src/HOL/ex/Tarski.thy
 author nipkow Fri Mar 06 17:38:47 2009 +0100 (2009-03-06) changeset 30313 b2441b0c8d38 parent 30198 922f944f03b2 child 31754 b5260f5272a4 permissions -rw-r--r--
```     1 (*  Title:      HOL/ex/Tarski.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Florian Kammüller, Cambridge University Computer Laboratory
```
```     4 *)
```
```     5
```
```     6 header {* The Full Theorem of Tarski *}
```
```     7
```
```     8 theory Tarski
```
```     9 imports Main FuncSet
```
```    10 begin
```
```    11
```
```    12 text {*
```
```    13   Minimal version of lattice theory plus the full theorem of Tarski:
```
```    14   The fixedpoints of a complete lattice themselves form a complete
```
```    15   lattice.
```
```    16
```
```    17   Illustrates first-class theories, using the Sigma representation of
```
```    18   structures.  Tidied and converted to Isar by lcp.
```
```    19 *}
```
```    20
```
```    21 record 'a potype =
```
```    22   pset  :: "'a set"
```
```    23   order :: "('a * 'a) set"
```
```    24
```
```    25 definition
```
```    26   monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool" where
```
```    27   "monotone f A r = (\<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r)"
```
```    28
```
```    29 definition
```
```    30   least :: "['a => bool, 'a potype] => 'a" where
```
```    31   "least P po = (SOME x. x: pset po & P x &
```
```    32                        (\<forall>y \<in> pset po. P y --> (x,y): order po))"
```
```    33
```
```    34 definition
```
```    35   greatest :: "['a => bool, 'a potype] => 'a" where
```
```    36   "greatest P po = (SOME x. x: pset po & P x &
```
```    37                           (\<forall>y \<in> pset po. P y --> (y,x): order po))"
```
```    38
```
```    39 definition
```
```    40   lub  :: "['a set, 'a potype] => 'a" where
```
```    41   "lub S po = least (%x. \<forall>y\<in>S. (y,x): order po) po"
```
```    42
```
```    43 definition
```
```    44   glb  :: "['a set, 'a potype] => 'a" where
```
```    45   "glb S po = greatest (%x. \<forall>y\<in>S. (x,y): order po) po"
```
```    46
```
```    47 definition
```
```    48   isLub :: "['a set, 'a potype, 'a] => bool" where
```
```    49   "isLub S po = (%L. (L: pset po & (\<forall>y\<in>S. (y,L): order po) &
```
```    50                    (\<forall>z\<in>pset po. (\<forall>y\<in>S. (y,z): order po) --> (L,z): order po)))"
```
```    51
```
```    52 definition
```
```    53   isGlb :: "['a set, 'a potype, 'a] => bool" where
```
```    54   "isGlb S po = (%G. (G: pset po & (\<forall>y\<in>S. (G,y): order po) &
```
```    55                  (\<forall>z \<in> pset po. (\<forall>y\<in>S. (z,y): order po) --> (z,G): order po)))"
```
```    56
```
```    57 definition
```
```    58   "fix"    :: "[('a => 'a), 'a set] => 'a set" where
```
```    59   "fix f A  = {x. x: A & f x = x}"
```
```    60
```
```    61 definition
```
```    62   interval :: "[('a*'a) set,'a, 'a ] => 'a set" where
```
```    63   "interval r a b = {x. (a,x): r & (x,b): r}"
```
```    64
```
```    65
```
```    66 definition
```
```    67   Bot :: "'a potype => 'a" where
```
```    68   "Bot po = least (%x. True) po"
```
```    69
```
```    70 definition
```
```    71   Top :: "'a potype => 'a" where
```
```    72   "Top po = greatest (%x. True) po"
```
```    73
```
```    74 definition
```
```    75   PartialOrder :: "('a potype) set" where
```
```    76   "PartialOrder = {P. refl_on (pset P) (order P) & antisym (order P) &
```
```    77                        trans (order P)}"
```
```    78
```
```    79 definition
```
```    80   CompleteLattice :: "('a potype) set" where
```
```    81   "CompleteLattice = {cl. cl: PartialOrder &
```
```    82                         (\<forall>S. S \<subseteq> pset cl --> (\<exists>L. isLub S cl L)) &
```
```    83                         (\<forall>S. S \<subseteq> pset cl --> (\<exists>G. isGlb S cl G))}"
```
```    84
```
```    85 definition
```
```    86   CLF_set :: "('a potype * ('a => 'a)) set" where
```
```    87   "CLF_set = (SIGMA cl: CompleteLattice.
```
```    88             {f. f: pset cl -> pset cl & monotone f (pset cl) (order cl)})"
```
```    89
```
```    90 definition
```
```    91   induced :: "['a set, ('a * 'a) set] => ('a *'a)set" where
```
```    92   "induced A r = {(a,b). a : A & b: A & (a,b): r}"
```
```    93
```
```    94
```
```    95 definition
```
```    96   sublattice :: "('a potype * 'a set)set" where
```
```    97   "sublattice =
```
```    98       (SIGMA cl: CompleteLattice.
