src/HOL/MetisExamples/Tarski.thy
 author wenzelm Mon Mar 16 18:24:30 2009 +0100 (2009-03-16) changeset 30549 d2d7874648bd parent 30198 922f944f03b2 child 32864 a226f29d4bdc permissions -rw-r--r--
simplified method setup;
```     1 (*  Title:      HOL/MetisTest/Tarski.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4
```
```     5 Testing the metis method
```
```     6 *)
```
```     7
```
```     8 header {* The Full Theorem of Tarski *}
```
```     9
```
```    10 theory Tarski
```
```    11 imports Main FuncSet
```
```    12 begin
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```    13
```
```    14 (*Many of these higher-order problems appear to be impossible using the
```
```    15 current linkup. They often seem to need either higher-order unification
```
```    16 or explicit reasoning about connectives such as conjunction. The numerous
```
```    17 set comprehensions are to blame.*)
```
```    18
```
```    19
```
```    20 record 'a potype =
```
```    21   pset  :: "'a set"
```
```    22   order :: "('a * 'a) set"
```
```    23
```
```    24 constdefs
```
```    25   monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool"
```
```    26   "monotone f A r == \<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r"
```
```    27
```
```    28   least :: "['a => bool, 'a potype] => 'a"
```
```    29   "least P po == @ x. x: pset po & P x &
```
```    30                        (\<forall>y \<in> pset po. P y --> (x,y): order po)"
```
```    31
```
```    32   greatest :: "['a => bool, 'a potype] => 'a"
```
```    33   "greatest P po == @ x. x: pset po & P x &
```
```    34                           (\<forall>y \<in> pset po. P y --> (y,x): order po)"
```
```    35
```
```    36   lub  :: "['a set, 'a potype] => 'a"
```
```    37   "lub S po == least (%x. \<forall>y\<in>S. (y,x): order po) po"
```
```    38
```
```    39   glb  :: "['a set, 'a potype] => 'a"
```
```    40   "glb S po == greatest (%x. \<forall>y\<in>S. (x,y): order po) po"
```
```    41
```
```    42   isLub :: "['a set, 'a potype, 'a] => bool"
```
```    43   "isLub S po == %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   isGlb :: "['a set, 'a potype, 'a] => bool"
```
```    47   "isGlb S po == %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   "fix"    :: "[('a => 'a), 'a set] => 'a set"
```
```    51   "fix f A  == {x. x: A & f x = x}"
```
```    52
```
```    53   interval :: "[('a*'a) set,'a, 'a ] => 'a set"
```
```    54   "interval r a b == {x. (a,x): r & (x,b): r}"
```
```    55
```
```    56 constdefs
```
```    57   Bot :: "'a potype => 'a"
```
```    58   "Bot po == least (%x. True) po"
```
```    59
```
```    60   Top :: "'a potype => 'a"
```
```    61   "Top po == greatest (%x. True) po"
```
```    62
```
```    63   PartialOrder :: "('a potype) set"
```
```    64   "PartialOrder == {P. refl_on (pset P) (order P) & antisym (order P) &
```
```    65                        trans (order P)}"
```
```    66
```
```    67   CompleteLattice :: "('a potype) set"
```
```    68   "CompleteLattice == {cl. cl: PartialOrder &
```
```    69                         (\<forall>S. S \<subseteq> pset cl --> (\<exists>L. isLub S cl L)) &
```
```    70                         (\<forall>S. S \<subseteq> pset cl --> (\<exists>G. isGlb S cl G))}"
```
```    71
```
```    72   induced :: "['a set, ('a * 'a) set] => ('a *'a)set"
```
```    73   "induced A r == {(a,b). a : A & b: A & (a,b): r}"
```
```    74
```
```    75 constdefs
```
```    76   sublattice :: "('a potype * 'a set)set"
```
```    77   "sublattice ==
```
```    78       SIGMA cl: CompleteLattice.
```
```    79           {S. S \<subseteq> pset cl &
```
```    80            (| pset = S, order = induced S (order cl) |): CompleteLattice }"
```
```    81
```
```    82 syntax
```
```    83   "@SL"  :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50)
```
```    84
```
```    85 translations
```
```    86   "S <<= cl" == "S : sublattice `` {cl}"
```
```    87
```
```    88 constdefs
```
```    89   dual :: "'a potype => 'a potype"
```
```    90   "dual po == (| pset = pset po, order = converse (order po) |)"
```
```    91
```
```    92 locale PO =
```
```    93   fixes cl :: "'a potype"
```
```    94     and A  :: "'a set"
```
```    95     and r  :: "('a * 'a) set"
```
```    96   assumes cl_po:  "cl : PartialOrder"
```
```    97   defines A_def: "A == pset cl"
```
```    98      and  r_def: "r == order cl"
```
```    99
```
```   100 locale CL = PO +
```
```   101   assumes cl_co:  "cl : CompleteLattice"
```
```   102
```
```   103 definition CLF_set :: "('a potype * ('a => 'a)) set" where
```
```   104   "CLF_set = (SIGMA cl: CompleteLattice.
```
```   105             {f. f: pset cl -> pset cl & monotone f (pset cl) (order cl)})"
```
```   106
```
```   107 locale CLF = CL +
```
```   108   fixes f :: "'a => 'a"
```
```   109     and P :: "'a set"
```
```   110   assumes f_cl:  "(cl,f) : CLF_set" (*was the equivalent "f : CLF``{cl}"*)
```
```   111   defines P_def: "P == fix f A"
```
```   112
```
```   113
```
```   114 locale Tarski = CLF +
```
```   115   fixes Y     :: "'a set"
```
```   116     and intY1 :: "'a set"
```
```   117     and v     :: "'a"
```
```   118   assumes
```
```   119     Y_ss: "Y \<subseteq> P"
```
```   120   defines
```
```   121     intY1_def: "intY1 == interval r (lub Y cl) (Top cl)"
```
```   122     and v_def: "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r &
```
```   123                              x: intY1}
```
```   124                       (| pset=intY1, order=induced intY1 r|)"
```
```   125
```
```   126
```
```   127 subsection {* Partial Order *}
```
```   128
```
```   129 lemma (in PO) PO_imp_refl_on: "refl_on A r"
```
```   130 apply (insert cl_po)
```
```   131 apply (simp add: PartialOrder_def A_def r_def)
```
```   132 done
```
```   133
```
```   134 lemma (in PO) PO_imp_sym: "antisym r"
```
```   135 apply (insert cl_po)
```
```   136 apply (simp add: PartialOrder_def r_def)
```
```   137 done
```
```   138
```
```   139 lemma (in PO) PO_imp_trans: "trans r"
```
```   140 apply (insert cl_po)
```
```   141 apply (simp add: PartialOrder_def r_def)
```
```   142 done
```
```   143
```
```   144 lemma (in PO) reflE: "x \<in> A ==> (x, x) \<in> r"
```
```   145 apply (insert cl_po)
```
```   146 apply (simp add: PartialOrder_def refl_on_def A_def r_def)
```
```   147 done
```
```   148
```
```   149 lemma (in PO) antisymE: "[| (a, b) \<in> r; (b, a) \<in> r |] ==> a = b"
```
```   150 apply (insert cl_po)
```
```   151 apply (simp add: PartialOrder_def antisym_def r_def)
```
```   152 done
```
```   153
```
