src/HOL/ex/Tarski.thy
 author wenzelm Tue Jul 16 18:52:26 2002 +0200 (2002-07-16) changeset 13383 041d78bf9403 parent 13115 0a6fbdedcde2 child 13585 db4005b40cc6 permissions -rw-r--r--
```     1 (*  Title:      HOL/ex/Tarski.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Florian Kammüller, Cambridge University Computer Laboratory
```
```     4     Copyright   1999  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header {* The full theorem of Tarski *}
```
```     8
```
```     9 theory Tarski = Main:
```
```    10
```
```    11 text {*
```
```    12   Minimal version of lattice theory plus the full theorem of Tarski:
```
```    13   The fixedpoints of a complete lattice themselves form a complete
```
```    14   lattice.
```
```    15
```
```    16   Illustrates first-class theories, using the Sigma representation of
```
```    17   structures.  Tidied and converted to Isar by lcp.
```
```    18 *}
```
```    19
```
```    20 record 'a potype =
```
```    21   pset  :: "'a set"
```
```    22   order :: "('a * 'a) set"
```
```    23
```
```    24 syntax
```
```    25   "@pset"  :: "'a potype => 'a set"            ("_ .<A>"  [90] 90)
```
```    26   "@order" :: "'a potype => ('a *'a)set"       ("_ .<r>"  [90] 90)
```
```    27
```
```    28 translations
```
```    29   "po.<A>" == "pset po"
```
```    30   "po.<r>" == "order po"
```
```    31
```
```    32 constdefs
```
```    33   monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool"
```
```    34   "monotone f A r == \<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r"
```
```    35
```
```    36   least :: "['a => bool, 'a potype] => 'a"
```
```    37   "least P po == @ x. x: po.<A> & P x &
```
```    38                        (\<forall>y \<in> po.<A>. P y --> (x,y): po.<r>)"
```
```    39
```
```    40   greatest :: "['a => bool, 'a potype] => 'a"
```
```    41   "greatest P po == @ x. x: po.<A> & P x &
```
```    42                           (\<forall>y \<in> po.<A>. P y --> (y,x): po.<r>)"
```
```    43
```
```    44   lub  :: "['a set, 'a potype] => 'a"
```
```    45   "lub S po == least (%x. \<forall>y\<in>S. (y,x): po.<r>) po"
```
```    46
```
```    47   glb  :: "['a set, 'a potype] => 'a"
```
```    48   "glb S po == greatest (%x. \<forall>y\<in>S. (x,y): po.<r>) po"
```
```    49
```
```    50   isLub :: "['a set, 'a potype, 'a] => bool"
```
```    51   "isLub S po == %L. (L: po.<A> & (\<forall>y\<in>S. (y,L): po.<r>) &
```
```    52                      (\<forall>z\<in>po.<A>. (\<forall>y\<in>S. (y,z): po.<r>) --> (L,z): po.<r>))"
```
```    53
```
```    54   isGlb :: "['a set, 'a potype, 'a] => bool"
```
```    55   "isGlb S po == %G. (G: po.<A> & (\<forall>y\<in>S. (G,y): po.<r>) &
```
```    56                      (\<forall>z \<in> po.<A>. (\<forall>y\<in>S. (z,y): po.<r>) --> (z,G): po.<r>))"
```
```    57
```
```    58   "fix"    :: "[('a => 'a), 'a set] => 'a set"
```
```    59   "fix f A  == {x. x: A & f x = x}"
```
```    60
```
```    61   interval :: "[('a*'a) set,'a, 'a ] => 'a set"
```
```    62   "interval r a b == {x. (a,x): r & (x,b): r}"
```
```    63
```
```    64
```
```    65 constdefs
```
```    66   Bot :: "'a potype => 'a"
```
```    67   "Bot po == least (%x. True) po"
```
```    68
```
```    69   Top :: "'a potype => 'a"
```
```    70   "Top po == greatest (%x. True) po"
```
```    71
```
```    72   PartialOrder :: "('a potype) set"
```
```    73   "PartialOrder == {P. refl (P.<A>) (P.<r>) & antisym (P.<r>) &
```
```    74                        trans (P.<r>)}"
```
```    75
```
```    76   CompleteLattice :: "('a potype) set"
```
```    77   "CompleteLattice == {cl. cl: PartialOrder &
```
```    78                         (\<forall>S. S <= cl.<A> --> (\<exists>L. isLub S cl L)) &
```
```    79                         (\<forall>S. S <= cl.<A> --> (\<exists>G. isGlb S cl G))}"
```
```    80
```
```    81   CLF :: "('a potype * ('a => 'a)) set"
```
```    82   "CLF == SIGMA cl: CompleteLattice.
```
```    83             {f. f: cl.<A> funcset cl.<A> & monotone f (cl.<A>) (cl.<r>)}"
```
```    84
```
```    85   induced :: "['a set, ('a * 'a) set] => ('a *'a)set"
```
```    86   "induced A r == {(a,b). a : A & b: A & (a,b): r}"
```
```    87
```
```    88
```
```    89 constdefs
```
```    90   sublattice :: "('a potype * 'a set)set"
```
```    91   "sublattice ==
```
```    92       SIGMA cl: CompleteLattice.
