src/HOL/Hyperreal/Filter.thy
author nipkow
Wed Aug 18 11:09:40 2004 +0200 (2004-08-18)
changeset 15140 322485b816ac
parent 15131 c69542757a4d
child 17290 a39d1430d271
permissions -rw-r--r--
import -> imports
     1 (*  Title       : Filter.thy
     2     ID          : $Id$
     3     Author      : Jacques D. Fleuriot
     4     Copyright   : 1998  University of Cambridge
     5     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
     6 *) 
     7 
     8 header{*Filters and Ultrafilters*}
     9 
    10 theory Filter
    11 imports Zorn
    12 begin
    13 
    14 constdefs
    15 
    16   is_Filter       :: "['a set set,'a set] => bool"
    17   "is_Filter F S == (F <= Pow(S) & S \<in> F & {} ~: F &
    18                    (\<forall>u \<in> F. \<forall>v \<in> F. u Int v \<in> F) &
    19                    (\<forall>u v. u \<in> F & u <= v & v <= S --> v \<in> F))" 
    20 
    21   Filter          :: "'a set => 'a set set set"
    22   "Filter S == {X. is_Filter X S}"
    23  
    24   (* free filter does not contain any finite set *)
    25 
    26   Freefilter      :: "'a set => 'a set set set"
    27   "Freefilter S == {X. X \<in> Filter S & (\<forall>x \<in> X. ~ finite x)}"
    28 
    29   Ultrafilter     :: "'a set => 'a set set set"
    30   "Ultrafilter S == {X. X \<in> Filter S & (\<forall>A \<in> Pow(S). A \<in> X | S - A \<in> X)}"
    31 
    32   FreeUltrafilter :: "'a set => 'a set set set"
    33   "FreeUltrafilter S == {X. X \<in> Ultrafilter S & (\<forall>x \<in> X. ~ finite x)}" 
    34 
    35   (* A locale makes proof of Ultrafilter Theorem more modular *)
    36 locale (open) UFT = 
    37   fixes frechet      :: "'a set => 'a set set"
    38     and superfrechet :: "'a set => 'a set set set"
    39   assumes not_finite_UNIV:  "~finite (UNIV :: 'a set)"
    40   defines frechet_def:  
    41 		"frechet S == {A. finite (S - A)}"
    42       and superfrechet_def:
    43 		"superfrechet S == {G.  G \<in> Filter S & frechet S <= G}"
    44 
    45 
    46 (*------------------------------------------------------------------
    47       Properties of Filters and Freefilters - 
    48       rules for intro, destruction etc.
    49  ------------------------------------------------------------------*)
    50 
    51 lemma is_FilterD1: "is_Filter X S ==> X <= Pow(S)"
    52 apply (simp add: is_Filter_def)
    53 done
    54 
    55 lemma is_FilterD2: "is_Filter X S ==> X ~= {}"
    56 apply (auto simp add: is_Filter_def)
    57 done
    58 
    59 lemma is_FilterD3: "is_Filter X S ==> {} ~: X"
    60 apply (simp add: is_Filter_def)
    61 done
    62 
    63 lemma mem_FiltersetI: "is_Filter X S ==> X \<in> Filter S"
    64 apply (simp add: Filter_def)
    65 done
    66 
    67 lemma mem_FiltersetD: "X \<in> Filter S ==> is_Filter X S"
    68 apply (simp add: Filter_def)
    69 done
    70 
    71 lemma Filter_empty_not_mem: "X \<in> Filter S ==> {} ~: X"
    72 apply (erule mem_FiltersetD [THEN is_FilterD3])
    73 done
    74 
    75 lemmas Filter_empty_not_memE = Filter_empty_not_mem [THEN notE, standard]
    76 
    77 lemma mem_FiltersetD1: "[| X \<in> Filter S; A \<in> X; B \<in> X |] ==> A Int B \<in> X"