```   154 lemma (in PO) transE: "[| (a, b) \<in> r; (b, c) \<in> r|] ==> (a,c) \<in> r"
```
```   155 apply (insert cl_po)
```
```   156 apply (simp add: PartialOrder_def r_def)
```
```   157 apply (unfold trans_def, fast)
```
```   158 done
```
```   159
```
```   160 lemma (in PO) monotoneE:
```
```   161      "[| monotone f A r;  x \<in> A; y \<in> A; (x, y) \<in> r |] ==> (f x, f y) \<in> r"
```
```   162 by (simp add: monotone_def)
```
```   163
```
```   164 lemma (in PO) po_subset_po:
```
```   165      "S \<subseteq> A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
```
```   166 apply (simp (no_asm) add: PartialOrder_def)
```
```   167 apply auto
```
```   168 -- {* refl *}
```
```   169 apply (simp add: refl_on_def induced_def)
```
```   170 apply (blast intro: reflE)
```
```   171 -- {* antisym *}
```
```   172 apply (simp add: antisym_def induced_def)
```
```   173 apply (blast intro: antisymE)
```
```   174 -- {* trans *}
```
```   175 apply (simp add: trans_def induced_def)
```
```   176 apply (blast intro: transE)
```
```   177 done
```
```   178
```
```   179 lemma (in PO) indE: "[| (x, y) \<in> induced S r; S \<subseteq> A |] ==> (x, y) \<in> r"
```
```   180 by (simp add: add: induced_def)
```
```   181
```
```   182 lemma (in PO) indI: "[| (x, y) \<in> r; x \<in> S; y \<in> S |] ==> (x, y) \<in> induced S r"
```
```   183 by (simp add: add: induced_def)
```
```   184
```
```   185 lemma (in CL) CL_imp_ex_isLub: "S \<subseteq> A ==> \<exists>L. isLub S cl L"
```
```   186 apply (insert cl_co)
```
```   187 apply (simp add: CompleteLattice_def A_def)
```
```   188 done
```
```   189
```
```   190 declare (in CL) cl_co [simp]
```
```   191
```
```   192 lemma isLub_lub: "(\<exists>L. isLub S cl L) = isLub S cl (lub S cl)"
```
```   193 by (simp add: lub_def least_def isLub_def some_eq_ex [symmetric])
```
```   194
```
```   195 lemma isGlb_glb: "(\<exists>G. isGlb S cl G) = isGlb S cl (glb S cl)"
```
```   196 by (simp add: glb_def greatest_def isGlb_def some_eq_ex [symmetric])
```
```   197
```
```   198 lemma isGlb_dual_isLub: "isGlb S cl = isLub S (dual cl)"
```
```   199 by (simp add: isLub_def isGlb_def dual_def converse_def)
```
```   200
```
```   201 lemma isLub_dual_isGlb: "isLub S cl = isGlb S (dual cl)"
```
```   202 by (simp add: isLub_def isGlb_def dual_def converse_def)
```
```   203
```
```   204 lemma (in PO) dualPO: "dual cl \<in> PartialOrder"
```
```   205 apply (insert cl_po)
```
```   206 apply (simp add: PartialOrder_def dual_def refl_on_converse
```
```   207                  trans_converse antisym_converse)
```
```   208 done
```
```   209
```
```   210 lemma Rdual:
```
```   211      "\<forall>S. (S \<subseteq> A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
```
```   212       ==> \<forall>S. (S \<subseteq> A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))"
```
```   213 apply safe
```
```   214 apply (rule_tac x = "lub {y. y \<in> A & (\<forall>k \<in> S. (y, k) \<in> r)}
```
```   215                       (|pset = A, order = r|) " in exI)
```
```   216 apply (drule_tac x = "{y. y \<in> A & (\<forall>k \<in> S. (y,k) \<in> r) }" in spec)
```
```   217 apply (drule mp, fast)
```
```   218 apply (simp add: isLub_lub isGlb_def)
```
```   219 apply (simp add: isLub_def, blast)
```
```   220 done
```
```   221
```
```   222 lemma lub_dual_glb: "lub S cl = glb S (dual cl)"
```
```   223 by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
```
```   224
```
```   225 lemma glb_dual_lub: "glb S cl = lub S (dual cl)"
```
```   226 by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
```
```   227
```
```   228 lemma CL_subset_PO: "CompleteLattice \<subseteq> PartialOrder"
```
```   229 by (simp add: PartialOrder_def CompleteLattice_def, fast)
```
```   230
```
```   231 lemmas CL_imp_PO = CL_subset_PO [THEN subsetD]
```
```   232
```
```   233 declare PO.PO_imp_refl_on  [OF PO.intro [OF CL_imp_PO], simp]
```
```   234 declare PO.PO_imp_sym   [OF PO.intro [OF CL_imp_PO], simp]
```
```   235 declare PO.PO_imp_trans [OF PO.intro [OF CL_imp_PO], simp]
```
```   236
```
```   237 lemma (in CL) CO_refl_on: "refl_on A r"
```
```   238 by (rule PO_imp_refl_on)
```
```   239
```
```   240 lemma (in CL) CO_antisym: "antisym r"
```
```   241 by (rule PO_imp_sym)
```
```   242
```
```   243 lemma (in CL) CO_trans: "trans r"
```
```   244 by (rule PO_imp_trans)
```
```   245
```
```   246 lemma CompleteLatticeI:
```
```   247      "[| po \<in> PartialOrder; (\<forall>S. S \<subseteq> pset po --> (\<exists>L. isLub S po L));
```
```   248          (\<forall>S. S \<subseteq> pset po --> (\<exists>G. isGlb S po G))|]
```
```   249       ==> po \<in> CompleteLattice"
```
```   250 apply (unfold CompleteLattice_def, blast)
```
```   251 done
```
```   252
```
```   253 lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice"
```
```   254 apply (insert cl_co)
```
```   255 apply (simp add: CompleteLattice_def dual_def)
```
```   256 apply (fold dual_def)
```
```   257 apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric]
```
```   258                  dualPO)
```
```   259 done
```
```   260
```
```   261 lemma (in PO) dualA_iff: "pset (dual cl) = pset cl"
```
```   262 by (simp add: dual_def)
```
```   263
```
```   264 lemma (in PO) dualr_iff: "((x, y) \<in> (order(dual cl))) = ((y, x) \<in> order cl)"
```
```   265 by (simp add: dual_def)
```
```   266
```
```   267 lemma (in PO) monotone_dual:
```
```   268      "monotone f (pset cl) (order cl)
```
```   269      ==> monotone f (pset (dual cl)) (order(dual cl))"
```
```   270 by (simp add: monotone_def dualA_iff dualr_iff)
```
```   271
```
```   272 lemma (in PO) interval_dual:
```
```   273      "[| x \<in> A; y \<in> A|] ==> interval r x y = interval (order(dual cl)) y x"
```
```   274 apply (simp add: interval_def dualr_iff)
```
```   275 apply (fold r_def, fast)
```
```   276 done
```
```   277
```
```   278 lemma (in PO) interval_not_empty:
```
```   279      "[| trans r; interval r a b \<noteq> {} |] ==> (a, b) \<in> r"
```
```   280 apply (simp add: interval_def)
```
```   281 apply (unfold trans_def, blast)
```
```   282 done
```
```   283
```
```   284 lemma (in PO) interval_imp_mem: "x \<in> interval r a b ==> (a, x) \<in> r"
```
```   285 by (simp add: interval_def)
```
```   286
```
```   287 lemma (in PO) left_in_interval:
```
```   288      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> a \<in> interval r a b"
```
```   289 apply (simp (no_asm_simp) add: interval_def)
```
```   290 apply (simp add: PO_imp_trans interval_not_empty)
```
```   291 apply (simp add: reflE)
```
```   292 done
```
```   293
```
```   294 lemma (in PO) right_in_interval:
```
```   295      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> b \<in> interval r a b"
```
```   296 apply (simp (no_asm_simp) add: interval_def)
```
```   297 apply (simp add: PO_imp_trans interval_not_empty)
```
```   298 apply (simp add: reflE)
```
```   299 done
```
```   300
```
```   301
```
```   302 subsection {* sublattice *}
```
```   303
```
```   304 lemma (in PO) sublattice_imp_CL:
```
```   305      "S <<= cl  ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
```
```   306 by (simp add: sublattice_def CompleteLattice_def A_def r_def)
```
```   307
```
```   308 lemma (in CL) sublatticeI:
```
```   309      "[| S \<subseteq> A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
```
```   310       ==> S <<= cl"
```
```   311 by (simp add: sublattice_def A_def r_def)
```
```   312
```
```   313
```
```   314 subsection {* lub *}
```
```   315
```
```   316 lemma (in CL) lub_unique: "[| S \<subseteq> A; isLub S cl x; isLub S cl L|] ==> x = L"
```
```   317 apply (rule antisymE)
```
```   318 apply (auto simp add: isLub_def r_def)
```
```   319 done
```
```   320
```
```   321 lemma (in CL) lub_upper: "[|S \<subseteq> A; x \<in> S|] ==> (x, lub S cl) \<in> r"
```
```   322 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   323 apply (unfold lub_def least_def)
```
```   324 apply (rule 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)
```
```   328 done
```
```   329
```
```   330 lemma (in CL) lub_least:
```
```   331      "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
```
```   332 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   333 apply (unfold lub_def least_def)
```
```   334 apply (rule_tac s=x in some_equality [THEN ssubst])
```
```   335   apply (simp add: isLub_def)
```
```   336  apply (simp add: lub_unique A_def isLub_def)
```
```   337 apply (simp add: isLub_def r_def A_def)
```
```   338 done
```
```   339
```
```   340 lemma (in CL) lub_in_lattice: "S \<subseteq> A ==> lub S cl \<in> A"
```
```   341 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
```
```   342 apply (unfold lub_def least_def)
```
```   343 apply (subst some_equality)
```
```   344 apply (simp add: isLub_def)
```
```   345 prefer 2 apply (simp add: isLub_def A_def)
```
```   346 apply (simp add: lub_unique A_def isLub_def)
```
```   347 done
```
```   348
```
```   349 lemma (in CL) lubI:
```
```   350      "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
```
```   351          \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"
```
```   352 apply (rule lub_unique, assumption)
```
```   353 apply (simp add: isLub_def A_def r_def)
```
```   354 apply (unfold isLub_def)
```
```   355 apply (rule conjI)
```
```   356 apply (fold A_def r_def)
```
```   357 apply (rule lub_in_lattice, assumption)
```
```   358 apply (simp add: lub_upper lub_least)
```
```   359 done
```
```   360
```
```   361 lemma (in CL) lubIa: "[| S \<subseteq> A; isLub S cl L |] ==> L = lub S cl"
```
```   362 by (simp add: lubI isLub_def A_def r_def)
```
```   363
```
```   364 lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"
```
```   365 by (simp add: isLub_def  A_def)
```
```   366
```
```   367 lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r"
```
```   368 by (simp add: isLub_def r_def)
```
```   369
```
```   370 lemma (in CL) isLub_least:
```
```   371      "[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r"
```
```   372 by (simp add: isLub_def A_def r_def)
```
```   373
```
```   374 lemma (in CL) isLubI:
```
```   375      "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;
```
```   376          (\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L"
```
```   377 by (simp add: isLub_def A_def r_def)
```
```   378
```
```   379
```
```   380
```
```   381 subsection {* glb *}
```
```   382
```
```   383 lemma (in CL) glb_in_lattice: "S \<subseteq> A ==> glb S cl \<in> A"
```
```   384 apply (subst glb_dual_lub)
```
```   385 apply (simp add: A_def)
```
```   386 apply (rule dualA_iff [THEN subst])
```
```   387 apply (rule CL.lub_in_lattice)
```
```   388 apply (rule CL.intro)
```
```   389 apply (rule PO.intro)
```
```   390 apply (rule dualPO)
```
```   391 apply (rule CL_axioms.intro)
```
```   392 apply (rule CL_dualCL)
```
```   393 apply (simp add: dualA_iff)
```
```   394 done
```
```   395
```
```   396 lemma (in CL) glb_lower: "[|S \<subseteq> A; x \<in> S|] ==> (glb S cl, x) \<in> r"
```
```   397 apply (subst glb_dual_lub)
```
```   398 apply (simp add: r_def)
```
```   399 apply (rule dualr_iff [THEN subst])
```
```   400 apply (rule CL.lub_upper)
```
```   401 apply (rule CL.intro)
```
```   402 apply (rule PO.intro)
```
```   403 apply (rule dualPO)
```
```   404 apply (rule CL_axioms.intro)
```
```   405 apply (rule CL_dualCL)
```
```   406 apply (simp add: dualA_iff A_def, assumption)
```
```   407 done
```
```   408
```
```   409 text {*
```
```   410   Reduce the sublattice property by using substructural properties;
```
```   411   abandoned see @{text "Tarski_4.ML"}.
```
```   412 *}
```
```   413
```
```   414 declare (in CLF) f_cl [simp]
```
```   415
```
```   416 (*never proved, 2007-01-22: Tarski__CLF_unnamed_lemma
```
```   417   NOT PROVABLE because of the conjunction used in the definition: we don't
```
```   418   allow reasoning with rules like conjE, which is essential here.*)
```
```   419 ML_command{*AtpWrapper.problem_name:="Tarski__CLF_unnamed_lemma"*}
```
```   420 lemma (in CLF) [simp]:
```
```   421     "f: pset cl -> pset cl & monotone f (pset cl) (order cl)"
```
```   422 apply (insert f_cl)
```
```   423 apply (unfold CLF_set_def)
```
```   424 apply (erule SigmaE2)
```
```   425 apply (erule CollectE)
```
```   426 apply assumption
```
```   427 done
```
```   428
```
```   429 lemma (in CLF) f_in_funcset: "f \<in> A -> A"
```
```   430 by (simp add: A_def)
```
```   431
```
```   432 lemma (in CLF) monotone_f: "monotone f A r"
```
```   433 by (simp add: A_def r_def)
```
```   434
```
```   435 (*never proved, 2007-01-22*)
```
```   436 ML_command{*AtpWrapper.problem_name:="Tarski__CLF_CLF_dual"*}
```
```   437 declare (in CLF) CLF_set_def [simp] CL_dualCL [simp] monotone_dual [simp] dualA_iff [simp]
```
```   438
```
```   439 lemma (in CLF) CLF_dual: "(dual cl, f) \<in> CLF_set"
```
```   440 apply (simp del: dualA_iff)
```
```   441 apply (simp)
```
```   442 done
```
```   443
```
```   444 declare (in CLF) CLF_set_def[simp del] CL_dualCL[simp del] monotone_dual[simp del]
```
```   445           dualA_iff[simp del]
```
```   446
```
```   447
```
```   448 subsection {* fixed points *}
```
```   449
```
```   450 lemma fix_subset: "fix f A \<subseteq> A"
