src/HOL/Metis_Examples/Tarski.thy
 author haftmann Mon Jun 05 15:59:41 2017 +0200 (2017-06-05) changeset 66010 2f7d39285a1a parent 64913 3a9eb793fa10 child 66453 cc19f7ca2ed6 permissions -rw-r--r--
executable domain membership checks
```     1 (*  Title:      HOL/Metis_Examples/Tarski.thy
```
```     2     Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
```
```     3     Author:     Jasmin Blanchette, TU Muenchen
```
```     4
```
```     5 Metis example featuring the full theorem of Tarski.
```
```     6 *)
```
```     7
```
```     8 section \<open>Metis Example Featuring the Full Theorem of Tarski\<close>
```
```     9
```
```    10 theory Tarski
```
```    11 imports Main "~~/src/HOL/Library/FuncSet"
```
```    12 begin
```
```    13
```
```    14 declare [[metis_new_skolem]]
```
```    15
```
```    16 (*Many of these higher-order problems appear to be impossible using the
```
```    17 current linkup. They often seem to need either higher-order unification
```
```    18 or explicit reasoning about connectives such as conjunction. The numerous
```
```    19 set comprehensions are to blame.*)
```
```    20
```
```    21 record 'a potype =
```
```    22   pset  :: "'a set"
```
```    23   order :: "('a * 'a) set"
```
```    24
```
```    25 definition monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool" where
```
```    26   "monotone f A r == \<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r"
```
```    27
```
```    28 definition least :: "['a => bool, 'a potype] => 'a" where
```
```    29   "least P po == @ x. x: pset po & P x &
```
```    30                        (\<forall>y \<in> pset po. P y --> (x,y): order po)"
```
```    31
```
```    32 definition greatest :: "['a => bool, 'a potype] => 'a" where
```
```    33   "greatest P po == @ x. x: pset po & P x &
```
```    34                           (\<forall>y \<in> pset po. P y --> (y,x): order po)"
```
```    35
```
```    36 definition lub  :: "['a set, 'a potype] => 'a" where
```
```    37   "lub S po == least (\<lambda>x. \<forall>y\<in>S. (y,x): order po) po"
```
```    38
```
```    39 definition glb  :: "['a set, 'a potype] => 'a" where
```
```    40   "glb S po == greatest (\<lambda>x. \<forall>y\<in>S. (x,y): order po) po"
```
```    41
```
```    42 definition isLub :: "['a set, 'a potype, 'a] => bool" where
```
```    43   "isLub S po == \<lambda>L. (L: pset po & (\<forall>y\<in>S. (y,L): order po) &
```
```    44                    (\<forall>z\<in>pset po. (\<forall>y\<in>S. (y,z): order po) --> (L,z): order po))"
```
```    45
```
```    46 definition isGlb :: "['a set, 'a potype, 'a] => bool" where
```
```    47   "isGlb S po == \<lambda>G. (G: pset po & (\<forall>y\<in>S. (G,y): order po) &
```
```    48                  (\<forall>z \<in> pset po. (\<forall>y\<in>S. (z,y): order po) --> (z,G): order po))"
```
```    49
```
```    50 definition "fix"    :: "[('a => 'a), 'a set] => 'a set" where
```
```    51   "fix f A  == {x. x: A & f x = x}"
```
```    52
```
```    53 definition interval :: "[('a*'a) set,'a, 'a ] => 'a set" where
```
```    54   "interval r a b == {x. (a,x): r & (x,b): r}"
```
```    55
```
```    56 definition Bot :: "'a potype => 'a" where
```
```    57   "Bot po == least (\<lambda>x. True) po"
```
```    58
```
```    59 definition Top :: "'a potype => 'a" where
```
```    60   "Top po == greatest (\<lambda>x. True) po"
```
```    61
```
```    62 definition PartialOrder :: "('a potype) set" where
```
```    63   "PartialOrder == {P. refl_on (pset P) (order P) & antisym (order P) &
```
```    64                        trans (order P)}"
```
```    65
```
```    66 definition CompleteLattice :: "('a potype) set" where
```
```    67   "CompleteLattice == {cl. cl: PartialOrder &
```
```    68                         (\<forall>S. S \<subseteq> pset cl --> (\<exists>L. isLub S cl L)) &
```
```    69                         (\<forall>S. S \<subseteq> pset cl --> (\<exists>G. isGlb S cl G))}"
```
```    70
```
```    71 definition induced :: "['a set, ('a * 'a) set] => ('a *'a)set" where
```
```    72   "induced A r == {(a,b). a : A & b: A & (a,b): r}"
```
```    73
```
```    74 definition sublattice :: "('a potype * 'a set)set" where
```
```    75   "sublattice ==
```
```    76       SIGMA cl: CompleteLattice.
```
```    77           {S. S \<subseteq> pset cl &
```
```    78            (| pset = S, order = induced S (order cl) |): CompleteLattice }"
```
```    79
```
```    80 abbreviation
```
```    81   sublattice_syntax :: "['a set, 'a potype] => bool" ("_ <<= _" [51, 50] 50)
```
```    82   where "S <<= cl \<equiv> S : sublattice `` {cl}"
```
```    83
```
```    84 definition dual :: "'a potype => 'a potype" where
```
```    85   "dual po == (| pset = pset po, order = converse (order po) |)"
```
```    86
```
```    87 locale PO =
```
```    88   fixes cl :: "'a potype"
```
```    89     and A  :: "'a set"
```
```    90     and r  :: "('a * 'a) set"
```
```    91   assumes cl_po:  "cl : PartialOrder"
```
```    92   defines A_def: "A == pset cl"
```
```    93      and  r_def: "r == order cl"
```
```    94
```
```    95 locale CL = PO +
```
```    96   assumes cl_co:  "cl : CompleteLattice"
```
```    97
```
```    98 definition CLF_set :: "('a potype * ('a => 'a)) set" where
```
```    99   "CLF_set = (SIGMA cl: CompleteLattice.
