src/HOL/Library/Zorn.thy
author wenzelm
Thu May 06 14:14:18 2004 +0200 (2004-05-06)
changeset 14706 71590b7733b7
parent 13652 172600c40793
child 15131 c69542757a4d
permissions -rw-r--r--
tuned document;
     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 = Main:
    10 
    11 text{*
    12   The lemma and section numbers refer to an unpublished article
    13   \cite{Abrial-Laffitte}.
    14 *}
    15 
    16 constdefs
    17   chain     ::  "'a set set => 'a set set set"
    18   "chain S  == {F. F \<subseteq> S & (\<forall>x \<in> F. \<forall>y \<in> F. x \<subseteq> y | y \<subseteq> x)}"
    19 
    20   super     ::  "['a set set,'a set set] => 'a set set set"
    21   "super S c == {d. d \<in> chain S & c \<subset> d}"
    22 
    23   maxchain  ::  "'a set set => 'a set set set"
    24   "maxchain S == {c. c \<in> chain S & super S c = {}}"
    25 
    26   succ      ::  "['a set set,'a set set] => 'a set set"
    27   "succ S c ==
    28     if c \<notin> chain S | c \<in> maxchain S
    29     then c else SOME c'. c' \<in> super S c"
    30 
    31 consts
    32   TFin :: "'a set set => 'a set set set"
    33 
    34 inductive "TFin S"
    35   intros
    36     succI:        "x \<in> TFin S ==> succ S x \<in> TFin S"
    37     Pow_UnionI:   "Y \<in> Pow(TFin S) ==> Union(Y) \<in> TFin S"
    38   monos          Pow_mono
    39 
    40 
    41 subsection{*Mathematical Preamble*}
    42 
    43 lemma Union_lemma0: "(\<forall>x \<in> C. x \<subseteq> A | B \<subseteq> x) ==> Union(C)<=A | B \<subseteq> Union(C)"
    44 by blast
    45 
    46 
    47 text{*This is theorem @{text increasingD2} of ZF/Zorn.thy*}
    48 lemma Abrial_axiom1: "x \<subseteq> succ S x"
    49 apply (unfold succ_def)
    50 apply (rule split_if [THEN iffD2])
    51 apply (auto simp add: super_def maxchain_def psubset_def)
    52 apply (rule swap, assumption)
    53 apply (rule someI2, blast+)
    54 done
    55 
    56 lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]
    57 
    58 lemma TFin_induct:
    59           "[| n \<in> TFin S;
    60              !!x. [| x \<in> TFin S; P(x) |] ==> P(succ S x);
    61              !!Y. [| Y \<subseteq> TFin S; Ball Y P |] ==> P(Union Y) |]
    62           ==> P(n)"
    63 apply (erule TFin.induct, blast+)
    64 done
    65 
    66 lemma succ_trans: "x \<subseteq> y ==> x \<subseteq> succ S y"
    67 apply (erule subset_trans)
    68 apply (rule Abrial_axiom1)
    69 done
    70 
    71 text{*Lemma 1 of section 3.1*}
    72 lemma TFin_linear_lemma1:
    73      "[| n \<in> TFin S;  m \<in> TFin S;
    74          \<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m
    75       |] ==> n \<subseteq> m | succ S m \<subseteq> n"
    76 apply (erule TFin_induct)
    77 apply (erule_tac [2] Union_lemma0) (*or just blast*)
    78 apply (blast del: subsetI intro: succ_trans)
    79 done
    80 
    81 text{* Lemma 2 of section 3.2 *}
    82 lemma TFin_linear_lemma2:
    83      "m \<in> TFin S ==> \<forall>n \<in> TFin S. n \<subseteq> m --> n=m | succ S n \<subseteq> m"
    84 apply (erule TFin_induct)
    85 apply (rule impI [THEN ballI])
    86 txt{*case split using @{text TFin_linear_lemma1}*}
    87 apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
    88        assumption+)
    89 apply (drule_tac x = n in bspec, assumption)
    90 apply (blast del: subsetI intro: succ_trans, blast)
    91 txt{*second induction step*}