    78 apply (unfold Filter_def is_Filter_def)
    79 apply blast
    80 done
    81 
    82 lemma mem_FiltersetD2: "[| X \<in> Filter S; A \<in> X; A <= B; B <= S|] ==> B \<in> X"
    83 apply (unfold Filter_def is_Filter_def)
    84 apply blast
    85 done
    86 
    87 lemma mem_FiltersetD3: "[| X \<in> Filter S; A \<in> X |] ==> A \<in> Pow S"
    88 apply (unfold Filter_def is_Filter_def)
    89 apply blast
    90 done
    91 
    92 lemma mem_FiltersetD4: "X \<in> Filter S  ==> S \<in> X"
    93 apply (unfold Filter_def is_Filter_def)
    94 apply blast
    95 done
    96 
    97 lemma is_FilterI: 
    98       "[| X <= Pow(S); 
    99                S \<in> X;  
   100                X ~= {};  
   101                {} ~: X;  
   102                \<forall>u \<in> X. \<forall>v \<in> X. u Int v \<in> X;  
   103                \<forall>u v. u \<in> X & u<=v & v<=S --> v \<in> X  
   104             |] ==> is_Filter X S"
   105 apply (unfold is_Filter_def)
   106 apply blast
   107 done
   108 
   109 lemma mem_FiltersetI2: "[| X <= Pow(S); 
   110                S \<in> X;  
   111                X ~= {};  
   112                {} ~: X;  
   113                \<forall>u \<in> X. \<forall>v \<in> X. u Int v \<in> X;  
   114                \<forall>u v. u \<in> X & u<=v & v<=S --> v \<in> X  
   115             |] ==> X \<in> Filter S"
   116 by (blast intro: mem_FiltersetI is_FilterI)
   117 
   118 lemma is_FilterE_lemma: 
   119       "is_Filter X S ==> X <= Pow(S) &  
   120                            S \<in> X &  
   121                            X ~= {} &  
   122                            {} ~: X  &  
   123                            (\<forall>u \<in> X. \<forall>v \<in> X. u Int v \<in> X) &  
   124                            (\<forall>u v. u \<in> X & u <= v & v<=S --> v \<in> X)"
   125 apply (unfold is_Filter_def)
   126 apply fast
   127 done
   128 
   129 lemma memFiltersetE_lemma: 
   130       "X \<in> Filter S ==> X <= Pow(S) & 
   131                            S \<in> X &  
   132                            X ~= {} &  
   133                            {} ~: X  &  
   134                            (\<forall>u \<in> X. \<forall>v \<in> X. u Int v \<in> X) &  
   135                            (\<forall>u v. u \<in> X & u <= v & v<=S --> v \<in> X)"
   136 by (erule mem_FiltersetD [THEN is_FilterE_lemma])
   137 
   138 lemma Freefilter_Filter: "X \<in> Freefilter S ==> X \<in> Filter S"
   139 apply (simp add: Filter_def Freefilter_def)
   140 done
   141 
   142 lemma mem_Freefilter_not_finite: "X \<in> Freefilter S ==> \<forall>y \<in> X. ~finite(y)"
   143 apply (simp add: Freefilter_def)
   144 done
   145 
   146 lemma mem_FreefiltersetD1: "[| X \<in> Freefilter S; x \<in> X |] ==> ~ finite x"
   147 apply (blast dest!: mem_Freefilter_not_finite)
   148 done
   149 
   150 lemmas mem_FreefiltersetE1 = mem_FreefiltersetD1 [THEN notE, standard]
   151 
   152 lemma mem_FreefiltersetD2: "[| X \<in> Freefilter S; finite x|] ==> x ~: X"
   153 apply (blast dest!: mem_Freefilter_not_finite)
   154 done
   155 
   156 lemma mem_FreefiltersetI1: 