```
```   100             {f. f: pset cl \<rightarrow> pset cl & monotone f (pset cl) (order cl)})"
```
```   101
```
```   102 locale CLF = CL +
```
```   103   fixes f :: "'a => 'a"
```
```   104     and P :: "'a set"
```
```   105   assumes f_cl:  "(cl,f) : CLF_set" (*was the equivalent "f : CLF``{cl}"*)
```
```   106   defines P_def: "P == fix f A"
```
```   107
```
```   108 locale Tarski = CLF +
```
```   109   fixes Y     :: "'a set"
```
```   110     and intY1 :: "'a set"
```
```   111     and v     :: "'a"
```
```   112   assumes
```
```   113     Y_ss: "Y \<subseteq> P"
```
```   114   defines
```
```   115     intY1_def: "intY1 == interval r (lub Y cl) (Top cl)"
```
```   116     and v_def: "v == glb {x. ((\<lambda>x \<in> intY1. f x) x, x): induced intY1 r &
```
```   117                              x: intY1}
```
```   118                       (| pset=intY1, order=induced intY1 r|)"
```
```   119
```
```   120 subsection \<open>Partial Order\<close>
```
```   121
```
```   122 lemma (in PO) PO_imp_refl_on: "refl_on A r"
```
```   123 apply (insert cl_po)
```
```   124 apply (simp add: PartialOrder_def A_def r_def)
```
```   125 done
```
```   126
```
```   127 lemma (in PO) PO_imp_sym: "antisym r"
```
```   128 apply (insert cl_po)
```
```   129 apply (simp add: PartialOrder_def r_def)
```
```   130 done
```
```   131
```
```   132 lemma (in PO) PO_imp_trans: "trans r"
```
```   133 apply (insert cl_po)
```
```   134 apply (simp add: PartialOrder_def r_def)
```
```   135 done
```
```   136
```
```   137 lemma (in PO) reflE: "x \<in> A ==> (x, x) \<in> r"
```
```   138 apply (insert cl_po)
```
```   139 apply (simp add: PartialOrder_def refl_on_def A_def r_def)
```
```   140 done
```
```   141
```
```   142 lemma (in PO) antisymE: "[| (a, b) \<in> r; (b, a) \<in> r |] ==> a = b"
```
```   143 apply (insert cl_po)
```
```   144 apply (simp add: PartialOrder_def antisym_def r_def)
```
```   145 done
```
```   146
```
```   147 lemma (in PO) transE: "[| (a, b) \<in> r; (b, c) \<in> r|] ==> (a,c) \<in> r"
```
```   148 apply (insert cl_po)
```
```   149 apply (simp add: PartialOrder_def r_def)
```
```   150 apply (unfold trans_def, fast)
```
```   151 done
```
```   152
```
```   153 lemma (in PO) monotoneE:
```
```   154      "[| monotone f A r;  x \<in> A; y \<in> A; (x, y) \<in> r |] ==> (f x, f y) \<in> r"
```
```   155 by (simp add: monotone_def)
```
```   156
```
```   157 lemma (in PO) po_subset_po:
```
```   158      "S \<subseteq> A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
```
```   159 apply (simp (no_asm) add: PartialOrder_def)
```
```   160 apply auto
```
```   161 \<comment> \<open>refl\<close>
```
```   162 apply (simp add: refl_on_def induced_def)
```
```   163 apply (blast intro: reflE)
```
```   164 \<comment> \<open>antisym\<close>
```
```   165 apply (simp add: antisym_def induced_def)
```
```   166 apply (blast intro: antisymE)
```
```   167 \<comment> \<open>trans\<close>
```
```   168 apply (simp add: trans_def induced_def)
```
```   169 apply (blast intro: transE)
```
```   170 done
```
```   171
```
```   172 lemma (in PO) indE: "[| (x, y) \<in> induced S r; S \<subseteq> A |] ==> (x, y) \<in> r"
```
```   173 by (simp add: add: induced_def)
```
```   174
```
```   175 lemma (in PO) indI: "[| (x, y) \<in> r; x \<in> S; y \<in> S |] ==> (x, y) \<in> induced S r"
```
```   176 by (simp add: add: induced_def)
```
```   177
```
```   178 lemma (in CL) CL_imp_ex_isLub: "S \<subseteq> A ==> \<exists>L. isLub S cl L"
```
```   179 apply (insert cl_co)
```
```   180 apply (simp add: CompleteLattice_def A_def)
```
```   181 done
```
```   182
```
```   183 declare (in CL) cl_co [simp]
```
```   184
```
```   185 lemma isLub_lub: "(\<exists>L. isLub S cl L) = isLub S cl (lub S cl)"
```
```   186 by (simp add: lub_def least_def isLub_def some_eq_ex [symmetric])
```
```   187
```
```   188 lemma isGlb_glb: "(\<exists>G. isGlb S cl G) = isGlb S cl (glb S cl)"
```
```   189 by (simp add: glb_def greatest_def isGlb_def some_eq_ex [symmetric])
```
```   190
```
```   191 lemma isGlb_dual_isLub: "isGlb S cl = isLub S (dual cl)"
```
```   192 by (simp add: isLub_def isGlb_def dual_def converse_unfold)
```
```   193
```
```   194 lemma isLub_dual_isGlb: "isLub S cl = isGlb S (dual cl)"
```
```   195 by (simp add: isLub_def isGlb_def dual_def converse_unfold)
```
```   196
```
```   197 lemma (in PO) dualPO: "dual cl \<in> PartialOrder"
```
```   198 apply (insert cl_po)
```
```   199 apply (simp add: PartialOrder_def dual_def)
```
```   200 done
```
```   201
```
```   202 lemma Rdual:
```
```   203      "\<forall>S. (S \<subseteq> A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
```
```   204       ==> \<forall>S. (S \<subseteq> A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))"
```
```   205 apply safe
```
```   206 apply (rule_tac x = "lub {y. y \<in> A & (\<forall>k \<in> S. (y, k) \<in> r)}
```
```   207                       (|pset = A, order = r|) " in exI)
```
```   208 apply (drule_tac x = "{y. y \<in> A & (\<forall>k \<in> S. (y,k) \<in> r) }" in spec)
```
```   209 apply (drule mp, fast)
```
```   210 apply (simp add: isLub_lub isGlb_def)
```
```   211 apply (simp add: isLub_def, blast)
```
```   212 done
```
```   213
```
```   214 lemma lub_dual_glb: "lub S cl = glb S (dual cl)"
```
```   215 by (simp add: lub_def glb_def least_def greatest_def dual_def converse_unfold)
```
```   216
```
```   217 lemma glb_dual_lub: "glb S cl = lub S (dual cl)"
```
```   218 by (simp add: lub_def glb_def least_def greatest_def dual_def converse_unfold)
```
```   219
```
```   220 lemma CL_subset_PO: "CompleteLattice \<subseteq> PartialOrder"
```
```   221 by (simp add: PartialOrder_def CompleteLattice_def, fast)
```
```   222
```
```   223 lemmas CL_imp_PO = CL_subset_PO [THEN subsetD]
```
```   224
```
```   225 declare PO.PO_imp_refl_on  [OF PO.intro [OF CL_imp_PO], simp]
```
```   226 declare PO.PO_imp_sym   [OF PO.intro [OF CL_imp_PO], simp]
```
```   227 declare PO.PO_imp_trans [OF PO.intro [OF CL_imp_PO], simp]
```
```   228
```
```   229 lemma (in CL) CO_refl_on: "refl_on A r"
```
```   230 by (rule PO_imp_refl_on)
```
```   231
```
```   232 lemma (in CL) CO_antisym: "antisym r"
```
```   233 by (rule PO_imp_sym)
```
```   234
```
```   235 lemma (in CL) CO_trans: "trans r"
```
```   236 by (rule PO_imp_trans)
```
```   237
```
```   238 lemma CompleteLatticeI:
```
```   239      "[| po \<in> PartialOrder; (\<forall>S. S \<subseteq> pset po --> (\<exists>L. isLub S po L));
```
```   240          (\<forall>S. S \<subseteq> pset po --> (\<exists>G. isGlb S po G))|]
```
```   241       ==> po \<in> CompleteLattice"
```
```   242 apply (unfold CompleteLattice_def, blast)
```
```   243 done
```
```   244
```
```   245 lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice"
```
```   246 apply (insert cl_co)
```