```
```   451 by (simp add: fix_def, fast)
```
```   452
```
```   453 lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"
```
```   454 by (simp add: fix_def)
```
```   455
```
```   456 lemma fixf_subset:
```
```   457      "[| A \<subseteq> B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
```
```   458 by (simp add: fix_def, auto)
```
```   459
```
```   460
```
```   461 subsection {* lemmas for Tarski, lub *}
```
```   462
```
```   463 (*never proved, 2007-01-22*)
```
```   464 ML{*AtpWrapper.problem_name:="Tarski__CLF_lubH_le_flubH"*}
```
```   465   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]
```
```   466 lemma (in CLF) lubH_le_flubH:
```
```   467      "H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r"
```
```   468 apply (rule lub_least, fast)
```
```   469 apply (rule f_in_funcset [THEN funcset_mem])
```
```   470 apply (rule lub_in_lattice, fast)
```
```   471 -- {* @{text "\<forall>x:H. (x, f (lub H r)) \<in> r"} *}
```
```   472 apply (rule ballI)
```
```   473 (*never proved, 2007-01-22*)
```
```   474 ML_command{*AtpWrapper.problem_name:="Tarski__CLF_lubH_le_flubH_simpler"*}
```
```   475 apply (rule transE)
```
```   476 -- {* instantiates @{text "(x, ?z) \<in> order cl to (x, f x)"}, *}
```
```   477 -- {* because of the def of @{text H} *}
```
```   478 apply fast
```
```   479 -- {* so it remains to show @{text "(f x, f (lub H cl)) \<in> r"} *}
```
```   480 apply (rule_tac f = "f" in monotoneE)
```
```   481 apply (rule monotone_f, fast)
```
```   482 apply (rule lub_in_lattice, fast)
```
```   483 apply (rule lub_upper, fast)
```
```   484 apply assumption
```
```   485 done
```
```   486   declare CL.lub_least[rule del] CLF.f_in_funcset[rule del]
```
```   487           funcset_mem[rule del] CL.lub_in_lattice[rule del]
```
```   488           PO.transE[rule del] PO.monotoneE[rule del]
```
```   489           CLF.monotone_f[rule del] CL.lub_upper[rule del]
```
```   490
```
```   491 (*never proved, 2007-01-22*)
```
```   492 ML{*AtpWrapper.problem_name:="Tarski__CLF_flubH_le_lubH"*}
```
```   493   declare CLF.f_in_funcset[intro] funcset_mem[intro] CL.lub_in_lattice[intro]
```
```   494        PO.monotoneE[intro] CLF.monotone_f[intro] CL.lub_upper[intro]
```
```   495        CLF.lubH_le_flubH[simp]
```
```   496 lemma (in CLF) flubH_le_lubH:
```
```   497      "[|  H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r"
```
```   498 apply (rule lub_upper, fast)
```
```   499 apply (rule_tac t = "H" in ssubst, assumption)
```
```   500 apply (rule CollectI)
```
```   501 apply (rule conjI)
```
```   502 ML_command{*AtpWrapper.problem_name:="Tarski__CLF_flubH_le_lubH_simpler"*}
```
```   503 (*??no longer terminates, with combinators
```
```   504 apply (metis CO_refl_on lubH_le_flubH monotone_def monotone_f reflD1 reflD2)
```
```   505 *)
```
```   506 apply (metis CO_refl_on lubH_le_flubH monotoneE [OF monotone_f] refl_onD1 refl_onD2)
```
```   507 apply (metis CO_refl_on lubH_le_flubH refl_onD2)
```
```   508 done
```
```   509   declare CLF.f_in_funcset[rule del] funcset_mem[rule del]
```
```   510           CL.lub_in_lattice[rule del] PO.monotoneE[rule del]
```
```   511           CLF.monotone_f[rule del] CL.lub_upper[rule del]
```
```   512           CLF.lubH_le_flubH[simp del]
```
```   513
```
```   514
```
```   515 (*never proved, 2007-01-22*)
```
```   516 ML{*AtpWrapper.problem_name:="Tarski__CLF_lubH_is_fixp"*}
```
```   517 (*Single-step version fails. The conjecture clauses refer to local abstraction
```
```   518 functions (Frees), which prevents expand_defs_tac from removing those
```
```   519 "definitions" at the end of the proof. *)
```
```   520 lemma (in CLF) lubH_is_fixp:
```
```   521      "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
```
```   522 apply (simp add: fix_def)
```
```   523 apply (rule conjI)
```
```   524 proof (neg_clausify)
```
```   525 assume 0: "H =
```
```   526 Collect
```
```   527  (COMBS (COMBB op \<and> (COMBC (COMBB op \<in> (COMBS Pair f)) r)) (COMBC op \<in> A))"
```
```   528 assume 1: "lub (Collect
```
```   529       (COMBS (COMBB op \<and> (COMBC (COMBB op \<in> (COMBS Pair f)) r))
```
```   530         (COMBC op \<in> A)))
```
```   531  cl
```
```   532 \<notin> A"
```
```   533 have 2: "lub H cl \<notin> A"
```
```   534   by (metis 1 0)
```
```   535 have 3: "(lub H cl, f (lub H cl)) \<in> r"
```
```   536   by (metis lubH_le_flubH 0)
```
```   537 have 4: "(f (lub H cl), lub H cl) \<in> r"
```
```   538   by (metis flubH_le_lubH 0)
```
```   539 have 5: "lub H cl = f (lub H cl) \<or> (lub H cl, f (lub H cl)) \<notin> r"
```
```   540   by (metis antisymE 4)
```
```   541 have 6: "lub H cl = f (lub H cl)"
```
```   542   by (metis 5 3)
```
```   543 have 7: "(lub H cl, lub H cl) \<in> r"
```
```   544   by (metis 6 4)
```
```   545 have 8: "\<And>X1. lub H cl \<in> X1 \<or> \<not> refl_on X1 r"
```
```   546   by (metis 7 refl_onD2)
```
```   547 have 9: "\<not> refl_on A r"
```
```   548   by (metis 8 2)
```
```   549 show "False"
```
```   550   by (metis CO_refl_on 9);
```
```   551 next --{*apparently the way to insert a second structured proof*}
```
```   552   show "H = {x. (x, f x) \<in> r \<and> x \<in> A} \<Longrightarrow>
```
```   553   f (lub {x. (x, f x) \<in> r \<and> x \<in> A} cl) = lub {x. (x, f x) \<in> r \<and> x \<in> A} cl"
```
```   554   proof (neg_clausify)
```
```   555   assume 0: "H =
```
```   556   Collect
```
```   557    (COMBS (COMBB op \<and> (COMBC (COMBB op \<in> (COMBS Pair f)) r)) (COMBC op \<in> A))"
```
```   558   assume 1: "f (lub (Collect
```
```   559 	   (COMBS (COMBB op \<and> (COMBC (COMBB op \<in> (COMBS Pair f)) r))
```
```   560 	     (COMBC op \<in> A)))
```
```   561       cl) \<noteq>
```
```   562   lub (Collect
```
```   563 	(COMBS (COMBB op \<and> (COMBC (COMBB op \<in> (COMBS Pair f)) r))
```
```   564 	  (COMBC op \<in> A)))
```
```   565    cl"
```
```   566   have 2: "f (lub H cl) \<noteq>
```
```   567   lub (Collect
```
```   568 	(COMBS (COMBB op \<and> (COMBC (COMBB op \<in> (COMBS Pair f)) r))
```
```   569 	  (COMBC op \<in> A)))
```
```   570    cl"
```
```   571     by (metis 1 0)
```
```   572   have 3: "f (lub H cl) \<noteq> lub H cl"
```
```   573     by (metis 2 0)
```
```   574   have 4: "(lub H cl, f (lub H cl)) \<in> r"
```
```   575     by (metis lubH_le_flubH 0)
```
```   576   have 5: "(f (lub H cl), lub H cl) \<in> r"
```
```   577     by (metis flubH_le_lubH 0)
```
```   578   have 6: "lub H cl = f (lub H cl) \<or> (lub H cl, f (lub H cl)) \<notin> r"
```
```   579     by (metis antisymE 5)
```
```   580   have 7: "lub H cl = f (lub H cl)"
