src/HOL/Library/Zorn.thy
author nipkow
Fri Mar 14 19:57:32 2008 +0100 (2008-03-14)
changeset 26272 d63776c3be97
parent 26191 ae537f315b34
child 26295 afc43168ed85
permissions -rw-r--r--
Added Order_Relation
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(*  Title       : HOL/Library/Zorn.thy
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    ID          : $Id$
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    Author      : Jacques D. Fleuriot, Tobias Nipkow
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    Description : Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF)
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                  The well-ordering theorem
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*)
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header {* Zorn's Lemma *}
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theory Zorn
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imports Order_Relation
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begin
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(* Define globally? In Set.thy? *)
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definition chain_subset :: "'a set set \<Rightarrow> bool" ("chain\<^bsub>\<subseteq>\<^esub>") where
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"chain\<^bsub>\<subseteq>\<^esub> C \<equiv> \<forall>A\<in>C.\<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A"
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text{*
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  The lemma and section numbers refer to an unpublished article
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  \cite{Abrial-Laffitte}.
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*}
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definition
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  chain     ::  "'a set set => 'a set set set" where
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  "chain S  = {F. F \<subseteq> S & chain\<^bsub>\<subseteq>\<^esub> F}"
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definition
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  super     ::  "['a set set,'a set set] => 'a set set set" where
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  "super S c = {d. d \<in> chain S & c \<subset> d}"
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definition
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  maxchain  ::  "'a set set => 'a set set set" where
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  "maxchain S = {c. c \<in> chain S & super S c = {}}"
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definition
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  succ      ::  "['a set set,'a set set] => 'a set set" where
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  "succ S c =
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    (if c \<notin> chain S | c \<in> maxchain S
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    then c else SOME c'. c' \<in> super S c)"
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inductive_set
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  TFin :: "'a set set => 'a set set set"
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  for S :: "'a set set"
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  where
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    succI:        "x \<in> TFin S ==> succ S x \<in> TFin S"
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  | Pow_UnionI:   "Y \<in> Pow(TFin S) ==> Union(Y) \<in> TFin S"
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  monos          Pow_mono
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subsection{*Mathematical Preamble*}
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lemma Union_lemma0:
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    "(\<forall>x \<in> C. x \<subseteq> A | B \<subseteq> x) ==> Union(C) \<subseteq> A | B \<subseteq> Union(C)"
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  by blast
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text{*This is theorem @{text increasingD2} of ZF/Zorn.thy*}
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lemma Abrial_axiom1: "x \<subseteq> succ S x"
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  apply (unfold succ_def)
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  apply (rule split_if [THEN iffD2])
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  apply (auto simp add: super_def maxchain_def psubset_def)
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  apply (rule contrapos_np, assumption)
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  apply (rule someI2, blast+)
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  done
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lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]
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lemma TFin_induct:
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          "[| n \<in> TFin S;
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             !!x. [| x \<in> TFin S; P(x) |] ==> P(succ S x);
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             !!Y. [| Y \<subseteq> TFin S; Ball Y P |] ==> P(Union Y) |]
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          ==> P(n)"
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  apply (induct set: TFin)
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   apply blast+
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  done
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lemma succ_trans: "x \<subseteq> y ==> x \<subseteq> succ S y"
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  apply (erule subset_trans)
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  apply (rule Abrial_axiom1)
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  done
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text{*Lemma 1 of section 3.1*}
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lemma TFin_linear_lemma1:
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     "[| n \<in> TFin S;  m \<in> TFin S;
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         \<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m
