src/HOL/ex/Tarski.thy
 author obua Mon Apr 10 16:00:34 2006 +0200 (2006-04-10) changeset 19404 9bf2cdc9e8e8 parent 19316 c04b75d482c4 child 19736 d8d0f8f51d69 permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
```     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 imports Main FuncSet begin
```
```     9
```
```    10 text {*
```
```    11   Minimal version of lattice theory plus the full theorem of Tarski:
```
```    12   The fixedpoints of a complete lattice themselves form a complete
```
```    13   lattice.
```
```    14
```
```    15   Illustrates first-class theories, using the Sigma representation of
```
```    16   structures.  Tidied and converted to Isar by lcp.
```
```    17 *}
```
```    18
```
```    19 record 'a potype =
```
```    20   pset  :: "'a set"
```
```    21   order :: "('a * 'a) set"
```
```    22
```
```    23 constdefs
```
```    24   monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool"
```
```    25   "monotone f A r == \<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r"
```
```    26
```
```    27   least :: "['a => bool, 'a potype] => 'a"
```
```    28   "least P po == @ x. x: pset po & P x &
```
```    29                        (\<forall>y \<in> pset po. P y --> (x,y): order po)"
```
```    30
```
```    31   greatest :: "['a => bool, 'a potype] => 'a"
```
```    32   "greatest P po == @ x. x: pset po & P x &
```
```    33                           (\<forall>y \<in> pset po. P y --> (y,x): order po)"
```
```    34
```
```    35   lub  :: "['a set, 'a potype] => 'a"
```
```    36   "lub S po == least (%x. \<forall>y\<in>S. (y,x): order po) po"
```
```    37
```
```    38   glb  :: "['a set, 'a potype] => 'a"
```
```    39   "glb S po == greatest (%x. \<forall>y\<in>S. (x,y): order po) po"
```
```    40
```
```    41   isLub :: "['a set, 'a potype, 'a] => bool"
```
```    42   "isLub S po == %L. (L: pset po & (\<forall>y\<in>S. (y,L): order po) &
```
```    43                    (\<forall>z\<in>pset po. (\<forall>y\<in>S. (y,z): order po) --> (L,z): order po))"
```
```    44
```
```    45   isGlb :: "['a set, 'a potype, 'a] => bool"
```
```    46   "isGlb S po == %G. (G: pset po & (\<forall>y\<in>S. (G,y): order po) &
```
```    47                  (\<forall>z \<in> pset po. (\<forall>y\<in>S. (z,y): order po) --> (z,G): order po))"
```
```    48
```
```    49   "fix"    :: "[('a => 'a), 'a set] => 'a set"
```
```    50   "fix f A  == {x. x: A & f x = x}"
```
```    51
```
```    52   interval :: "[('a*'a) set,'a, 'a ] => 'a set"
```
```    53   "interval r a b == {x. (a,x): r & (x,b): r}"
```
```    54
```
```    55
```
```    56 constdefs
```
```    57   Bot :: "'a potype => 'a"
```
```    58   "Bot po == least (%x. True) po"
```
```    59
```
```    60   Top :: "'a potype => 'a"
```
```    61   "Top po == greatest (%x. True) po"
```
```    62
```
```    63   PartialOrder :: "('a potype) set"
```
```    64   "PartialOrder == {P. refl (pset P) (order P) & antisym (order P) &
```
```    65                        trans (order P)}"
```
```    66
```
```    67   CompleteLattice :: "('a potype) set"
```
```    68   "CompleteLattice == {cl. cl: PartialOrder &
```
```    69                         (\<forall>S. S \<subseteq> pset cl --> (\<exists>L. isLub S cl L)) &
```
```    70                         (\<forall>S. S \<subseteq> pset cl --> (\<exists>G. isGlb S cl G))}"
```
```    71
```
```    72   CLF :: "('a potype * ('a => 'a)) set"
```
```    73   "CLF == SIGMA cl: CompleteLattice.
```
```    74             {f. f: pset cl -> pset cl & monotone f (pset cl) (order cl)}"
```
```    75
```
```    76   induced :: "['a set, ('a * 'a) set] => ('a *'a)set"
```
```    77   "induced A r == {(a,b). a : A & b: A & (a,b): r}"
```
```    78
```
```    79
```
```    80 constdefs
```
```    81   sublattice :: "('a potype * 'a set)set"
```
```    82   "sublattice ==
```
```    83       SIGMA cl: CompleteLattice.