```
```    93           {S. S <= cl.<A> &
```
```    94            (| pset = S, order = induced S (cl.<r>) |): CompleteLattice }"
```
```    95
```
```    96 syntax
```
```    97   "@SL"  :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50)
```
```    98
```
```    99 translations
```
```   100   "S <<= cl" == "S : sublattice `` {cl}"
```
```   101
```
```   102 constdefs
```
```   103   dual :: "'a potype => 'a potype"
```
```   104   "dual po == (| pset = po.<A>, order = converse (po.<r>) |)"
```
```   105
```
```   106 locale (open) PO =
```
```   107   fixes cl :: "'a potype"
```
```   108     and A  :: "'a set"
```
```   109     and r  :: "('a * 'a) set"
```
```   110   assumes cl_po:  "cl : PartialOrder"
```
```   111   defines A_def: "A == cl.<A>"
```
```   112      and  r_def: "r == cl.<r>"
```
```   113
```
```   114 locale (open) CL = PO +
```
```   115   assumes cl_co:  "cl : CompleteLattice"
```
```   116
```
```   117 locale (open) CLF = CL +
```
```   118   fixes f :: "'a => 'a"
```
```   119     and P :: "'a set"
```
```   120   assumes f_cl:  "(cl,f) : CLF" (*was the equivalent "f : CLF``{cl}"*)
```
```   121   defines P_def: "P == fix f A"
```
```   122
```
```   123
```
```   124 locale (open) Tarski = CLF +
```
```   125   fixes Y     :: "'a set"
```
```   126     and intY1 :: "'a set"
```
```   127     and v     :: "'a"
```
```   128   assumes
```
```   129     Y_ss: "Y <= P"
```
```   130   defines
```
```   131     intY1_def: "intY1 == interval r (lub Y cl) (Top cl)"
```
```   132     and v_def: "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r &
```
```   133                              x: intY1}
```
```   134                       (| pset=intY1, order=induced intY1 r|)"
```
```   135
```
```   136
```
```   137 subsubsection {* Partial Order *}
```
```   138
```
```   139 lemma (in PO) PO_imp_refl: "refl A r"
```
```   140 apply (insert cl_po)
```
```   141 apply (simp add: PartialOrder_def A_def r_def)
```
```   142 done
```
```   143
```
```   144 lemma (in PO) PO_imp_sym: "antisym r"
```
```   145 apply (insert cl_po)
```
```   146 apply (simp add: PartialOrder_def A_def r_def)
```
```   147 done
```
```   148
```
```   149 lemma (in PO) PO_imp_trans: "trans r"
```
```   150 apply (insert cl_po)
```
```   151 apply (simp add: PartialOrder_def A_def r_def)
```
```   152 done
```
```   153
```
```   154 lemma (in PO) reflE: "[| refl A r; x \<in> A|] ==> (x, x) \<in> r"
```
```   155 apply (insert cl_po)
```
```   156 apply (simp add: PartialOrder_def refl_def)
```
```   157 done
```
```   158
```
```   159 lemma (in PO) antisymE: "[| antisym r; (a, b) \<in> r; (b, a) \<in> r |] ==> a = b"
```
```   160 apply (insert cl_po)
```
```   161 apply (simp add: PartialOrder_def antisym_def)
```
```   162 done
```
```   163
```
```   164 lemma (in PO) transE: "[| trans r; (a, b) \<in> r; (b, c) \<in> r|] ==> (a,c) \<in> r"
```
```   165 apply (insert cl_po)
```
```   166 apply (simp add: PartialOrder_def)
```
```   167 apply (unfold trans_def, fast)
```
```   168 done
```
```   169
```
```   170 lemma (in PO) monotoneE:
```
```   171      "[| monotone f A r;  x \<in> A; y \<in> A; (x, y) \<in> r |] ==> (f x, f y) \<in> r"
```
```   172 by (simp add: monotone_def)
```
```   173
```
```   174 lemma (in PO) po_subset_po:
```
```   175      "S <= A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
```
```   176 apply (simp (no_asm) add: PartialOrder_def)
```
```   177 apply auto
```
```   178 -- {* refl *}
```
```   179 apply (simp add: refl_def induced_def)
```
```   180 apply (blast intro: PO_imp_refl [THEN reflE])
```
```   181 -- {* antisym *}
```
```   182 apply (simp add: antisym_def induced_def)
```
```   183 apply (blast intro: PO_imp_sym [THEN antisymE])
```
```   184 -- {* trans *}
```
```   185 apply (simp add: trans_def induced_def)
```
```   186 apply (blast intro: PO_imp_trans [THEN transE])
```
```   187 done
```
```   188
```
```   189 lemma (in PO) indE: "[| (x, y) \<in> induced S r; S <= A |] ==> (x, y) \<in> r"
```
```   190 by (simp add: add: induced_def)
```
```   191
```
```   192 lemma (in PO) indI: "[| (x, y) \<in> r; x \<in> S; y \<in> S |] ==> (x, y) \<in> induced S r"
```
```   193 by (simp add: add: induced_def)
```
```   194
```
```   195 lemma (in CL) CL_imp_ex_isLub: "S <= A ==> \<exists>L. isLub S cl L"
```
```   196 apply (insert cl_co)
```
```   197 apply (simp add: CompleteLattice_def A_def)
```
```   198 done
```
```   199
```
```   200 declare (in CL) cl_co [simp]
```
```   201
```
```   202 lemma isLub_lub: "(\<exists>L. isLub S cl L) = isLub S cl (lub S cl)"
```
```   203 by (simp add: lub_def least_def isLub_def some_eq_ex [symmetric])
```
```   204
```
```   205 lemma isGlb_glb: "(\<exists>G. isGlb S cl G) = isGlb S cl (glb S cl)"
