"
```
```   566 unfolding rank_lt_def rank_eq_def rank_le_def by auto
```
```   567
```
```   568 lemma rank_lt_Un_rank_eq: "rank_lt x \<union> rank_eq x = rank_le x"
```
```   569 unfolding rank_lt_def rank_eq_def rank_le_def by auto
```
```   570
```
```   571 subsubsection {* Sequencing basis elements *}
```
```   572
```
```   573 definition
```
```   574   place :: "'a compact_basis \<Rightarrow> nat"
```
```   575 where
```
```   576   "place x = card (rank_lt x) + choose_pos (rank_eq x) x"
```
```   577
```
```   578 lemma place_bounded: "place x < card (rank_le x)"
```
```   579 unfolding place_def
```
```   580  apply (rule ord_less_eq_trans)
```
```   581   apply (rule add_strict_left_mono)
```
```   582   apply (rule choose_pos_bounded)
```
```   583    apply (rule finite_rank_eq)
```
```   584   apply (simp add: rank_eq_def)
```
```   585  apply (subst card_Un_disjoint [symmetric])
```
```   586     apply (rule finite_rank_lt)
```
```   587    apply (rule finite_rank_eq)
```
```   588   apply (rule rank_lt_Int_rank_eq)
```
```   589  apply (simp add: rank_lt_Un_rank_eq)
```
```   590 done
```
```   591
```
```   592 lemma place_ge: "card (rank_lt x) \<le> place x"
```
```   593 unfolding place_def by simp
```
```   594
```
```   595 lemma place_rank_mono:
```
```   596   fixes x y :: "'a compact_basis"
```
```   597   shows "rank x < rank y \<Longrightarrow> place x < place y"
```
```   598 apply (rule less_le_trans [OF place_bounded])
```
```   599 apply (rule order_trans [OF _ place_ge])
```
```   600 apply (rule card_mono)
```
```   601 apply (rule finite_rank_lt)
```
```   602 apply (simp add: rank_le_def rank_lt_def subset_eq)
```
```   603 done
```
```   604
```
```   605 lemma place_eqD: "place x = place y \<Longrightarrow> x = y"
```
```   606  apply (rule linorder_cases [where x="rank x" and y="rank y"])
```
```   607    apply (drule place_rank_mono, simp)
```
```   608   apply (simp add: place_def)
```
```   609   apply (rule inj_on_choose_pos [where A="rank_eq x", THEN inj_onD])
```
```   610      apply (rule finite_rank_eq)
```
```   611     apply (simp cong: rank_lt_cong rank_eq_cong)
```
```   612    apply (simp add: rank_eq_def)
```
```   613   apply (simp add: rank_eq_def)
```
```   614  apply (drule place_rank_mono, simp)
```
```   615 done
```
```   616
```
```   617 lemma inj_place: "inj place"
```
```   618 by (rule inj_onI, erule place_eqD)
```
```   619
```
```   620 subsubsection {* Embedding and projection on basis elements *}
```
```   621
```
```   622 definition
```
```   623   sub :: "'a compact_basis \<Rightarrow> 'a compact_basis"
```
```   624 where
```
```   625   "sub x = (case rank x of 0 \<Rightarrow> compact_bot | Suc k \<Rightarrow> cb_take k x)"
```
```   626
```
```   627 lemma rank_sub_less: "x \<noteq> compact_bot \<Longrightarrow> rank (sub x) < rank x"
```
```   628 unfolding sub_def
```
```   629 apply (cases "rank x", simp)
```
```   630 apply (simp add: less_Suc_eq_le)
```
```   631 apply (rule rank_leI)
```
```   632 apply (rule cb_take_idem)
```