```
```    99           {S. S \<subseteq> pset cl &
```
```   100            (| pset = S, order = induced S (order cl) |): CompleteLattice})"
```
```   101
```
```   102 abbreviation
```
```   103   sublat :: "['a set, 'a potype] => bool"  ("_ <<= _" [51,50]50) where
```
```   104   "S <<= cl == S : sublattice `` {cl}"
```
```   105
```
```   106 definition
```
```   107   dual :: "'a potype => 'a potype" where
```
```   108   "dual po = (| pset = pset po, order = converse (order po) |)"
```
```   109
```
```   110 locale S =
```
```   111   fixes cl :: "'a potype"
```
```   112     and A  :: "'a set"
```
```   113     and r  :: "('a * 'a) set"
```
```   114   defines A_def: "A == pset cl"
```
```   115      and  r_def: "r == order cl"
```
```   116
```
```   117 locale PO = S +
```
```   118   assumes cl_po:  "cl : PartialOrder"
```
```   119
```
```   120 locale CL = S +
```
```   121   assumes cl_co:  "cl : CompleteLattice"
```
```   122
```
```   123 sublocale CL < PO
```
```   124 apply (simp_all add: A_def r_def)
```
```   125 apply unfold_locales
```
```   126 using cl_co unfolding CompleteLattice_def by auto
```
```   127
```
```   128 locale CLF = S +
```
```   129   fixes f :: "'a => 'a"
```
```   130     and P :: "'a set"
```
```   131   assumes f_cl:  "(cl,f) : CLF_set" (*was the equivalent "f : CLF_set``{cl}"*)
```
```   132   defines P_def: "P == fix f A"
```
```   133
```
```   134 sublocale CLF < CL
```
```   135 apply (simp_all add: A_def r_def)
```
```   136 apply unfold_locales
```
```   137 using f_cl unfolding CLF_set_def by auto
```
```   138
```
```   139 locale Tarski = CLF +
```
```   140   fixes Y     :: "'a set"
```
```   141     and intY1 :: "'a set"
```
```   142     and v     :: "'a"
```
```   143   assumes
```
```   144     Y_ss: "Y \<subseteq> P"
```
```   145   defines
```
```   146     intY1_def: "intY1 == interval r (lub Y cl) (Top cl)"
```
```   147     and v_def: "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r &
```
```   148                              x: intY1}
```
```   149                       (| pset=intY1, order=induced intY1 r|)"
```
```   150
```
```   151
```
```   152 subsection {* Partial Order *}
```
```   153
```
```   154 lemma (in PO) dual:
```
```   155   "PO (dual cl)"
```
```   156 apply unfold_locales
```
```   157 using cl_po
```
```   158 unfolding PartialOrder_def dual_def
```
```   159 by auto
```
```   160
```
```   161 lemma (in PO) PO_imp_refl_on [simp]: "refl_on A r"
```
```   162 apply (insert cl_po)
```
```   163 apply (simp add: PartialOrder_def A_def r_def)
```
```   164 done
```
```   165
```
```   166 lemma (in PO) PO_imp_sym [simp]: "antisym r"
```
```   167 apply (insert cl_po)
```
```   168 apply (simp add: PartialOrder_def r_def)
```
```   169 done
```
```   170
```
```   171 lemma (in PO) PO_imp_trans [simp]: "trans r"
```
```   172 apply (insert cl_po)
```
```   173 apply (simp add: PartialOrder_def r_def)
```
```   174 done
```
```   175
```
```   176 lemma (in PO) reflE: "x \<in> A ==> (x, x) \<in> r"
```
```   177 apply (insert cl_po)
```
```   178 apply (simp add: PartialOrder_def refl_on_def A_def r_def)
```
```   179 done
```
```   180
```
```   181 lemma (in PO) antisymE: "[| (a, b) \<in> r; (b, a) \<in> r |] ==> a = b"
```
```   182 apply (insert cl_po)
```
```   183 apply (simp add: PartialOrder_def antisym_def r_def)
```
```   184 done
```
```   185
```
```   186 lemma (in PO) transE: "[| (a, b) \<in> r; (b, c) \<in> r|] ==> (a,c) \<in> r"
```
```   187 apply (insert cl_po)
```
```   188 apply (simp add: PartialOrder_def r_def)
```
```   189 apply (unfold trans_def, fast)
```
```   190 done
```
```   191
```
```   192 lemma (in PO) monotoneE:
```
```   193      "[| monotone f A r;  x \<in> A; y \<in> A; (x, y) \<in> r |] ==> (f x, f y) \<in> r"
```
```   194 by (simp add: monotone_def)
```
```   195
```
```   196 lemma (in PO) po_subset_po:
```
```   197      "S \<subseteq> A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
```
```   198 apply (simp (no_asm) add: PartialOrder_def)
```
```   199 apply auto
```
```   200 -- {* refl *}
```
```   201 apply (simp add: refl_on_def induced_def)
```
```   202 apply (blast intro: reflE)
```
```   203 -- {* antisym *}
```
```   204 apply (simp add: antisym_def induced_def)
```
```   205 apply (blast intro: antisymE)
```
```   206 -- {* trans *}
```
```   207 apply (simp add: trans_def induced_def)
```
```   208 apply (blast intro: transE)
```
```   209 done
```
```   210
```
```   211 lemma (in PO) indE: "[| (x, y) \<in> induced S r; S \<subseteq> A |] ==> (x, y) \<in> r"
```
```   212 by (simp add: add: induced_def)
```
```   213
```
```   214 lemma (in PO) indI: "[| (x, y) \<in> r; x \<in> S; y \<in> S |] ==> (x, y) \<in> induced S r"
```
```   215 by (simp add: add: induced_def)
```
```   216
```
```   217 lemma (in CL) CL_imp_ex_isLub: "S \<subseteq> A ==> \<exists>L. isLub S cl L"
```
```   218 apply (insert cl_co)
```
```   219 apply (simp add: CompleteLattice_def A_def)
```
```   220 done
```
```   221
```
```   222 declare (in CL) cl_co [simp]
```
```   223
```