```
```   206 by (simp add: glb_def greatest_def isGlb_def some_eq_ex [symmetric])
```
```   207
```
```   208 lemma isGlb_dual_isLub: "isGlb S cl = isLub S (dual cl)"
```
```   209 by (simp add: isLub_def isGlb_def dual_def converse_def)
```
```   210
```
```   211 lemma isLub_dual_isGlb: "isLub S cl = isGlb S (dual cl)"
```
```   212 by (simp add: isLub_def isGlb_def dual_def converse_def)
```
```   213
```
```   214 lemma (in PO) dualPO: "dual cl \<in> PartialOrder"
```
```   215 apply (insert cl_po)
```
```   216 apply (simp add: PartialOrder_def dual_def refl_converse
```
```   217                  trans_converse antisym_converse)
```
```   218 done
```
```   219
```
```   220 lemma Rdual:
```
```   221      "\<forall>S. (S <= A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
```
```   222       ==> \<forall>S. (S <= A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))"
```
```   223 apply safe
```
```   224 apply (rule_tac x = "lub {y. y \<in> A & (\<forall>k \<in> S. (y, k) \<in> r)}
```
```   225                       (|pset = A, order = r|) " in exI)
```
```   226 apply (drule_tac x = "{y. y \<in> A & (\<forall>k \<in> S. (y,k) \<in> r) }" in spec)
```
```   227 apply (drule mp, fast)
```
```   228 apply (simp add: isLub_lub isGlb_def)
```
```   229 apply (simp add: isLub_def, blast)
```
```   230 done
```
```   231
```
```   232 lemma lub_dual_glb: "lub S cl = glb S (dual cl)"
```
```   233 by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
```
```   234
```
```   235 lemma glb_dual_lub: "glb S cl = lub S (dual cl)"
```
```   236 by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
```
```   237
```
```   238 lemma CL_subset_PO: "CompleteLattice <= PartialOrder"
```
```   239 by (simp add: PartialOrder_def CompleteLattice_def, fast)
```
```   240
```
```   241 lemmas CL_imp_PO = CL_subset_PO [THEN subsetD]
```
```   242
```
```   243 declare CL_imp_PO [THEN Tarski.PO_imp_refl, simp]
```
```   244 declare CL_imp_PO [THEN Tarski.PO_imp_sym, simp]
```
```   245 declare CL_imp_PO [THEN Tarski.PO_imp_trans, simp]
```
```   246
```
```   247 lemma (in CL) CO_refl: "refl A r"
```
```   248 by (rule PO_imp_refl)
```
```   249
```
```   250 lemma (in CL) CO_antisym: "antisym r"
```
```   251 by (rule PO_imp_sym)
```
```   252
```
```   253 lemma (in CL) CO_trans: "trans r"
```
```   254 by (rule PO_imp_trans)
```
```   255
```
```   256 lemma CompleteLatticeI:
```
```   257      "[| po \<in> PartialOrder; (\<forall>S. S <= po.<A> --> (\<exists>L. isLub S po L));
```
```   258          (\<forall>S. S <= po.<A> --> (\<exists>G. isGlb S po G))|]
```
```   259       ==> po \<in> CompleteLattice"
```
```   260 apply (unfold CompleteLattice_def, blast)
```
```   261 done
```
```   262
```
```   263 lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice"
```
```   264 apply (insert cl_co)
```
```   265 apply (simp add: CompleteLattice_def dual_def)
```
```   266 apply (fold dual_def)
```
```   267 apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric]
```
```   268                  dualPO)
```
```   269 done
```
```   270
```
```   271 lemma (in PO) dualA_iff: "(dual cl.<A>) = cl.<A>"
```
```   272 by (simp add: dual_def)
```
```   273
```
```   274 lemma (in PO) dualr_iff: "((x, y) \<in> (dual cl.<r>)) = ((y, x) \<in> cl.<r>)"
```
```   275 by (simp add: dual_def)
```
```   276
```
```   277 lemma (in PO) monotone_dual:
```
```   278      "monotone f (cl.<A>) (cl.<r>) ==> monotone f (dual cl.<A>) (dual cl.<r>)"
```
```   279 apply (simp add: monotone_def dualA_iff dualr_iff)
```
```   280 done
```
```   281
```
```   282 lemma (in PO) interval_dual:
```
```   283      "[| x \<in> A; y \<in> A|] ==> interval r x y = interval (dual cl.<r>) y x"
```
```   284 apply (simp add: interval_def dualr_iff)
```
```   285 apply (fold r_def, fast)
```
```   286 done
```
```   287
```
```   288 lemma (in PO) interval_not_empty:
```
```   289      "[| trans r; interval r a b \<noteq> {} |] ==> (a, b) \<in> r"
```
```   290 apply (simp add: interval_def)
```
```   291 apply (unfold trans_def, blast)
```
```   292 done
```
```   293
```
```   294 lemma (in PO) interval_imp_mem: "x \<in> interval r a b ==> (a, x) \<in> r"
```
```   295 by (simp add: interval_def)
```
```   296
```
```   297 lemma (in PO) left_in_interval:
```
```   298      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> a \<in> interval r a b"
```
```   299 apply (simp (no_asm_simp) add: interval_def)
```
```   300 apply (simp add: PO_imp_trans interval_not_empty)
```
```   301 apply (simp add: PO_imp_refl [THEN reflE])
```
```   302 done
```
```   303
```
```   304 lemma (in PO) right_in_interval:
```
```   305      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> b \<in> interval r a b"