```   633 done
```
```   634
```
```   635 lemma place_sub_less: "x \<noteq> compact_bot \<Longrightarrow> place (sub x) < place x"
```
```   636 apply (rule place_rank_mono)
```
```   637 apply (erule rank_sub_less)
```
```   638 done
```
```   639
```
```   640 lemma sub_below: "sub x \<sqsubseteq> x"
```
```   641 unfolding sub_def by (cases "rank x", simp_all add: cb_take_less)
```
```   642
```
```   643 lemma rank_less_imp_below_sub: "\<lbrakk>x \<sqsubseteq> y; rank x < rank y\<rbrakk> \<Longrightarrow> x \<sqsubseteq> sub y"
```
```   644 unfolding sub_def
```
```   645 apply (cases "rank y", simp)
```
```   646 apply (simp add: less_Suc_eq_le)
```
```   647 apply (subgoal_tac "cb_take nat x \<sqsubseteq> cb_take nat y")
```
```   648 apply (simp add: rank_leD)
```
```   649 apply (erule cb_take_mono)
```
```   650 done
```
```   651
```
```   652 function
```
```   653   basis_emb :: "'a compact_basis \<Rightarrow> ubasis"
```
```   654 where
```
```   655   "basis_emb x = (if x = compact_bot then 0 else
```
```   656     node (place x) (basis_emb (sub x))
```
```   657       (basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y}))"
```
```   658 by auto
```
```   659
```
```   660 termination basis_emb
```
```   661 apply (relation "measure place", simp)
```
```   662 apply (simp add: place_sub_less)
```
```   663 apply simp
```
```   664 done
```
```   665
```
```   666 declare basis_emb.simps [simp del]
```
```   667
```
```   668 lemma basis_emb_compact_bot [simp]: "basis_emb compact_bot = 0"
```
```   669 by (simp add: basis_emb.simps)
```
```   670
```
```   671 lemma fin1: "finite {y. place y < place x \<and> x \<sqsubseteq> y}"
```
```   672 apply (subst Collect_conj_eq)
```
```   673 apply (rule finite_Int)
```
```   674 apply (rule disjI1)
```
```   675 apply (subgoal_tac "finite (place -` {n. n < place x})", simp)
```
```   676 apply (rule finite_vimageI [OF _ inj_place])
```
```   677 apply (simp add: lessThan_def [symmetric])
```
```   678 done
```
```   679
```
```   680 lemma fin2: "finite (basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y})"
```
```   681 by (rule finite_imageI [OF fin1])
```
```   682
```
```   683 lemma rank_place_mono:
```
```   684   "\<lbrakk>place x < place y; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> rank x < rank y"
```
```   685 apply (rule linorder_cases, assumption)
```
```   686 apply (simp add: place_def cong: rank_lt_cong rank_eq_cong)
```
```   687 apply (drule choose_pos_lessD)
```
```   688 apply (rule finite_rank_eq)
```
```   689 apply (simp add: rank_eq_def)
```
```   690 apply (simp add: rank_eq_def)
```
```   691 apply simp
```
```   692 apply (drule place_rank_mono, simp)
```
```   693 done
```
```   694
```
```   695 lemma basis_emb_mono:
```
```   696   "x \<sqsubseteq> y \<Longrightarrow> ubasis_le (basis_emb x) (basis_emb y)"
```
```   697 proof (induct n \<equiv> "max (place x) (place y)" arbitrary: x y rule: less_induct)
```
```   698   case (less n)
```
```   699   hence IH:
```
```   700     "\<And>(a::'a compact_basis) b.