```   224 lemma isLub_lub: "(\<exists>L. isLub S cl L) = isLub S cl (lub S cl)"
```
```   225 by (simp add: lub_def least_def isLub_def some_eq_ex [symmetric])
```
```   226
```
```   227 lemma isGlb_glb: "(\<exists>G. isGlb S cl G) = isGlb S cl (glb S cl)"
```
```   228 by (simp add: glb_def greatest_def isGlb_def some_eq_ex [symmetric])
```
```   229
```
```   230 lemma isGlb_dual_isLub: "isGlb S cl = isLub S (dual cl)"
```
```   231 by (simp add: isLub_def isGlb_def dual_def converse_def)
```
```   232
```
```   233 lemma isLub_dual_isGlb: "isLub S cl = isGlb S (dual cl)"
```
```   234 by (simp add: isLub_def isGlb_def dual_def converse_def)
```
```   235
```
```   236 lemma (in PO) dualPO: "dual cl \<in> PartialOrder"
```
```   237 apply (insert cl_po)
```
```   238 apply (simp add: PartialOrder_def dual_def refl_on_converse
```
```   239                  trans_converse antisym_converse)
```
```   240 done
```
```   241
```
```   242 lemma Rdual:
```
```   243      "\<forall>S. (S \<subseteq> A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
```
```   244       ==> \<forall>S. (S \<subseteq> A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))"
```
```   245 apply safe
```
```   246 apply (rule_tac x = "lub {y. y \<in> A & (\<forall>k \<in> S. (y, k) \<in> r)}
```
```   247                       (|pset = A, order = r|) " in exI)
```
```   248 apply (drule_tac x = "{y. y \<in> A & (\<forall>k \<in> S. (y,k) \<in> r) }" in spec)
```
```   249 apply (drule mp, fast)
```
```   250 apply (simp add: isLub_lub isGlb_def)
```
```   251 apply (simp add: isLub_def, blast)
```
```   252 done
```
```   253
```
```   254 lemma lub_dual_glb: "lub S cl = glb S (dual cl)"
```
```   255 by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
```
```   256
```
```   257 lemma glb_dual_lub: "glb S cl = lub S (dual cl)"
```
```   258 by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
```
```   259
```
```   260 lemma CL_subset_PO: "CompleteLattice \<subseteq> PartialOrder"
```
```   261 by (simp add: PartialOrder_def CompleteLattice_def, fast)
```
```   262
```
```   263 lemmas CL_imp_PO = CL_subset_PO [THEN subsetD]
```
```   264
```
```   265 (*declare CL_imp_PO [THEN PO.PO_imp_refl, simp]
```
```   266 declare CL_imp_PO [THEN PO.PO_imp_sym, simp]
```
```   267 declare CL_imp_PO [THEN PO.PO_imp_trans, simp]*)
```
```   268
```
```   269 lemma (in CL) CO_refl_on: "refl_on A r"
```
```   270 by (rule PO_imp_refl_on)
```
```   271
```
```   272 lemma (in CL) CO_antisym: "antisym r"
```
```   273 by (rule PO_imp_sym)
```
```   274
```
```   275 lemma (in CL) CO_trans: "trans r"
```
```   276 by (rule PO_imp_trans)
```
```   277
```
```   278 lemma CompleteLatticeI:
```
```   279      "[| po \<in> PartialOrder; (\<forall>S. S \<subseteq> pset po --> (\<exists>L. isLub S po L));
```
```   280          (\<forall>S. S \<subseteq> pset po --> (\<exists>G. isGlb S po G))|]
```
```   281       ==> po \<in> CompleteLattice"
```
```   282 apply (unfold CompleteLattice_def, blast)
```
```   283 done
```
```   284
```
```   285 lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice"
```
```   286 apply (insert cl_co)
```
```   287 apply (simp add: CompleteLattice_def dual_def)
```
```   288 apply (fold dual_def)
```
```   289 apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric]
```
```   290                  dualPO)
```
```   291 done
```
```   292
```
```   293 lemma (in PO) dualA_iff: "pset (dual cl) = pset cl"
```
```   294 by (simp add: dual_def)
```
```   295
```
```   296 lemma (in PO) dualr_iff: "((x, y) \<in> (order(dual cl))) = ((y, x) \<in> order cl)"
```
```   297 by (simp add: dual_def)
```
```   298
```
```   299 lemma (in PO) monotone_dual:
```
```   300      "monotone f (pset cl) (order cl)
```
```   301      ==> monotone f (pset (dual cl)) (order(dual cl))"
```
```   302 by (simp add: monotone_def dualA_iff dualr_iff)
```
```   303
```
```   304 lemma (in PO) interval_dual:
```
```   305      "[| x \<in> A; y \<in> A|] ==> interval r x y = interval (order(dual cl)) y x"
```
```   306 apply (simp add: interval_def dualr_iff)
```
```   307 apply (fold r_def, fast)
```
```   308 done
```
```   309
```
```   310 lemma (in PO) trans:
```
```   311   "(x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r"
```
```   312 using cl_po apply (auto simp add: PartialOrder_def r_def)
```
```   313 unfolding trans_def by blast
```
```   314
```
```   315 lemma (in PO) interval_not_empty:
```
```   316   "interval r a b \<noteq> {} ==> (a, b) \<in> r"
```
```   317 apply (simp add: interval_def)
```
```   318 using trans by blast
```
```   319
```
```   320 lemma (in PO) interval_imp_mem: "x \<in> interval r a b ==> (a, x) \<in> r"
```
```   321 by (simp add: interval_def)
```
```   322
```
```   323 lemma (in PO) left_in_interval:
```
```   324      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> a \<in> interval r a b"
```
```   325 apply (simp (no_asm_simp) add: interval_def)
```
```   326 apply (simp add: PO_imp_trans interval_not_empty)
```
```   327 apply (simp add: reflE)
```
```   328 done
```
```   329
```
```   330 lemma (in PO) right_in_interval:
```
```   331      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> b \<in> interval r a b"
```
```   332 apply (simp (no_asm_simp) add: interval_def)
```
```   333 apply (simp add: PO_imp_trans interval_not_empty)
```
```   334 apply (simp add: reflE)
```
```   335 done
```
```   336
```
```   337
```
```   338 subsection {* sublattice *}
```
```   339
```
```   340 lemma (in PO) sublattice_imp_CL:
```
```   341      "S <<= cl  ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
```
```   342 by (simp add: sublattice_def CompleteLattice_def r_def)
```
```   343
```
```   344 lemma (in CL) sublatticeI:
```
```   345      "[| S \<subseteq> A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
```
```   346       ==> S <<= cl"
```
```   347 by (simp add: sublattice_def A_def r_def)
```
```   348
```
```   349 lemma (in CL) dual:
```
```   350   "CL (dual cl)"
```
```   351 apply unfold_locales
```
```   352 using cl_co unfolding CompleteLattice_def
```
```   353 apply (simp add: dualPO isGlb_dual_isLub [symmetric] isLub_dual_isGlb [symmetric] dualA_iff)
```
```   354 done
```
```   355
```
```   356
```
```   357 subsection {* lub *}
```
```   358
```
```   359 lemma (in CL) lub_unique: "[| S \<subseteq> A; isLub S cl x; isLub S cl L|] ==> x = L"
```
```   360 apply (rule antisymE)
```
```   361 apply (auto simp add: isLub_def r_def)
```
```   362 done
```
```   363
```
```   364 lemma (in CL) lub_upper: "[|S \<subseteq> A; x \<in> S|] ==> (x, lub S cl) \<in> r"
```
```   365 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   366 apply (unfold lub_def least_def)
```
```   367 apply (rule some_equality [THEN ssubst])
```
```   368   apply (simp add: isLub_def)
```
```   369  apply (simp add: lub_unique A_def isLub_def)
```
```   370 apply (simp add: isLub_def r_def)
```
```   371 done
```
```   372
```
```   373 lemma (in CL) lub_least:
```
```   374      "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
```
```   375 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   376 apply (unfold lub_def least_def)
```
```   377 apply (rule_tac s=x in some_equality [THEN ssubst])
```
```   378   apply (simp add: isLub_def)
```
```   379  apply (simp add: lub_unique A_def isLub_def)
```
```   380 apply (simp add: isLub_def r_def A_def)
```
```   381 done
```
```   382
```
```   383 lemma (in CL) lub_in_lattice: "S \<subseteq> A ==> lub S cl \<in> A"
```
```   384 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   385 apply (unfold lub_def least_def)
```
```   386 apply (subst some_equality)
```
```   387 apply (simp add: isLub_def)
```
```   388 prefer 2 apply (simp add: isLub_def A_def)
```
```   389 apply (simp add: lub_unique A_def isLub_def)
```
```   390 done
```
```   391
```
```   392 lemma (in CL) lubI:
```
```   393      "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
```
```   394          \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"
```
```   395 apply (rule lub_unique, assumption)
```
```   396 apply (simp add: isLub_def A_def r_def)
```
```   397 apply (unfold isLub_def)
```
```   398 apply (rule conjI)
```
```   399 apply (fold A_def r_def)
```
```   400 apply (rule lub_in_lattice, assumption)
```
```   401 apply (simp add: lub_upper lub_least)
```
```   402 done
```
```   403
```
```   404 lemma (in CL) lubIa: "[| S \<subseteq> A; isLub S cl L |] ==> L = lub S cl"
```
```   405 by (simp add: lubI isLub_def A_def r_def)
```
```   406
```
```   407 lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"
```
```   408 by (simp add: isLub_def  A_def)
```
```   409
```
```   410 lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r"
```
```   411 by (simp add: isLub_def r_def)
```
```   412
```
```   413 lemma (in CL) isLub_least:
```
```   414      "[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r"
```
```   415 by (simp add: isLub_def A_def r_def)
```
```   416
```
```   417 lemma (in CL) isLubI:
```
```   418      "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;
```
```   419          (\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L"
```
```   420 by (simp add: isLub_def A_def r_def)
```
```   421
```
```   422
```
```   423 subsection {* glb *}
```
```   424
```
```   425 lemma (in CL) glb_in_lattice: "S \<subseteq> A ==> glb S cl \<in> A"
```
```   426 apply (subst glb_dual_lub)
```
```   427 apply (simp add: A_def)
```
```   428 apply (rule dualA_iff [THEN subst])
```
```   429 apply (rule CL.lub_in_lattice)
```
```   430 apply (rule dual)
```
```   431 apply (simp add: dualA_iff)
```
```   432 done
```
```   433
```
```   434 lemma (in CL) glb_lower: "[|S \<subseteq> A; x \<in> S|] ==> (glb S cl, x) \<in> r"
```
```   435 apply (subst glb_dual_lub)
```
```   436 apply (simp add: r_def)
```
```   437 apply (rule dualr_iff [THEN subst])
```
```   438 apply (rule CL.lub_upper)
```
```   439 apply (rule dual)
```
```   440 apply (simp add: dualA_iff A_def, assumption)
```
```   441 done
```
```   442
```
```   443 text {*
```
```   444   Reduce the sublattice property by using substructural properties;
```
```   445   abandoned see @{text "Tarski_4.ML"}.