```
```    84           {S. S \<subseteq> pset cl &
```
```    85            (| pset = S, order = induced S (order cl) |): CompleteLattice }"
```
```    86
```
```    87 syntax
```
```    88   "@SL"  :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50)
```
```    89
```
```    90 translations
```
```    91   "S <<= cl" == "S : sublattice `` {cl}"
```
```    92
```
```    93 constdefs
```
```    94   dual :: "'a potype => 'a potype"
```
```    95   "dual po == (| pset = pset po, order = converse (order po) |)"
```
```    96
```
```    97 locale (open) PO =
```
```    98   fixes cl :: "'a potype"
```
```    99     and A  :: "'a set"
```
```   100     and r  :: "('a * 'a) set"
```
```   101   assumes cl_po:  "cl : PartialOrder"
```
```   102   defines A_def: "A == pset cl"
```
```   103      and  r_def: "r == order cl"
```
```   104
```
```   105 locale (open) CL = PO +
```
```   106   assumes cl_co:  "cl : CompleteLattice"
```
```   107
```
```   108 locale (open) CLF = CL +
```
```   109   fixes f :: "'a => 'a"
```
```   110     and P :: "'a set"
```
```   111   assumes f_cl:  "(cl,f) : CLF" (*was the equivalent "f : CLF``{cl}"*)
```
```   112   defines P_def: "P == fix f A"
```
```   113
```
```   114
```
```   115 locale (open) Tarski = CLF +
```
```   116   fixes Y     :: "'a set"
```
```   117     and intY1 :: "'a set"
```
```   118     and v     :: "'a"
```
```   119   assumes
```
```   120     Y_ss: "Y \<subseteq> P"
```
```   121   defines
```
```   122     intY1_def: "intY1 == interval r (lub Y cl) (Top cl)"
```
```   123     and v_def: "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r &
```
```   124                              x: intY1}
```
```   125                       (| pset=intY1, order=induced intY1 r|)"
```
```   126
```
```   127
```
```   128 subsection {* Partial Order *}
```
```   129
```
```   130 lemma (in PO) PO_imp_refl: "refl A r"
```
```   131 apply (insert cl_po)
```
```   132 apply (simp add: PartialOrder_def A_def r_def)
```
```   133 done
```
```   134
```
```   135 lemma (in PO) PO_imp_sym: "antisym r"
```
```   136 apply (insert cl_po)
```
```   137 apply (simp add: PartialOrder_def r_def)
```
```   138 done
```
```   139
```
```   140 lemma (in PO) PO_imp_trans: "trans r"
```
```   141 apply (insert cl_po)
```
```   142 apply (simp add: PartialOrder_def r_def)
```
```   143 done
```
```   144
```
```   145 lemma (in PO) reflE: "x \<in> A ==> (x, x) \<in> r"
```
```   146 apply (insert cl_po)
```
```   147 apply (simp add: PartialOrder_def refl_def A_def r_def)
```
```   148 done
```
```   149
```
```   150 lemma (in PO) antisymE: "[| (a, b) \<in> r; (b, a) \<in> r |] ==> a = b"
```
```   151 apply (insert cl_po)
```
```   152 apply (simp add: PartialOrder_def antisym_def r_def)
```
```   153 done
```
```   154
```
```   155 lemma (in PO) transE: "[| (a, b) \<in> r; (b, c) \<in> r|] ==> (a,c) \<in> r"
```
```   156 apply (insert cl_po)
```
```   157 apply (simp add: PartialOrder_def r_def)
```
```   158 apply (unfold trans_def, fast)
```
```   159 done
```
```   160
```
```   161 lemma (in PO) monotoneE:
```
```   162      "[| monotone f A r;  x \<in> A; y \<in> A; (x, y) \<in> r |] ==> (f x, f y) \<in> r"
```
```   163 by (simp add: monotone_def)
```
```   164
```
```   165 lemma (in PO) po_subset_po:
```
```   166      "S \<subseteq> A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
```
```   167 apply (simp (no_asm) add: PartialOrder_def)
```
```   168 apply auto
```
```   169 -- {* refl *}
```
```   170 apply (simp add: refl_def induced_def)
```
```   171 apply (blast intro: reflE)
```
```   172 -- {* antisym *}
```
```   173 apply (simp add: antisym_def induced_def)
```
```   174 apply (blast intro: antisymE)
```
```   175 -- {* trans *}
```
```   176 apply (simp add: trans_def induced_def)
```
```   177 apply (blast intro: transE)
```
```   178 done
```
```   179
```
```   180 lemma (in PO) indE: "[| (x, y) \<in> induced S r; S \<subseteq> A |] ==> (x, y) \<in> r"
```
```   181 by (simp add: add: induced_def)
```
```   182
```
```   183 lemma (in PO) indI: "[| (x, y) \<in> r; x \<in> S; y \<in> S |] ==> (x, y) \<in> induced S r"
```
```   184 by (simp add: add: induced_def)
```
```   185
```
```   186 lemma (in CL) CL_imp_ex_isLub: "S \<subseteq> A ==> \<exists>L. isLub S cl L"
```
```   187 apply (insert cl_co)
```
```   188 apply (simp add: CompleteLattice_def A_def)
```
```   189 done
```
```   190
```
```   191 declare (in CL) cl_co [simp]
```
```   192
```
```   193 lemma isLub_lub: "(\<exists>L. isLub S cl L) = isLub S cl (lub S cl)"
```
```   194 by (simp add: lub_def least_def isLub_def some_eq_ex [symmetric])
```
```   195
```
```   196 lemma isGlb_glb: "(\<exists>G. isGlb S cl G) = isGlb S cl (glb S cl)"
```