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      |] ==> n \<subseteq> m | succ S m \<subseteq> n"
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  apply (erule TFin_induct)
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   apply (erule_tac [2] Union_lemma0)
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  apply (blast del: subsetI intro: succ_trans)
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  done
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text{* Lemma 2 of section 3.2 *}
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lemma TFin_linear_lemma2:
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     "m \<in> TFin S ==> \<forall>n \<in> TFin S. n \<subseteq> m --> n=m | succ S n \<subseteq> m"
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  apply (erule TFin_induct)
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   apply (rule impI [THEN ballI])
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   txt{*case split using @{text TFin_linear_lemma1}*}
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   apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
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     assumption+)
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    apply (drule_tac x = n in bspec, assumption)
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    apply (blast del: subsetI intro: succ_trans, blast)
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  txt{*second induction step*}
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  apply (rule impI [THEN ballI])
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  apply (rule Union_lemma0 [THEN disjE])
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    apply (rule_tac [3] disjI2)
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    prefer 2 apply blast
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   apply (rule ballI)
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   apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
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     assumption+, auto)
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  apply (blast intro!: Abrial_axiom1 [THEN subsetD])
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  done
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text{*Re-ordering the premises of Lemma 2*}
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lemma TFin_subsetD:
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     "[| n \<subseteq> m;  m \<in> TFin S;  n \<in> TFin S |] ==> n=m | succ S n \<subseteq> m"
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  by (rule TFin_linear_lemma2 [rule_format])
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text{*Consequences from section 3.3 -- Property 3.2, the ordering is total*}
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lemma TFin_subset_linear: "[| m \<in> TFin S;  n \<in> TFin S|] ==> n \<subseteq> m | m \<subseteq> n"
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  apply (rule disjE)
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    apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2])
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      apply (assumption+, erule disjI2)
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  apply (blast del: subsetI
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    intro: subsetI Abrial_axiom1 [THEN subset_trans])
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  done
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text{*Lemma 3 of section 3.3*}
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lemma eq_succ_upper: "[| n \<in> TFin S;  m \<in> TFin S;  m = succ S m |] ==> n \<subseteq> m"
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  apply (erule TFin_induct)
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   apply (drule TFin_subsetD)
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     apply (assumption+, force, blast)
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  done
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text{*Property 3.3 of section 3.3*}
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lemma equal_succ_Union: "m \<in> TFin S ==> (m = succ S m) = (m = Union(TFin S))"
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  apply (rule iffI)
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   apply (rule Union_upper [THEN equalityI])
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    apply assumption
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   apply (rule eq_succ_upper [THEN Union_least], assumption+)
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  apply (erule ssubst)
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  apply (rule Abrial_axiom1 [THEN equalityI])
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  apply (blast del: subsetI intro: subsetI TFin_UnionI TFin.succI)
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  done
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subsection{*Hausdorff's Theorem: Every Set Contains a Maximal Chain.*}
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text{*NB: We assume the partial ordering is @{text "\<subseteq>"},
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 the subset relation!*}
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lemma empty_set_mem_chain: "({} :: 'a set set) \<in> chain S"
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by (unfold chain_def chain_subset_def) auto
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lemma super_subset_chain: "super S c \<subseteq> chain S"
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  by (unfold super_def) blast
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lemma maxchain_subset_chain: "maxchain S \<subseteq> chain S"
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  by (unfold maxchain_def) blast
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lemma mem_super_Ex: "c \<in> chain S - maxchain S ==> EX d. d \<in> super S c"
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  by (unfold super_def maxchain_def) auto
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lemma select_super:
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     "c \<in> chain S - maxchain S ==> (\<some>c'. c': super S c): super S c"