```
```   446 *}
```
```   447
```
```   448 lemma (in CLF) [simp]:
```
```   449     "f: pset cl -> pset cl & monotone f (pset cl) (order cl)"
```
```   450 apply (insert f_cl)
```
```   451 apply (simp add: CLF_set_def)
```
```   452 done
```
```   453
```
```   454 declare (in CLF) f_cl [simp]
```
```   455
```
```   456
```
```   457 lemma (in CLF) f_in_funcset: "f \<in> A -> A"
```
```   458 by (simp add: A_def)
```
```   459
```
```   460 lemma (in CLF) monotone_f: "monotone f A r"
```
```   461 by (simp add: A_def r_def)
```
```   462
```
```   463 lemma (in CLF) CLF_dual: "(dual cl, f) \<in> CLF_set"
```
```   464 apply (simp add: CLF_set_def  CL_dualCL monotone_dual)
```
```   465 apply (simp add: dualA_iff)
```
```   466 done
```
```   467
```
```   468 lemma (in CLF) dual:
```
```   469   "CLF (dual cl) f"
```
```   470 apply (rule CLF.intro)
```
```   471 apply (rule CLF_dual)
```
```   472 done
```
```   473
```
```   474
```
```   475 subsection {* fixed points *}
```
```   476
```
```   477 lemma fix_subset: "fix f A \<subseteq> A"
```
```   478 by (simp add: fix_def, fast)
```
```   479
```
```   480 lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"
```
```   481 by (simp add: fix_def)
```
```   482
```
```   483 lemma fixf_subset:
```
```   484      "[| A \<subseteq> B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
```
```   485 by (simp add: fix_def, auto)
```
```   486
```
```   487
```
```   488 subsection {* lemmas for Tarski, lub *}
```
```   489 lemma (in CLF) lubH_le_flubH:
```
```   490      "H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r"
```
```   491 apply (rule lub_least, fast)
```
```   492 apply (rule f_in_funcset [THEN funcset_mem])
```
```   493 apply (rule lub_in_lattice, fast)
```
```   494 -- {* @{text "\<forall>x:H. (x, f (lub H r)) \<in> r"} *}
```
```   495 apply (rule ballI)
```
```   496 apply (rule transE)
```
```   497 -- {* instantiates @{text "(x, ???z) \<in> order cl to (x, f x)"}, *}
```
```   498 -- {* because of the def of @{text H} *}
```
```   499 apply fast
```
```   500 -- {* so it remains to show @{text "(f x, f (lub H cl)) \<in> r"} *}
```
```   501 apply (rule_tac f = "f" in monotoneE)
```
```   502 apply (rule monotone_f, fast)
```
```   503 apply (rule lub_in_lattice, fast)
```
```   504 apply (rule lub_upper, fast)
```
```   505 apply assumption
```
```   506 done
```
```   507
```
```   508 lemma (in CLF) flubH_le_lubH:
```
```   509      "[|  H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r"
```
```   510 apply (rule lub_upper, fast)
```
```   511 apply (rule_tac t = "H" in ssubst, assumption)
```
```   512 apply (rule CollectI)
```
```   513 apply (rule conjI)
```
```   514 apply (rule_tac [2] f_in_funcset [THEN funcset_mem])
```
```   515 apply (rule_tac [2] lub_in_lattice)
```
```   516 prefer 2 apply fast
```
```   517 apply (rule_tac f = "f" in monotoneE)
```
```   518 apply (rule monotone_f)
```
```   519   apply (blast intro: lub_in_lattice)
```
```   520  apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem])
```
```   521 apply (simp add: lubH_le_flubH)
```
```   522 done
```
```   523
```
```   524 lemma (in CLF) lubH_is_fixp:
```
```   525      "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
```
```   526 apply (simp add: fix_def)
```
```   527 apply (rule conjI)
```
```   528 apply (rule lub_in_lattice, fast)
```
```   529 apply (rule antisymE)
```
```   530 apply (simp add: flubH_le_lubH)
```
```   531 apply (simp add: lubH_le_flubH)
```
```   532 done
```
```   533
```
```   534 lemma (in CLF) fix_in_H:
```
```   535      "[| H = {x. (x, f x) \<in> r & x \<in> A};  x \<in> P |] ==> x \<in> H"
```
```   536 by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl_on
```
```   537                     fix_subset [of f A, THEN subsetD])
```
```   538
```
```   539 lemma (in CLF) fixf_le_lubH:
```
```   540      "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
```
```   541 apply (rule ballI)
```
```   542 apply (rule lub_upper, fast)
```
```   543 apply (rule fix_in_H)
```
```   544 apply (simp_all add: P_def)
```
```   545 done
```
```   546
```
```   547 lemma (in CLF) lubH_least_fixf:
```
```   548      "H = {x. (x, f x) \<in> r & x \<in> A}
```
```   549       ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"
```
```   550 apply (rule allI)
```
```   551 apply (rule impI)
```
```   552 apply (erule bspec)
```
```   553 apply (rule lubH_is_fixp, assumption)
```
```   554 done
```
```   555
```
```   556 subsection {* Tarski fixpoint theorem 1, first part *}
```
```   557 lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"
```
```   558 apply (rule sym)
```
```   559 apply (simp add: P_def)
```
```   560 apply (rule lubI)
```
```   561 apply (rule fix_subset)
```
```   562 apply (rule lub_in_lattice, fast)
```
```   563 apply (simp add: fixf_le_lubH)
```
```   564 apply (simp add: lubH_least_fixf)
```
```   565 done
```
```   566
```
```   567 lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P"
```
```   568   -- {* Tarski for glb *}
```
```   569 apply (simp add: glb_dual_lub P_def A_def r_def)
```
```   570 apply (rule dualA_iff [THEN subst])
```
```   571 apply (rule CLF.lubH_is_fixp)
```
```   572 apply (rule dual)
```
```   573 apply (simp add: dualr_iff dualA_iff)
```
```   574 done
```
```   575
```
```   576 lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl"
```