```
```   581     by (metis 6 4)
```
```   582   show "False"
```
```   583     by (metis 3 7)
```
```   584   qed
```
```   585 qed
```
```   586
```
```   587 lemma (in CLF) (*lubH_is_fixp:*)
```
```   588      "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
```
```   589 apply (simp add: fix_def)
```
```   590 apply (rule conjI)
```
```   591 ML_command{*AtpWrapper.problem_name:="Tarski__CLF_lubH_is_fixp_simpler"*}
```
```   592 apply (metis CO_refl_on lubH_le_flubH refl_onD1)
```
```   593 apply (metis antisymE flubH_le_lubH lubH_le_flubH)
```
```   594 done
```
```   595
```
```   596 lemma (in CLF) fix_in_H:
```
```   597      "[| H = {x. (x, f x) \<in> r & x \<in> A};  x \<in> P |] ==> x \<in> H"
```
```   598 by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl_on
```
```   599                     fix_subset [of f A, THEN subsetD])
```
```   600
```
```   601
```
```   602 lemma (in CLF) fixf_le_lubH:
```
```   603      "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
```
```   604 apply (rule ballI)
```
```   605 apply (rule lub_upper, fast)
```
```   606 apply (rule fix_in_H)
```
```   607 apply (simp_all add: P_def)
```
```   608 done
```
```   609
```
```   610 ML{*AtpWrapper.problem_name:="Tarski__CLF_lubH_least_fixf"*}
```
```   611 lemma (in CLF) lubH_least_fixf:
```
```   612      "H = {x. (x, f x) \<in> r & x \<in> A}
```
```   613       ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"
```
```   614 apply (metis P_def lubH_is_fixp)
```
```   615 done
```
```   616
```
```   617 subsection {* Tarski fixpoint theorem 1, first part *}
```
```   618 ML{*AtpWrapper.problem_name:="Tarski__CLF_T_thm_1_lub"*}
```
```   619   declare CL.lubI[intro] fix_subset[intro] CL.lub_in_lattice[intro]
```
```   620           CLF.fixf_le_lubH[simp] CLF.lubH_least_fixf[simp]
```
```   621 lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"
```
```   622 (*sledgehammer;*)
```
```   623 apply (rule sym)
```
```   624 apply (simp add: P_def)
```
```   625 apply (rule lubI)
```
```   626 ML_command{*AtpWrapper.problem_name:="Tarski__CLF_T_thm_1_lub_simpler"*}
```
```   627 apply (metis P_def fix_subset)
```
```   628 apply (metis Collect_conj_eq Collect_mem_eq Int_commute Int_lower1 lub_in_lattice vimage_def)
```
```   629 (*??no longer terminates, with combinators
```
```   630 apply (metis P_def fix_def fixf_le_lubH)
```
```   631 apply (metis P_def fix_def lubH_least_fixf)
```
```   632 *)
```
```   633 apply (simp add: fixf_le_lubH)
```
```   634 apply (simp add: lubH_least_fixf)
```
```   635 done
```
```   636   declare CL.lubI[rule del] fix_subset[rule del] CL.lub_in_lattice[rule del]
```
```   637           CLF.fixf_le_lubH[simp del] CLF.lubH_least_fixf[simp del]
```
```   638
```
```   639
```
```   640 (*never proved, 2007-01-22*)
```
```   641 ML{*AtpWrapper.problem_name:="Tarski__CLF_glbH_is_fixp"*}
```
```   642   declare glb_dual_lub[simp] PO.dualA_iff[intro] CLF.lubH_is_fixp[intro]
```
```   643           PO.dualPO[intro] CL.CL_dualCL[intro] PO.dualr_iff[simp]
```
```   644 lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P"
```
```   645   -- {* Tarski for glb *}
```
```   646 (*sledgehammer;*)
```
```   647 apply (simp add: glb_dual_lub P_def A_def r_def)
```
```   648 apply (rule dualA_iff [THEN subst])
```
```   649 apply (rule CLF.lubH_is_fixp)
```
```   650 apply (rule CLF.intro)
```
```   651 apply (rule CL.intro)
```
```   652 apply (rule PO.intro)
```
```   653 apply (rule dualPO)
```
```   654 apply (rule CL_axioms.intro)
```
```   655 apply (rule CL_dualCL)
```
```   656 apply (rule CLF_axioms.intro)
```
```   657 apply (rule CLF_dual)
```
```   658 apply (simp add: dualr_iff dualA_iff)
```
```   659 done
```
```   660   declare glb_dual_lub[simp del] PO.dualA_iff[rule del] CLF.lubH_is_fixp[rule del]
```
```   661           PO.dualPO[rule del] CL.CL_dualCL[rule del] PO.dualr_iff[simp del]
```
```   662
```
```   663
```
```   664 (*never proved, 2007-01-22*)
```
```   665 ML{*AtpWrapper.problem_name:="Tarski__T_thm_1_glb"*}  (*ALL THEOREMS*)
```
```   666 lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl"
```
```   667 (*sledgehammer;*)
```
```   668 apply (simp add: glb_dual_lub P_def A_def r_def)
```
```   669 apply (rule dualA_iff [THEN subst])
```
```   670 (*never proved, 2007-01-22*)
```
```   671 ML_command{*AtpWrapper.problem_name:="Tarski__T_thm_1_glb_simpler"*}  (*ALL THEOREMS*)
```
```   672 (*sledgehammer;*)
```
```   673 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,
```
```   674   OF dualPO CL_dualCL] dualPO CL_dualCL CLF_dual dualr_iff)
```
```   675 done
```
```   676
```
```   677 subsection {* interval *}
```
```   678
```
```   679
```
```   680 ML{*AtpWrapper.problem_name:="Tarski__rel_imp_elem"*}
```
```   681   declare (in CLF) CO_refl_on[simp] refl_on_def [simp]
```
```   682 lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"
```
```   683 by (metis CO_refl_on refl_onD1)
```
```   684   declare (in CLF) CO_refl_on[simp del]  refl_on_def [simp del]
```
```   685
```
```   686 ML{*AtpWrapper.problem_name:="Tarski__interval_subset"*}
```
```   687   declare (in CLF) rel_imp_elem[intro]
```
```   688   declare interval_def [simp]
```
```   689 lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b \<subseteq> A"
```
```   690 by (metis CO_refl_on interval_imp_mem refl_onD refl_onD2 rel_imp_elem subset_eq)
```
```   691   declare (in CLF) rel_imp_elem[rule del]
```
```   692   declare interval_def [simp del]
```
```   693
```
```   694
```
```   695 lemma (in CLF) intervalI:
```
```   696      "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"
```
```   697 by (simp add: interval_def)
```
```   698
```
```   699 lemma (in CLF) interval_lemma1:
```
```   700      "[| S \<subseteq> interval r a b; x \<in> S |] ==> (a, x) \<in> r"
```
```   701 by (unfold interval_def, fast)
```
```   702
```
```   703 lemma (in CLF) interval_lemma2:
```
```   704      "[| S \<subseteq> interval r a b; x \<in> S |] ==> (x, b) \<in> r"
```
```   705 by (unfold interval_def, fast)
```
```   706
```
```   707 lemma (in CLF) a_less_lub:
```
```   708      "[| S \<subseteq> A; S \<noteq> {};
```
```   709          \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"
```
```   710 by (blast intro: transE)
```
```   711
```
```   712 lemma (in CLF) glb_less_b:
```
```   713      "[| S \<subseteq> A; S \<noteq> {};
```
```   714          \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"
```
```   715 by (blast intro: transE)
```
```   716
```
```   717 lemma (in CLF) S_intv_cl:
```
```   718      "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b |]==> S \<subseteq> A"