```   247 apply (simp add: CompleteLattice_def dual_def)
```
```   248 apply (fold dual_def)
```
```   249 apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric]
```
```   250                  dualPO)
```
```   251 done
```
```   252
```
```   253 lemma (in PO) dualA_iff: "pset (dual cl) = pset cl"
```
```   254 by (simp add: dual_def)
```
```   255
```
```   256 lemma (in PO) dualr_iff: "((x, y) \<in> (order(dual cl))) = ((y, x) \<in> order cl)"
```
```   257 by (simp add: dual_def)
```
```   258
```
```   259 lemma (in PO) monotone_dual:
```
```   260      "monotone f (pset cl) (order cl)
```
```   261      ==> monotone f (pset (dual cl)) (order(dual cl))"
```
```   262 by (simp add: monotone_def dualA_iff dualr_iff)
```
```   263
```
```   264 lemma (in PO) interval_dual:
```
```   265      "[| x \<in> A; y \<in> A|] ==> interval r x y = interval (order(dual cl)) y x"
```
```   266 apply (simp add: interval_def dualr_iff)
```
```   267 apply (fold r_def, fast)
```
```   268 done
```
```   269
```
```   270 lemma (in PO) interval_not_empty:
```
```   271      "[| trans r; interval r a b \<noteq> {} |] ==> (a, b) \<in> r"
```
```   272 apply (simp add: interval_def)
```
```   273 apply (unfold trans_def, blast)
```
```   274 done
```
```   275
```
```   276 lemma (in PO) interval_imp_mem: "x \<in> interval r a b ==> (a, x) \<in> r"
```
```   277 by (simp add: interval_def)
```
```   278
```
```   279 lemma (in PO) left_in_interval:
```
```   280      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> a \<in> interval r a b"
```
```   281 apply (simp (no_asm_simp) add: interval_def)
```
```   282 apply (simp add: PO_imp_trans interval_not_empty)
```
```   283 apply (simp add: reflE)
```
```   284 done
```
```   285
```
```   286 lemma (in PO) right_in_interval:
```
```   287      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> b \<in> interval r a b"
```
```   288 apply (simp (no_asm_simp) add: interval_def)
```
```   289 apply (simp add: PO_imp_trans interval_not_empty)
```
```   290 apply (simp add: reflE)
```
```   291 done
```
```   292
```
```   293 subsection \<open>sublattice\<close>
```
```   294
```
```   295 lemma (in PO) sublattice_imp_CL:
```
```   296      "S <<= cl  ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
```
```   297 by (simp add: sublattice_def CompleteLattice_def A_def r_def)
```
```   298
```
```   299 lemma (in CL) sublatticeI:
```
```   300      "[| S \<subseteq> A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
```
```   301       ==> S <<= cl"
```
```   302 by (simp add: sublattice_def A_def r_def)
```
```   303
```
```   304 subsection \<open>lub\<close>
```
```   305
```
```   306 lemma (in CL) lub_unique: "[| S \<subseteq> A; isLub S cl x; isLub S cl L|] ==> x = L"
```
```   307 apply (rule antisymE)
```
```   308 apply (auto simp add: isLub_def r_def)
```
```   309 done
```
```   310
```
```   311 lemma (in CL) lub_upper: "[|S \<subseteq> A; x \<in> S|] ==> (x, lub S cl) \<in> r"
```
```   312 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   313 apply (unfold lub_def least_def)
```
```   314 apply (rule some_equality [THEN ssubst])
```
```   315   apply (simp add: isLub_def)
```
```   316  apply (simp add: lub_unique A_def isLub_def)
```
```   317 apply (simp add: isLub_def r_def)
```
```   318 done
```
```   319
```
```   320 lemma (in CL) lub_least:
```
```   321      "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
```
```   322 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   323 apply (unfold lub_def least_def)
```
```   324 apply (rule_tac s=x in some_equality [THEN ssubst])
```
```   325   apply (simp add: isLub_def)
```
```   326  apply (simp add: lub_unique A_def isLub_def)
```
```   327 apply (simp add: isLub_def r_def A_def)
```
```   328 done
```
```   329
```
```   330 lemma (in CL) lub_in_lattice: "S \<subseteq> A ==> lub S cl \<in> A"
```
```   331 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   332 apply (unfold lub_def least_def)
```
```   333 apply (subst some_equality)
```
```   334 apply (simp add: isLub_def)
```
```   335 prefer 2 apply (simp add: isLub_def A_def)
```
```   336 apply (simp add: lub_unique A_def isLub_def)
```
```   337 done
```
```   338
```
```   339 lemma (in CL) lubI:
```
```   340      "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
```
```   341          \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"
```
```   342 apply (rule lub_unique, assumption)
```
```   343 apply (simp add: isLub_def A_def r_def)
```
```   344 apply (unfold isLub_def)
```
```   345 apply (rule conjI)
```
```   346 apply (fold A_def r_def)
```
```   347 apply (rule lub_in_lattice, assumption)
```
```   348 apply (simp add: lub_upper lub_least)
```
```   349 done
```
```   350
```
```   351 lemma (in CL) lubIa: "[| S \<subseteq> A; isLub S cl L |] ==> L = lub S cl"
```
```   352 by (simp add: lubI isLub_def A_def r_def)
```
```   353
```
```   354 lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"
```
```   355 by (simp add: isLub_def  A_def)
```
```   356
```
```   357 lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r"
```
```   358 by (simp add: isLub_def r_def)
```
```   359
```
```   360 lemma (in CL) isLub_least:
```
```   361      "[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r"
```
```   362 by (simp add: isLub_def A_def r_def)
```
```   363
```
```   364 lemma (in CL) isLubI:
```
```   365      "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;
```
```   366          (\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L"
```
```   367 by (simp add: isLub_def A_def r_def)
```
```   368
```
```   369 subsection \<open>glb\<close>
```
```   370
```
```   371 lemma (in CL) glb_in_lattice: "S \<subseteq> A ==> glb S cl \<in> A"
```
```   372 apply (subst glb_dual_lub)
```
```   373 apply (simp add: A_def)
```
```   374 apply (rule dualA_iff [THEN subst])
```
```   375 apply (rule CL.lub_in_lattice)
```
```   376 apply (rule CL.intro)
```
```   377 apply (rule PO.intro)
```
```   378 apply (rule dualPO)
```
```   379 apply (rule CL_axioms.intro)
```
```   380 apply (rule CL_dualCL)
```
```   381 apply (simp add: dualA_iff)
```
```   382 done
```
```   383
```
```   384 lemma (in CL) glb_lower: "[|S \<subseteq> A; x \<in> S|] ==> (glb S cl, x) \<in> r"
```
```   385 apply (subst glb_dual_lub)
```
```   386 apply (simp add: r_def)
```
```   387 apply (rule dualr_iff [THEN subst])
```
```   388 apply (rule CL.lub_upper)
```
```   389 apply (rule CL.intro)
```
```   390 apply (rule PO.intro)
```
```   391 apply (rule dualPO)
```
```   392 apply (rule CL_axioms.intro)
```
```   393 apply (rule CL_dualCL)
```
```   394 apply (simp add: dualA_iff A_def, assumption)
```
```   395 done
```
```   396
```
```   397 text \<open>
```
```   398   Reduce the sublattice property by using substructural properties;
```
```   399   abandoned see \<open>Tarski_4.ML\<close>.