```   197 by (simp add: glb_def greatest_def isGlb_def some_eq_ex [symmetric])
```
```   198
```
```   199 lemma isGlb_dual_isLub: "isGlb S cl = isLub S (dual cl)"
```
```   200 by (simp add: isLub_def isGlb_def dual_def converse_def)
```
```   201
```
```   202 lemma isLub_dual_isGlb: "isLub S cl = isGlb S (dual cl)"
```
```   203 by (simp add: isLub_def isGlb_def dual_def converse_def)
```
```   204
```
```   205 lemma (in PO) dualPO: "dual cl \<in> PartialOrder"
```
```   206 apply (insert cl_po)
```
```   207 apply (simp add: PartialOrder_def dual_def refl_converse
```
```   208                  trans_converse antisym_converse)
```
```   209 done
```
```   210
```
```   211 lemma Rdual:
```
```   212      "\<forall>S. (S \<subseteq> A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
```
```   213       ==> \<forall>S. (S \<subseteq> A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))"
```
```   214 apply safe
```
```   215 apply (rule_tac x = "lub {y. y \<in> A & (\<forall>k \<in> S. (y, k) \<in> r)}
```
```   216                       (|pset = A, order = r|) " in exI)
```
```   217 apply (drule_tac x = "{y. y \<in> A & (\<forall>k \<in> S. (y,k) \<in> r) }" in spec)
```
```   218 apply (drule mp, fast)
```
```   219 apply (simp add: isLub_lub isGlb_def)
```
```   220 apply (simp add: isLub_def, blast)
```
```   221 done
```
```   222
```
```   223 lemma lub_dual_glb: "lub S cl = glb S (dual cl)"
```
```   224 by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
```
```   225
```
```   226 lemma glb_dual_lub: "glb S cl = lub S (dual cl)"
```
```   227 by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
```
```   228
```
```   229 lemma CL_subset_PO: "CompleteLattice \<subseteq> PartialOrder"
```
```   230 by (simp add: PartialOrder_def CompleteLattice_def, fast)
```
```   231
```
```   232 lemmas CL_imp_PO = CL_subset_PO [THEN subsetD]
```
```   233
```
```   234 declare CL_imp_PO [THEN Tarski.PO_imp_refl, simp]
```
```   235 declare CL_imp_PO [THEN Tarski.PO_imp_sym, simp]
```
```   236 declare CL_imp_PO [THEN Tarski.PO_imp_trans, simp]
```
```   237
```
```   238 lemma (in CL) CO_refl: "refl A r"
```
```   239 by (rule PO_imp_refl)
```
```   240
```
```   241 lemma (in CL) CO_antisym: "antisym r"
```
```   242 by (rule PO_imp_sym)
```
```   243
```
```   244 lemma (in CL) CO_trans: "trans r"
```
```   245 by (rule PO_imp_trans)
```
```   246
```
```   247 lemma CompleteLatticeI:
```
```   248      "[| po \<in> PartialOrder; (\<forall>S. S \<subseteq> pset po --> (\<exists>L. isLub S po L));
```
```   249          (\<forall>S. S \<subseteq> pset po --> (\<exists>G. isGlb S po G))|]
```
```   250       ==> po \<in> CompleteLattice"
```
```   251 apply (unfold CompleteLattice_def, blast)
```
```   252 done
```
```   253
```
```   254 lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice"
```
```   255 apply (insert cl_co)
```
```   256 apply (simp add: CompleteLattice_def dual_def)
```
```   257 apply (fold dual_def)
```
```   258 apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric]
```
```   259                  dualPO)
```
```   260 done
```
```   261
```
```   262 lemma (in PO) dualA_iff: "pset (dual cl) = pset cl"
```
```   263 by (simp add: dual_def)
```
```   264
```
```   265 lemma (in PO) dualr_iff: "((x, y) \<in> (order(dual cl))) = ((y, x) \<in> order cl)"
```
```   266 by (simp add: dual_def)
```
```   267
```
```   268 lemma (in PO) monotone_dual:
```
```   269      "monotone f (pset cl) (order cl)
```
```   270      ==> monotone f (pset (dual cl)) (order(dual cl))"
```
```   271 by (simp add: monotone_def dualA_iff dualr_iff)
```
```   272
```
```   273 lemma (in PO) interval_dual:
```
```   274      "[| x \<in> A; y \<in> A|] ==> interval r x y = interval (order(dual cl)) y x"
```
```   275 apply (simp add: interval_def dualr_iff)
```
```   276 apply (fold r_def, fast)
```
```   277 done
```
```   278
```
```   279 lemma (in PO) interval_not_empty:
```
```   280      "[| trans r; interval r a b \<noteq> {} |] ==> (a, b) \<in> r"
```
```   281 apply (simp add: interval_def)
```
```   282 apply (unfold trans_def, blast)
```
```   283 done
```
```   284
```
```   285 lemma (in PO) interval_imp_mem: "x \<in> interval r a b ==> (a, x) \<in> r"
```
```   286 by (simp add: interval_def)
```
```   287
```
```   288 lemma (in PO) left_in_interval:
```
```   289      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> a \<in> interval r a b"
```
```   290 apply (simp (no_asm_simp) add: interval_def)
```
```   291 apply (simp add: PO_imp_trans interval_not_empty)
```
```   292 apply (simp add: reflE)
```
```   293 done
```
```   294
```
```   295 lemma (in PO) right_in_interval:
```
```   296      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> b \<in> interval r a b"
```