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  apply (erule mem_super_Ex [THEN exE])
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  apply (rule someI2, auto)
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  done
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lemma select_not_equals:
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     "c \<in> chain S - maxchain S ==> (\<some>c'. c': super S c) \<noteq> c"
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  apply (rule notI)
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  apply (drule select_super)
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  apply (simp add: super_def psubset_def)
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  done
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lemma succI3: "c \<in> chain S - maxchain S ==> succ S c = (\<some>c'. c': super S c)"
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  by (unfold succ_def) (blast intro!: if_not_P)
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lemma succ_not_equals: "c \<in> chain S - maxchain S ==> succ S c \<noteq> c"
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  apply (frule succI3)
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  apply (simp (no_asm_simp))
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  apply (rule select_not_equals, assumption)
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  done
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lemma TFin_chain_lemma4: "c \<in> TFin S ==> (c :: 'a set set): chain S"
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  apply (erule TFin_induct)
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   apply (simp add: succ_def select_super [THEN super_subset_chain[THEN subsetD]])
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  apply (unfold chain_def chain_subset_def)
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  apply (rule CollectI, safe)
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   apply (drule bspec, assumption)
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   apply (rule_tac [2] m1 = Xa and n1 = X in TFin_subset_linear [THEN disjE],
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     blast+)
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  done
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theorem Hausdorff: "\<exists>c. (c :: 'a set set): maxchain S"
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  apply (rule_tac x = "Union (TFin S)" in exI)
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  apply (rule classical)
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  apply (subgoal_tac "succ S (Union (TFin S)) = Union (TFin S) ")
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   prefer 2
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   apply (blast intro!: TFin_UnionI equal_succ_Union [THEN iffD2, symmetric])
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  apply (cut_tac subset_refl [THEN TFin_UnionI, THEN TFin_chain_lemma4])
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  apply (drule DiffI [THEN succ_not_equals], blast+)
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  done
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subsection{*Zorn's Lemma: If All Chains Have Upper Bounds Then
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                               There Is  a Maximal Element*}
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lemma chain_extend:
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  "[| c \<in> chain S; z \<in> S; \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chain S"
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by (unfold chain_def chain_subset_def) blast
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lemma chain_Union_upper: "[| c \<in> chain S; x \<in> c |] ==> x \<subseteq> Union(c)"
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by auto
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lemma chain_ball_Union_upper: "c \<in> chain S ==> \<forall>x \<in> c. x \<subseteq> Union(c)"
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by auto
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lemma maxchain_Zorn:
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  "[| c \<in> maxchain S; u \<in> S; Union(c) \<subseteq> u |] ==> Union(c) = u"
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apply (rule ccontr)
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apply (simp add: maxchain_def)
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apply (erule conjE)
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apply (subgoal_tac "({u} Un c) \<in> super S c")
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 apply simp
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apply (unfold super_def psubset_def)
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apply (blast intro: chain_extend dest: chain_Union_upper)
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done
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theorem Zorn_Lemma:
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  "\<forall>c \<in> chain S. Union(c): S ==> \<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z --> y = z"
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apply (cut_tac Hausdorff maxchain_subset_chain)
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apply (erule exE)
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apply (drule subsetD, assumption)
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apply (drule bspec, assumption)
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apply (rule_tac x = "Union(c)" in bexI)
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 apply (rule ballI, rule impI)
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 apply (blast dest!: maxchain_Zorn, assumption)
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done
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subsection{*Alternative version of Zorn's Lemma*}
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lemma Zorn_Lemma2:
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  "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>x \<in> c. x \<subseteq> y
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    ==> \<exists>y \<in> S. \<forall>x \<in> S. (y :: 'a set) \<subseteq> x --> y = x"