```
```   719 by (simp add: subset_trans [OF _ interval_subset])
```
```   720
```
```   721 ML{*AtpWrapper.problem_name:="Tarski__L_in_interval"*}  (*ALL THEOREMS*)
```
```   722 lemma (in CLF) L_in_interval:
```
```   723      "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b;
```
```   724          S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"
```
```   725 (*WON'T TERMINATE
```
```   726 apply (metis CO_trans intervalI interval_lemma1 interval_lemma2 isLub_least isLub_upper subset_empty subset_iff trans_def)
```
```   727 *)
```
```   728 apply (rule intervalI)
```
```   729 apply (rule a_less_lub)
```
```   730 prefer 2 apply assumption
```
```   731 apply (simp add: S_intv_cl)
```
```   732 apply (rule ballI)
```
```   733 apply (simp add: interval_lemma1)
```
```   734 apply (simp add: isLub_upper)
```
```   735 -- {* @{text "(L, b) \<in> r"} *}
```
```   736 apply (simp add: isLub_least interval_lemma2)
```
```   737 done
```
```   738
```
```   739 (*never proved, 2007-01-22*)
```
```   740 ML{*AtpWrapper.problem_name:="Tarski__G_in_interval"*}  (*ALL THEOREMS*)
```
```   741 lemma (in CLF) G_in_interval:
```
```   742      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S \<subseteq> interval r a b; isGlb S cl G;
```
```   743          S \<noteq> {} |] ==> G \<in> interval r a b"
```
```   744 apply (simp add: interval_dual)
```
```   745 apply (simp add: CLF.L_in_interval [of _ f, OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro]
```
```   746                  dualA_iff A_def dualPO CL_dualCL CLF_dual isGlb_dual_isLub)
```
```   747 done
```
```   748
```
```   749 ML{*AtpWrapper.problem_name:="Tarski__intervalPO"*}  (*ALL THEOREMS*)
```
```   750 lemma (in CLF) intervalPO:
```
```   751      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   752       ==> (| pset = interval r a b, order = induced (interval r a b) r |)
```
```   753           \<in> PartialOrder"
```
```   754 proof (neg_clausify)
```
```   755 assume 0: "a \<in> A"
```
```   756 assume 1: "b \<in> A"
```
```   757 assume 2: "\<lparr>pset = interval r a b, order = induced (interval r a b) r\<rparr> \<notin> PartialOrder"
```
```   758 have 3: "\<not> interval r a b \<subseteq> A"
```
```   759   by (metis 2 po_subset_po)
```
```   760 have 4: "b \<notin> A \<or> a \<notin> A"
```
```   761   by (metis 3 interval_subset)
```
```   762 have 5: "a \<notin> A"
```
```   763   by (metis 4 1)
```
```   764 show "False"
```
```   765   by (metis 5 0)
```
```   766 qed
```
```   767
```
```   768 lemma (in CLF) intv_CL_lub:
```
```   769  "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   770   ==> \<forall>S. S \<subseteq> interval r a b -->
```
```   771           (\<exists>L. isLub S (| pset = interval r a b,
```
```   772                           order = induced (interval r a b) r |)  L)"
```
```   773 apply (intro strip)
```
```   774 apply (frule S_intv_cl [THEN CL_imp_ex_isLub])
```
```   775 prefer 2 apply assumption
```
```   776 apply assumption
```
```   777 apply (erule exE)
```
```   778 -- {* define the lub for the interval as *}
```
```   779 apply (rule_tac x = "if S = {} then a else L" in exI)
```
```   780 apply (simp (no_asm_simp) add: isLub_def split del: split_if)
```
```   781 apply (intro impI conjI)
```
```   782 -- {* @{text "(if S = {} then a else L) \<in> interval r a b"} *}
```
```   783 apply (simp add: CL_imp_PO L_in_interval)
```
```   784 apply (simp add: left_in_interval)
```
```   785 -- {* lub prop 1 *}
```
```   786 apply (case_tac "S = {}")
```
```   787 -- {* @{text "S = {}, y \<in> S = False => everything"} *}
```
```   788 apply fast
```
```   789 -- {* @{text "S \<noteq> {}"} *}
```
```   790 apply simp
```
```   791 -- {* @{text "\<forall>y:S. (y, L) \<in> induced (interval r a b) r"} *}
```
```   792 apply (rule ballI)
```
```   793 apply (simp add: induced_def  L_in_interval)
```
```   794 apply (rule conjI)
```
```   795 apply (rule subsetD)
```
```   796 apply (simp add: S_intv_cl, assumption)
```
```   797 apply (simp add: isLub_upper)
```
```   798 -- {* @{text "\<forall>z:interval r a b. (\<forall>y:S. (y, z) \<in> induced (interval r a b) r \<longrightarrow> (if S = {} then a else L, z) \<in> induced (interval r a b) r"} *}
```
```   799 apply (rule ballI)
```
```   800 apply (rule impI)
```
```   801 apply (case_tac "S = {}")
```
```   802 -- {* @{text "S = {}"} *}
```
```   803 apply simp
```
```   804 apply (simp add: induced_def  interval_def)
```
```   805 apply (rule conjI)
```
```   806 apply (rule reflE, assumption)
```
```   807 apply (rule interval_not_empty)
```
```   808 apply (rule CO_trans)
```
```   809 apply (simp add: interval_def)
```
```   810 -- {* @{text "S \<noteq> {}"} *}
```
```   811 apply simp
```
```   812 apply (simp add: induced_def  L_in_interval)
```
```   813 apply (rule isLub_least, assumption)
```
```   814 apply (rule subsetD)
```
```   815 prefer 2 apply assumption
```
```   816 apply (simp add: S_intv_cl, fast)
```
```   817 done
```
```   818
```
```   819 lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual]
```
```   820
```
```   821 (*never proved, 2007-01-22*)
```
```   822 ML{*AtpWrapper.problem_name:="Tarski__interval_is_sublattice"*}  (*ALL THEOREMS*)
```
```   823 lemma (in CLF) interval_is_sublattice:
```
```   824      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
```
```   825         ==> interval r a b <<= cl"
```
```   826 (*sledgehammer *)
```
```   827 apply (rule sublatticeI)
```
```   828 apply (simp add: interval_subset)
```
```   829 (*never proved, 2007-01-22*)
```
```   830 ML_command{*AtpWrapper.problem_name:="Tarski__interval_is_sublattice_simpler"*}
```
```   831 (*sledgehammer *)
```
```   832 apply (rule CompleteLatticeI)
```
```   833 apply (simp add: intervalPO)
```
```   834  apply (simp add: intv_CL_lub)
```
```   835 apply (simp add: intv_CL_glb)
```
```   836 done
```
```   837
```
```   838 lemmas (in CLF) interv_is_compl_latt =
```
```   839     interval_is_sublattice [THEN sublattice_imp_CL]
```
```   840
```
```   841
```
```   842 subsection {* Top and Bottom *}
```
```   843 lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"
```
```   844 by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
```
```   845
```
```   846 lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)"
```
```   847 by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
```
```   848
```
```   849 ML_command{*AtpWrapper.problem_name:="Tarski__Bot_in_lattice"*}  (*ALL THEOREMS*)
```
```   850 lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"
```
```   851 (*sledgehammer; *)
```
