src/HOL/Library/Zorn.thy
author wenzelm
Thu Jan 05 22:29:55 2006 +0100 (2006-01-05)
changeset 18585 5d379fe2eb74
parent 18143 fe14f0282c60
child 19736 d8d0f8f51d69
permissions -rw-r--r--
replaced swap by contrapos_np;
     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    14   The lemma and section numbers refer to an unpublished article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:
    46     "(\<forall>x \<in> C. x \<subseteq> A | B \<subseteq> x) ==> Union(C) \<subseteq> A | B \<subseteq> Union(C)"
    47   by blast
    48 
    49 
    50 text{*This is theorem @{text increasingD2} of ZF/Zorn.thy*}
    51 
    52 lemma Abrial_axiom1: "x \<subseteq> succ S x"
    53   apply (unfold succ_def)
    54   apply (rule split_if [THEN iffD2])
    55   apply (auto simp add: super_def maxchain_def psubset_def)
    56   apply (rule contrapos_np, assumption)
    57   apply (rule someI2, blast+)
    58   done
    59 
    60 lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]
    61 
    62 lemma TFin_induct:
    63           "[| n \<in> TFin S;
    64              !!x. [| x \<in> TFin S; P(x) |] ==> P(succ S x);
    65              !!Y. [| Y \<subseteq> TFin S; Ball Y P |] ==> P(Union Y) |]
    66           ==> P(n)"
    67   apply (erule TFin.induct)
    68    apply blast+
    69   done
    70 
    71 lemma succ_trans: "x \<subseteq> y ==> x \<subseteq> succ S y"
    72   apply (erule subset_trans)
    73   apply (rule Abrial_axiom1)
    74   done
    75 
    76 text{*Lemma 1 of section 3.1*}
    77 lemma TFin_linear_lemma1:
    78      "[| n \<in> TFin S;  m \<in> TFin S;
    79          \<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m
    80       |] ==> n \<subseteq> m | succ S m \<subseteq> n"
    81   apply (erule TFin_induct)
    82    apply (erule_tac [2] Union_lemma0)
    83   apply (blast del: subsetI intro: succ_trans)
    84   done
    85 
    86 text{* Lemma 2 of section 3.2 *}
    87 lemma TFin_linear_lemma2:
    88      "m \<in> TFin S ==> \<forall>n \<in> TFin S. n \<subseteq> m --> n=m | succ S n \<subseteq> m"
    89   apply (erule TFin_induct)
    90    apply (rule impI [THEN ballI])
    91    txt{*case split using @{text TFin_linear_lemma1}*}
    92    apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
    93      assumption+)
    94     apply (drule_tac x = n in bspec, assumption)
    95     apply (blast del: subsetI intro: succ_trans, blast)
    96   txt{*second induction step*}
    97   apply (rule impI [THEN ballI])
    98   apply (rule Union_lemma0 [THEN disjE])
    99     apply (rule_tac [3] disjI2)
   100     prefer 2 apply blast
   101    apply (rule ballI)
   102    apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
   103      assumption+, auto)
   104   apply (blast intro!: Abrial_axiom1 [THEN subsetD])
   105   done
   106 
   107 text{*Re-ordering the premises of Lemma 2*}
   108 lemma TFin_subsetD:
   109      "[| n \<subseteq> m;  m \<in> TFin S;  n \<in> TFin S |] ==> n=m | succ S n \<subseteq> m"
   110   by (rule TFin_linear_lemma2 [rule_format])
   111 
   112 text{*Consequences from section 3.3 -- Property 3.2, the ordering is total*}
   113 lemma TFin_subset_linear: "[| m \<in> TFin S;  n \<in> TFin S|] ==> n \<subseteq> m | m \<subseteq> n"
   114   apply (rule disjE)
   115     apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2])
   116       apply (assumption+, erule disjI2)
   117   apply (blast del: subsetI
   118     intro: subsetI Abrial_axiom1 [THEN subset_trans])
   119   done
   120 
   121 text{*Lemma 3 of section 3.3*}
   122 lemma eq_succ_upper: "[| n \<in> TFin S;  m \<in> TFin S;  m = succ S m |] ==> n \<subseteq> m"
   123   apply (erule TFin_induct)
   124    apply (drule TFin_subsetD)
   125      apply (assumption+, force, blast)
   126   done
   127 
   128 text{*Property 3.3 of section 3.3*}
   129 lemma equal_succ_Union: "m \<in> TFin S ==> (m = succ S m) = (m = Union(TFin S))"
   130   apply (rule iffI)
   131    apply (rule Union_upper [THEN equalityI])
   132     apply assumption
   133    apply (rule eq_succ_upper [THEN Union_least], assumption+)
   134   apply (erule ssubst)
   135   apply (rule Abrial_axiom1 [THEN equalityI])
   136   apply (blast del: subsetI 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) blast
   149 
   150 lemma maxchain_subset_chain: "maxchain S \<subseteq> chain S"
   151   by (unfold maxchain_def) blast
   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:
   157      "c \<in> chain S - maxchain S ==> (\<some>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:
   163      "c \<in> chain S - maxchain S ==> (\<some>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 = (\<some>c'. c': super S c)"
   170   by (unfold succ_def) (blast intro!: if_not_P)
   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 \<subseteq> (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