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apply (cut_tac Hausdorff maxchain_subset_chain)
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apply (erule exE)
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apply (drule subsetD, assumption)
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apply (drule bspec, assumption, erule bexE)
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apply (rule_tac x = y in bexI)
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 prefer 2 apply assumption
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apply clarify
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apply (rule ccontr)
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apply (frule_tac z = x in chain_extend)
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  apply (assumption, blast)
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apply (unfold maxchain_def super_def psubset_def)
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apply (blast elim!: equalityCE)
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done
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text{*Various other lemmas*}
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lemma chainD: "[| c \<in> chain S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
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by (unfold chain_def chain_subset_def) blast
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lemma chainD2: "!!(c :: 'a set set). c \<in> chain S ==> c \<subseteq> S"
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by (unfold chain_def) blast
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(* Define globally? In Relation.thy? *)
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definition Chain :: "('a*'a)set \<Rightarrow> 'a set set" where
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"Chain r \<equiv> {A. \<forall>a\<in>A.\<forall>b\<in>A. (a,b) : r \<or> (b,a) \<in> r}"
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lemma mono_Chain: "r \<subseteq> s \<Longrightarrow> Chain r \<subseteq> Chain s"
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unfolding Chain_def by blast
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text{* Zorn's lemma for partial orders: *}
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lemma Zorns_po_lemma:
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assumes po: "Partial_order r" and u: "\<forall>C\<in>Chain r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a,u):r"
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shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m,a):r \<longrightarrow> a=m"
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proof-
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  have "Preorder r" using po by(simp add:Partial_order_def)
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--{* Mirror r in the set of subsets below (wrt r) elements of A*}
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  let ?B = "%x. r^-1 `` {x}" let ?S = "?B ` Field r"
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  have "\<forall>C \<in> chain ?S. EX U:?S. ALL A:C. A\<subseteq>U"
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  proof (auto simp:chain_def chain_subset_def)
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    fix C assume 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C.\<forall>B\<in>C. A\<subseteq>B | B\<subseteq>A"
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    let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
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    have "C = ?B ` ?A" using 1 by(auto simp: image_def)
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    have "?A\<in>Chain r"
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    proof (simp add:Chain_def, intro allI impI, elim conjE)
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      fix a b
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      assume "a \<in> Field r" "?B a \<in> C" "b \<in> Field r" "?B b \<in> C"
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      hence "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" using 2 by auto
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      thus "(a, b) \<in> r \<or> (b, a) \<in> r" using `Preorder r` `a:Field r` `b:Field r`
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	by(simp add:subset_Image1_Image1_iff)
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    qed
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    then obtain u where uA: "u:Field r" "\<forall>a\<in>?A. (a,u) : r" using u by auto
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    have "\<forall>A\<in>C. A \<subseteq> r^-1 `` {u}" (is "?P u")
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    proof auto
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      fix a B assume aB: "B:C" "a:B"
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      with 1 obtain x where "x:Field r" "B = r^-1 `` {x}" by auto
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      thus "(a,u) : r" using uA aB `Preorder r`
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	by (auto simp add: Preorder_def Refl_def) (metis transD)
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    qed
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    thus "EX u:Field r. ?P u" using `u:Field r` by blast
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  qed
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  from Zorn_Lemma2[OF this]
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  obtain m B where "m:Field r" "B = r^-1 `` {m}"
nipkow@26191
   310
    "\<forall>x\<in>Field r. B \<subseteq> r^-1 `` {x} \<longrightarrow> B = r^-1 `` {x}"
nipkow@26191
   311
    by(auto simp:image_def) blast
nipkow@26191
   312
  hence "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m" using po `Preorder r` `m:Field r`
nipkow@26191
   313
    by(auto simp:subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
nipkow@26191
   314
  thus ?thesis using `m:Field r` by blast
nipkow@26191
   315
qed
nipkow@26191
   316
nipkow@26191
   317
(* The initial segment of a relation appears generally useful.
nipkow@26191
   318
   Move to Relation.thy?
nipkow@26191
   319
   Definition correct/most general?
nipkow@26191
   320
   Naming?