```   297 apply (simp (no_asm_simp) add: interval_def)
```
```   298 apply (simp add: PO_imp_trans interval_not_empty)
```
```   299 apply (simp add: reflE)
```
```   300 done
```
```   301
```
```   302
```
```   303 subsection {* sublattice *}
```
```   304
```
```   305 lemma (in PO) sublattice_imp_CL:
```
```   306      "S <<= cl  ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
```
```   307 by (simp add: sublattice_def CompleteLattice_def r_def)
```
```   308
```
```   309 lemma (in CL) sublatticeI:
```
```   310      "[| S \<subseteq> A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
```
```   311       ==> S <<= cl"
```
```   312 by (simp add: sublattice_def A_def r_def)
```
```   313
```
```   314
```
```   315 subsection {* lub *}
```
```   316
```
```   317 lemma (in CL) lub_unique: "[| S \<subseteq> A; isLub S cl x; isLub S cl L|] ==> x = L"
```
```   318 apply (rule antisymE)
```
```   319 apply (auto simp add: isLub_def r_def)
```
```   320 done
```
```   321
```
```   322 lemma (in CL) lub_upper: "[|S \<subseteq> A; x \<in> S|] ==> (x, lub S cl) \<in> r"
```
```   323 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   324 apply (unfold lub_def least_def)
```
```   325 apply (rule some_equality [THEN ssubst])
```
```   326   apply (simp add: isLub_def)
```
```   327  apply (simp add: lub_unique A_def isLub_def)
```
```   328 apply (simp add: isLub_def r_def)
```
```   329 done
```
```   330
```
```   331 lemma (in CL) lub_least:
```
```   332      "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
```
```   333 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   334 apply (unfold lub_def least_def)
```
```   335 apply (rule_tac s=x in some_equality [THEN ssubst])
```
```   336   apply (simp add: isLub_def)
```
```   337  apply (simp add: lub_unique A_def isLub_def)
```
```   338 apply (simp add: isLub_def r_def A_def)
```
```   339 done
```
```   340
```
```   341 lemma (in CL) lub_in_lattice: "S \<subseteq> A ==> lub S cl \<in> A"
```
```   342 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   343 apply (unfold lub_def least_def)
```
```   344 apply (subst some_equality)
```
```   345 apply (simp add: isLub_def)
```
```   346 prefer 2 apply (simp add: isLub_def A_def)
```
```   347 apply (simp add: lub_unique A_def isLub_def)
```
```   348 done
```
```   349
```
```   350 lemma (in CL) lubI:
```
```   351      "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
```
```   352          \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"
```
```   353 apply (rule lub_unique, assumption)
```
```   354 apply (simp add: isLub_def A_def r_def)
```
```   355 apply (unfold isLub_def)
```
```   356 apply (rule conjI)
```
```   357 apply (fold A_def r_def)
```
```   358 apply (rule lub_in_lattice, assumption)
```
```   359 apply (simp add: lub_upper lub_least)
```
```   360 done
```
```   361
```
```   362 lemma (in CL) lubIa: "[| S \<subseteq> A; isLub S cl L |] ==> L = lub S cl"
```
```   363 by (simp add: lubI isLub_def A_def r_def)
```
```   364
```
```   365 lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"
```
```   366 by (simp add: isLub_def  A_def)
```
```   367
```
```   368 lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r"
```
```   369 by (simp add: isLub_def r_def)
```
```   370
```
```   371 lemma (in CL) isLub_least:
```
```   372      "[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r"
```
```   373 by (simp add: isLub_def A_def r_def)
```
```   374
```
```   375 lemma (in CL) isLubI:
```
```   376      "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;
```
```   377          (\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L"
```
```   378 by (simp add: isLub_def A_def r_def)
```
```   379
```
```   380
```
```   381 subsection {* glb *}
```
```   382
```
```   383 lemma (in CL) glb_in_lattice: "S \<subseteq> A ==> glb S cl \<in> A"
```
```   384 apply (subst glb_dual_lub)
```
```   385 apply (simp add: A_def)
```
```   386 apply (rule dualA_iff [THEN subst])
```
```   387 apply (rule Tarski.lub_in_lattice)
```
```   388 apply (rule dualPO)
```
```   389 apply (rule CL_dualCL)
```
```   390 apply (simp add: dualA_iff)
```
```   391 done
```
```   392
```
```   393 lemma (in CL) glb_lower: "[|S \<subseteq> A; x \<in> S|] ==> (glb S cl, x) \<in> r"
```
```   394 apply (subst glb_dual_lub)
```
```   395 apply (simp add: r_def)
```
```   396 apply (rule dualr_iff [THEN subst])
```
```   397 apply (rule Tarski.lub_upper)
```
```   398 apply (rule dualPO)
```
```   399 apply (rule CL_dualCL)
```
```   400 apply (simp add: dualA_iff A_def, assumption)
```
```   401 done
```
```   402
```
```   403 text {*
```
```   404   Reduce the sublattice property by using substructural properties;
```
```   405   abandoned see @{text "Tarski_4.ML"}.
