src/HOL/Library/Zorn.thy
author paulson
Tue Jun 28 15:27:45 2005 +0200 (2005-06-28)
changeset 16587 b34c8aa657a5
parent 15140 322485b816ac
child 17200 3a4d03d1a31b
permissions -rw-r--r--
Constant "If" is now local
     1 (*  Title       : HOL/Library/Zorn.thy
     2     ID          : $Id$
     3     Author      : Jacques D. Fleuriot
     4     Description : Zorn's Lemma -- see Larry Paulson's Zorn.thy in ZF
     5 *)
     6 
     7 header {* Zorn's Lemma *}
     8 
     9 theory Zorn
    10 imports Main
    11 begin
    12 
    13 text{*
    14   The lemma and section numbers refer to an unpublished article
    15   \cite{Abrial-Laffitte}.
    16 *}
    17 
    18 constdefs
    19   chain     ::  "'a set set => 'a set set set"
    20   "chain S  == {F. F \<subseteq> S & (\<forall>x \<in> F. \<forall>y \<in> F. x \<subseteq> y | y \<subseteq> x)}"
    21 
    22   super     ::  "['a set set,'a set set] => 'a set set set"
    23   "super S c == {d. d \<in> chain S & c \<subset> d}"
    24 
    25   maxchain  ::  "'a set set => 'a set set set"
    26   "maxchain S == {c. c \<in> chain S & super S c = {}}"
    27 
    28   succ      ::  "['a set set,'a set set] => 'a set set"
    29   "succ S c ==
    30     if c \<notin> chain S | c \<in> maxchain S
    31     then c else SOME c'. c' \<in> super S c"
    32 
    33 consts
    34   TFin :: "'a set set => 'a set set set"
    35 
    36 inductive "TFin S"
    37   intros
    38     succI:        "x \<in> TFin S ==> succ S x \<in> TFin S"
    39     Pow_UnionI:   "Y \<in> Pow(TFin S) ==> Union(Y) \<in> TFin S"
    40   monos          Pow_mono
    41 
    42 
    43 subsection{*Mathematical Preamble*}
    44 
    45 lemma Union_lemma0: "(\<forall>x \<in> C. x \<subseteq> A | B \<subseteq> x) ==> Union(C)<=A | B \<subseteq> Union(C)"
    46 by blast
    47 
    48 
    49 text{*This is theorem @{text increasingD2} of ZF/Zorn.thy*}
    50 lemma Abrial_axiom1: "x \<subseteq> succ S x"
    51 apply (unfold succ_def)
    52 apply (rule split_if [THEN iffD2])
    53 apply (auto simp add: super_def maxchain_def psubset_def)
    54 apply (rule swap, assumption)
    55 apply (rule someI2, blast+)
    56 done
    57 
    58 lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]
    59 
    60 lemma TFin_induct:
    61           "[| n \<in> TFin S;
    62              !!x. [| x \<in> TFin S; P(x) |] ==> P(succ S x);
    63              !!Y. [| Y \<subseteq> TFin S; Ball Y P |] ==> P(Union Y) |]
    64           ==> P(n)"
    65 apply (erule TFin.induct, blast+)
    66 done
    67 
    68 lemma succ_trans: "x \<subseteq> y ==> x \<subseteq> succ S y"
    69 apply (erule subset_trans)
    70 apply (rule Abrial_axiom1)
    71 done
    72 
    73 text{*Lemma 1 of section 3.1*}
    74 lemma TFin_linear_lemma1:
    75      "[| n \<in> TFin S;  m \<in> TFin S;
    76          \<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m
    77       |] ==> n \<subseteq> m | succ S m \<subseteq> n"
    78 apply (erule TFin_induct)
    79 apply (erule_tac [2] Union_lemma0) (*or just blast*)
    80 apply (blast del: subsetI intro: succ_trans)
    81 done
    82 
    83 text{* Lemma 2 of section 3.2 *}
    84 lemma TFin_linear_lemma2:
    85      "m \<in> TFin S ==> \<forall>n \<in> TFin S. n \<subseteq> m --> n=m | succ S n \<subseteq> m"
    86 apply (erule TFin_induct)
    87 apply (rule impI [THEN ballI])
    88 txt{*case split using @{text TFin_linear_lemma1}*}
    89 apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
    90        assumption+)