   157       "[| X \<in> Filter S; \<forall>x. ~(x \<in> X & finite x) |] ==> X \<in> Freefilter S"
   158 by (simp add: Freefilter_def)
   159 
   160 lemma mem_FreefiltersetI2: 
   161       "[| X \<in> Filter S; \<forall>x. (x ~: X | ~ finite x) |] ==> X \<in> Freefilter S"
   162 by (simp add: Freefilter_def)
   163 
   164 lemma Filter_Int_not_empty: "[| X \<in> Filter S; A \<in> X; B \<in> X |] ==> A Int B ~= {}"
   165 apply (frule_tac A = "A" and B = "B" in mem_FiltersetD1)
   166 apply (auto dest!: Filter_empty_not_mem)
   167 done
   168 
   169 lemmas Filter_Int_not_emptyE = Filter_Int_not_empty [THEN notE, standard]
   170 
   171 subsection{*Ultrafilters and Free Ultrafilters*}
   172 
   173 lemma Ultrafilter_Filter: "X \<in> Ultrafilter S ==> X \<in> Filter S"
   174 apply (simp add: Ultrafilter_def)
   175 done
   176 
   177 lemma mem_UltrafiltersetD2: 
   178       "X \<in> Ultrafilter S ==> \<forall>A \<in> Pow(S). A \<in> X | S - A \<in> X"
   179 by (auto simp add: Ultrafilter_def)
   180 
   181 lemma mem_UltrafiltersetD3: 
   182       "[|X \<in> Ultrafilter S; A <= S; A ~: X |] ==> S - A \<in> X"
   183 by (auto simp add: Ultrafilter_def)
   184 
   185 lemma mem_UltrafiltersetD4: 
   186       "[|X \<in> Ultrafilter S; A <= S; S - A ~: X |] ==> A \<in> X"
   187 by (auto simp add: Ultrafilter_def)
   188 
   189 lemma mem_UltrafiltersetI: 
   190      "[| X \<in> Filter S;  
   191          \<forall>A \<in> Pow(S). A \<in> X | S - A \<in> X |] ==> X \<in> Ultrafilter S"
   192 by (simp add: Ultrafilter_def)
   193 
   194 lemma FreeUltrafilter_Ultrafilter: 
   195      "X \<in> FreeUltrafilter S ==> X \<in> Ultrafilter S"
   196 by (auto simp add: Ultrafilter_def FreeUltrafilter_def)
   197 
   198 lemma mem_FreeUltrafilter_not_finite: 
   199      "X \<in> FreeUltrafilter S ==> \<forall>y \<in> X. ~finite(y)"
   200 by (simp add: FreeUltrafilter_def)
   201 
   202 lemma mem_FreeUltrafiltersetD1: "[| X \<in> FreeUltrafilter S; x \<in> X |] ==> ~ finite x"
   203 apply (blast dest!: mem_FreeUltrafilter_not_finite)
   204 done
   205 
   206 lemmas mem_FreeUltrafiltersetE1 = mem_FreeUltrafiltersetD1 [THEN notE, standard]
   207 
   208 lemma mem_FreeUltrafiltersetD2: "[| X \<in> FreeUltrafilter S; finite x|] ==> x ~: X"
   209 apply (blast dest!: mem_FreeUltrafilter_not_finite)
   210 done
   211 
   212 lemma mem_FreeUltrafiltersetI1: 
   213       "[| X \<in> Ultrafilter S;  
   214           \<forall>x. ~(x \<in> X & finite x) |] ==> X \<in> FreeUltrafilter S"
   215 by (simp add: FreeUltrafilter_def)
   216 
   217 lemma mem_FreeUltrafiltersetI2: 
   218       "[| X \<in> Ultrafilter S;  
   219           \<forall>x. (x ~: X | ~ finite x) |] ==> X \<in> FreeUltrafilter S"
   220 by (simp add: FreeUltrafilter_def)
   221 
   222 lemma FreeUltrafilter_iff: 
   223      "(X \<in> FreeUltrafilter S) = (X \<in> Freefilter S & (\<forall>x \<in> Pow(S). x \<in> X | S - x \<in> X))"
   224 by (auto simp add: FreeUltrafilter_def Freefilter_def Ultrafilter_def)
   225 
   226 
   227 (*-------------------------------------------------------------------
   228    A Filter F on S is an ultrafilter iff it is a maximal filter 