```   852 apply (simp add: Bot_def least_def)
```
```   853 apply (rule_tac a="glb A cl" in someI2)
```
```   854 apply (simp_all add: glb_in_lattice glb_lower
```
```   855                      r_def [symmetric] A_def [symmetric])
```
```   856 done
```
```   857
```
```   858 (*first proved 2007-01-25 after relaxing relevance*)
```
```   859 ML_command{*AtpWrapper.problem_name:="Tarski__Top_in_lattice"*}  (*ALL THEOREMS*)
```
```   860 lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
```
```   861 (*sledgehammer;*)
```
```   862 apply (simp add: Top_dual_Bot A_def)
```
```   863 (*first proved 2007-01-25 after relaxing relevance*)
```
```   864 ML_command{*AtpWrapper.problem_name:="Tarski__Top_in_lattice_simpler"*}  (*ALL THEOREMS*)
```
```   865 (*sledgehammer*)
```
```   866 apply (rule dualA_iff [THEN subst])
```
```   867 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)
```
```   868 done
```
```   869
```
```   870 lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
```
```   871 apply (simp add: Top_def greatest_def)
```
```   872 apply (rule_tac a="lub A cl" in someI2)
```
```   873 apply (rule someI2)
```
```   874 apply (simp_all add: lub_in_lattice lub_upper
```
```   875                      r_def [symmetric] A_def [symmetric])
```
```   876 done
```
```   877
```
```   878 (*never proved, 2007-01-22*)
```
```   879 ML_command{*AtpWrapper.problem_name:="Tarski__Bot_prop"*}  (*ALL THEOREMS*)
```
```   880 lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
```
```   881 (*sledgehammer*)
```
```   882 apply (simp add: Bot_dual_Top r_def)
```
```   883 apply (rule dualr_iff [THEN subst])
```
```   884 apply (simp add: CLF.Top_prop [of _ f, OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro]
```
```   885                  dualA_iff A_def dualPO CL_dualCL CLF_dual)
```
```   886 done
```
```   887
```
```   888 ML_command{*AtpWrapper.problem_name:="Tarski__Bot_in_lattice"*}  (*ALL THEOREMS*)
```
```   889 lemma (in CLF) Top_intv_not_empty: "x \<in> A  ==> interval r x (Top cl) \<noteq> {}"
```
```   890 apply (metis Top_in_lattice Top_prop empty_iff intervalI reflE)
```
```   891 done
```
```   892
```
```   893 ML_command{*AtpWrapper.problem_name:="Tarski__Bot_intv_not_empty"*}  (*ALL THEOREMS*)
```
```   894 lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}"
```
```   895 apply (metis Bot_prop ex_in_conv intervalI reflE rel_imp_elem)
```
```   896 done
```
```   897
```
```   898
```
```   899 subsection {* fixed points form a partial order *}
```
```   900
```
```   901 lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
```
```   902 by (simp add: P_def fix_subset po_subset_po)
```
```   903
```
```   904 (*first proved 2007-01-25 after relaxing relevance*)
```
```   905 ML_command{*AtpWrapper.problem_name:="Tarski__Y_subset_A"*}
```
```   906   declare (in Tarski) P_def[simp] Y_ss [simp]
```
```   907   declare fix_subset [intro] subset_trans [intro]
```
```   908 lemma (in Tarski) Y_subset_A: "Y \<subseteq> A"
```
```   909 (*sledgehammer*)
```
```   910 apply (rule subset_trans [OF _ fix_subset])
```
```   911 apply (rule Y_ss [simplified P_def])
```
```   912 done
```
```   913   declare (in Tarski) P_def[simp del] Y_ss [simp del]
```
```   914   declare fix_subset [rule del] subset_trans [rule del]
```
```   915
```
```   916
```
```   917 lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A"
```
```   918   by (rule Y_subset_A [THEN lub_in_lattice])
```
```   919
```
```   920 (*never proved, 2007-01-22*)
```
```   921 ML_command{*AtpWrapper.problem_name:="Tarski__lubY_le_flubY"*}  (*ALL THEOREMS*)
```
```   922 lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"
```
```   923 (*sledgehammer*)
```
```   924 apply (rule lub_least)
```
```   925 apply (rule Y_subset_A)
```
```   926 apply (rule f_in_funcset [THEN funcset_mem])
```
```   927 apply (rule lubY_in_A)
```
```   928 -- {* @{text "Y \<subseteq> P ==> f x = x"} *}
```
```   929 apply (rule ballI)
```
```   930 ML_command{*AtpWrapper.problem_name:="Tarski__lubY_le_flubY_simpler"*}  (*ALL THEOREMS*)
```
```   931 (*sledgehammer *)
```
```   932 apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
```
```   933 apply (erule Y_ss [simplified P_def, THEN subsetD])
```
```   934 -- {* @{text "reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r"} by monotonicity *}
```
```   935 ML_command{*AtpWrapper.problem_name:="Tarski__lubY_le_flubY_simplest"*}  (*ALL THEOREMS*)
```
```   936 (*sledgehammer*)
```
```   937 apply (rule_tac f = "f" in monotoneE)
```
```   938 apply (rule monotone_f)
```
```   939 apply (simp add: Y_subset_A [THEN subsetD])
```
```   940 apply (rule lubY_in_A)
```
```   941 apply (simp add: lub_upper Y_subset_A)
```
```   942 done
```
```   943
```
```   944 (*first proved 2007-01-25 after relaxing relevance*)
```
```   945 ML_command{*AtpWrapper.problem_name:="Tarski__intY1_subset"*}  (*ALL THEOREMS*)
```
```   946 lemma (in Tarski) intY1_subset: "intY1 \<subseteq> A"
```
```   947 (*sledgehammer*)
```
```   948 apply (unfold intY1_def)
```
```   949 apply (rule interval_subset)
```
```   950 apply (rule lubY_in_A)
```
```   951 apply (rule Top_in_lattice)
```
```   952 done
```
```   953
```
```   954 lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD]
```
```   955
```
```   956 (*never proved, 2007-01-22*)
```
```   957 ML_command{*AtpWrapper.problem_name:="Tarski__intY1_f_closed"*}  (*ALL THEOREMS*)
```
```   958 lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"
```
```   959 (*sledgehammer*)
```
```   960 apply (simp add: intY1_def  interval_def)
```
```   961 apply (rule conjI)
```
```   962 apply (rule transE)
```
```   963 apply (rule lubY_le_flubY)
```
```   964 -- {* @{text "(f (lub Y cl), f x) \<in> r"} *}
```
```   965 ML_command{*AtpWrapper.problem_name:="Tarski__intY1_f_closed_simpler"*}  (*ALL THEOREMS*)
```
```   966 (*sledgehammer [has been proved before now...]*)
```
```   967 apply (rule_tac f=f in monotoneE)
```
```   968 apply (rule monotone_f)
```
```   969 apply (rule lubY_in_A)
```
```   970 apply (simp add: intY1_def interval_def  intY1_elem)
```
```   971 apply (simp add: intY1_def  interval_def)
```
```   972 -- {* @{text "(f x, Top cl) \<in> r"} *}
```
```   973 apply (rule Top_prop)
```
```   974 apply (rule f_in_funcset [THEN funcset_mem])
```
```   975 apply (simp add: intY1_def interval_def  intY1_elem)
```
```   976 done
```
```   977
```
```   978 ML_command{*AtpWrapper.problem_name:="Tarski__intY1_func"*}  (*ALL THEOREMS*)
```
```   979 lemma (in Tarski) intY1_func: "(%x: intY1. f x) \<in> intY1 -> intY1"
```
```   980 apply (rule restrict_in_funcset)
```