```   577 apply (simp add: glb_dual_lub P_def A_def r_def)
```
```   578 apply (rule dualA_iff [THEN subst])
```
```   579 apply (simp add: CLF.T_thm_1_lub [of _ f, OF dual]
```
```   580                  dualPO CL_dualCL CLF_dual dualr_iff)
```
```   581 done
```
```   582
```
```   583 subsection {* interval *}
```
```   584
```
```   585 lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"
```
```   586 apply (insert CO_refl_on)
```
```   587 apply (simp add: refl_on_def, blast)
```
```   588 done
```
```   589
```
```   590 lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b \<subseteq> A"
```
```   591 apply (simp add: interval_def)
```
```   592 apply (blast intro: rel_imp_elem)
```
```   593 done
```
```   594
```
```   595 lemma (in CLF) intervalI:
```
```   596      "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"
```
```   597 by (simp add: interval_def)
```
```   598
```
```   599 lemma (in CLF) interval_lemma1:
```
```   600      "[| S \<subseteq> interval r a b; x \<in> S |] ==> (a, x) \<in> r"
```
```   601 by (unfold interval_def, fast)
```
```   602
```
```   603 lemma (in CLF) interval_lemma2:
```
```   604      "[| S \<subseteq> interval r a b; x \<in> S |] ==> (x, b) \<in> r"
```
```   605 by (unfold interval_def, fast)
```
```   606
```
```   607 lemma (in CLF) a_less_lub:
```
```   608      "[| S \<subseteq> A; S \<noteq> {};
```
```   609          \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"
```
```   610 by (blast intro: transE)
```
```   611
```
```   612 lemma (in CLF) glb_less_b:
```
```   613      "[| S \<subseteq> A; S \<noteq> {};
```
```   614          \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"
```
```   615 by (blast intro: transE)
```
```   616
```
```   617 lemma (in CLF) S_intv_cl:
```
```   618      "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b |]==> S \<subseteq> A"
```
```   619 by (simp add: subset_trans [OF _ interval_subset])
```
```   620
```
```   621 lemma (in CLF) L_in_interval:
```
```   622      "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b;
```
```   623          S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"
```
```   624 apply (rule intervalI)
```
```   625 apply (rule a_less_lub)
```
```   626 prefer 2 apply assumption
```
```   627 apply (simp add: S_intv_cl)
```
```   628 apply (rule ballI)
```
```   629 apply (simp add: interval_lemma1)
```
```   630 apply (simp add: isLub_upper)
```
```   631 -- {* @{text "(L, b) \<in> r"} *}
```
```   632 apply (simp add: isLub_least interval_lemma2)
```
```   633 done
```
```   634
```
```   635 lemma (in CLF) G_in_interval:
```
```   636      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S \<subseteq> interval r a b; isGlb S cl G;
```
```   637          S \<noteq> {} |] ==> G \<in> interval r a b"
```
```   638 apply (simp add: interval_dual)
```
```   639 apply (simp add: CLF.L_in_interval [of _ f, OF dual]
```
```   640                  dualA_iff A_def isGlb_dual_isLub)
```
```   641 done
```
```   642
```
```   643 lemma (in CLF) intervalPO:
```
```   644      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   645       ==> (| pset = interval r a b, order = induced (interval r a b) r |)
```
```   646           \<in> PartialOrder"
```
```   647 apply (rule po_subset_po)
```
```   648 apply (simp add: interval_subset)
```
```   649 done
```
```   650
```
```   651 lemma (in CLF) intv_CL_lub:
```
```   652  "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   653   ==> \<forall>S. S \<subseteq> interval r a b -->
```
```   654           (\<exists>L. isLub S (| pset = interval r a b,
```
```   655                           order = induced (interval r a b) r |)  L)"
```
```   656 apply (intro strip)
```
```   657 apply (frule S_intv_cl [THEN CL_imp_ex_isLub])
```
```   658 prefer 2 apply assumption
```
```   659 apply assumption
```
```   660 apply (erule exE)
```
```   661 -- {* define the lub for the interval as *}
```
```   662 apply (rule_tac x = "if S = {} then a else L" in exI)
```
```   663 apply (simp (no_asm_simp) add: isLub_def split del: split_if)
```
```   664 apply (intro impI conjI)
```
```   665 -- {* @{text "(if S = {} then a else L) \<in> interval r a b"} *}
```
```   666 apply (simp add: CL_imp_PO L_in_interval)
```
```   667 apply (simp add: left_in_interval)
```
```   668 -- {* lub prop 1 *}
```
```   669 apply (case_tac "S = {}")
```
```   670 -- {* @{text "S = {}, y \<in> S = False => everything"} *}
```
```   671 apply fast
```
```   672 -- {* @{text "S \<noteq> {}"} *}
```
```   673 apply simp
```
```   674 -- {* @{text "\<forall>y:S. (y, L) \<in> induced (interval r a b) r"} *}
```
```   675 apply (rule ballI)
```
```   676 apply (simp add: induced_def  L_in_interval)
```
```   677 apply (rule conjI)
```
```   678 apply (rule subsetD)
```
```   679 apply (simp add: S_intv_cl, assumption)
```
```   680 apply (simp add: isLub_upper)
```
```   681 -- {* @{text "\<forall>z:interval r a b. (\<forall>y:S. (y, z) \<in> induced (interval r a b) r \<longrightarrow> (if S = {} then a else L, z) \<in> induced (interval r a b) r"} *}
```
```   682 apply (rule ballI)
```
```   683 apply (rule impI)
```
```   684 apply (case_tac "S = {}")
```
```   685 -- {* @{text "S = {}"} *}
```
```   686 apply simp
```
```   687 apply (simp add: induced_def  interval_def)
```
```   688 apply (rule conjI)
```
```   689 apply (rule reflE, assumption)
```
```   690 apply (rule interval_not_empty)