nipkow@26191
   321
*)
nipkow@26191
   322
definition init_seg_of :: "(('a*'a)set * ('a*'a)set)set" where
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   323
"init_seg_of == {(r,s). r \<subseteq> s \<and> (\<forall>a b c. (a,b):s \<and> (b,c):r \<longrightarrow> (a,b):r)}"
nipkow@26191
   324
nipkow@26191
   325
abbreviation initialSegmentOf :: "('a*'a)set \<Rightarrow> ('a*'a)set \<Rightarrow> bool"
nipkow@26191
   326
             (infix "initial'_segment'_of" 55) where
nipkow@26191
   327
"r initial_segment_of s == (r,s):init_seg_of"
nipkow@26191
   328
nipkow@26191
   329
lemma refl_init_seg_of[simp]: "r initial_segment_of r"
nipkow@26191
   330
by(simp add:init_seg_of_def)
nipkow@26191
   331
nipkow@26191
   332
lemma trans_init_seg_of:
nipkow@26191
   333
  "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
nipkow@26191
   334
by(simp (no_asm_use) add: init_seg_of_def)
nipkow@26191
   335
  (metis Domain_iff UnCI Un_absorb2 subset_trans)
nipkow@26191
   336
nipkow@26191
   337
lemma antisym_init_seg_of:
nipkow@26191
   338
  "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r=s"
nipkow@26191
   339
by(auto simp:init_seg_of_def)
nipkow@26191
   340
nipkow@26191
   341
lemma Chain_init_seg_of_Union:
nipkow@26191
   342
  "R \<in> Chain init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
nipkow@26191
   343
by(auto simp add:init_seg_of_def Chain_def Ball_def) blast
nipkow@26191
   344
nipkow@26272
   345
lemma chain_subset_trans_Union:
nipkow@26272
   346
  "chain\<^bsub>\<subseteq>\<^esub> R \<Longrightarrow> \<forall>r\<in>R. trans r \<Longrightarrow> trans(\<Union>R)"
nipkow@26272
   347
apply(auto simp add:chain_subset_def)
nipkow@26191
   348
apply(simp (no_asm_use) add:trans_def)
nipkow@26191
   349
apply (metis subsetD)
nipkow@26191
   350
done
nipkow@26191
   351
nipkow@26272
   352
lemma chain_subset_antisym_Union:
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   353
  "chain\<^bsub>\<subseteq>\<^esub> R \<Longrightarrow> \<forall>r\<in>R. antisym r \<Longrightarrow> antisym(\<Union>R)"
nipkow@26272
   354
apply(auto simp add:chain_subset_def antisym_def)
nipkow@26191
   355
apply (metis subsetD)
nipkow@26191
   356
done
nipkow@26191
   357
nipkow@26272
   358
lemma chain_subset_Total_Union:
nipkow@26272
   359
assumes "chain\<^bsub>\<subseteq>\<^esub> R" "\<forall>r\<in>R. Total r"
nipkow@26191
   360
shows "Total (\<Union>R)"
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   361
proof (simp add: Total_def Ball_def, auto del:disjCI)
nipkow@26191
   362
  fix r s a b assume A: "r:R" "s:R" "a:Field r" "b:Field s" "a\<noteq>b"
nipkow@26272
   363
  from `chain\<^bsub>\<subseteq>\<^esub> R` `r:R` `s:R` have "r\<subseteq>s \<or> s\<subseteq>r"
nipkow@26272
   364
    by(simp add:chain_subset_def)
nipkow@26191
   365
  thus "(\<exists>r\<in>R. (a,b) \<in> r) \<or> (\<exists>r\<in>R. (b,a) \<in> r)"
nipkow@26191
   366
  proof
nipkow@26191
   367
    assume "r\<subseteq>s" hence "(a,b):s \<or> (b,a):s" using assms(2) A
nipkow@26191
   368
      by(simp add:Total_def)(metis mono_Field subsetD)
nipkow@26191
   369
    thus ?thesis using `s:R` by blast
nipkow@26191
   370
  next
nipkow@26191
   371
    assume "s\<subseteq>r" hence "(a,b):r \<or> (b,a):r" using assms(2) A
nipkow@26191
   372
      by(simp add:Total_def)(metis mono_Field subsetD)
nipkow@26191