```
```   406 *}
```
```   407
```
```   408 lemma (in CLF) [simp]:
```
```   409     "f: pset cl -> pset cl & monotone f (pset cl) (order cl)"
```
```   410 apply (insert f_cl)
```
```   411 apply (simp add: CLF_def)
```
```   412 done
```
```   413
```
```   414 declare (in CLF) f_cl [simp]
```
```   415
```
```   416
```
```   417 lemma (in CLF) f_in_funcset: "f \<in> A -> A"
```
```   418 by (simp add: A_def)
```
```   419
```
```   420 lemma (in CLF) monotone_f: "monotone f A r"
```
```   421 by (simp add: A_def r_def)
```
```   422
```
```   423 lemma (in CLF) CLF_dual: "(cl,f) \<in> CLF ==> (dual cl, f) \<in> CLF"
```
```   424 apply (simp add: CLF_def  CL_dualCL monotone_dual)
```
```   425 apply (simp add: dualA_iff)
```
```   426 done
```
```   427
```
```   428
```
```   429 subsection {* fixed points *}
```
```   430
```
```   431 lemma fix_subset: "fix f A \<subseteq> A"
```
```   432 by (simp add: fix_def, fast)
```
```   433
```
```   434 lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"
```
```   435 by (simp add: fix_def)
```
```   436
```
```   437 lemma fixf_subset:
```
```   438      "[| A \<subseteq> B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
```
```   439 by (simp add: fix_def, auto)
```
```   440
```
```   441
```
```   442 subsection {* lemmas for Tarski, lub *}
```
```   443 lemma (in CLF) lubH_le_flubH:
```
```   444      "H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r"
```
```   445 apply (rule lub_least, fast)
```
```   446 apply (rule f_in_funcset [THEN funcset_mem])
```
```   447 apply (rule lub_in_lattice, fast)
```
```   448 -- {* @{text "\<forall>x:H. (x, f (lub H r)) \<in> r"} *}
```
```   449 apply (rule ballI)
```
```   450 apply (rule transE)
```
```   451 -- {* instantiates @{text "(x, ???z) \<in> order cl to (x, f x)"}, *}
```
```   452 -- {* because of the def of @{text H} *}
```
```   453 apply fast
```
```   454 -- {* so it remains to show @{text "(f x, f (lub H cl)) \<in> r"} *}
```
```   455 apply (rule_tac f = "f" in monotoneE)
```
```   456 apply (rule monotone_f, fast)
```
```   457 apply (rule lub_in_lattice, fast)
```
```   458 apply (rule lub_upper, fast)
```
```   459 apply assumption
```
```   460 done
```
```   461
```
```   462 lemma (in CLF) flubH_le_lubH:
```
```   463      "[|  H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r"
```
```   464 apply (rule lub_upper, fast)
```
```   465 apply (rule_tac t = "H" in ssubst, assumption)
```
```   466 apply (rule CollectI)
```
```   467 apply (rule conjI)
```
```   468 apply (rule_tac [2] f_in_funcset [THEN funcset_mem])
```
```   469 apply (rule_tac [2] lub_in_lattice)
```
```   470 prefer 2 apply fast
```
```   471 apply (rule_tac f = "f" in monotoneE)
```
```   472 apply (rule monotone_f)
```
```   473   apply (blast intro: lub_in_lattice)
```
```   474  apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem])
```
```   475 apply (simp add: lubH_le_flubH)
```
```   476 done
```
```   477
```
```   478 lemma (in CLF) lubH_is_fixp:
```
```   479      "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
```
```   480 apply (simp add: fix_def)
```
```   481 apply (rule conjI)
```
```   482 apply (rule lub_in_lattice, fast)
```
```   483 apply (rule antisymE)
```
```   484 apply (simp add: flubH_le_lubH)
```
```   485 apply (simp add: lubH_le_flubH)
```
```   486 done
```
```   487
```
```   488 lemma (in CLF) fix_in_H:
```
```   489      "[| H = {x. (x, f x) \<in> r & x \<in> A};  x \<in> P |] ==> x \<in> H"
```
```   490 by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl
```
```   491                     fix_subset [of f A, THEN subsetD])
```
```   492
```
```   493 lemma (in CLF) fixf_le_lubH:
```
```   494      "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
```
```   495 apply (rule ballI)
```
```   496 apply (rule lub_upper, fast)
```
```   497 apply (rule fix_in_H)
```
```   498 apply (simp_all add: P_def)
```
```   499 done
```
```   500
```
```   501 lemma (in CLF) lubH_least_fixf:
```
```   502      "H = {x. (x, f x) \<in> r & x \<in> A}
```
```   503       ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"
```
```   504 apply (rule allI)
```
```   505 apply (rule impI)
```
```   506 apply (erule bspec)
```
```   507 apply (rule lubH_is_fixp, assumption)
```
```   508 done
```
```   509
```
```   510 subsection {* Tarski fixpoint theorem 1, first part *}
```
```   511 lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"
```
```   512 apply (rule sym)
```
```   513 apply (simp add: P_def)
```
```   514 apply (rule lubI)
```
```   515 apply (rule fix_subset)
```
```   516 apply (rule lub_in_lattice, fast)
```
```   517 apply (simp add: fixf_le_lubH)
```
```   518 apply (simp add: lubH_least_fixf)
```
```   519 done
```
```   520
```
```   521 lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P"
```
```   522   -- {* Tarski for glb *}
```
```   523 apply (simp add: glb_dual_lub P_def A_def r_def)
```
```   524 apply (rule dualA_iff [THEN subst])
```
```   525 apply (rule Tarski.lubH_is_fixp)
```
```   526 apply (rule dualPO)
```
```   527 apply (rule CL_dualCL)
```
```   528 apply (rule f_cl [THEN CLF_dual])
```
```   529 apply (simp add: dualr_iff dualA_iff)
```
```   530 done
```
```   531
```
```   532 lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl"
```
```   533 apply (simp add: glb_dual_lub P_def A_def r_def)
```