```
```   400 \<close>
```
```   401
```
```   402 declare (in CLF) f_cl [simp]
```
```   403
```
```   404 lemma (in CLF) [simp]:
```
```   405     "f: pset cl \<rightarrow> pset cl & monotone f (pset cl) (order cl)"
```
```   406 proof -
```
```   407   have "\<forall>u v. (v, u) \<in> CLF_set \<longrightarrow> u \<in> {R \<in> pset v \<rightarrow> pset v. monotone R (pset v) (order v)}"
```
```   408     unfolding CLF_set_def using SigmaE2 by blast
```
```   409   hence F1: "\<forall>u v. (v, u) \<in> CLF_set \<longrightarrow> u \<in> pset v \<rightarrow> pset v \<and> monotone u (pset v) (order v)"
```
```   410     using CollectE by blast
```
```   411   hence "Tarski.monotone f (pset cl) (order cl)" by (metis f_cl)
```
```   412   hence "(cl, f) \<in> CLF_set \<and> Tarski.monotone f (pset cl) (order cl)"
```
```   413     by (metis f_cl)
```
```   414   thus "f \<in> pset cl \<rightarrow> pset cl \<and> Tarski.monotone f (pset cl) (order cl)"
```
```   415     using F1 by metis
```
```   416 qed
```
```   417
```
```   418 lemma (in CLF) f_in_funcset: "f \<in> A \<rightarrow> A"
```
```   419 by (simp add: A_def)
```
```   420
```
```   421 lemma (in CLF) monotone_f: "monotone f A r"
```
```   422 by (simp add: A_def r_def)
```
```   423
```
```   424 (*never proved, 2007-01-22*)
```
```   425
```
```   426 declare (in CLF) CLF_set_def [simp] CL_dualCL [simp] monotone_dual [simp] dualA_iff [simp]
```
```   427
```
```   428 lemma (in CLF) CLF_dual: "(dual cl, f) \<in> CLF_set"
```
```   429 apply (simp del: dualA_iff)
```
```   430 apply (simp)
```
```   431 done
```
```   432
```
```   433 declare (in CLF) CLF_set_def[simp del] CL_dualCL[simp del] monotone_dual[simp del]
```
```   434           dualA_iff[simp del]
```
```   435
```
```   436 subsection \<open>fixed points\<close>
```
```   437
```
```   438 lemma fix_subset: "fix f A \<subseteq> A"
```
```   439 by (simp add: fix_def, fast)
```
```   440
```
```   441 lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"
```
```   442 by (simp add: fix_def)
```
```   443
```
```   444 lemma fixf_subset:
```
```   445      "[| A \<subseteq> B; x \<in> fix (\<lambda>y \<in> A. f y) A |] ==> x \<in> fix f B"
```
```   446 by (simp add: fix_def, auto)
```
```   447
```
```   448 subsection \<open>lemmas for Tarski, lub\<close>
```
```   449
```
```   450 (*never proved, 2007-01-22*)
```
```   451
```
```   452 declare CL.lub_least[intro] CLF.f_in_funcset[intro] funcset_mem[intro] CL.lub_in_lattice[intro] PO.transE[intro] PO.monotoneE[intro] CLF.monotone_f[intro] CL.lub_upper[intro]
```
```   453
```
```   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 \<comment> \<open>\<open>\<forall>x:H. (x, f (lub H r)) \<in> r\<close>\<close>
```
```   460 apply (rule ballI)
```
```   461 (*never proved, 2007-01-22*)
```
```   462 apply (rule transE)
```
```   463 \<comment> \<open>instantiates \<open>(x, ?z) \<in> order cl to (x, f x)\<close>,\<close>
```
```   464 \<comment> \<open>because of the definition of \<open>H\<close>\<close>
```
```   465 apply fast
```
```   466 \<comment> \<open>so it remains to show \<open>(f x, f (lub H cl)) \<in> r\<close>\<close>
```
```   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 declare CL.lub_least[rule del] CLF.f_in_funcset[rule del]
```
```   475         funcset_mem[rule del] CL.lub_in_lattice[rule del]
```
```   476         PO.transE[rule del] PO.monotoneE[rule del]
```
```   477         CLF.monotone_f[rule del] CL.lub_upper[rule del]
```
```   478
```
```   479 (*never proved, 2007-01-22*)
```
```   480
```
```   481 declare CLF.f_in_funcset[intro] funcset_mem[intro] CL.lub_in_lattice[intro]
```
```   482      PO.monotoneE[intro] CLF.monotone_f[intro] CL.lub_upper[intro]
```
```   483      CLF.lubH_le_flubH[simp]
```
```   484
```
```   485 lemma (in CLF) flubH_le_lubH:
```
```   486      "[|  H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r"
```
```   487 apply (rule lub_upper, fast)
```
```   488 apply (rule_tac t = "H" in ssubst, assumption)
```
```   489 apply (rule CollectI)
```
```   490 by (metis (lifting) CO_refl_on lubH_le_flubH monotone_def monotone_f refl_onD1 refl_onD2)
```
```   491
```
```   492 declare CLF.f_in_funcset[rule del] funcset_mem[rule del]
```
```   493         CL.lub_in_lattice[rule del] PO.monotoneE[rule del]
```
```   494         CLF.monotone_f[rule del] CL.lub_upper[rule del]
```
```   495         CLF.lubH_le_flubH[simp del]
```
```   496
```
```   497 (*never proved, 2007-01-22*)
```
```   498
```
```   499 (* Single-step version fails. The conjecture clauses refer to local abstraction
```
```   500 functions (Frees). *)
```
```   501 lemma (in CLF) lubH_is_fixp:
```
```   502      "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
```
```   503 apply (simp add: fix_def)
```
```   504 apply (rule conjI)
```
```   505 proof -
```
```   506   assume A1: "H = {x. (x, f x) \<in> r \<and> x \<in> A}"
```
```   507   have F1: "\<forall>u v. v \<inter> u \<subseteq> u" by (metis Int_commute Int_lower1)
```
```   508   have "{R. (R, f R) \<in> r} \<inter> {R. R \<in> A} = H" using A1 by (metis Collect_conj_eq)
```
```   509   hence "H \<subseteq> {R. R \<in> A}" using F1 by metis
```
```   510   hence "H \<subseteq> A" by (metis Collect_mem_eq)
```
```   511   hence "lub H cl \<in> A" by (metis lub_in_lattice)
```
```   512   thus "lub {x. (x, f x) \<in> r \<and> x \<in> A} cl \<in> A" using A1 by metis
```
```   513 next
```
```   514   assume A1: "H = {x. (x, f x) \<in> r \<and> x \<in> A}"
```
```   515   have F1: "\<forall>v. {R. R \<in> v} = v" by (metis Collect_mem_eq)
```
```   516   have F2: "\<forall>w u. {R. R \<in> u \<and> R \<in> w} = u \<inter> w"
```
```   517     by (metis Collect_conj_eq Collect_mem_eq)
```
```   518   have F3: "\<forall>x v. {R. v R \<in> x} = v -` x" by (metis vimage_def)
```
```   519   hence F4: "A \<inter> (\<lambda>R. (R, f R)) -` r = H" using A1 by auto
```
```   520   hence F5: "(f (lub H cl), lub H cl) \<in> r"
```
```   521     by (metis A1 flubH_le_lubH)