   229    i.e. whenever G is a filter on I and F <= F then F = G
   230  --------------------------------------------------------------------*)
   231 (*---------------------------------------------------------------------
   232   lemmas that shows existence of an extension to what was assumed to
   233   be a maximal filter. Will be used to derive contradiction in proof of
   234   property of ultrafilter 
   235  ---------------------------------------------------------------------*)
   236 lemma lemma_set_extend: "[| F ~= {}; A <= S |] ==> \<exists>x. x \<in> {X. X <= S & (\<exists>f \<in> F. A Int f <= X)}"
   237 apply blast
   238 done
   239 
   240 lemma lemma_set_not_empty: "a \<in> X ==> X ~= {}"
   241 apply (safe)
   242 done
   243 
   244 lemma lemma_empty_Int_subset_Compl: "x Int F <= {} ==> F <= - x"
   245 apply blast
   246 done
   247 
   248 lemma mem_Filterset_disjI: 
   249       "[| F \<in> Filter S; A ~: F; A <= S|]  
   250            ==> \<forall>B. B ~: F | ~ B <= A"
   251 apply (unfold Filter_def is_Filter_def)
   252 apply blast
   253 done
   254 
   255 lemma Ultrafilter_max_Filter: "F \<in> Ultrafilter S ==>  
   256           (F \<in> Filter S & (\<forall>G \<in> Filter S. F <= G --> F = G))"
   257 apply (auto simp add: Ultrafilter_def)
   258 apply (drule_tac x = "x" in bspec)
   259 apply (erule mem_FiltersetD3 , assumption)
   260 apply (safe)
   261 apply (drule subsetD , assumption)
   262 apply (blast dest!: Filter_Int_not_empty)
   263 done
   264 
   265 
   266 (*--------------------------------------------------------------------------------
   267      This is a very long and tedious proof; need to break it into parts.
   268      Have proof that {X. X <= S & (\<exists>f \<in> F. A Int f <= X)} is a filter as 
   269      a lemma
   270 --------------------------------------------------------------------------------*)
   271 lemma max_Filter_Ultrafilter: 
   272       "[| F \<in> Filter S;  
   273           \<forall>G \<in> Filter S. F <= G --> F = G |] ==> F \<in> Ultrafilter S"
   274 apply (simp add: Ultrafilter_def)
   275 apply (safe)
   276 apply (rule ccontr)
   277 apply (frule mem_FiltersetD [THEN is_FilterD2])
   278 apply (frule_tac x = "{X. X <= S & (\<exists>f \<in> F. A Int f <= X) }" in bspec)
   279 apply (rule mem_FiltersetI2) 
   280 apply (blast intro: elim:); 
   281 apply (simp add: ); 
   282 apply (blast dest: mem_FiltersetD3)
   283 apply (erule lemma_set_extend [THEN exE])
   284 apply (assumption , erule lemma_set_not_empty)
   285 txt{*First we prove @{term "{} \<notin> {X. X \<subseteq> S \<and> (\<exists>f\<in>F. A \<inter> f \<subseteq> X)}"}*}
   286    apply (clarify ); 
   287    apply (drule lemma_empty_Int_subset_Compl)
   288    apply (frule mem_Filterset_disjI) 
   289    apply assumption; 
   290    apply (blast intro: elim:); 
   291    apply (fast dest: mem_FiltersetD3 elim:) 
   292 txt{*Next case: @{term "u \<inter> v"} is an element*}
   293   apply (intro ballI) 
   294 apply (simp add: ); 
   295   apply (rule conjI, blast) 
   296 apply (clarify ); 
   297   apply (rule_tac x = "f Int fa" in bexI)
   298    apply (fast intro: elim:); 
   299   apply (blast dest: mem_FiltersetD1 elim:)