```
```   306 apply (simp (no_asm_simp) add: interval_def)
```
```   307 apply (simp add: PO_imp_trans interval_not_empty)
```
```   308 apply (simp add: PO_imp_refl [THEN reflE])
```
```   309 done
```
```   310
```
```   311
```
```   312 subsubsection {* sublattice *}
```
```   313
```
```   314 lemma (in PO) sublattice_imp_CL:
```
```   315      "S <<= cl  ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
```
```   316 by (simp add: sublattice_def CompleteLattice_def A_def r_def)
```
```   317
```
```   318 lemma (in CL) sublatticeI:
```
```   319      "[| S <= A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
```
```   320       ==> S <<= cl"
```
```   321 by (simp add: sublattice_def A_def r_def)
```
```   322
```
```   323
```
```   324 subsubsection {* lub *}
```
```   325
```
```   326 lemma (in CL) lub_unique: "[| S <= A; isLub S cl x; isLub S cl L|] ==> x = L"
```
```   327 apply (rule antisymE)
```
```   328 apply (rule CO_antisym)
```
```   329 apply (auto simp add: isLub_def r_def)
```
```   330 done
```
```   331
```
```   332 lemma (in CL) lub_upper: "[|S <= A; x \<in> S|] ==> (x, lub S cl) \<in> r"
```
```   333 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   334 apply (unfold lub_def least_def)
```
```   335 apply (rule 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)
```
```   339 done
```
```   340
```
```   341 lemma (in CL) lub_least:
```
```   342      "[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
```
```   343 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   344 apply (unfold lub_def least_def)
```
```   345 apply (rule_tac s=x in some_equality [THEN ssubst])
```
```   346   apply (simp add: isLub_def)
```
```   347  apply (simp add: lub_unique A_def isLub_def)
```
```   348 apply (simp add: isLub_def r_def A_def)
```
```   349 done
```
```   350
```
```   351 lemma (in CL) lub_in_lattice: "S <= A ==> lub S cl \<in> A"
```
```   352 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   353 apply (unfold lub_def least_def)
```
```   354 apply (subst some_equality)
```
```   355 apply (simp add: isLub_def)
```
```   356 prefer 2 apply (simp add: isLub_def A_def)
```
```   357 apply (simp add: lub_unique A_def isLub_def)
```
```   358 done
```
```   359
```
```   360 lemma (in CL) lubI:
```
```   361      "[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
```
```   362          \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"
```
```   363 apply (rule lub_unique, assumption)
```
```   364 apply (simp add: isLub_def A_def r_def)
```
```   365 apply (unfold isLub_def)
```
```   366 apply (rule conjI)
```
```   367 apply (fold A_def r_def)
```
```   368 apply (rule lub_in_lattice, assumption)
```
```   369 apply (simp add: lub_upper lub_least)
```
```   370 done
```
```   371
```
```   372 lemma (in CL) lubIa: "[| S <= A; isLub S cl L |] ==> L = lub S cl"
```
```   373 by (simp add: lubI isLub_def A_def r_def)
```
```   374
```
```   375 lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"
```
```   376 by (simp add: isLub_def  A_def)
```
```   377
```
```   378 lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r"
```
```   379 by (simp add: isLub_def r_def)
```
```   380
```
```   381 lemma (in CL) isLub_least:
```
```   382      "[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r"
```
```   383 by (simp add: isLub_def A_def r_def)
```
```   384
```
```   385 lemma (in CL) isLubI:
```
```   386      "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;
```
```   387          (\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L"
```
```   388 by (simp add: isLub_def A_def r_def)
```
```   389
```
```   390
```
```   391 subsubsection {* glb *}
```
```   392
```
```   393 lemma (in CL) glb_in_lattice: "S <= A ==> glb S cl \<in> A"
```
```   394 apply (subst glb_dual_lub)
```
```   395 apply (simp add: A_def)
```
```   396 apply (rule dualA_iff [THEN subst])
```
```   397 apply (rule Tarski.lub_in_lattice)
```
```   398 apply (rule dualPO)
```
```   399 apply (rule CL_dualCL)
```
```   400 apply (simp add: dualA_iff)
```
```   401 done
```
```   402
```
```   403 lemma (in CL) glb_lower: "[|S <= A; x \<in> S|] ==> (glb S cl, x) \<in> r"
```
```   404 apply (subst glb_dual_lub)
```
```   405 apply (simp add: r_def)
```
```   406 apply (rule dualr_iff [THEN subst])
```
```   407 apply (rule Tarski.lub_upper [rule_format])
```
```   408 apply (rule dualPO)
```
```   409 apply (rule CL_dualCL)
```
```   410 apply (simp add: dualA_iff A_def, assumption)
```
```   411 done
```
```   412
```
```   413 text {*
```
```   414   Reduce the sublattice property by using substructural properties;
```
```   415   abandoned see @{text "Tarski_4.ML"}.