```
```   522   have F6: "(lub H cl, f (lub H cl)) \<in> r"
```
```   523     by (metis A1 lubH_le_flubH)
```
```   524   have "(lub H cl, f (lub H cl)) \<in> r \<longrightarrow> f (lub H cl) = lub H cl"
```
```   525     using F5 by (metis antisymE)
```
```   526   hence "f (lub H cl) = lub H cl" using F6 by metis
```
```   527   thus "H = {x. (x, f x) \<in> r \<and> x \<in> A}
```
```   528         \<Longrightarrow> f (lub {x. (x, f x) \<in> r \<and> x \<in> A} cl) =
```
```   529            lub {x. (x, f x) \<in> r \<and> x \<in> A} cl"
```
```   530     by metis
```
```   531 qed
```
```   532
```
```   533 lemma (in CLF) (*lubH_is_fixp:*)
```
```   534      "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
```
```   535 apply (simp add: fix_def)
```
```   536 apply (rule conjI)
```
```   537 apply (metis CO_refl_on lubH_le_flubH refl_onD1)
```
```   538 apply (metis antisymE flubH_le_lubH lubH_le_flubH)
```
```   539 done
```
```   540
```
```   541 lemma (in CLF) fix_in_H:
```
```   542      "[| H = {x. (x, f x) \<in> r & x \<in> A};  x \<in> P |] ==> x \<in> H"
```
```   543 by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl_on
```
```   544                     fix_subset [of f A, THEN subsetD])
```
```   545
```
```   546 lemma (in CLF) fixf_le_lubH:
```
```   547      "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
```
```   548 apply (rule ballI)
```
```   549 apply (rule lub_upper, fast)
```
```   550 apply (rule fix_in_H)
```
```   551 apply (simp_all add: P_def)
```
```   552 done
```
```   553
```
```   554 lemma (in CLF) lubH_least_fixf:
```
```   555      "H = {x. (x, f x) \<in> r & x \<in> A}
```
```   556       ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"
```
```   557 apply (metis P_def lubH_is_fixp)
```
```   558 done
```
```   559
```
```   560 subsection \<open>Tarski fixpoint theorem 1, first part\<close>
```
```   561
```
```   562 declare CL.lubI[intro] fix_subset[intro] CL.lub_in_lattice[intro]
```
```   563         CLF.fixf_le_lubH[simp] CLF.lubH_least_fixf[simp]
```
```   564
```
```   565 lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"
```
```   566 (*sledgehammer;*)
```
```   567 apply (rule sym)
```
```   568 apply (simp add: P_def)
```
```   569 apply (rule lubI)
```
```   570 apply (simp add: fix_subset)
```
```   571 using fix_subset lubH_is_fixp apply fastforce
```
```   572 apply (simp add: fixf_le_lubH)
```
```   573 using lubH_is_fixp apply blast
```
```   574 done
```
```   575
```
```   576 declare CL.lubI[rule del] fix_subset[rule del] CL.lub_in_lattice[rule del]
```
```   577         CLF.fixf_le_lubH[simp del] CLF.lubH_least_fixf[simp del]
```
```   578
```
```   579 (*never proved, 2007-01-22*)
```
```   580
```
```   581 declare glb_dual_lub[simp] PO.dualA_iff[intro] CLF.lubH_is_fixp[intro]
```
```   582         PO.dualPO[intro] CL.CL_dualCL[intro] PO.dualr_iff[simp]
```
```   583
```
```   584 lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P"
```
```   585   \<comment> \<open>Tarski for glb\<close>
```
```   586 (*sledgehammer;*)
```
```   587 apply (simp add: glb_dual_lub P_def A_def r_def)
```
```   588 apply (rule dualA_iff [THEN subst])
```
```   589 apply (rule CLF.lubH_is_fixp)
```
```   590 apply (rule CLF.intro)
```
```   591 apply (rule CL.intro)
```
```   592 apply (rule PO.intro)
```
```   593 apply (rule dualPO)
```
```   594 apply (rule CL_axioms.intro)
```
```   595 apply (rule CL_dualCL)
```
```   596 apply (rule CLF_axioms.intro)
```
```   597 apply (rule CLF_dual)
```
```   598 apply (simp add: dualr_iff dualA_iff)
```
```   599 done
```
```   600
```
```   601 declare glb_dual_lub[simp del] PO.dualA_iff[rule del] CLF.lubH_is_fixp[rule del]
```
```   602         PO.dualPO[rule del] CL.CL_dualCL[rule del] PO.dualr_iff[simp del]
```
```   603
```
```   604 (*never proved, 2007-01-22*)
```
```   605
```
```   606 lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl"
```
```   607 (*sledgehammer;*)
```
```   608 apply (simp add: glb_dual_lub P_def A_def r_def)
```
```   609 apply (rule dualA_iff [THEN subst])
```
```   610 (*never proved, 2007-01-22*)
```
```   611 (*sledgehammer;*)
```
```   612 apply (simp add: CLF.T_thm_1_lub [of _ f, OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro,
```
```   613   OF dualPO CL_dualCL] dualPO CL_dualCL CLF_dual dualr_iff)
```
```   614 done
```
```   615
```
```   616 subsection \<open>interval\<close>
```
```   617
```
```   618 declare (in CLF) CO_refl_on[simp] refl_on_def [simp]
```
```   619
```
```   620 lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"
```
```   621 by (metis CO_refl_on refl_onD1)
```
```   622
```
```   623 declare (in CLF) CO_refl_on[simp del]  refl_on_def [simp del]
```
```   624
```
```   625 declare (in CLF) rel_imp_elem[intro]
```
```   626 declare interval_def [simp]
```
```   627
```
```   628 lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b \<subseteq> A"
```
```   629 by (metis CO_refl_on interval_imp_mem refl_onD refl_onD2 rel_imp_elem subset_eq)
```
```   630
```
```   631 declare (in CLF) rel_imp_elem[rule del]
```
```   632 declare interval_def [simp del]
```
```   633
```
```   634 lemma (in CLF) intervalI:
```
```   635      "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"
```
```   636 by (simp add: interval_def)
```
```   637
```
```   638 lemma (in CLF) interval_lemma1:
```
```   639      "[| S \<subseteq> interval r a b; x \<in> S |] ==> (a, x) \<in> r"
```
```   640 by (unfold interval_def, fast)
```
```   641
```
```   642 lemma (in CLF) interval_lemma2:
```
```   643      "[| S \<subseteq> interval r a b; x \<in> S |] ==> (x, b) \<in> r"
```
```   644 by (unfold interval_def, fast)
```
```   645
```
```   646 lemma (in CLF) a_less_lub:
```
```   647      "[| S \<subseteq> A; S \<noteq> {};
```
```   648          \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"
```
```   649 by (blast intro: transE)
```
```   650
```
```   651 lemma (in CLF) glb_less_b:
```
```   652      "[| S \<subseteq> A; S \<noteq> {};