```
```   691 apply (simp add: interval_def)
```
```   692 -- {* @{text "S \<noteq> {}"} *}
```
```   693 apply simp
```
```   694 apply (simp add: induced_def  L_in_interval)
```
```   695 apply (rule isLub_least, assumption)
```
```   696 apply (rule subsetD)
```
```   697 prefer 2 apply assumption
```
```   698 apply (simp add: S_intv_cl, fast)
```
```   699 done
```
```   700
```
```   701 lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual]
```
```   702
```
```   703 lemma (in CLF) interval_is_sublattice:
```
```   704      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   705         ==> interval r a b <<= cl"
```
```   706 apply (rule sublatticeI)
```
```   707 apply (simp add: interval_subset)
```
```   708 apply (rule CompleteLatticeI)
```
```   709 apply (simp add: intervalPO)
```
```   710  apply (simp add: intv_CL_lub)
```
```   711 apply (simp add: intv_CL_glb)
```
```   712 done
```
```   713
```
```   714 lemmas (in CLF) interv_is_compl_latt =
```
```   715     interval_is_sublattice [THEN sublattice_imp_CL]
```
```   716
```
```   717
```
```   718 subsection {* Top and Bottom *}
```
```   719 lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"
```
```   720 by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
```
```   721
```
```   722 lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)"
```
```   723 by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
```
```   724
```
```   725 lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"
```
```   726 apply (simp add: Bot_def least_def)
```
```   727 apply (rule_tac a="glb A cl" in someI2)
```
```   728 apply (simp_all add: glb_in_lattice glb_lower
```
```   729                      r_def [symmetric] A_def [symmetric])
```
```   730 done
```
```   731
```
```   732 lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
```
```   733 apply (simp add: Top_dual_Bot A_def)
```
```   734 apply (rule dualA_iff [THEN subst])
```
```   735 apply (rule CLF.Bot_in_lattice [OF dual])
```
```   736 done
```
```   737
```
```   738 lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
```
```   739 apply (simp add: Top_def greatest_def)
```
```   740 apply (rule_tac a="lub A cl" in someI2)
```
```   741 apply (rule someI2)
```
```   742 apply (simp_all add: lub_in_lattice lub_upper
```
```   743                      r_def [symmetric] A_def [symmetric])
```
```   744 done
```
```   745
```
```   746 lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
```
```   747 apply (simp add: Bot_dual_Top r_def)
```
```   748 apply (rule dualr_iff [THEN subst])
```
```   749 apply (rule CLF.Top_prop [OF dual])
```
```   750 apply (simp add: dualA_iff A_def)
```
```   751 done
```
```   752
```
```   753 lemma (in CLF) Top_intv_not_empty: "x \<in> A  ==> interval r x (Top cl) \<noteq> {}"
```
```   754 apply (rule notI)
```
```   755 apply (drule_tac a = "Top cl" in equals0D)
```
```   756 apply (simp add: interval_def)
```
```   757 apply (simp add: refl_on_def Top_in_lattice Top_prop)
```
```   758 done
```
```   759
```
```   760 lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}"
```
```   761 apply (simp add: Bot_dual_Top)
```
```   762 apply (subst interval_dual)
```
```   763 prefer 2 apply assumption
```
```   764 apply (simp add: A_def)
```
```   765 apply (rule dualA_iff [THEN subst])
```
```   766 apply (rule CLF.Top_in_lattice [OF dual])
```
```   767 apply (rule CLF.Top_intv_not_empty [OF dual])
```
```   768 apply (simp add: dualA_iff A_def)
```
```   769 done
```
```   770
```
```   771 subsection {* fixed points form a partial order *}
```
```   772
```
```   773 lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
```
```   774 by (simp add: P_def fix_subset po_subset_po)
```
```   775
```
```   776 lemma (in Tarski) Y_subset_A: "Y \<subseteq> A"
```
```   777 apply (rule subset_trans [OF _ fix_subset])
```
```   778 apply (rule Y_ss [simplified P_def])
```
```   779 done
```
```   780
```
```   781 lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A"
```
```   782   by (rule Y_subset_A [THEN lub_in_lattice])
```
```   783
```
```   784 lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"
```
```   785 apply (rule lub_least)
```
```   786 apply (rule Y_subset_A)
```
```   787 apply (rule f_in_funcset [THEN funcset_mem])
```
```   788 apply (rule lubY_in_A)
```
```   789 -- {* @{text "Y \<subseteq> P ==> f x = x"} *}
```
```   790 apply (rule ballI)
```
```   791 apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
```
```   792 apply (erule Y_ss [simplified P_def, THEN subsetD])
```
```   793 -- {* @{text "reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r"} by monotonicity *}
```
```   794 apply (rule_tac f = "f" in monotoneE)
```
```   795 apply (rule monotone_f)
```
```   796 apply (simp add: Y_subset_A [THEN subsetD])
```
```   797 apply (rule lubY_in_A)
```
```   798 apply (simp add: lub_upper Y_subset_A)
```
```   799 done
```
```   800
```
```   801 lemma (in Tarski) intY1_subset: "intY1 \<subseteq> A"
```
```   802 apply (unfold intY1_def)
```
```   803 apply (rule interval_subset)
```
```   804 apply (rule lubY_in_A)
```
```   805 apply (rule Top_in_lattice)
```
```   806 done
```
```   807
```
```   808 lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD]
```
```   809
```
```   810 lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"
```
```   811 apply (simp add: intY1_def  interval_def)