   373
    thus ?thesis using `r:R` by blast
nipkow@26191
   374
  qed
nipkow@26191
   375
qed
nipkow@26191
   376
nipkow@26191
   377
lemma wf_Union_wf_init_segs:
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   378
assumes "R \<in> Chain init_seg_of" and "\<forall>r\<in>R. wf r" shows "wf(\<Union>R)"
nipkow@26191
   379
proof(simp add:wf_iff_no_infinite_down_chain, rule ccontr, auto)
nipkow@26191
   380
  fix f assume 1: "\<forall>i. \<exists>r\<in>R. (f(Suc i), f i) \<in> r"
nipkow@26191
   381
  then obtain r where "r:R" and "(f(Suc 0), f 0) : r" by auto
nipkow@26191
   382
  { fix i have "(f(Suc i), f i) \<in> r"
nipkow@26191
   383
    proof(induct i)
nipkow@26191
   384
      case 0 show ?case by fact
nipkow@26191
   385
    next
nipkow@26191
   386
      case (Suc i)
nipkow@26191
   387
      moreover obtain s where "s\<in>R" and "(f(Suc(Suc i)), f(Suc i)) \<in> s"
nipkow@26191
   388
	using 1 by auto
nipkow@26191
   389
      moreover hence "s initial_segment_of r \<or> r initial_segment_of s"
nipkow@26191
   390
	using assms(1) `r:R` by(simp add: Chain_def)
nipkow@26191
   391
      ultimately show ?case by(simp add:init_seg_of_def) blast
nipkow@26191
   392
    qed
nipkow@26191
   393
  }
nipkow@26191
   394
  thus False using assms(2) `r:R`
nipkow@26191
   395
    by(simp add:wf_iff_no_infinite_down_chain) blast
nipkow@26191
   396
qed
nipkow@26191
   397
nipkow@26191
   398
lemma Chain_inits_DiffI:
nipkow@26191
   399
  "R \<in> Chain init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chain init_seg_of"
nipkow@26191
   400
apply(auto simp:Chain_def init_seg_of_def)
nipkow@26191
   401
apply (metis subsetD)
nipkow@26191
   402
apply (metis subsetD)
nipkow@26191
   403
done
nipkow@26191
   404
nipkow@26272
   405
theorem well_ordering: "\<exists>r::('a*'a)set. Well_order r \<and> Field r = UNIV"
nipkow@26191
   406
proof-
nipkow@26191
   407
-- {*The initial segment relation on well-orders: *}
nipkow@26191
   408
  let ?WO = "{r::('a*'a)set. Well_order r}"
nipkow@26191
   409
  def I \<equiv> "init_seg_of \<inter> ?WO \<times> ?WO"
nipkow@26191
   410
  have I_init: "I \<subseteq> init_seg_of" by(auto simp:I_def)
nipkow@26272
   411
  hence subch: "!!R. R : Chain I \<Longrightarrow> chain\<^bsub>\<subseteq>\<^esub> R"
nipkow@26272
   412
    by(auto simp:init_seg_of_def chain_subset_def Chain_def)
nipkow@26191
   413
  have Chain_wo: "!!R r. R \<in> Chain I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
nipkow@26191
   414
    by(simp add:Chain_def I_def) blast
nipkow@26191
   415
  have FI: "Field I = ?WO" by(auto simp add:I_def init_seg_of_def Field_def)
nipkow@26191
   416
  hence 0: "Partial_order I"
nipkow@26191
   417
    by(auto simp add: Partial_order_def Preorder_def antisym_def antisym_init_seg_of Refl_def trans_def I_def)(metis trans_init_seg_of)
nipkow@26191
   418
-- {*I-chains have upper bounds in ?WO wrt I: their Union*}
nipkow@26191
   419
  { fix R assume "R \<in> Chain I"
nipkow@26191
   420
    hence Ris: "R \<in> Chain init_seg_of" using mono_Chain[OF I_init] by blast
nipkow@26272
   421
    have subch: "chain\<^bsub>\<subseteq>\<^esub> R" using `R : Chain I` I_init
nipkow@26272
   422
      by(auto simp:init_seg_of_def chain_subset_def Chain_def)
nipkow@26191
   423
    have "\<forall>r\<in>R. Refl r" "\<forall>r\<in>R. trans r" "\<forall>r\<in>R. antisym r" "\<forall>r\<in>R. Total r"