```
```   653          \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"
```
```   654 by (blast intro: transE)
```
```   655
```
```   656 lemma (in CLF) S_intv_cl:
```
```   657      "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b |]==> S \<subseteq> A"
```
```   658 by (simp add: subset_trans [OF _ interval_subset])
```
```   659
```
```   660
```
```   661 lemma (in CLF) L_in_interval:
```
```   662      "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b;
```
```   663          S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"
```
```   664 (*WON'T TERMINATE
```
```   665 apply (metis CO_trans intervalI interval_lemma1 interval_lemma2 isLub_least isLub_upper subset_empty subset_iff trans_def)
```
```   666 *)
```
```   667 apply (rule intervalI)
```
```   668 apply (rule a_less_lub)
```
```   669 prefer 2 apply assumption
```
```   670 apply (simp add: S_intv_cl)
```
```   671 apply (rule ballI)
```
```   672 apply (simp add: interval_lemma1)
```
```   673 apply (simp add: isLub_upper)
```
```   674 \<comment> \<open>\<open>(L, b) \<in> r\<close>\<close>
```
```   675 apply (simp add: isLub_least interval_lemma2)
```
```   676 done
```
```   677
```
```   678 (*never proved, 2007-01-22*)
```
```   679
```
```   680 lemma (in CLF) G_in_interval:
```
```   681      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S \<subseteq> interval r a b; isGlb S cl G;
```
```   682          S \<noteq> {} |] ==> G \<in> interval r a b"
```
```   683 apply (simp add: interval_dual)
```
```   684 apply (simp add: CLF.L_in_interval [of _ f, OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro]
```
```   685                  dualA_iff A_def dualPO CL_dualCL CLF_dual isGlb_dual_isLub)
```
```   686 done
```
```   687
```
```   688
```
```   689 lemma (in CLF) intervalPO:
```
```   690      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   691       ==> (| pset = interval r a b, order = induced (interval r a b) r |)
```
```   692           \<in> PartialOrder"
```
```   693 proof -
```
```   694   assume A1: "a \<in> A"
```
```   695   assume "b \<in> A"
```
```   696   hence "\<forall>u. u \<in> A \<longrightarrow> interval r u b \<subseteq> A" by (metis interval_subset)
```
```   697   hence "interval r a b \<subseteq> A" using A1 by metis
```
```   698   hence "interval r a b \<subseteq> A" by metis
```
```   699   thus ?thesis by (metis po_subset_po)
```
```   700 qed
```
```   701
```
```   702 lemma (in CLF) intv_CL_lub:
```
```   703  "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   704   ==> \<forall>S. S \<subseteq> interval r a b -->
```
```   705           (\<exists>L. isLub S (| pset = interval r a b,
```
```   706                           order = induced (interval r a b) r |)  L)"
```
```   707 apply (intro strip)
```
```   708 apply (frule S_intv_cl [THEN CL_imp_ex_isLub])
```
```   709 prefer 2 apply assumption
```
```   710 apply assumption
```
```   711 apply (erule exE)
```
```   712 \<comment> \<open>define the lub for the interval as\<close>
```
```   713 apply (rule_tac x = "if S = {} then a else L" in exI)
```
```   714 apply (simp (no_asm_simp) add: isLub_def split del: if_split)
```
```   715 apply (intro impI conjI)
```
```   716 \<comment> \<open>\<open>(if S = {} then a else L) \<in> interval r a b\<close>\<close>
```
```   717 apply (simp add: CL_imp_PO L_in_interval)
```
```   718 apply (simp add: left_in_interval)
```
```   719 \<comment> \<open>lub prop 1\<close>
```
```   720 apply (case_tac "S = {}")
```
```   721 \<comment> \<open>\<open>S = {}, y \<in> S = False => everything\<close>\<close>
```
```   722 apply fast
```
```   723 \<comment> \<open>\<open>S \<noteq> {}\<close>\<close>
```
```   724 apply simp
```
```   725 \<comment> \<open>\<open>\<forall>y:S. (y, L) \<in> induced (interval r a b) r\<close>\<close>
```
```   726 apply (rule ballI)
```
```   727 apply (simp add: induced_def  L_in_interval)
```
```   728 apply (rule conjI)
```
```   729 apply (rule subsetD)
```
```   730 apply (simp add: S_intv_cl, assumption)
```
```   731 apply (simp add: isLub_upper)
```
```   732 \<comment> \<open>\<open>\<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\<close>\<close>
```
```   733 apply (rule ballI)
```
```   734 apply (rule impI)
```
```   735 apply (case_tac "S = {}")
```
```   736 \<comment> \<open>\<open>S = {}\<close>\<close>
```
```   737 apply simp
```
```   738 apply (simp add: induced_def  interval_def)
```
```   739 apply (rule conjI)
```
```   740 apply (rule reflE, assumption)
```
```   741 apply (rule interval_not_empty)
```
```   742 apply (rule CO_trans)
```
```   743 apply (simp add: interval_def)
```
```   744 \<comment> \<open>\<open>S \<noteq> {}\<close>\<close>
```
```   745 apply simp
```
```   746 apply (simp add: induced_def  L_in_interval)
```
```   747 apply (rule isLub_least, assumption)
```
```   748 apply (rule subsetD)
```
```   749 prefer 2 apply assumption
```
```   750 apply (simp add: S_intv_cl, fast)
```
```   751 done
```
```   752
```
```   753 lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual]
```
```   754
```
```   755 (*never proved, 2007-01-22*)
```
```   756
```
```   757 lemma (in CLF) interval_is_sublattice:
```
```   758      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   759         ==> interval r a b <<= cl"
```
```   760 (*sledgehammer *)
```
```   761 apply (rule sublatticeI)
```
```   762 apply (simp add: interval_subset)
```
```   763 (*never proved, 2007-01-22*)
```
```   764 (*sledgehammer *)
```
```   765 apply (rule CompleteLatticeI)
```
```   766 apply (simp add: intervalPO)
```
```   767  apply (simp add: intv_CL_lub)
```
```   768 apply (simp add: intv_CL_glb)
```
```   769 done
```
```   770
```
```   771 lemmas (in CLF) interv_is_compl_latt =
```
```   772     interval_is_sublattice [THEN sublattice_imp_CL]
```
```   773
```
```   774 subsection \<open>Top and Bottom\<close>
```
```   775 lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"
```