```   981 apply (metis intY1_f_closed restrict_in_funcset)
```
```   982 done
```
```   983
```
```   984 ML_command{*AtpWrapper.problem_name:="Tarski__intY1_mono"*}  (*ALL THEOREMS*)
```
```   985 lemma (in Tarski) intY1_mono:
```
```   986      "monotone (%x: intY1. f x) intY1 (induced intY1 r)"
```
```   987 (*sledgehammer *)
```
```   988 apply (auto simp add: monotone_def induced_def intY1_f_closed)
```
```   989 apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
```
```   990 done
```
```   991
```
```   992 (*proof requires relaxing relevance: 2007-01-25*)
```
```   993 ML_command{*AtpWrapper.problem_name:="Tarski__intY1_is_cl"*}  (*ALL THEOREMS*)
```
```   994 lemma (in Tarski) intY1_is_cl:
```
```   995     "(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
```
```   996 (*sledgehammer*)
```
```   997 apply (unfold intY1_def)
```
```   998 apply (rule interv_is_compl_latt)
```
```   999 apply (rule lubY_in_A)
```
```  1000 apply (rule Top_in_lattice)
```
```  1001 apply (rule Top_intv_not_empty)
```
```  1002 apply (rule lubY_in_A)
```
```  1003 done
```
```  1004
```
```  1005 (*never proved, 2007-01-22*)
```
```  1006 ML_command{*AtpWrapper.problem_name:="Tarski__v_in_P"*}  (*ALL THEOREMS*)
```
```  1007 lemma (in Tarski) v_in_P: "v \<in> P"
```
```  1008 (*sledgehammer*)
```
```  1009 apply (unfold P_def)
```
```  1010 apply (rule_tac A = "intY1" in fixf_subset)
```
```  1011 apply (rule intY1_subset)
```
```  1012 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]
```
```  1013                  v_def CL_imp_PO intY1_is_cl CLF_set_def intY1_func intY1_mono)
```
```  1014 done
```
```  1015
```
```  1016 ML_command{*AtpWrapper.problem_name:="Tarski__z_in_interval"*}  (*ALL THEOREMS*)
```
```  1017 lemma (in Tarski) z_in_interval:
```
```  1018      "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1"
```
```  1019 (*sledgehammer *)
```
```  1020 apply (unfold intY1_def P_def)
```
```  1021 apply (rule intervalI)
```
```  1022 prefer 2
```
```  1023  apply (erule fix_subset [THEN subsetD, THEN Top_prop])
```
```  1024 apply (rule lub_least)
```
```  1025 apply (rule Y_subset_A)
```
```  1026 apply (fast elim!: fix_subset [THEN subsetD])
```
```  1027 apply (simp add: induced_def)
```
```  1028 done
```
```  1029
```
```  1030 ML_command{*AtpWrapper.problem_name:="Tarski__fz_in_int_rel"*}  (*ALL THEOREMS*)
```
```  1031 lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |]
```
```  1032       ==> ((%x: intY1. f x) z, z) \<in> induced intY1 r"
```
```  1033 apply (metis P_def acc_def fix_imp_eq fix_subset indI reflE restrict_apply subset_eq z_in_interval)
```
```  1034 done
```
```  1035
```
```  1036 (*never proved, 2007-01-22*)
```
```  1037 ML_command{*AtpWrapper.problem_name:="Tarski__tarski_full_lemma"*}  (*ALL THEOREMS*)
```
```  1038 lemma (in Tarski) tarski_full_lemma:
```
```  1039      "\<exists>L. isLub Y (| pset = P, order = induced P r |) L"
```
```  1040 apply (rule_tac x = "v" in exI)
```
```  1041 apply (simp add: isLub_def)
```
```  1042 -- {* @{text "v \<in> P"} *}
```
```  1043 apply (simp add: v_in_P)
```
```  1044 apply (rule conjI)
```
```  1045 (*sledgehammer*)
```
```  1046 -- {* @{text v} is lub *}
```
```  1047 -- {* @{text "1. \<forall>y:Y. (y, v) \<in> induced P r"} *}
```
```  1048 apply (rule ballI)
```
```  1049 apply (simp add: induced_def subsetD v_in_P)
```
```  1050 apply (rule conjI)
```
```  1051 apply (erule Y_ss [THEN subsetD])
```
```  1052 apply (rule_tac b = "lub Y cl" in transE)
```
```  1053 apply (rule lub_upper)
```
```  1054 apply (rule Y_subset_A, assumption)
```
```  1055 apply (rule_tac b = "Top cl" in interval_imp_mem)
```
```  1056 apply (simp add: v_def)
```
```  1057 apply (fold intY1_def)
```
```  1058 apply (rule CL.glb_in_lattice [OF CL.intro, OF PO.intro CL_axioms.intro, OF _ intY1_is_cl, simplified])
```
```  1059  apply (simp add: CL_imp_PO intY1_is_cl, force)
```
```  1060 -- {* @{text v} is LEAST ub *}
```
```  1061 apply clarify
```
```  1062 apply (rule indI)
```
```  1063   prefer 3 apply assumption
```
```  1064  prefer 2 apply (simp add: v_in_P)
```
```  1065 apply (unfold v_def)
```
```  1066 (*never proved, 2007-01-22*)
```
```  1067 ML_command{*AtpWrapper.problem_name:="Tarski__tarski_full_lemma_simpler"*}
```
```  1068 (*sledgehammer*)
```
```  1069 apply (rule indE)
```
```  1070 apply (rule_tac [2] intY1_subset)
```
```  1071 (*never proved, 2007-01-22*)
```
```  1072 ML_command{*AtpWrapper.problem_name:="Tarski__tarski_full_lemma_simplest"*}
```
```  1073 (*sledgehammer*)
```
```  1074 apply (rule CL.glb_lower [OF CL.intro, OF PO.intro CL_axioms.intro, OF _ intY1_is_cl, simplified])
```
```  1075   apply (simp add: CL_imp_PO intY1_is_cl)
```
```  1076  apply force
```
```  1077 apply (simp add: induced_def intY1_f_closed z_in_interval)
```
```  1078 apply (simp add: P_def fix_imp_eq [of _ f A] reflE
```
```  1079                  fix_subset [of f A, THEN subsetD])
```
```  1080 done
```
```  1081
```
```  1082 lemma CompleteLatticeI_simp:
```
```  1083      "[| (| pset = A, order = r |) \<in> PartialOrder;
```
```  1084          \<forall>S. S \<subseteq> A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]
```
```  1085     ==> (| pset = A, order = r |) \<in> CompleteLattice"
```
```  1086 by (simp add: CompleteLatticeI Rdual)
```
```  1087
```
```  1088
```
```  1089 (*never proved, 2007-01-22*)
```
```  1090 ML_command{*AtpWrapper.problem_name:="Tarski__Tarski_full"*}
```
```  1091   declare (in CLF) fixf_po[intro] P_def [simp] A_def [simp] r_def [simp]
```
```  1092                Tarski.tarski_full_lemma [intro] cl_po [intro] cl_co [intro]
```
```  1093                CompleteLatticeI_simp [intro]
```
```  1094 theorem (in CLF) Tarski_full:
```
```  1095      "(| pset = P, order = induced P r|) \<in> CompleteLattice"
```
```  1096 (*sledgehammer*)
```
```  1097 apply (rule CompleteLatticeI_simp)
```
```  1098 apply (rule fixf_po, clarify)
```
```  1099 (*never proved, 2007-01-22*)
```
```  1100 ML_command{*AtpWrapper.problem_name:="Tarski__Tarski_full_simpler"*}
```
```  1101 (*sledgehammer*)
```
```  1102 apply (simp add: P_def A_def r_def)
```
```  1103 apply (blast intro!: Tarski.tarski_full_lemma [OF Tarski.intro, OF CLF.intro Tarski_axioms.intro,
```
```  1104   OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro] cl_po cl_co f_cl)
```
```  1105 done
```
```  1106   declare (in CLF) fixf_po[rule del] P_def [simp del] A_def [simp del] r_def [simp del]
```
```  1107          Tarski.tarski_full_lemma [rule del] cl_po [rule del] cl_co [rule del]
```
```  1108          CompleteLatticeI_simp [rule del]
```
```  1109
```
```  1110
```
```  1111 end
```