```   534 apply (rule dualA_iff [THEN subst])
```
```   535 apply (simp add: Tarski.T_thm_1_lub [of _ f, OF dualPO CL_dualCL]
```
```   536                  dualPO CL_dualCL CLF_dual dualr_iff)
```
```   537 done
```
```   538
```
```   539 subsection {* interval *}
```
```   540
```
```   541 lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"
```
```   542 apply (insert CO_refl)
```
```   543 apply (simp add: refl_def, blast)
```
```   544 done
```
```   545
```
```   546 lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b \<subseteq> A"
```
```   547 apply (simp add: interval_def)
```
```   548 apply (blast intro: rel_imp_elem)
```
```   549 done
```
```   550
```
```   551 lemma (in CLF) intervalI:
```
```   552      "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"
```
```   553 by (simp add: interval_def)
```
```   554
```
```   555 lemma (in CLF) interval_lemma1:
```
```   556      "[| S \<subseteq> interval r a b; x \<in> S |] ==> (a, x) \<in> r"
```
```   557 by (unfold interval_def, fast)
```
```   558
```
```   559 lemma (in CLF) interval_lemma2:
```
```   560      "[| S \<subseteq> interval r a b; x \<in> S |] ==> (x, b) \<in> r"
```
```   561 by (unfold interval_def, fast)
```
```   562
```
```   563 lemma (in CLF) a_less_lub:
```
```   564      "[| S \<subseteq> A; S \<noteq> {};
```
```   565          \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"
```
```   566 by (blast intro: transE)
```
```   567
```
```   568 lemma (in CLF) glb_less_b:
```
```   569      "[| S \<subseteq> A; S \<noteq> {};
```
```   570          \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"
```
```   571 by (blast intro: transE)
```
```   572
```
```   573 lemma (in CLF) S_intv_cl:
```
```   574      "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b |]==> S \<subseteq> A"
```
```   575 by (simp add: subset_trans [OF _ interval_subset])
```
```   576
```
```   577 lemma (in CLF) L_in_interval:
```
```   578      "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b;
```
```   579          S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"
```
```   580 apply (rule intervalI)
```
```   581 apply (rule a_less_lub)
```
```   582 prefer 2 apply assumption
```
```   583 apply (simp add: S_intv_cl)
```
```   584 apply (rule ballI)
```
```   585 apply (simp add: interval_lemma1)
```
```   586 apply (simp add: isLub_upper)
```
```   587 -- {* @{text "(L, b) \<in> r"} *}
```
```   588 apply (simp add: isLub_least interval_lemma2)
```
```   589 done
```
```   590
```
```   591 lemma (in CLF) G_in_interval:
```
```   592      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S \<subseteq> interval r a b; isGlb S cl G;
```
```   593          S \<noteq> {} |] ==> G \<in> interval r a b"
```
```   594 apply (simp add: interval_dual)
```
```   595 apply (simp add: Tarski.L_in_interval [of _ f]
```
```   596                  dualA_iff A_def dualPO CL_dualCL CLF_dual isGlb_dual_isLub)
```
```   597 done
```
```   598
```
```   599 lemma (in CLF) intervalPO:
```
```   600      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   601       ==> (| pset = interval r a b, order = induced (interval r a b) r |)
```
```   602           \<in> PartialOrder"
```
```   603 apply (rule po_subset_po)
```
```   604 apply (simp add: interval_subset)
```
```   605 done
```
```   606
```
```   607 lemma (in CLF) intv_CL_lub:
```
```   608  "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   609   ==> \<forall>S. S \<subseteq> interval r a b -->
```
```   610           (\<exists>L. isLub S (| pset = interval r a b,
```
```   611                           order = induced (interval r a b) r |)  L)"
```
```   612 apply (intro strip)
```
```   613 apply (frule S_intv_cl [THEN CL_imp_ex_isLub])
```
```   614 prefer 2 apply assumption
```
```   615 apply assumption
```
```   616 apply (erule exE)
```
```   617 -- {* define the lub for the interval as *}
```
```   618 apply (rule_tac x = "if S = {} then a else L" in exI)
```
```   619 apply (simp (no_asm_simp) add: isLub_def split del: split_if)
```
```   620 apply (intro impI conjI)
```
```   621 -- {* @{text "(if S = {} then a else L) \<in> interval r a b"} *}
```
```   622 apply (simp add: CL_imp_PO L_in_interval)
```
```   623 apply (simp add: left_in_interval)
```
```   624 -- {* lub prop 1 *}
```
```   625 apply (case_tac "S = {}")
```
```   626 -- {* @{text "S = {}, y \<in> S = False => everything"} *}
```
```   627 apply fast
```
```   628 -- {* @{text "S \<noteq> {}"} *}
```
```   629 apply simp
```
```   630 -- {* @{text "\<forall>y:S. (y, L) \<in> induced (interval r a b) r"} *}
```
```   631 apply (rule ballI)
```
```   632 apply (simp add: induced_def  L_in_interval)
```
```   633 apply (rule conjI)
```
```   634 apply (rule subsetD)
```
```   635 apply (simp add: S_intv_cl, assumption)
```
```   636 apply (simp add: isLub_upper)
```
```   637 -- {* @{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"} *}
```
```   638 apply (rule ballI)
```
```   639 apply (rule impI)
```
```   640 apply (case_tac "S = {}")
```
```   641 -- {* @{text "S = {}"} *}
```
```   642 apply simp
```
```   643 apply (simp add: induced_def  interval_def)
```
```   644 apply (rule conjI)
```
```   645 apply (rule reflE, assumption)
```
```   646 apply (rule interval_not_empty)
```
```   647 apply (rule CO_trans)
```
```   648 apply (simp add: interval_def)
```
```   649 -- {* @{text "S \<noteq> {}"} *}