```
```   701      \<lbrakk>max (place a) (place b) < max (place x) (place y); a \<sqsubseteq> b\<rbrakk>
```
```   702         \<Longrightarrow> ubasis_le (basis_emb a) (basis_emb b)"
```
```   703     by simp
```
```   704   show ?case proof (rule linorder_cases)
```
```   705     assume "place x < place y"
```
```   706     then have "rank x < rank y"
```
```   707       using `x \<sqsubseteq> y` by (rule rank_place_mono)
```
```   708     with `place x < place y` show ?case
```
```   709       apply (case_tac "y = compact_bot", simp)
```
```   710       apply (simp add: basis_emb.simps [of y])
```
```   711       apply (rule ubasis_le_trans [OF _ ubasis_le_lower [OF fin2]])
```
```   712       apply (rule IH)
```
```   713        apply (simp add: less_max_iff_disj)
```
```   714        apply (erule place_sub_less)
```
```   715       apply (erule rank_less_imp_below_sub [OF `x \<sqsubseteq> y`])
```
```   716       done
```
```   717   next
```
```   718     assume "place x = place y"
```
```   719     hence "x = y" by (rule place_eqD)
```
```   720     thus ?case by (simp add: ubasis_le_refl)
```
```   721   next
```
```   722     assume "place x > place y"
```
```   723     with `x \<sqsubseteq> y` show ?case
```
```   724       apply (case_tac "x = compact_bot", simp add: ubasis_le_minimal)
```
```   725       apply (simp add: basis_emb.simps [of x])
```
```   726       apply (rule ubasis_le_upper [OF fin2], simp)
```
```   727       apply (rule IH)
```
```   728        apply (simp add: less_max_iff_disj)
```
```   729        apply (erule place_sub_less)
```
```   730       apply (erule rev_below_trans)
```
```   731       apply (rule sub_below)
```
```   732       done
```
```   733   qed
```
```   734 qed
```
```   735
```
```   736 lemma inj_basis_emb: "inj basis_emb"
```
```   737  apply (rule inj_onI)
```
```   738  apply (case_tac "x = compact_bot")
```
```   739   apply (case_tac [!] "y = compact_bot")
```
```   740     apply simp
```
```   741    apply (simp add: basis_emb.simps)
```
```   742   apply (simp add: basis_emb.simps)
```
```   743  apply (simp add: basis_emb.simps)
```
```   744  apply (simp add: fin2 inj_eq [OF inj_place])
```
```   745 done
```
```   746
```
```   747 definition
```
```   748   basis_prj :: "ubasis \<Rightarrow> 'a compact_basis"
```
```   749 where
```
```   750   "basis_prj x = inv basis_emb
```
```   751     (ubasis_until (\<lambda>x. x \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> ubasis)) x)"
```
```   752
```
```   753 lemma basis_prj_basis_emb: "\<And>x. basis_prj (basis_emb x) = x"
```
```   754 unfolding basis_prj_def
```
```   755  apply (subst ubasis_until_same)
```
```   756   apply (rule rangeI)
```
```   757  apply (rule inv_f_f)
```
```   758  apply (rule inj_basis_emb)
```
```   759 done
```
```   760
```
```   761 lemma basis_prj_node:
```
```   762   "\<lbrakk>finite S; node i a S \<notin> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)\<rbrakk>
```
```   763     \<Longrightarrow> basis_prj (node i a S) = (basis_prj a :: 'a compact_basis)"
```
```   764 unfolding basis_prj_def by simp
```
```   765
```
```   766 lemma basis_prj_0: "basis_prj 0 = compact_bot"
```
```   767 apply (subst basis_emb_compact_bot [symmetric])
```
```   768 apply (rule basis_prj_basis_emb)
```
```   769 done
```
```   770
```
```   771 lemma node_eq_basis_emb_iff:
```
```   772   "finite S \<Longrightarrow> node i a S = basis_emb x \<longleftrightarrow>
```
```   773     x \<noteq> compact_bot \<and> i = place x \<and> a = basis_emb (sub x) \<and>
```
```   774         S = basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y}"
```
```   775 apply (cases "x = compact_bot", simp)
```
```   776 apply (simp add: basis_emb.simps [of x])