```
```   416 *}
```
```   417
```
```   418 lemma (in CLF) [simp]:
```
```   419     "f: cl.<A> funcset cl.<A> & monotone f (cl.<A>) (cl.<r>)"
```
```   420 apply (insert f_cl)
```
```   421 apply (simp add: CLF_def)
```
```   422 done
```
```   423
```
```   424 declare (in CLF) f_cl [simp]
```
```   425
```
```   426
```
```   427 lemma (in CLF) f_in_funcset: "f \<in> A funcset A"
```
```   428 by (simp add: A_def)
```
```   429
```
```   430 lemma (in CLF) monotone_f: "monotone f A r"
```
```   431 by (simp add: A_def r_def)
```
```   432
```
```   433 lemma (in CLF) CLF_dual: "(cl,f) \<in> CLF ==> (dual cl, f) \<in> CLF"
```
```   434 apply (simp add: CLF_def  CL_dualCL monotone_dual)
```
```   435 apply (simp add: dualA_iff)
```
```   436 done
```
```   437
```
```   438
```
```   439 subsubsection {* fixed points *}
```
```   440
```
```   441 lemma fix_subset: "fix f A <= A"
```
```   442 by (simp add: fix_def, fast)
```
```   443
```
```   444 lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"
```
```   445 by (simp add: fix_def)
```
```   446
```
```   447 lemma fixf_subset:
```
```   448      "[| A <= B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
```
```   449 apply (simp add: fix_def, auto)
```
```   450 done
```
```   451
```
```   452
```
```   453 subsubsection {* lemmas for Tarski, lub *}
```
```   454 lemma (in CLF) lubH_le_flubH:
```
```   455      "H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r"
```
```   456 apply (rule lub_least, fast)
```
```   457 apply (rule f_in_funcset [THEN funcset_mem])
```
```   458 apply (rule lub_in_lattice, fast)
```
```   459 -- {* @{text "\<forall>x:H. (x, f (lub H r)) \<in> r"} *}
```
```   460 apply (rule ballI)
```
```   461 apply (rule transE)
```
```   462 apply (rule CO_trans)
```
```   463 -- {* instantiates @{text "(x, ???z) \<in> cl.<r> to (x, f x)"}, *}
```
```   464 -- {* because of the def of @{text H} *}
```
```   465 apply fast
```
```   466 -- {* so it remains to show @{text "(f x, f (lub H cl)) \<in> r"} *}
```
```   467 apply (rule_tac f = "f" in monotoneE)
```
```   468 apply (rule monotone_f, fast)
```
```   469 apply (rule lub_in_lattice, fast)
```
```   470 apply (rule lub_upper, fast)
```
```   471 apply assumption
```
```   472 done
```
```   473
```
```   474 lemma (in CLF) flubH_le_lubH:
```
```   475      "[|  H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r"
```
```   476 apply (rule lub_upper, fast)
```
```   477 apply (rule_tac t = "H" in ssubst, assumption)
```
```   478 apply (rule CollectI)
```
```   479 apply (rule conjI)
```
```   480 apply (rule_tac [2] f_in_funcset [THEN funcset_mem])
```
```   481 apply (rule_tac [2] lub_in_lattice)
```
```   482 prefer 2 apply fast
```
```   483 apply (rule_tac f = "f" in monotoneE)
```
```   484 apply (rule monotone_f)
```
```   485   apply (blast intro: lub_in_lattice)
```
```   486  apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem])
```
```   487 apply (simp add: lubH_le_flubH)
```
```   488 done
```
```   489
```
```   490 lemma (in CLF) lubH_is_fixp:
```
```   491      "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
```
```   492 apply (simp add: fix_def)
```
```   493 apply (rule conjI)
```
```   494 apply (rule lub_in_lattice, fast)
```
```   495 apply (rule antisymE)
```
```   496 apply (rule CO_antisym)
```
```   497 apply (simp add: flubH_le_lubH)
```
```   498 apply (simp add: lubH_le_flubH)
```
```   499 done
```
```   500
```
```   501 lemma (in CLF) fix_in_H:
```
```   502      "[| H = {x. (x, f x) \<in> r & x \<in> A};  x \<in> P |] ==> x \<in> H"
```
```   503 by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl
```
```   504                     fix_subset [of f A, THEN subsetD])
```
```   505
```
```   506 lemma (in CLF) fixf_le_lubH:
```
```   507      "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
```
```   508 apply (rule ballI)
```
```   509 apply (rule lub_upper, fast)
```
```   510 apply (rule fix_in_H)
```
```   511 apply (simp_all add: P_def)
```
```   512 done
```
```   513
```
```   514 lemma (in CLF) lubH_least_fixf:
```
```   515      "H = {x. (x, f x) \<in> r & x \<in> A}
```
```   516       ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"
```
```   517 apply (rule allI)
```
```   518 apply (rule impI)
```
```   519 apply (erule bspec)
```
```   520 apply (rule lubH_is_fixp, assumption)
```
```   521 done
```
```   522
```
```   523 subsubsection {* Tarski fixpoint theorem 1, first part *}
```
```   524 lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"
```
```   525 apply (rule sym)
```
```   526 apply (simp add: P_def)
```
```   527 apply (rule lubI)
```
```   528 apply (rule fix_subset)
```
```   529 apply (rule lub_in_lattice, fast)
```
```   530 apply (simp add: fixf_le_lubH)
```
```   531 apply (simp add: lubH_least_fixf)
```
```   532 done
```
```   533
```
```   534 lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P"
```
```   535   -- {* Tarski for glb *}
```
```   536 apply (simp add: glb_dual_lub P_def A_def r_def)
```
```   537 apply (rule dualA_iff [THEN subst])
```
```   538 apply (rule Tarski.lubH_is_fixp)
```
```   539 apply (rule dualPO)
```
```   540 apply (rule CL_dualCL)
```
```   541 apply (rule f_cl [THEN CLF_dual])
```
```   542 apply (simp add: dualr_iff dualA_iff)
```
```   543 done
```
```   544
```
```   545 lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl"
```
```   546 apply (simp add: glb_dual_lub P_def A_def r_def)
```
```   547 apply (rule dualA_iff [THEN subst])