```
```   650 apply simp
```
```   651 apply (simp add: induced_def  L_in_interval)
```
```   652 apply (rule isLub_least, assumption)
```
```   653 apply (rule subsetD)
```
```   654 prefer 2 apply assumption
```
```   655 apply (simp add: S_intv_cl, fast)
```
```   656 done
```
```   657
```
```   658 lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual]
```
```   659
```
```   660 lemma (in CLF) interval_is_sublattice:
```
```   661      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   662         ==> interval r a b <<= cl"
```
```   663 apply (rule sublatticeI)
```
```   664 apply (simp add: interval_subset)
```
```   665 apply (rule CompleteLatticeI)
```
```   666 apply (simp add: intervalPO)
```
```   667  apply (simp add: intv_CL_lub)
```
```   668 apply (simp add: intv_CL_glb)
```
```   669 done
```
```   670
```
```   671 lemmas (in CLF) interv_is_compl_latt =
```
```   672     interval_is_sublattice [THEN sublattice_imp_CL]
```
```   673
```
```   674
```
```   675 subsection {* Top and Bottom *}
```
```   676 lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"
```
```   677 by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
```
```   678
```
```   679 lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)"
```
```   680 by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
```
```   681
```
```   682 lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"
```
```   683 apply (simp add: Bot_def least_def)
```
```   684 apply (rule_tac a="glb A cl" in someI2)
```
```   685 apply (simp_all add: glb_in_lattice glb_lower
```
```   686                      r_def [symmetric] A_def [symmetric])
```
```   687 done
```
```   688
```
```   689 lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
```
```   690 apply (simp add: Top_dual_Bot A_def)
```
```   691 apply (rule dualA_iff [THEN subst])
```
```   692 apply (blast intro!: Tarski.Bot_in_lattice dualPO CL_dualCL CLF_dual f_cl)
```
```   693 done
```
```   694
```
```   695 lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
```
```   696 apply (simp add: Top_def greatest_def)
```
```   697 apply (rule_tac a="lub A cl" in someI2)
```
```   698 apply (rule someI2)
```
```   699 apply (simp_all add: lub_in_lattice lub_upper
```
```   700                      r_def [symmetric] A_def [symmetric])
```
```   701 done
```
```   702
```
```   703 lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
```
```   704 apply (simp add: Bot_dual_Top r_def)
```
```   705 apply (rule dualr_iff [THEN subst])
```
```   706 apply (simp add: Tarski.Top_prop [of _ f]
```
```   707                  dualA_iff A_def dualPO CL_dualCL CLF_dual)
```
```   708 done
```
```   709
```
```   710 lemma (in CLF) Top_intv_not_empty: "x \<in> A  ==> interval r x (Top cl) \<noteq> {}"
```
```   711 apply (rule notI)
```
```   712 apply (drule_tac a = "Top cl" in equals0D)
```
```   713 apply (simp add: interval_def)
```
```   714 apply (simp add: refl_def Top_in_lattice Top_prop)
```
```   715 done
```
```   716
```
```   717 lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}"
```
```   718 apply (simp add: Bot_dual_Top)
```
```   719 apply (subst interval_dual)
```
```   720 prefer 2 apply assumption
```
```   721 apply (simp add: A_def)
```
```   722 apply (rule dualA_iff [THEN subst])
```
```   723 apply (blast intro!: Tarski.Top_in_lattice
```
```   724                  f_cl dualPO CL_dualCL CLF_dual)
```
```   725 apply (simp add: Tarski.Top_intv_not_empty [of _ f]
```
```   726                  dualA_iff A_def dualPO CL_dualCL CLF_dual)
```
```   727 done
```
```   728
```
```   729 subsection {* fixed points form a partial order *}
```
```   730
```
```   731 lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
```
```   732 by (simp add: P_def fix_subset po_subset_po)
```
```   733
```
```   734 lemma (in Tarski) Y_subset_A: "Y \<subseteq> A"
```
```   735 apply (rule subset_trans [OF _ fix_subset])
```
```   736 apply (rule Y_ss [simplified P_def])
```
```   737 done
```
```   738
```
```   739 lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A"
```
```   740   by (rule Y_subset_A [THEN lub_in_lattice])
```
```   741
```
```   742 lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"
```
```   743 apply (rule lub_least)
```
```   744 apply (rule Y_subset_A)
```
```   745 apply (rule f_in_funcset [THEN funcset_mem])
```
```   746 apply (rule lubY_in_A)
```
```   747 -- {* @{text "Y \<subseteq> P ==> f x = x"} *}
```
```   748 apply (rule ballI)
```
```   749 apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
```
```   750 apply (erule Y_ss [simplified P_def, THEN subsetD])
```
```   751 -- {* @{text "reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r"} by monotonicity *}
```
```   752 apply (rule_tac f = "f" in monotoneE)
```
```   753 apply (rule monotone_f)
```
```   754 apply (simp add: Y_subset_A [THEN subsetD])
```
```   755 apply (rule lubY_in_A)
```
```   756 apply (simp add: lub_upper Y_subset_A)
```
```   757 done
```
```   758
```
```   759 lemma (in Tarski) intY1_subset: "intY1 \<subseteq> A"
```
```   760 apply (unfold intY1_def)
```
```   761 apply (rule interval_subset)
```
```   762 apply (rule lubY_in_A)
```
```   763 apply (rule Top_in_lattice)
```
```   764 done
```
```   765
```
```   766 lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD]
```
```   767
```
```   768 lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"