```   776 by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
```
```   777
```
```   778 lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)"
```
```   779 by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
```
```   780
```
```   781
```
```   782 lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"
```
```   783 (*sledgehammer; *)
```
```   784 apply (simp add: Bot_def least_def)
```
```   785 apply (rule_tac a="glb A cl" in someI2)
```
```   786 apply (simp_all add: glb_in_lattice glb_lower
```
```   787                      r_def [symmetric] A_def [symmetric])
```
```   788 done
```
```   789
```
```   790 (*first proved 2007-01-25 after relaxing relevance*)
```
```   791
```
```   792 lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
```
```   793 (*sledgehammer;*)
```
```   794 apply (simp add: Top_dual_Bot A_def)
```
```   795 (*first proved 2007-01-25 after relaxing relevance*)
```
```   796 (*sledgehammer*)
```
```   797 apply (rule dualA_iff [THEN subst])
```
```   798 apply (blast intro!: CLF.Bot_in_lattice [OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro] dualPO CL_dualCL CLF_dual)
```
```   799 done
```
```   800
```
```   801 lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
```
```   802 apply (simp add: Top_def greatest_def)
```
```   803 apply (rule_tac a="lub A cl" in someI2)
```
```   804 apply (rule someI2)
```
```   805 apply (simp_all add: lub_in_lattice lub_upper
```
```   806                      r_def [symmetric] A_def [symmetric])
```
```   807 done
```
```   808
```
```   809 (*never proved, 2007-01-22*)
```
```   810
```
```   811 lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
```
```   812 (*sledgehammer*)
```
```   813 apply (simp add: Bot_dual_Top r_def)
```
```   814 apply (rule dualr_iff [THEN subst])
```
```   815 apply (simp add: CLF.Top_prop [of _ f, OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro]
```
```   816                  dualA_iff A_def dualPO CL_dualCL CLF_dual)
```
```   817 done
```
```   818
```
```   819
```
```   820 lemma (in CLF) Top_intv_not_empty: "x \<in> A  ==> interval r x (Top cl) \<noteq> {}"
```
```   821 apply (metis Top_in_lattice Top_prop empty_iff intervalI reflE)
```
```   822 done
```
```   823
```
```   824
```
```   825 lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}"
```
```   826 apply (metis Bot_prop ex_in_conv intervalI reflE rel_imp_elem)
```
```   827 done
```
```   828
```
```   829 subsection \<open>fixed points form a partial order\<close>
```
```   830
```
```   831 lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
```
```   832 by (simp add: P_def fix_subset po_subset_po)
```
```   833
```
```   834 (*first proved 2007-01-25 after relaxing relevance*)
```
```   835
```
```   836 declare (in Tarski) P_def[simp] Y_ss [simp]
```
```   837 declare fix_subset [intro] subset_trans [intro]
```
```   838
```
```   839 lemma (in Tarski) Y_subset_A: "Y \<subseteq> A"
```
```   840 (*sledgehammer*)
```
```   841 apply (rule subset_trans [OF _ fix_subset])
```
```   842 apply (rule Y_ss [simplified P_def])
```
```   843 done
```
```   844
```
```   845 declare (in Tarski) P_def[simp del] Y_ss [simp del]
```
```   846 declare fix_subset [rule del] subset_trans [rule del]
```
```   847
```
```   848 lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A"
```
```   849   by (rule Y_subset_A [THEN lub_in_lattice])
```
```   850
```
```   851 (*never proved, 2007-01-22*)
```
```   852
```
```   853 lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"
```
```   854 (*sledgehammer*)
```
```   855 apply (rule lub_least)
```
```   856 apply (rule Y_subset_A)
```
```   857 apply (rule f_in_funcset [THEN funcset_mem])
```
```   858 apply (rule lubY_in_A)
```
```   859 \<comment> \<open>\<open>Y \<subseteq> P ==> f x = x\<close>\<close>
```
```   860 apply (rule ballI)
```
```   861 (*sledgehammer *)
```
```   862 apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
```
```   863 apply (erule Y_ss [simplified P_def, THEN subsetD])
```
```   864 \<comment> \<open>\<open>reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r\<close> by monotonicity\<close>
```
```   865 (*sledgehammer*)
```
```   866 apply (rule_tac f = "f" in monotoneE)
```
```   867 apply (rule monotone_f)
```
```   868 apply (simp add: Y_subset_A [THEN subsetD])
```
```   869 apply (rule lubY_in_A)
```
```   870 apply (simp add: lub_upper Y_subset_A)
```
```   871 done
```
```   872
```
```   873 (*first proved 2007-01-25 after relaxing relevance*)
```
```   874
```
```   875 lemma (in Tarski) intY1_subset: "intY1 \<subseteq> A"
```
```   876 (*sledgehammer*)
```
```   877 apply (unfold intY1_def)
```
```   878 apply (rule interval_subset)
```
```   879 apply (rule lubY_in_A)
```
```   880 apply (rule Top_in_lattice)
```
```   881 done
```
```   882
```
```   883 lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD]
```
```   884
```
```   885 (*never proved, 2007-01-22*)
```
```   886
```
```   887 lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"
```
```   888 (*sledgehammer*)
```
```   889 apply (simp add: intY1_def  interval_def)
```
```   890 apply (rule conjI)
```
```   891 apply (rule transE)
```
```   892 apply (rule lubY_le_flubY)
```
```   893 \<comment> \<open>\<open>(f (lub Y cl), f x) \<in> r\<close>\<close>
```
```   894 (*sledgehammer [has been proved before now...]*)
```
```   895 apply (rule_tac f=f in monotoneE)
```
```   896 apply (rule monotone_f)
```
```   897 apply (rule lubY_in_A)
```
```   898 apply (simp add: intY1_def interval_def  intY1_elem)
```
```   899 apply (simp add: intY1_def  interval_def)
```
```   900 \<comment> \<open>\<open>(f x, Top cl) \<in> r\<close>\<close>
```
```   901 apply (rule Top_prop)
```
```   902 apply (rule f_in_funcset [THEN funcset_mem])
```
```   903 apply (simp add: intY1_def interval_def  intY1_elem)
```
```   904 done
```
```   905
```
```   906
```
```   907 lemma (in Tarski) intY1_func: "(\<lambda>x \<in> intY1. f x) \<in> intY1 \<rightarrow> intY1"
```
```   908 apply (rule restrict_in_funcset)