```
```   548 apply (simp add: Tarski.T_thm_1_lub [of _ f, OF dualPO CL_dualCL]
```
```   549                  dualPO CL_dualCL CLF_dual dualr_iff)
```
```   550 done
```
```   551
```
```   552 subsubsection {* interval *}
```
```   553
```
```   554 lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"
```
```   555 apply (insert CO_refl)
```
```   556 apply (simp add: refl_def, blast)
```
```   557 done
```
```   558
```
```   559 lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b <= A"
```
```   560 apply (simp add: interval_def)
```
```   561 apply (blast intro: rel_imp_elem)
```
```   562 done
```
```   563
```
```   564 lemma (in CLF) intervalI:
```
```   565      "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"
```
```   566 apply (simp add: interval_def)
```
```   567 done
```
```   568
```
```   569 lemma (in CLF) interval_lemma1:
```
```   570      "[| S <= interval r a b; x \<in> S |] ==> (a, x) \<in> r"
```
```   571 apply (unfold interval_def, fast)
```
```   572 done
```
```   573
```
```   574 lemma (in CLF) interval_lemma2:
```
```   575      "[| S <= interval r a b; x \<in> S |] ==> (x, b) \<in> r"
```
```   576 apply (unfold interval_def, fast)
```
```   577 done
```
```   578
```
```   579 lemma (in CLF) a_less_lub:
```
```   580      "[| S <= A; S \<noteq> {};
```
```   581          \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"
```
```   582 by (blast intro: transE PO_imp_trans)
```
```   583
```
```   584 lemma (in CLF) glb_less_b:
```
```   585      "[| S <= A; S \<noteq> {};
```
```   586          \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"
```
```   587 by (blast intro: transE PO_imp_trans)
```
```   588
```
```   589 lemma (in CLF) S_intv_cl:
```
```   590      "[| a \<in> A; b \<in> A; S <= interval r a b |]==> S <= A"
```
```   591 by (simp add: subset_trans [OF _ interval_subset])
```
```   592
```
```   593 lemma (in CLF) L_in_interval:
```
```   594      "[| a \<in> A; b \<in> A; S <= interval r a b;
```
```   595          S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"
```
```   596 apply (rule intervalI)
```
```   597 apply (rule a_less_lub)
```
```   598 prefer 2 apply assumption
```
```   599 apply (simp add: S_intv_cl)
```
```   600 apply (rule ballI)
```
```   601 apply (simp add: interval_lemma1)
```
```   602 apply (simp add: isLub_upper)
```
```   603 -- {* @{text "(L, b) \<in> r"} *}
```
```   604 apply (simp add: isLub_least interval_lemma2)
```
```   605 done
```
```   606
```
```   607 lemma (in CLF) G_in_interval:
```
```   608      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S <= interval r a b; isGlb S cl G;
```
```   609          S \<noteq> {} |] ==> G \<in> interval r a b"
```
```   610 apply (simp add: interval_dual)
```
```   611 apply (simp add: Tarski.L_in_interval [of _ f]
```
```   612                  dualA_iff A_def dualPO CL_dualCL CLF_dual isGlb_dual_isLub)
```
```   613 done
```
```   614
```
```   615 lemma (in CLF) intervalPO:
```
```   616      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   617       ==> (| pset = interval r a b, order = induced (interval r a b) r |)
```
```   618           \<in> PartialOrder"
```
```   619 apply (rule po_subset_po)
```
```   620 apply (simp add: interval_subset)
```
```   621 done
```
```   622
```
```   623 lemma (in CLF) intv_CL_lub:
```
```   624  "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   625   ==> \<forall>S. S <= interval r a b -->
```
```   626           (\<exists>L. isLub S (| pset = interval r a b,
```
```   627                           order = induced (interval r a b) r |)  L)"
```
```   628 apply (intro strip)
```
```   629 apply (frule S_intv_cl [THEN CL_imp_ex_isLub])
```
```   630 prefer 2 apply assumption
```
```   631 apply assumption
```
```   632 apply (erule exE)
```
```   633 -- {* define the lub for the interval as *}
```
```   634 apply (rule_tac x = "if S = {} then a else L" in exI)
```
```   635 apply (simp (no_asm_simp) add: isLub_def split del: split_if)
```
```   636 apply (intro impI conjI)
```
```   637 -- {* @{text "(if S = {} then a else L) \<in> interval r a b"} *}
```
```   638 apply (simp add: CL_imp_PO L_in_interval)
```
```   639 apply (simp add: left_in_interval)
```
```   640 -- {* lub prop 1 *}
```
```   641 apply (case_tac "S = {}")
```
```   642 -- {* @{text "S = {}, y \<in> S = False => everything"} *}
```
```   643 apply fast
```
```   644 -- {* @{text "S \<noteq> {}"} *}
```
```   645 apply simp
```
```   646 -- {* @{text "\<forall>y:S. (y, L) \<in> induced (interval r a b) r"} *}
```
```   647 apply (rule ballI)
```
```   648 apply (simp add: induced_def  L_in_interval)
```
```   649 apply (rule conjI)
```
```   650 apply (rule subsetD)
```
```   651 apply (simp add: S_intv_cl, assumption)
```
```   652 apply (simp add: isLub_upper)
```
```   653 -- {* @{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"} *}
```
```   654 apply (rule ballI)
```
```   655 apply (rule impI)
```
```   656 apply (case_tac "S = {}")
```
```   657 -- {* @{text "S = {}"} *}
```
```   658 apply simp
```
```   659 apply (simp add: induced_def  interval_def)
```
```   660 apply (rule conjI)
```
```   661 apply (rule reflE)
```
```   662 apply (rule CO_refl, assumption)
```
```   663 apply (rule interval_not_empty)
```
```   664 apply (rule CO_trans)
```
```   665 apply (simp add: interval_def)
```
```   666 -- {* @{text "S \<noteq> {}"} *}
```
```   667 apply simp
```