```
```   769 apply (simp add: intY1_def  interval_def)
```
```   770 apply (rule conjI)
```
```   771 apply (rule transE)
```
```   772 apply (rule lubY_le_flubY)
```
```   773 -- {* @{text "(f (lub Y cl), f x) \<in> r"} *}
```
```   774 apply (rule_tac f=f in monotoneE)
```
```   775 apply (rule monotone_f)
```
```   776 apply (rule lubY_in_A)
```
```   777 apply (simp add: intY1_def interval_def  intY1_elem)
```
```   778 apply (simp add: intY1_def  interval_def)
```
```   779 -- {* @{text "(f x, Top cl) \<in> r"} *}
```
```   780 apply (rule Top_prop)
```
```   781 apply (rule f_in_funcset [THEN funcset_mem])
```
```   782 apply (simp add: intY1_def interval_def  intY1_elem)
```
```   783 done
```
```   784
```
```   785 lemma (in Tarski) intY1_func: "(%x: intY1. f x) \<in> intY1 -> intY1"
```
```   786 apply (rule restrictI)
```
```   787 apply (erule intY1_f_closed)
```
```   788 done
```
```   789
```
```   790 lemma (in Tarski) intY1_mono:
```
```   791      "monotone (%x: intY1. f x) intY1 (induced intY1 r)"
```
```   792 apply (auto simp add: monotone_def induced_def intY1_f_closed)
```
```   793 apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
```
```   794 done
```
```   795
```
```   796 lemma (in Tarski) intY1_is_cl:
```
```   797     "(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
```
```   798 apply (unfold intY1_def)
```
```   799 apply (rule interv_is_compl_latt)
```
```   800 apply (rule lubY_in_A)
```
```   801 apply (rule Top_in_lattice)
```
```   802 apply (rule Top_intv_not_empty)
```
```   803 apply (rule lubY_in_A)
```
```   804 done
```
```   805
```
```   806 lemma (in Tarski) v_in_P: "v \<in> P"
```
```   807 apply (unfold P_def)
```
```   808 apply (rule_tac A = "intY1" in fixf_subset)
```
```   809 apply (rule intY1_subset)
```
```   810 apply (simp add: Tarski.glbH_is_fixp [OF _ intY1_is_cl, simplified]
```
```   811                  v_def CL_imp_PO intY1_is_cl CLF_def intY1_func intY1_mono)
```
```   812 done
```
```   813
```
```   814 lemma (in Tarski) z_in_interval:
```
```   815      "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1"
```
```   816 apply (unfold intY1_def P_def)
```
```   817 apply (rule intervalI)
```
```   818 prefer 2
```
```   819  apply (erule fix_subset [THEN subsetD, THEN Top_prop])
```
```   820 apply (rule lub_least)
```
```   821 apply (rule Y_subset_A)
```
```   822 apply (fast elim!: fix_subset [THEN subsetD])
```
```   823 apply (simp add: induced_def)
```
```   824 done
```
```   825
```
```   826 lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |]
```
```   827       ==> ((%x: intY1. f x) z, z) \<in> induced intY1 r"
```
```   828 apply (simp add: induced_def  intY1_f_closed z_in_interval P_def)
```
```   829 apply (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD]
```
```   830                  reflE)
```
```   831 done
```
```   832
```
```   833 lemma (in Tarski) tarski_full_lemma:
```
```   834      "\<exists>L. isLub Y (| pset = P, order = induced P r |) L"
```
```   835 apply (rule_tac x = "v" in exI)
```
```   836 apply (simp add: isLub_def)
```
```   837 -- {* @{text "v \<in> P"} *}
```
```   838 apply (simp add: v_in_P)
```
```   839 apply (rule conjI)
```
```   840 -- {* @{text v} is lub *}
```
```   841 -- {* @{text "1. \<forall>y:Y. (y, v) \<in> induced P r"} *}
```
```   842 apply (rule ballI)
```
```   843 apply (simp add: induced_def subsetD v_in_P)
```
```   844 apply (rule conjI)
```
```   845 apply (erule Y_ss [THEN subsetD])
```
```   846 apply (rule_tac b = "lub Y cl" in transE)
```
```   847 apply (rule lub_upper)
```
```   848 apply (rule Y_subset_A, assumption)
```
```   849 apply (rule_tac b = "Top cl" in interval_imp_mem)
```
```   850 apply (simp add: v_def)
```
```   851 apply (fold intY1_def)
```
```   852 apply (rule Tarski.glb_in_lattice [OF _ intY1_is_cl, simplified])
```
```   853  apply (simp add: CL_imp_PO intY1_is_cl, force)
```
```   854 -- {* @{text v} is LEAST ub *}
```
```   855 apply clarify
```
```   856 apply (rule indI)
```
```   857   prefer 3 apply assumption
```
```   858  prefer 2 apply (simp add: v_in_P)
```
```   859 apply (unfold v_def)
```
```   860 apply (rule indE)
```
```   861 apply (rule_tac [2] intY1_subset)
```
```   862 apply (rule Tarski.glb_lower [OF _ intY1_is_cl, simplified])
```
```   863   apply (simp add: CL_imp_PO intY1_is_cl)
```
```   864  apply force
```
```   865 apply (simp add: induced_def intY1_f_closed z_in_interval)
```
```   866 apply (simp add: P_def fix_imp_eq [of _ f A] reflE
```
```   867                  fix_subset [of f A, THEN subsetD])
```
```   868 done
```
```   869
```
```   870 lemma CompleteLatticeI_simp:
```
```   871      "[| (| pset = A, order = r |) \<in> PartialOrder;
```
```   872          \<forall>S. S \<subseteq> A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]
```
```   873     ==> (| pset = A, order = r |) \<in> CompleteLattice"
```
```   874 by (simp add: CompleteLatticeI Rdual)
```
```   875
```
```   876 theorem (in CLF) Tarski_full:
```
```   877      "(| pset = P, order = induced P r|) \<in> CompleteLattice"
```
```   878 apply (rule CompleteLatticeI_simp)
```
```   879 apply (rule fixf_po, clarify)
```
```   880 apply (simp add: P_def A_def r_def)
```
```   881 apply (blast intro!: Tarski.tarski_full_lemma cl_po cl_co f_cl)
```
```   882 done
```
```   883
```
```   884 end
```