```
```   777 apply (simp add: fin2)
```
```   778 done
```
```   779
```
```   780 lemma basis_prj_mono: "ubasis_le a b \<Longrightarrow> basis_prj a \<sqsubseteq> basis_prj b"
```
```   781 proof (induct a b rule: ubasis_le.induct)
```
```   782   case (ubasis_le_refl a) show ?case by (rule below_refl)
```
```   783 next
```
```   784   case (ubasis_le_trans a b c) thus ?case by - (rule below_trans)
```
```   785 next
```
```   786   case (ubasis_le_lower S a i) thus ?case
```
```   787     apply (cases "node i a S \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)")
```
```   788      apply (erule rangeE, rename_tac x)
```
```   789      apply (simp add: basis_prj_basis_emb)
```
```   790      apply (simp add: node_eq_basis_emb_iff)
```
```   791      apply (simp add: basis_prj_basis_emb)
```
```   792      apply (rule sub_below)
```
```   793     apply (simp add: basis_prj_node)
```
```   794     done
```
```   795 next
```
```   796   case (ubasis_le_upper S b a i) thus ?case
```
```   797     apply (cases "node i a S \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)")
```
```   798      apply (erule rangeE, rename_tac x)
```
```   799      apply (simp add: basis_prj_basis_emb)
```
```   800      apply (clarsimp simp add: node_eq_basis_emb_iff)
```
```   801      apply (simp add: basis_prj_basis_emb)
```
```   802     apply (simp add: basis_prj_node)
```
```   803     done
```
```   804 qed
```
```   805
```
```   806 lemma basis_emb_prj_less: "ubasis_le (basis_emb (basis_prj x)) x"
```
```   807 unfolding basis_prj_def
```
```   808  apply (subst f_inv_f [where f=basis_emb])
```
```   809   apply (rule ubasis_until)
```
```   810   apply (rule range_eqI [where x=compact_bot])
```
```   811   apply simp
```
```   812  apply (rule ubasis_until_less)
```
```   813 done
```
```   814
```
```   815 hide (open) const
```
```   816   node
```
```   817   choose
```
```   818   choose_pos
```
```   819   place
```
```   820   sub
```
```   821
```
```   822 subsubsection {* EP-pair from any bifinite domain into @{typ udom} *}
```
```   823
```
```   824 definition
```
```   825   udom_emb :: "'a::bifinite \<rightarrow> udom"
```
```   826 where
```
```   827   "udom_emb = compact_basis.basis_fun (\<lambda>x. udom_principal (basis_emb x))"
```
```   828
```
```   829 definition
```
```   830   udom_prj :: "udom \<rightarrow> 'a::bifinite"
```
```   831 where
```
```   832   "udom_prj = udom.basis_fun (\<lambda>x. Rep_compact_basis (basis_prj x))"
```
```   833
```
```   834 lemma udom_emb_principal:
```
```   835   "udom_emb\<cdot>(Rep_compact_basis x) = udom_principal (basis_emb x)"
```
```   836 unfolding udom_emb_def
```
```   837 apply (rule compact_basis.basis_fun_principal)
```
```   838 apply (rule udom.principal_mono)
```
```   839 apply (erule basis_emb_mono)
```
```   840 done
```
```   841
```
```   842 lemma udom_prj_principal:
```
```   843   "udom_prj\<cdot>(udom_principal x) = Rep_compact_basis (basis_prj x)"
```
```   844 unfolding udom_prj_def
```
```   845 apply (rule udom.basis_fun_principal)
```
```   846 apply (rule compact_basis.principal_mono)
```
```   847 apply (erule basis_prj_mono)
```
```   848 done
```
```   849
```
```   850 lemma ep_pair_udom: "ep_pair udom_emb udom_prj"
```
```   851  apply default
```
```   852   apply (rule compact_basis.principal_induct, simp)
```
```   853   apply (simp add: udom_emb_principal udom_prj_principal)
```
```   854   apply (simp add: basis_prj_basis_emb)
```
```   855  apply (rule udom.principal_induct, simp)
```
```   856  apply (simp add: udom_emb_principal udom_prj_principal)
```
```   857  apply (rule basis_emb_prj_less)
```
```   858 done
```
```   859
```
```   860 end
```