   300  apply force;
   301 apply (blast dest: mem_FiltersetD3 elim:) 
   302 done
   303 
   304 lemma Ultrafilter_iff: "(F \<in> Ultrafilter S) = (F \<in> Filter S & (\<forall>G \<in> Filter S. F <= G --> F = G))"
   305 apply (blast intro!: Ultrafilter_max_Filter max_Filter_Ultrafilter)
   306 done
   307 
   308 
   309 subsection{* A Few Properties of Freefilters*}
   310 
   311 lemma lemma_Compl_cancel_eq: "F1 Int F2 = ((F1 Int Y) Int F2) Un ((F2 Int (- Y)) Int F1)"
   312 apply auto
   313 done
   314 
   315 lemma finite_IntI1: "finite X ==> finite (X Int Y)"
   316 apply (erule Int_lower1 [THEN finite_subset])
   317 done
   318 
   319 lemma finite_IntI2: "finite Y ==> finite (X Int Y)"
   320 apply (erule Int_lower2 [THEN finite_subset])
   321 done
   322 
   323 lemma finite_Int_Compl_cancel: "[| finite (F1 Int Y);  
   324                   finite (F2 Int (- Y))  
   325                |] ==> finite (F1 Int F2)"
   326 apply (rule_tac Y1 = "Y" in lemma_Compl_cancel_eq [THEN ssubst])
   327 apply (rule finite_UnI)
   328 apply (auto intro!: finite_IntI1 finite_IntI2)
   329 done
   330 
   331 lemma Freefilter_lemma_not_finite: "U \<in> Freefilter S  ==>  
   332           ~ (\<exists>f1 \<in> U. \<exists>f2 \<in> U. finite (f1 Int x)  
   333                              & finite (f2 Int (- x)))"
   334 apply (safe)
   335 apply (frule_tac A = "f1" and B = "f2" in Freefilter_Filter [THEN mem_FiltersetD1])
   336 apply (drule_tac [3] x = "f1 Int f2" in mem_FreefiltersetD1)
   337 apply (drule_tac [4] finite_Int_Compl_cancel)
   338 apply auto
   339 done
   340 
   341 (* the lemmas below follow *)
   342 lemma Freefilter_Compl_not_finite_disjI: "U \<in> Freefilter S ==>  
   343            \<forall>f \<in> U. ~ finite (f Int x) | ~finite (f Int (- x))"
   344 by (blast dest!: Freefilter_lemma_not_finite bspec)
   345 
   346 lemma Freefilter_Compl_not_finite_disjI2: "U \<in> Freefilter S ==> (\<forall>f \<in> U. ~ finite (f Int x)) | (\<forall>f \<in> U. ~finite (f Int (- x)))"
   347 apply (blast dest!: Freefilter_lemma_not_finite bspec)
   348 done
   349 
   350 lemma cofinite_Filter: "~ finite (UNIV:: 'a set) ==> {A:: 'a set. finite (- A)} \<in> Filter UNIV"
   351 apply (rule mem_FiltersetI2)
   352 apply (auto simp del: Collect_empty_eq)
   353 apply (erule_tac c = "UNIV" in equalityCE)
   354 apply auto
   355 apply (erule Compl_anti_mono [THEN finite_subset])
   356 apply assumption
   357 done
   358 
   359 lemma not_finite_UNIV_disjI: "~finite(UNIV :: 'a set) ==> ~finite (X :: 'a set) | ~finite (- X)" 
   360 apply (drule_tac A1 = "X" in Compl_partition [THEN ssubst])
   361 apply simp
   362 done
   363 
   364 lemma not_finite_UNIV_Compl: "[| ~finite(UNIV :: 'a set); finite (X :: 'a set) |] ==>  ~finite (- X)"
   365 apply (drule_tac X = "X" in not_finite_UNIV_disjI)
   366 apply blast
   367 done
   368 
   369 lemma mem_cofinite_Filter_not_finite:
   370      "~ finite (UNIV:: 'a set) 
   371       ==> \<forall>X \<in> {A:: 'a set. finite (- A)}. ~ finite X"
   372 by (auto dest: not_finite_UNIV_disjI)
   373 
   374 lemma cofinite_Freefilter:
   375     "~ finite (UNIV:: 'a set) ==> {A:: 'a set. finite (- A)} \<in> Freefilter UNIV"
   376 apply (rule mem_FreefiltersetI2)
   377 apply (rule cofinite_Filter , assumption)
   378 apply (blast dest!: mem_cofinite_Filter_not_finite)
   379 done
   380 
   381 (*????Set.thy*)
   382 lemma UNIV_diff_Compl [simp]: "UNIV - x = - x"
   383 apply auto
   384 done
   385 
   386 lemma FreeUltrafilter_contains_cofinite_set: 
   387      "[| ~finite(UNIV :: 'a set); (U :: 'a set set): FreeUltrafilter UNIV 
   388           |] ==> {X. finite(- X)} <= U"
   389 by (auto simp add: Ultrafilter_def FreeUltrafilter_def)
   390 
   391 (*--------------------------------------------------------------------
   392    We prove: 1. Existence of maximal filter i.e. ultrafilter
   393              2. Freeness property i.e ultrafilter is free
   394              Use a locale to prove various lemmas and then 
   395              export main result: The Ultrafilter Theorem
   396  -------------------------------------------------------------------*)
   397 
   398 lemma (in UFT) chain_Un_subset_Pow: 
   399    "!!(c :: 'a set set set). c \<in> chain (superfrechet S) ==>  Union c <= Pow S"
   400 apply (simp add: chain_def superfrechet_def frechet_def)
   401 apply (blast dest: mem_FiltersetD3 elim:) 
   402 done
   403 
   404 lemma (in UFT) mem_chain_psubset_empty: 
   405           "!!(c :: 'a set set set). c: chain (superfrechet S)  
   406           ==> !x: c. {} < x"
   407 by (auto simp add: chain_def Filter_def is_Filter_def superfrechet_def frechet_def)
   408 
   409 lemma (in UFT) chain_Un_not_empty: "!!(c :: 'a set set set).  
   410              [| c: chain (superfrechet S); 
   411                 c ~= {} |] 
   412              ==> Union(c) ~= {}"
   413 apply (drule mem_chain_psubset_empty)
   414 apply (safe)
   415 apply (drule bspec , assumption)
   416 apply (auto dest: Union_upper bspec simp add: psubset_def)
   417 done
   418 
   419 lemma (in UFT) Filter_empty_not_mem_Un: 
   420        "!!(c :: 'a set set set). c \<in> chain (superfrechet S) ==> {} ~: Union(c)"
   421 by (auto simp add: is_Filter_def Filter_def chain_def superfrechet_def)
   422 
   423 lemma (in UFT) Filter_Un_Int: "c \<in> chain (superfrechet S)  
   424           ==> \<forall>u \<in> Union(c). \<forall>v \<in> Union(c). u Int v \<in> Union(c)"
   425 apply (safe)
   426 apply (frule_tac x = "X" and y = "Xa" in chainD)
   427 apply (assumption)+
   428 apply (drule chainD2)
   429 apply (erule disjE)
   430  apply (rule_tac [2] X = "X" in UnionI)
   431   apply (rule_tac X = "Xa" in UnionI)
   432 apply (auto intro: mem_FiltersetD1 simp add: superfrechet_def)
   433 done
   434 
   435 lemma (in UFT) Filter_Un_subset: "c \<in> chain (superfrechet S)  
   436           ==> \<forall>u v. u \<in> Union(c) &  
   437                   (u :: 'a set) <= v & v <= S --> v \<in> Union(c)"
   438 apply (safe)
   439 apply (drule chainD2)
   440 apply (drule subsetD , assumption)
   441 apply (rule UnionI , assumption)
   442 apply (auto intro: mem_FiltersetD2 simp add: superfrechet_def)
   443 done
   444 
   445 lemma (in UFT) lemma_mem_chain_Filter:
   446       "!!(c :: 'a set set set).  