```
```   909 apply (metis intY1_f_closed restrict_in_funcset)
```
```   910 done
```
```   911
```
```   912
```
```   913 lemma (in Tarski) intY1_mono:
```
```   914      "monotone (\<lambda>x \<in> intY1. f x) intY1 (induced intY1 r)"
```
```   915 (*sledgehammer *)
```
```   916 apply (auto simp add: monotone_def induced_def intY1_f_closed)
```
```   917 apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
```
```   918 done
```
```   919
```
```   920 (*proof requires relaxing relevance: 2007-01-25*)
```
```   921
```
```   922 lemma (in Tarski) intY1_is_cl:
```
```   923     "(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
```
```   924 (*sledgehammer*)
```
```   925 apply (unfold intY1_def)
```
```   926 apply (rule interv_is_compl_latt)
```
```   927 apply (rule lubY_in_A)
```
```   928 apply (rule Top_in_lattice)
```
```   929 apply (rule Top_intv_not_empty)
```
```   930 apply (rule lubY_in_A)
```
```   931 done
```
```   932
```
```   933 (*never proved, 2007-01-22*)
```
```   934
```
```   935 lemma (in Tarski) v_in_P: "v \<in> P"
```
```   936 (*sledgehammer*)
```
```   937 apply (unfold P_def)
```
```   938 apply (rule_tac A = "intY1" in fixf_subset)
```
```   939 apply (rule intY1_subset)
```
```   940 apply (simp add: CLF.glbH_is_fixp [OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro, OF _ intY1_is_cl, simplified]
```
```   941                  v_def CL_imp_PO intY1_is_cl CLF_set_def intY1_func intY1_mono)
```
```   942 done
```
```   943
```
```   944
```
```   945 lemma (in Tarski) z_in_interval:
```
```   946      "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1"
```
```   947 (*sledgehammer *)
```
```   948 apply (unfold intY1_def P_def)
```
```   949 apply (rule intervalI)
```
```   950 prefer 2
```
```   951  apply (erule fix_subset [THEN subsetD, THEN Top_prop])
```
```   952 apply (rule lub_least)
```
```   953 apply (rule Y_subset_A)
```
```   954 apply (fast elim!: fix_subset [THEN subsetD])
```
```   955 apply (simp add: induced_def)
```
```   956 done
```
```   957
```
```   958
```
```   959 lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |]
```
```   960       ==> ((\<lambda>x \<in> intY1. f x) z, z) \<in> induced intY1 r"
```
```   961 using P_def fix_imp_eq indI intY1_elem reflE z_in_interval by fastforce
```
```   962
```
```   963 (*never proved, 2007-01-22*)
```
```   964
```
```   965 lemma (in Tarski) tarski_full_lemma:
```
```   966      "\<exists>L. isLub Y (| pset = P, order = induced P r |) L"
```
```   967 apply (rule_tac x = "v" in exI)
```
```   968 apply (simp add: isLub_def)
```
```   969 \<comment> \<open>\<open>v \<in> P\<close>\<close>
```
```   970 apply (simp add: v_in_P)
```
```   971 apply (rule conjI)
```
```   972 (*sledgehammer*)
```
```   973 \<comment> \<open>\<open>v\<close> is lub\<close>
```
```   974 \<comment> \<open>\<open>1. \<forall>y:Y. (y, v) \<in> induced P r\<close>\<close>
```
```   975 apply (rule ballI)
```
```   976 apply (simp add: induced_def subsetD v_in_P)
```
```   977 apply (rule conjI)
```
```   978 apply (erule Y_ss [THEN subsetD])
```
```   979 apply (rule_tac b = "lub Y cl" in transE)
```
```   980 apply (rule lub_upper)
```
```   981 apply (rule Y_subset_A, assumption)
```
```   982 apply (rule_tac b = "Top cl" in interval_imp_mem)
```
```   983 apply (simp add: v_def)
```
```   984 apply (fold intY1_def)
```
```   985 apply (rule CL.glb_in_lattice [OF CL.intro, OF PO.intro CL_axioms.intro, OF _ intY1_is_cl, simplified])
```
```   986  apply (simp add: CL_imp_PO intY1_is_cl, force)
```
```   987 \<comment> \<open>\<open>v\<close> is LEAST ub\<close>
```
```   988 apply clarify
```
```   989 apply (rule indI)
```
```   990   prefer 3 apply assumption
```
```   991  prefer 2 apply (simp add: v_in_P)
```
```   992 apply (unfold v_def)
```
```   993 (*never proved, 2007-01-22*)
```
```   994 (*sledgehammer*)
```
```   995 apply (rule indE)
```
```   996 apply (rule_tac [2] intY1_subset)
```
```   997 (*never proved, 2007-01-22*)
```
```   998 (*sledgehammer*)
```
```   999 apply (rule CL.glb_lower [OF CL.intro, OF PO.intro CL_axioms.intro, OF _ intY1_is_cl, simplified])
```
```  1000   apply (simp add: CL_imp_PO intY1_is_cl)
```
```  1001  apply force
```
```  1002 apply (simp add: induced_def intY1_f_closed z_in_interval)
```
```  1003 apply (simp add: P_def fix_imp_eq [of _ f A] reflE
```
```  1004                  fix_subset [of f A, THEN subsetD])
```
```  1005 done
```
```  1006
```
```  1007 lemma CompleteLatticeI_simp:
```
```  1008      "[| (| pset = A, order = r |) \<in> PartialOrder;
```
```  1009          \<forall>S. S \<subseteq> A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]
```
```  1010     ==> (| pset = A, order = r |) \<in> CompleteLattice"
```
```  1011 by (simp add: CompleteLatticeI Rdual)
```
```  1012
```
```  1013 (*never proved, 2007-01-22*)
```
```  1014
```
```  1015 declare (in CLF) fixf_po[intro] P_def [simp] A_def [simp] r_def [simp]
```
```  1016              Tarski.tarski_full_lemma [intro] cl_po [intro] cl_co [intro]
```
```  1017              CompleteLatticeI_simp [intro]
```
```  1018
```
```  1019 theorem (in CLF) Tarski_full:
```
```  1020      "(| pset = P, order = induced P r|) \<in> CompleteLattice"
```
```  1021 (*sledgehammer*)
```
```  1022 apply (rule CompleteLatticeI_simp)
```
```  1023 apply (rule fixf_po, clarify)
```
```  1024 (*never proved, 2007-01-22*)
```
```  1025 (*sledgehammer*)
```
```  1026 apply (simp add: P_def A_def r_def)
```
```  1027 apply (blast intro!: Tarski.tarski_full_lemma [OF Tarski.intro, OF CLF.intro Tarski_axioms.intro,
```
```  1028   OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro] cl_po cl_co f_cl)
```
```  1029 done
```
```  1030
```
```  1031 declare (in CLF) fixf_po [rule del] P_def [simp del] A_def [simp del] r_def [simp del]
```
```  1032          Tarski.tarski_full_lemma [rule del] cl_po [rule del] cl_co [rule del]
```
```  1033          CompleteLatticeI_simp [rule del]
```
```  1034
```
```  1035 end
```