```   668 apply (simp add: induced_def  L_in_interval)
```
```   669 apply (rule isLub_least, assumption)
```
```   670 apply (rule subsetD)
```
```   671 prefer 2 apply assumption
```
```   672 apply (simp add: S_intv_cl, fast)
```
```   673 done
```
```   674
```
```   675 lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual]
```
```   676
```
```   677 lemma (in CLF) interval_is_sublattice:
```
```   678      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   679         ==> interval r a b <<= cl"
```
```   680 apply (rule sublatticeI)
```
```   681 apply (simp add: interval_subset)
```
```   682 apply (rule CompleteLatticeI)
```
```   683 apply (simp add: intervalPO)
```
```   684  apply (simp add: intv_CL_lub)
```
```   685 apply (simp add: intv_CL_glb)
```
```   686 done
```
```   687
```
```   688 lemmas (in CLF) interv_is_compl_latt =
```
```   689     interval_is_sublattice [THEN sublattice_imp_CL]
```
```   690
```
```   691
```
```   692 subsubsection {* Top and Bottom *}
```
```   693 lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"
```
```   694 by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
```
```   695
```
```   696 lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)"
```
```   697 by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
```
```   698
```
```   699 lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"
```
```   700 apply (simp add: Bot_def least_def)
```
```   701 apply (rule someI2)
```
```   702 apply (fold A_def)
```
```   703 apply (erule_tac [2] conjunct1)
```
```   704 apply (rule conjI)
```
```   705 apply (rule glb_in_lattice)
```
```   706 apply (rule subset_refl)
```
```   707 apply (fold r_def)
```
```   708 apply (simp add: glb_lower)
```
```   709 done
```
```   710
```
```   711 lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
```
```   712 apply (simp add: Top_dual_Bot A_def)
```
```   713 apply (rule dualA_iff [THEN subst])
```
```   714 apply (blast intro!: Tarski.Bot_in_lattice dualPO CL_dualCL CLF_dual f_cl)
```
```   715 done
```
```   716
```
```   717 lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
```
```   718 apply (simp add: Top_def greatest_def)
```
```   719 apply (rule someI2)
```
```   720 apply (fold r_def  A_def)
```
```   721 prefer 2 apply fast
```
```   722 apply (intro conjI ballI)
```
```   723 apply (rule_tac [2] lub_upper)
```
```   724 apply (auto simp add: lub_in_lattice)
```
```   725 done
```
```   726
```
```   727 lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
```
```   728 apply (simp add: Bot_dual_Top r_def)
```
```   729 apply (rule dualr_iff [THEN subst])
```
```   730 apply (simp add: Tarski.Top_prop [of _ f]
```
```   731                  dualA_iff A_def dualPO CL_dualCL CLF_dual)
```
```   732 done
```
```   733
```
```   734 lemma (in CLF) Top_intv_not_empty: "x \<in> A  ==> interval r x (Top cl) \<noteq> {}"
```
```   735 apply (rule notI)
```
```   736 apply (drule_tac a = "Top cl" in equals0D)
```
```   737 apply (simp add: interval_def)
```
```   738 apply (simp add: refl_def Top_in_lattice Top_prop)
```
```   739 done
```
```   740
```
```   741 lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}"
```
```   742 apply (simp add: Bot_dual_Top)
```
```   743 apply (subst interval_dual)
```
```   744 prefer 2 apply assumption
```
```   745 apply (simp add: A_def)
```
```   746 apply (rule dualA_iff [THEN subst])
```
```   747 apply (blast intro!: Tarski.Top_in_lattice
```
```   748                  f_cl dualPO CL_dualCL CLF_dual)
```
```   749 apply (simp add: Tarski.Top_intv_not_empty [of _ f]
```
```   750                  dualA_iff A_def dualPO CL_dualCL CLF_dual)
```
```   751 done
```
```   752
```
```   753 subsubsection {* fixed points form a partial order *}
```
```   754
```
```   755 lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
```
```   756 by (simp add: P_def fix_subset po_subset_po)
```
```   757
```
```   758 lemma (in Tarski) Y_subset_A: "Y <= A"
```
```   759 apply (rule subset_trans [OF _ fix_subset])
```
```   760 apply (rule Y_ss [simplified P_def])
```
```   761 done
```
```   762
```
```   763 lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A"
```
```   764 by (simp add: Y_subset_A [THEN lub_in_lattice])
```
```   765
```
```   766 lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"
```
```   767 apply (rule lub_least)
```
```   768 apply (rule Y_subset_A)
```
```   769 apply (rule f_in_funcset [THEN funcset_mem])
```
```   770 apply (rule lubY_in_A)
```
```   771 -- {* @{text "Y <= P ==> f x = x"} *}
```
```   772 apply (rule ballI)
```
```   773 apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
```
```   774 apply (erule Y_ss [simplified P_def, THEN subsetD])
```
```   775 -- {* @{text "reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r"} by monotonicity *}
```
```   776 apply (rule_tac f = "f" in monotoneE)
```
```   777 apply (rule monotone_f)
```
```   778 apply (simp add: Y_subset_A [THEN subsetD])
```
```   779 apply (rule lubY_in_A)
```
```   780 apply (simp add: lub_upper Y_subset_A)
```
```   781 done
```
```   782
```
```   783 lemma (in Tarski) intY1_subset: "intY1 <= A"
```
```   784 apply (unfold intY1_def)
```
```   785 apply (rule interval_subset)
```
```   786 apply (rule lubY_in_A)
```
```   787 apply (rule Top_in_lattice)
```
```   788 done
```
```   789
```
```   790 lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD]
```
```   791
```
```   792 lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"
```