   447              [| c \<in> chain (superfrechet S); 
   448                 x \<in> c  
   449              |] ==> x \<in> Filter S"
   450 by (auto simp add: chain_def superfrechet_def)
   451 
   452 lemma (in UFT) lemma_mem_chain_frechet_subset: 
   453      "!!(c :: 'a set set set).  
   454              [| c \<in> chain (superfrechet S); 
   455                 x \<in> c  
   456              |] ==> frechet S <= x"
   457 by (auto simp add: chain_def superfrechet_def)
   458 
   459 lemma (in UFT) Un_chain_mem_cofinite_Filter_set: "!!(c :: 'a set set set).  
   460           [| c ~= {};  
   461              c \<in> chain (superfrechet (UNIV :: 'a set)) 
   462           |] ==> Union c \<in> superfrechet (UNIV)"
   463 apply (simp (no_asm) add: superfrechet_def frechet_def)
   464 apply (safe)
   465 apply (rule mem_FiltersetI2)
   466 apply (erule chain_Un_subset_Pow)
   467 apply (rule UnionI , assumption)
   468 apply (erule lemma_mem_chain_Filter [THEN mem_FiltersetD4] , assumption)
   469 apply (erule chain_Un_not_empty)
   470 apply (erule_tac [2] Filter_empty_not_mem_Un)
   471 apply (erule_tac [2] Filter_Un_Int)
   472 apply (erule_tac [2] Filter_Un_subset)
   473 apply (subgoal_tac [2] "xa \<in> frechet (UNIV) ")
   474 apply (blast intro: elim:); 
   475 apply (rule UnionI)
   476 apply assumption; 
   477 apply (rule lemma_mem_chain_frechet_subset [THEN subsetD])
   478 apply (auto simp add: frechet_def)
   479 done
   480 
   481 lemma (in UFT) max_cofinite_Filter_Ex: "\<exists>U \<in> superfrechet (UNIV).  
   482                 \<forall>G \<in> superfrechet (UNIV). U <= G --> U = G"
   483 apply (rule Zorn_Lemma2)
   484 apply (insert not_finite_UNIV [THEN cofinite_Filter])
   485 apply (safe)
   486 apply (rule_tac Q = "c={}" in excluded_middle [THEN disjE])
   487 apply (rule_tac x = "Union c" in bexI , blast)
   488 apply (rule Un_chain_mem_cofinite_Filter_set);
   489 apply (auto simp add: superfrechet_def frechet_def)
   490 done
   491 
   492 lemma (in UFT) max_cofinite_Freefilter_Ex: "\<exists>U \<in> superfrechet UNIV. ( 
   493                 \<forall>G \<in> superfrechet UNIV. U <= G --> U = G)   
   494                               & (\<forall>x \<in> U. ~finite x)"
   495 apply (insert not_finite_UNIV [THEN UFT.max_cofinite_Filter_Ex]);
   496 apply (safe)
   497 apply (rule_tac x = "U" in bexI)
   498 apply (auto simp add: superfrechet_def frechet_def)
   499 apply (drule_tac c = "- x" in subsetD)
   500 apply (simp (no_asm_simp))
   501 apply (frule_tac A = "x" and B = "- x" in mem_FiltersetD1)
   502 apply (drule_tac [3] Filter_empty_not_mem)
   503 apply (auto ); 
   504 done
   505 
   506 text{*There exists a free ultrafilter on any infinite set*}
   507 
   508 theorem (in UFT) FreeUltrafilter_ex: "\<exists>U. U \<in> FreeUltrafilter (UNIV :: 'a set)"
   509 apply (simp add: FreeUltrafilter_def)
   510 apply (insert not_finite_UNIV [THEN UFT.max_cofinite_Freefilter_Ex])
   511 apply (simp add: superfrechet_def Ultrafilter_iff frechet_def)
   512 apply (safe)
   513 apply (rule_tac x = "U" in exI)
   514 apply (safe)
   515 apply blast
   516 done
   517 
   518 theorems FreeUltrafilter_Ex = UFT.FreeUltrafilter_ex
   519 
   520 end