nipkow@26191
   424
         "\<forall>r\<in>R. wf(r-Id)"
nipkow@26191
   425
      using Chain_wo[OF `R \<in> Chain I`] by(simp_all add:Order_defs)
nipkow@26191
   426
    have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` by(auto simp:Refl_def)
nipkow@26191
   427
    moreover have "trans (\<Union>R)"
nipkow@26272
   428
      by(rule chain_subset_trans_Union[OF subch `\<forall>r\<in>R. trans r`])
nipkow@26191
   429
    moreover have "antisym(\<Union>R)"
nipkow@26272
   430
      by(rule chain_subset_antisym_Union[OF subch `\<forall>r\<in>R. antisym r`])
nipkow@26191
   431
    moreover have "Total (\<Union>R)"
nipkow@26272
   432
      by(rule chain_subset_Total_Union[OF subch `\<forall>r\<in>R. Total r`])
nipkow@26191
   433
    moreover have "wf((\<Union>R)-Id)"
nipkow@26191
   434
    proof-
nipkow@26191
   435
      have "(\<Union>R)-Id = \<Union>{r-Id|r. r \<in> R}" by blast
nipkow@26191
   436
      with `\<forall>r\<in>R. wf(r-Id)` wf_Union_wf_init_segs[OF Chain_inits_DiffI[OF Ris]]
nipkow@26191
   437
      show ?thesis by (simp (no_asm_simp)) blast
nipkow@26191
   438
    qed
nipkow@26191
   439
    ultimately have "Well_order (\<Union>R)" by(simp add:Order_defs)
nipkow@26191
   440
    moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R" using Ris
nipkow@26191
   441
      by(simp add: Chain_init_seg_of_Union)
nipkow@26191
   442
    ultimately have "\<Union>R : ?WO \<and> (\<forall>r\<in>R. (r,\<Union>R) : I)"
nipkow@26191
   443
      using mono_Chain[OF I_init] `R \<in> Chain I`
nipkow@26191
   444
      by(simp (no_asm) add:I_def del:Field_Union)(metis Chain_wo subsetD)
nipkow@26191
   445
  }
nipkow@26191
   446
  hence 1: "\<forall>R \<in> Chain I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r,u) : I" by (subst FI) blast
nipkow@26191
   447
--{*Zorn's Lemma yields a maximal well-order m:*}
nipkow@26191
   448
  then obtain m::"('a*'a)set" where "Well_order m" and
nipkow@26191
   449
    max: "\<forall>r. Well_order r \<and> (m,r):I \<longrightarrow> r=m"
nipkow@26191
   450
    using Zorns_po_lemma[OF 0 1] by (auto simp:FI)
nipkow@26191
   451
--{*Now show by contradiction that m covers the whole type:*}
nipkow@26191
   452
  { fix x::'a assume "x \<notin> Field m"
nipkow@26191
   453
--{*We assume that x is not covered and extend m at the top with x*}
nipkow@26191
   454
    have "m \<noteq> {}"
nipkow@26191
   455
    proof
nipkow@26191
   456
      assume "m={}"
nipkow@26191
   457
      moreover have "Well_order {(x,x)}"
nipkow@26191
   458
	by(simp add:Order_defs Refl_def trans_def antisym_def Total_def Field_def Domain_def Range_def)
nipkow@26191
   459
      ultimately show False using max
nipkow@26191
   460
	by (auto simp:I_def init_seg_of_def simp del:Field_insert)
nipkow@26191
   461
    qed
nipkow@26191
   462
    hence "Field m \<noteq> {}" by(auto simp:Field_def)
nipkow@26191
   463
    moreover have "wf(m-Id)" using `Well_order m` by(simp add:Well_order_def)
nipkow@26191
   464
--{*The extension of m by x:*}
nipkow@26191
   465
    let ?s = "{(a,x)|a. a : Field m}" let ?m = "insert (x,x) m Un ?s"
nipkow@26191
   466
    have Fm: "Field ?m = insert x (Field m)"
nipkow@26191
   467
      apply(simp add:Field_insert Field_Un)
nipkow@26191
   468
      unfolding Field_def by auto
nipkow@26191
   469