```
```   812 apply (rule conjI)
```
```   813 apply (rule transE)
```
```   814 apply (rule lubY_le_flubY)
```
```   815 -- {* @{text "(f (lub Y cl), f x) \<in> r"} *}
```
```   816 apply (rule_tac f=f in monotoneE)
```
```   817 apply (rule monotone_f)
```
```   818 apply (rule lubY_in_A)
```
```   819 apply (simp add: intY1_def interval_def  intY1_elem)
```
```   820 apply (simp add: intY1_def  interval_def)
```
```   821 -- {* @{text "(f x, Top cl) \<in> r"} *}
```
```   822 apply (rule Top_prop)
```
```   823 apply (rule f_in_funcset [THEN funcset_mem])
```
```   824 apply (simp add: intY1_def interval_def  intY1_elem)
```
```   825 done
```
```   826
```
```   827 lemma (in Tarski) intY1_func: "(%x: intY1. f x) \<in> intY1 -> intY1"
```
```   828 apply (rule restrictI)
```
```   829 apply (erule intY1_f_closed)
```
```   830 done
```
```   831
```
```   832 lemma (in Tarski) intY1_mono:
```
```   833      "monotone (%x: intY1. f x) intY1 (induced intY1 r)"
```
```   834 apply (auto simp add: monotone_def induced_def intY1_f_closed)
```
```   835 apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
```
```   836 done
```
```   837
```
```   838 lemma (in Tarski) intY1_is_cl:
```
```   839     "(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
```
```   840 apply (unfold intY1_def)
```
```   841 apply (rule interv_is_compl_latt)
```
```   842 apply (rule lubY_in_A)
```
```   843 apply (rule Top_in_lattice)
```
```   844 apply (rule Top_intv_not_empty)
```
```   845 apply (rule lubY_in_A)
```
```   846 done
```
```   847
```
```   848 lemma (in Tarski) v_in_P: "v \<in> P"
```
```   849 apply (unfold P_def)
```
```   850 apply (rule_tac A = "intY1" in fixf_subset)
```
```   851 apply (rule intY1_subset)
```
```   852 unfolding v_def
```
```   853 apply (rule CLF.glbH_is_fixp [OF CLF.intro, unfolded CLF_set_def, of "\<lparr>pset = intY1, order = induced intY1 r\<rparr>", simplified])
```
```   854 apply auto
```
```   855 apply (rule intY1_is_cl)
```
```   856 apply (rule intY1_func)
```
```   857 apply (rule intY1_mono)
```
```   858 done
```
```   859
```
```   860 lemma (in Tarski) z_in_interval:
```
```   861      "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1"
```
```   862 apply (unfold intY1_def P_def)
```
```   863 apply (rule intervalI)
```
```   864 prefer 2
```
```   865  apply (erule fix_subset [THEN subsetD, THEN Top_prop])
```
```   866 apply (rule lub_least)
```
```   867 apply (rule Y_subset_A)
```
```   868 apply (fast elim!: fix_subset [THEN subsetD])
```
```   869 apply (simp add: induced_def)
```
```   870 done
```
```   871
```
```   872 lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |]
```
```   873       ==> ((%x: intY1. f x) z, z) \<in> induced intY1 r"
```
```   874 apply (simp add: induced_def  intY1_f_closed z_in_interval P_def)
```
```   875 apply (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD]
```
```   876                  reflE)
```
```   877 done
```
```   878
```
```   879 lemma (in Tarski) tarski_full_lemma:
```
```   880      "\<exists>L. isLub Y (| pset = P, order = induced P r |) L"
```
```   881 apply (rule_tac x = "v" in exI)
```
```   882 apply (simp add: isLub_def)
```
```   883 -- {* @{text "v \<in> P"} *}
```
```   884 apply (simp add: v_in_P)
```
```   885 apply (rule conjI)
```
```   886 -- {* @{text v} is lub *}
```
```   887 -- {* @{text "1. \<forall>y:Y. (y, v) \<in> induced P r"} *}
```
```   888 apply (rule ballI)
```
```   889 apply (simp add: induced_def subsetD v_in_P)
```
```   890 apply (rule conjI)
```
```   891 apply (erule Y_ss [THEN subsetD])
```
```   892 apply (rule_tac b = "lub Y cl" in transE)
```
```   893 apply (rule lub_upper)
```
```   894 apply (rule Y_subset_A, assumption)
```
```   895 apply (rule_tac b = "Top cl" in interval_imp_mem)
```
```   896 apply (simp add: v_def)
```
```   897 apply (fold intY1_def)
```
```   898 apply (rule CL.glb_in_lattice [OF CL.intro [OF intY1_is_cl], simplified])
```
```   899 apply auto
```
```   900 apply (rule indI)
```
```   901   prefer 3 apply assumption
```
```   902  prefer 2 apply (simp add: v_in_P)
```
```   903 apply (unfold v_def)
```
```   904 apply (rule indE)
```
```   905 apply (rule_tac [2] intY1_subset)
```
```   906 apply (rule CL.glb_lower [OF CL.intro [OF intY1_is_cl], simplified])
```
```   907   apply (simp add: CL_imp_PO intY1_is_cl)
```
```   908  apply force
```
```   909 apply (simp add: induced_def intY1_f_closed z_in_interval)
```
```   910 apply (simp add: P_def fix_imp_eq [of _ f A] reflE
```
```   911                  fix_subset [of f A, THEN subsetD])
```
```   912 done
```
```   913
```
```   914 lemma CompleteLatticeI_simp:
```
```   915      "[| (| pset = A, order = r |) \<in> PartialOrder;
```
```   916          \<forall>S. S \<subseteq> A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]
```
```   917     ==> (| pset = A, order = r |) \<in> CompleteLattice"
```
```   918 by (simp add: CompleteLatticeI Rdual)
```
```   919
```
```   920 theorem (in CLF) Tarski_full:
```
```   921      "(| pset = P, order = induced P r|) \<in> CompleteLattice"
```
```   922 apply (rule CompleteLatticeI_simp)
```
```   923 apply (rule fixf_po, clarify)
```
```   924 apply (simp add: P_def A_def r_def)
```
```   925 apply (rule Tarski.tarski_full_lemma [OF Tarski.intro [OF _ Tarski_axioms.intro]])
```
```   926 proof - show "CLF cl f" .. qed
```
```   927
```
```   928 end
```