    91 apply (drule_tac x = n in bspec, assumption)
    92 apply (blast del: subsetI intro: succ_trans, blast)
    93 txt{*second induction step*}
    94 apply (rule impI [THEN ballI])
    95 apply (rule Union_lemma0 [THEN disjE])
    96 apply (rule_tac [3] disjI2)
    97  prefer 2 apply blast
    98 apply (rule ballI)
    99 apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
   100        assumption+, auto)
   101 apply (blast intro!: Abrial_axiom1 [THEN subsetD])
   102 done
   103 
   104 text{*Re-ordering the premises of Lemma 2*}
   105 lemma TFin_subsetD:
   106      "[| n \<subseteq> m;  m \<in> TFin S;  n \<in> TFin S |] ==> n=m | succ S n \<subseteq> m"
   107 apply (rule TFin_linear_lemma2 [rule_format])
   108 apply (assumption+)
   109 done
   110 
   111 text{*Consequences from section 3.3 -- Property 3.2, the ordering is total*}
   112 lemma TFin_subset_linear: "[| m \<in> TFin S;  n \<in> TFin S|] ==> n \<subseteq> m | m \<subseteq> n"
   113 apply (rule disjE)
   114 apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2])
   115 apply (assumption+, erule disjI2)
   116 apply (blast del: subsetI
   117              intro: subsetI Abrial_axiom1 [THEN subset_trans])
   118 done
   119 
   120 text{*Lemma 3 of section 3.3*}
   121 lemma eq_succ_upper: "[| n \<in> TFin S;  m \<in> TFin S;  m = succ S m |] ==> n \<subseteq> m"
   122 apply (erule TFin_induct)
   123 apply (drule TFin_subsetD)
   124 apply (assumption+, force, blast)
   125 done
   126 
   127 text{*Property 3.3 of section 3.3*}
   128 lemma equal_succ_Union: "m \<in> TFin S ==> (m = succ S m) = (m = Union(TFin S))"
   129 apply (rule iffI)
   130 apply (rule Union_upper [THEN equalityI])
   131 apply (rule_tac [2] eq_succ_upper [THEN Union_least])
   132 apply (assumption+)
   133 apply (erule ssubst)
   134 apply (rule Abrial_axiom1 [THEN equalityI])
   135 apply (blast del: subsetI
   136              intro: subsetI TFin_UnionI TFin.succI)
   137 done
   138 
   139 subsection{*Hausdorff's Theorem: Every Set Contains a Maximal Chain.*}
   140 
   141 text{*NB: We assume the partial ordering is @{text "\<subseteq>"},
   142  the subset relation!*}
   143 
   144 lemma empty_set_mem_chain: "({} :: 'a set set) \<in> chain S"
   145 by (unfold chain_def, auto)
   146 
   147 lemma super_subset_chain: "super S c \<subseteq> chain S"
   148 by (unfold super_def, fast)
   149 
   150 lemma maxchain_subset_chain: "maxchain S \<subseteq> chain S"
   151 by (unfold maxchain_def, fast)
   152 
   153 lemma mem_super_Ex: "c \<in> chain S - maxchain S ==> ? d. d \<in> super S c"
   154 by (unfold super_def maxchain_def, auto)
   155 
   156 lemma select_super: "c \<in> chain S - maxchain S ==>
   157                           (@c'. c': super S c): super S c"
   158 apply (erule mem_super_Ex [THEN exE])
   159 apply (rule someI2, auto)
   160 done
   161 
   162 lemma select_not_equals: "c \<in> chain S - maxchain S ==>
   163                           (@c'. c': super S c) \<noteq> c"
   164 apply (rule notI)
   165 apply (drule select_super)
   166 apply (simp add: super_def psubset_def)
   167 done
   168 
   169 lemma succI3: "c \<in> chain S - maxchain S ==> succ S c = (@c'. c': super S c)"
   170 apply (unfold succ_def)
   171 apply (fast intro!: if_not_P)
   172 done
   173 
   174 lemma succ_not_equals: "c \<in> chain S - maxchain S ==> succ S c \<noteq> c"
   175 apply (frule succI3)
   176 apply (simp (no_asm_simp))
   177 apply (rule select_not_equals, assumption)
   178 done