    have "Refl m" "trans m" "antisym m" "Total m" "wf(m-Id)"
nipkow@26191
   470
      using `Well_order m` by(simp_all add:Order_defs)
nipkow@26191
   471
--{*We show that the extension is a well-order*}
nipkow@26191
   472
    have "Refl ?m" using `Refl m` Fm by(auto simp:Refl_def)
nipkow@26191
   473
    moreover have "trans ?m" using `trans m` `x \<notin> Field m`
nipkow@26191
   474
      unfolding trans_def Field_def Domain_def Range_def by blast
nipkow@26191
   475
    moreover have "antisym ?m" using `antisym m` `x \<notin> Field m`
nipkow@26191
   476
      unfolding antisym_def Field_def Domain_def Range_def by blast
nipkow@26191
   477
    moreover have "Total ?m" using `Total m` Fm by(auto simp: Total_def)
nipkow@26191
   478
    moreover have "wf(?m-Id)"
nipkow@26191
   479
    proof-
nipkow@26191
   480
      have "wf ?s" using `x \<notin> Field m`
nipkow@26191
   481
	by(auto simp add:wf_eq_minimal Field_def Domain_def Range_def) metis
nipkow@26191
   482
      thus ?thesis using `wf(m-Id)` `x \<notin> Field m`
nipkow@26191
   483
	wf_subset[OF `wf ?s` Diff_subset]
nipkow@26191
   484
	by (fastsimp intro!: wf_Un simp add: Un_Diff Field_def)
nipkow@26191
   485
    qed
nipkow@26191
   486
    ultimately have "Well_order ?m" by(simp add:Order_defs)
nipkow@26191
   487
--{*We show that the extension is above m*}
nipkow@26191
   488
    moreover hence "(m,?m) : I" using `Well_order m` `x \<notin> Field m`
nipkow@26191
   489
      by(fastsimp simp:I_def init_seg_of_def Field_def Domain_def Range_def)
nipkow@26191
   490
    ultimately
nipkow@26191
   491
--{*This contradicts maximality of m:*}
nipkow@26191
   492
    have False using max `x \<notin> Field m` unfolding Field_def by blast
nipkow@26191
   493
  }
nipkow@26191
   494
  hence "Field m = UNIV" by auto
nipkow@26272
   495
  moreover with `Well_order m` have "Well_order m" by simp
nipkow@26272
   496
  ultimately show ?thesis by blast
nipkow@26272
   497
qed
nipkow@26272
   498
nipkow@26272
   499
corollary well_ordering_set: "\<exists>r::('a*'a)set. Well_order r \<and> Field r = A"
nipkow@26272
   500
proof -
nipkow@26272
   501
  obtain r::"('a*'a)set" where wo: "Well_order r" and univ: "Field r = UNIV"
nipkow@26272
   502
    using well_ordering[where 'a = "'a"] by blast
nipkow@26272
   503
  let ?r = "{(x,y). x:A & y:A & (x,y):r}"
nipkow@26272
   504
  have 1: "Field ?r = A" using wo univ
nipkow@26272
   505
    by(fastsimp simp: Field_def Domain_def Range_def Order_defs Refl_def)
nipkow@26272
   506
  have "Refl r" "trans r" "antisym r" "Total r" "wf(r-Id)"
nipkow@26272
   507
    using `Well_order r` by(simp_all add:Order_defs)
nipkow@26272
   508
  have "Refl ?r" using `Refl r` by(auto simp:Refl_def 1 univ)
nipkow@26272
   509
  moreover have "trans ?r" using `trans r`
nipkow@26272
   510
    unfolding trans_def by blast
nipkow@26272
   511
  moreover have "antisym ?r" using `antisym r`
nipkow@26272
   512
    unfolding antisym_def by blast
nipkow@26272
   513
  moreover have "Total ?r" using `Total r` by(simp add:Total_def 1 univ)
nipkow@26272
   514
  moreover have "wf(?r - Id)" by(rule wf_subset[OF `wf(r-Id)`]) blast
nipkow@26272
   515
  ultimately have "Well_order ?r" by(simp add:Order_defs)
nipkow@26272
   516
  with 1 show ?thesis by blast
nipkow@26191
   517
qed
nipkow@26191
   518
paulson@13551
   519
end