    92 apply (rule impI [THEN ballI])
    93 apply (rule Union_lemma0 [THEN disjE])
    94 apply (rule_tac [3] disjI2)
    95  prefer 2 apply blast
    96 apply (rule ballI)
    97 apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
    98        assumption+, auto)
    99 apply (blast intro!: Abrial_axiom1 [THEN subsetD])
   100 done
   101 
   102 text{*Re-ordering the premises of Lemma 2*}
   103 lemma TFin_subsetD:
   104      "[| n \<subseteq> m;  m \<in> TFin S;  n \<in> TFin S |] ==> n=m | succ S n \<subseteq> m"
   105 apply (rule TFin_linear_lemma2 [rule_format])
   106 apply (assumption+)
   107 done
   108 
   109 text{*Consequences from section 3.3 -- Property 3.2, the ordering is total*}
   110 lemma TFin_subset_linear: "[| m \<in> TFin S;  n \<in> TFin S|] ==> n \<subseteq> m | m \<subseteq> n"
   111 apply (rule disjE)
   112 apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2])
   113 apply (assumption+, erule disjI2)
   114 apply (blast del: subsetI
   115              intro: subsetI Abrial_axiom1 [THEN subset_trans])
   116 done
   117 
   118 text{*Lemma 3 of section 3.3*}
   119 lemma eq_succ_upper: "[| n \<in> TFin S;  m \<in> TFin S;  m = succ S m |] ==> n \<subseteq> m"
   120 apply (erule TFin_induct)
   121 apply (drule TFin_subsetD)
   122 apply (assumption+, force, blast)
   123 done
   124 
   125 text{*Property 3.3 of section 3.3*}
   126 lemma equal_succ_Union: "m \<in> TFin S ==> (m = succ S m) = (m = Union(TFin S))"
   127 apply (rule iffI)
   128 apply (rule Union_upper [THEN equalityI])
   129 apply (rule_tac [2] eq_succ_upper [THEN Union_least])
   130 apply (assumption+)
   131 apply (erule ssubst)
   132 apply (rule Abrial_axiom1 [THEN equalityI])
   133 apply (blast del: subsetI
   134              intro: subsetI TFin_UnionI TFin.succI)
   135 done
   136 
   137 subsection{*Hausdorff's Theorem: Every Set Contains a Maximal Chain.*}
   138 
   139 text{*NB: We assume the partial ordering is @{text "\<subseteq>"},
   140  the subset relation!*}
   141 
   142 lemma empty_set_mem_chain: "({} :: 'a set set) \<in> chain S"
   143 by (unfold chain_def, auto)
   144 
   145 lemma super_subset_chain: "super S c \<subseteq> chain S"
   146 by (unfold super_def, fast)
   147 
   148 lemma maxchain_subset_chain: "maxchain S \<subseteq> chain S"
   149 by (unfold maxchain_def, fast)
   150 
   151 lemma mem_super_Ex: "c \<in> chain S - maxchain S ==> ? d. d \<in> super S c"
   152 by (unfold super_def maxchain_def, auto)
   153 
   154 lemma select_super: "c \<in> chain S - maxchain S ==>
   155                           (@c'. c': super S c): super S c"
   156 apply (erule mem_super_Ex [THEN exE])
   157 apply (rule someI2, auto)
   158 done
   159 
   160 lemma select_not_equals: "c \<in> chain S - maxchain S ==>
   161                           (@c'. c': super S c) \<noteq> c"
   162 apply (rule notI)
   163 apply (drule select_super)
   164 apply (simp add: super_def psubset_def)
   165 done
   166 
   167 lemma succI3: "c \<in> chain S - maxchain S ==> succ S c = (@c'. c': super S c)"
   168 apply (unfold succ_def)
   169 apply (fast intro!: if_not_P)
   170 done
   171 
   172 lemma succ_not_equals: "c \<in> chain S - maxchain S ==> succ S c \<noteq> c"
   173 apply (frule succI3)
   174 apply (simp (no_asm_simp))
   175 apply (rule select_not_equals, assumption)
   176 done
   177 
   178 lemma TFin_chain_lemma4: "c \<in> TFin S ==> (c :: 'a set set): chain S"