   179 
   180 lemma TFin_chain_lemma4: "c \<in> TFin S ==> (c :: 'a set set): chain S"
   181 apply (erule TFin_induct)
   182 apply (simp add: succ_def select_super [THEN super_subset_chain[THEN subsetD]])
   183 apply (unfold chain_def)
   184 apply (rule CollectI, safe)
   185 apply (drule bspec, assumption)
   186 apply (rule_tac [2] m1 = Xa and n1 = X in TFin_subset_linear [THEN disjE],
   187        blast+)
   188 done
   189 
   190 theorem Hausdorff: "\<exists>c. (c :: 'a set set): maxchain S"
   191 apply (rule_tac x = "Union (TFin S) " in exI)
   192 apply (rule classical)
   193 apply (subgoal_tac "succ S (Union (TFin S)) = Union (TFin S) ")
   194  prefer 2
   195  apply (blast intro!: TFin_UnionI equal_succ_Union [THEN iffD2, symmetric])
   196 apply (cut_tac subset_refl [THEN TFin_UnionI, THEN TFin_chain_lemma4])
   197 apply (drule DiffI [THEN succ_not_equals], blast+)
   198 done
   199 
   200 
   201 subsection{*Zorn's Lemma: If All Chains Have Upper Bounds Then
   202                                There Is  a Maximal Element*}
   203 
   204 lemma chain_extend:
   205     "[| c \<in> chain S; z \<in> S;
   206         \<forall>x \<in> c. x<=(z:: 'a set) |] ==> {z} Un c \<in> chain S"
   207 by (unfold chain_def, blast)
   208 
   209 lemma chain_Union_upper: "[| c \<in> chain S; x \<in> c |] ==> x \<subseteq> Union(c)"
   210 by (unfold chain_def, auto)
   211 
   212 lemma chain_ball_Union_upper: "c \<in> chain S ==> \<forall>x \<in> c. x \<subseteq> Union(c)"
   213 by (unfold chain_def, auto)
   214 
   215 lemma maxchain_Zorn:
   216      "[| c \<in> maxchain S; u \<in> S; Union(c) \<subseteq> u |] ==> Union(c) = u"
   217 apply (rule ccontr)
   218 apply (simp add: maxchain_def)
   219 apply (erule conjE)
   220 apply (subgoal_tac " ({u} Un c) \<in> super S c")
   221 apply simp
   222 apply (unfold super_def psubset_def)
   223 apply (blast intro: chain_extend dest: chain_Union_upper)
   224 done
   225 
   226 theorem Zorn_Lemma:
   227      "\<forall>c \<in> chain S. Union(c): S ==> \<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z --> y = z"
   228 apply (cut_tac Hausdorff maxchain_subset_chain)
   229 apply (erule exE)
   230 apply (drule subsetD, assumption)
   231 apply (drule bspec, assumption)
   232 apply (rule_tac x = "Union (c) " in bexI)
   233 apply (rule ballI, rule impI)
   234 apply (blast dest!: maxchain_Zorn, assumption)
   235 done
   236 
   237 subsection{*Alternative version of Zorn's Lemma*}
   238 
   239 lemma Zorn_Lemma2:
   240      "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>x \<in> c. x \<subseteq> y
   241       ==> \<exists>y \<in> S. \<forall>x \<in> S. (y :: 'a set) \<subseteq> x --> y = x"
   242 apply (cut_tac Hausdorff maxchain_subset_chain)
   243 apply (erule exE)
   244 apply (drule subsetD, assumption)
   245 apply (drule bspec, assumption, erule bexE)
   246 apply (rule_tac x = y in bexI)
   247  prefer 2 apply assumption
   248 apply clarify
   249 apply (rule ccontr)
   250 apply (frule_tac z = x in chain_extend)
   251 apply (assumption, blast)
   252 apply (unfold maxchain_def super_def psubset_def)
   253 apply (blast elim!: equalityCE)
   254 done
   255 
   256 text{*Various other lemmas*}
   257 
   258 lemma chainD: "[| c \<in> chain S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
   259 by (unfold chain_def, blast)
   260 
   261 lemma chainD2: "!!(c :: 'a set set). c \<in> chain S ==> c \<subseteq> S"
   262 by (unfold chain_def, blast)
   263 
   264 end