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