   179 apply (erule TFin_induct)
   180 apply (simp add: succ_def select_super [THEN super_subset_chain[THEN subsetD]])
   181 apply (unfold chain_def)
   182 apply (rule CollectI, safe)
   183 apply (drule bspec, assumption)
   184 apply (rule_tac [2] m1 = Xa and n1 = X in TFin_subset_linear [THEN disjE],
   185        blast+)
   186 done
   187 
   188 theorem Hausdorff: "\<exists>c. (c :: 'a set set): maxchain S"
   189 apply (rule_tac x = "Union (TFin S) " in exI)
   190 apply (rule classical)
   191 apply (subgoal_tac "succ S (Union (TFin S)) = Union (TFin S) ")
   192  prefer 2
   193  apply (blast intro!: TFin_UnionI equal_succ_Union [THEN iffD2, symmetric])
   194 apply (cut_tac subset_refl [THEN TFin_UnionI, THEN TFin_chain_lemma4])
   195 apply (drule DiffI [THEN succ_not_equals], blast+)
   196 done
   197 
   198 
   199 subsection{*Zorn's Lemma: If All Chains Have Upper Bounds Then
   200                                There Is  a Maximal Element*}
   201 
   202 lemma chain_extend:
   203     "[| c \<in> chain S; z \<in> S;
   204         \<forall>x \<in> c. x<=(z:: 'a set) |] ==> {z} Un c \<in> chain S"
   205 by (unfold chain_def, blast)
   206 
   207 lemma chain_Union_upper: "[| c \<in> chain S; x \<in> c |] ==> x \<subseteq> Union(c)"
   208 by (unfold chain_def, auto)
   209 
   210 lemma chain_ball_Union_upper: "c \<in> chain S ==> \<forall>x \<in> c. x \<subseteq> Union(c)"
   211 by (unfold chain_def, auto)
   212 
   213 lemma maxchain_Zorn:
   214      "[| c \<in> maxchain S; u \<in> S; Union(c) \<subseteq> u |] ==> Union(c) = u"
   215 apply (rule ccontr)
   216 apply (simp add: maxchain_def)
   217 apply (erule conjE)
   218 apply (subgoal_tac " ({u} Un c) \<in> super S c")
   219 apply simp
   220 apply (unfold super_def psubset_def)
   221 apply (blast intro: chain_extend dest: chain_Union_upper)
   222 done
   223 
   224 theorem Zorn_Lemma:
   225      "\<forall>c \<in> chain S. Union(c): S ==> \<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z --> y = z"
   226 apply (cut_tac Hausdorff maxchain_subset_chain)
   227 apply (erule exE)
   228 apply (drule subsetD, assumption)
   229 apply (drule bspec, assumption)
   230 apply (rule_tac x = "Union (c) " in bexI)
   231 apply (rule ballI, rule impI)
   232 apply (blast dest!: maxchain_Zorn, assumption)
   233 done
   234 
   235 subsection{*Alternative version of Zorn's Lemma*}
   236 
   237 lemma Zorn_Lemma2:
   238      "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>x \<in> c. x \<subseteq> y
   239       ==> \<exists>y \<in> S. \<forall>x \<in> S. (y :: 'a set) \<subseteq> x --> y = x"
   240 apply (cut_tac Hausdorff maxchain_subset_chain)
   241 apply (erule exE)
   242 apply (drule subsetD, assumption)
   243 apply (drule bspec, assumption, erule bexE)
   244 apply (rule_tac x = y in bexI)
   245  prefer 2 apply assumption
   246 apply clarify
   247 apply (rule ccontr)
   248 apply (frule_tac z = x in chain_extend)
   249 apply (assumption, blast)
   250 apply (unfold maxchain_def super_def psubset_def)
   251 apply (blast elim!: equalityCE)
   252 done
   253 
   254 text{*Various other lemmas*}
   255 
   256 lemma chainD: "[| c \<in> chain S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
   257 by (unfold chain_def, blast)
   258 
   259 lemma chainD2: "!!(c :: 'a set set). c \<in> chain S ==> c \<subseteq> S"
   260 by (unfold chain_def, blast)
   261 
   262 end