```   793 apply (simp add: intY1_def  interval_def)
```
```   794 apply (rule conjI)
```
```   795 apply (rule transE)
```
```   796 apply (rule CO_trans)
```
```   797 apply (rule lubY_le_flubY)
```
```   798 -- {* @{text "(f (lub Y cl), f x) \<in> r"} *}
```
```   799 apply (rule_tac f=f in monotoneE)
```
```   800 apply (rule monotone_f)
```
```   801 apply (rule lubY_in_A)
```
```   802 apply (simp add: intY1_def interval_def  intY1_elem)
```
```   803 apply (simp add: intY1_def  interval_def)
```
```   804 -- {* @{text "(f x, Top cl) \<in> r"} *}
```
```   805 apply (rule Top_prop)
```
```   806 apply (rule f_in_funcset [THEN funcset_mem])
```
```   807 apply (simp add: intY1_def interval_def  intY1_elem)
```
```   808 done
```
```   809
```
```   810 lemma (in Tarski) intY1_func: "(%x: intY1. f x) \<in> intY1 funcset intY1"
```
```   811 apply (rule restrictI)
```
```   812 apply (erule intY1_f_closed)
```
```   813 done
```
```   814
```
```   815 lemma (in Tarski) intY1_mono:
```
```   816      "monotone (%x: intY1. f x) intY1 (induced intY1 r)"
```
```   817 apply (auto simp add: monotone_def induced_def intY1_f_closed)
```
```   818 apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
```
```   819 done
```
```   820
```
```   821 lemma (in Tarski) intY1_is_cl:
```
```   822     "(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
```
```   823 apply (unfold intY1_def)
```
```   824 apply (rule interv_is_compl_latt)
```
```   825 apply (rule lubY_in_A)
```
```   826 apply (rule Top_in_lattice)
```
```   827 apply (rule Top_intv_not_empty)
```
```   828 apply (rule lubY_in_A)
```
```   829 done
```
```   830
```
```   831 lemma (in Tarski) v_in_P: "v \<in> P"
```
```   832 apply (unfold P_def)
```
```   833 apply (rule_tac A = "intY1" in fixf_subset)
```
```   834 apply (rule intY1_subset)
```
```   835 apply (simp add: Tarski.glbH_is_fixp [OF _ intY1_is_cl, simplified]
```
```   836                  v_def CL_imp_PO intY1_is_cl CLF_def intY1_func intY1_mono)
```
```   837 done
```
```   838
```
```   839 lemma (in Tarski) z_in_interval:
```
```   840      "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1"
```
```   841 apply (unfold intY1_def P_def)
```
```   842 apply (rule intervalI)
```
```   843 prefer 2
```
```   844  apply (erule fix_subset [THEN subsetD, THEN Top_prop])
```
```   845 apply (rule lub_least)
```
```   846 apply (rule Y_subset_A)
```
```   847 apply (fast elim!: fix_subset [THEN subsetD])
```
```   848 apply (simp add: induced_def)
```
```   849 done
```
```   850
```
```   851 lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |]
```
```   852       ==> ((%x: intY1. f x) z, z) \<in> induced intY1 r"
```
```   853 apply (simp add: induced_def  intY1_f_closed z_in_interval P_def)
```
```   854 apply (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD]
```
```   855                  CO_refl [THEN reflE])
```
```   856 done
```
```   857
```
```   858 lemma (in Tarski) tarski_full_lemma:
```
```   859      "\<exists>L. isLub Y (| pset = P, order = induced P r |) L"
```
```   860 apply (rule_tac x = "v" in exI)
```
```   861 apply (simp add: isLub_def)
```
```   862 -- {* @{text "v \<in> P"} *}
```
```   863 apply (simp add: v_in_P)
```
```   864 apply (rule conjI)
```
```   865 -- {* @{text v} is lub *}
```
```   866 -- {* @{text "1. \<forall>y:Y. (y, v) \<in> induced P r"} *}
```
```   867 apply (rule ballI)
```
```   868 apply (simp add: induced_def subsetD v_in_P)
```
```   869 apply (rule conjI)
```
```   870 apply (erule Y_ss [THEN subsetD])
```
```   871 apply (rule_tac b = "lub Y cl" in transE)
```
```   872 apply (rule CO_trans)
```
```   873 apply (rule lub_upper)
```
```   874 apply (rule Y_subset_A, assumption)
```
```   875 apply (rule_tac b = "Top cl" in interval_imp_mem)
```
```   876 apply (simp add: v_def)
```
```   877 apply (fold intY1_def)
```
```   878 apply (rule Tarski.glb_in_lattice [OF _ intY1_is_cl, simplified])
```
```   879  apply (simp add: CL_imp_PO intY1_is_cl, force)
```
```   880 -- {* @{text v} is LEAST ub *}
```
```   881 apply clarify
```
```   882 apply (rule indI)
```
```   883   prefer 3 apply assumption
```
```   884  prefer 2 apply (simp add: v_in_P)
```
```   885 apply (unfold v_def)
```
```   886 apply (rule indE)
```
```   887 apply (rule_tac [2] intY1_subset)
```
```   888 apply (rule Tarski.glb_lower [OF _ intY1_is_cl, simplified])
```
```   889   apply (simp add: CL_imp_PO intY1_is_cl)
```
```   890  apply force
```
```   891 apply (simp add: induced_def intY1_f_closed z_in_interval)
```
```   892 apply (simp add: P_def fix_imp_eq [of _ f A]
```
```   893                  fix_subset [of f A, THEN subsetD]
```
```   894                  CO_refl [THEN reflE])
```
```   895 done
```
```   896
```
```   897 lemma CompleteLatticeI_simp:
```
```   898      "[| (| pset = A, order = r |) \<in> PartialOrder;
```
```   899          \<forall>S. S <= A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]
```
```   900     ==> (| pset = A, order = r |) \<in> CompleteLattice"
```
```   901 by (simp add: CompleteLatticeI Rdual)
```
```   902
```
```   903 theorem (in CLF) Tarski_full:
```
```   904      "(| pset = P, order = induced P r|) \<in> CompleteLattice"
```
```   905 apply (rule CompleteLatticeI_simp)
```
```   906 apply (rule fixf_po, clarify)
```
```   907 apply (simp add: P_def A_def r_def)
```
```   908 apply (blast intro!: Tarski.tarski_full_lemma cl_po cl_co f_cl)
```
```   909 done
```
```   910
```
```   911 end
```