src/ZF/Zorn.thy
author wenzelm
Thu Sep 02 00:48:07 2010 +0200 (2010-09-02)
changeset 38980 af73cf0dc31f
parent 32960 69916a850301
child 45602 2a858377c3d2
permissions -rw-r--r--
turned show_question_marks into proper configuration option;
show_question_marks only affects regular type/term pretty printing, not raw Term.string_of_vname;
tuned;
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(*  Title:      ZF/Zorn.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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*)
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header{*Zorn's Lemma*}
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theory Zorn imports OrderArith AC Inductive_ZF begin
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text{*Based upon the unpublished article ``Towards the Mechanization of the
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Proofs of Some Classical Theorems of Set Theory,'' by Abrial and Laffitte.*}
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definition
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  Subset_rel :: "i=>i"  where
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   "Subset_rel(A) == {z \<in> A*A . \<exists>x y. z=<x,y> & x<=y & x\<noteq>y}"
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definition
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  chain      :: "i=>i"  where
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   "chain(A)      == {F \<in> Pow(A). \<forall>X\<in>F. \<forall>Y\<in>F. X<=Y | Y<=X}"
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definition
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  super      :: "[i,i]=>i"  where
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   "super(A,c)    == {d \<in> chain(A). c<=d & c\<noteq>d}"
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definition
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  maxchain   :: "i=>i"  where
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   "maxchain(A)   == {c \<in> chain(A). super(A,c)=0}"
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definition
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  increasing :: "i=>i"  where
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    "increasing(A) == {f \<in> Pow(A)->Pow(A). \<forall>x. x<=A --> x<=f`x}"
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text{*Lemma for the inductive definition below*}
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lemma Union_in_Pow: "Y \<in> Pow(Pow(A)) ==> Union(Y) \<in> Pow(A)"
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by blast
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text{*We could make the inductive definition conditional on
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    @{term "next \<in> increasing(S)"}
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    but instead we make this a side-condition of an introduction rule.  Thus
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    the induction rule lets us assume that condition!  Many inductive proofs
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    are therefore unconditional.*}
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consts
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  "TFin" :: "[i,i]=>i"
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inductive
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  domains       "TFin(S,next)" <= "Pow(S)"
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  intros
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    nextI:       "[| x \<in> TFin(S,next);  next \<in> increasing(S) |]
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                  ==> next`x \<in> TFin(S,next)"
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    Pow_UnionI: "Y \<in> Pow(TFin(S,next)) ==> Union(Y) \<in> TFin(S,next)"
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  monos         Pow_mono
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  con_defs      increasing_def
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  type_intros   CollectD1 [THEN apply_funtype] Union_in_Pow
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subsection{*Mathematical Preamble *}
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lemma Union_lemma0: "(\<forall>x\<in>C. x<=A | B<=x) ==> Union(C)<=A | B<=Union(C)"
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by blast
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lemma Inter_lemma0:
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     "[| c \<in> C; \<forall>x\<in>C. A<=x | x<=B |] ==> A <= Inter(C) | Inter(C) <= B"
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by blast
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subsection{*The Transfinite Construction *}
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lemma increasingD1: "f \<in> increasing(A) ==> f \<in> Pow(A)->Pow(A)"
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apply (unfold increasing_def)
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apply (erule CollectD1)
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done
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lemma increasingD2: "[| f \<in> increasing(A); x<=A |] ==> x <= f`x"
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by (unfold increasing_def, blast)
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lemmas TFin_UnionI = PowI [THEN TFin.Pow_UnionI, standard]
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lemmas TFin_is_subset = TFin.dom_subset [THEN subsetD, THEN PowD, standard]
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text{*Structural induction on @{term "TFin(S,next)"} *}
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lemma TFin_induct:
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  "[| n \<in> TFin(S,next);
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      !!x. [| x \<in> TFin(S,next);  P(x);  next \<in> increasing(S) |] ==> P(next`x);
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      !!Y. [| Y <= TFin(S,next);  \<forall>y\<in>Y. P(y) |] ==> P(Union(Y))
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   |] ==> P(n)"
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by (erule TFin.induct, blast+)
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subsection{*Some Properties of the Transfinite Construction *}
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lemmas increasing_trans = subset_trans [OF _ increasingD2,
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                                        OF _ _ TFin_is_subset]
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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,next);  m \<in> TFin(S,next);
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         \<forall>x \<in> TFin(S,next) . x<=m --> x=m | next`x<=m |]
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      ==> n<=m | next`m<=n"
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apply (erule TFin_induct)
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apply (erule_tac [2] Union_lemma0) (*or just Blast_tac*)
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(*downgrade subsetI from intro! to intro*)
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apply (blast dest: increasing_trans)
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done
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text{*Lemma 2 of section 3.2.  Interesting in its own right!
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  Requires @{term "next \<in> increasing(S)"} in the second induction step.*}
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lemma TFin_linear_lemma2:
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    "[| m \<in> TFin(S,next);  next \<in> increasing(S) |]
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     ==> \<forall>n \<in> TFin(S,next). n<=m --> n=m | next`n <= 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 (blast del: subsetI
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             intro: increasing_trans subsetI, 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 (erule_tac [3] disjI2)
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prefer 2 apply blast
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apply (rule ballI)
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apply (drule bspec, assumption)
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apply (drule subsetD, assumption)
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apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
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       assumption+, blast)
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apply (erule increasingD2 [THEN subset_trans, THEN disjI1])
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apply (blast dest: TFin_is_subset)+
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done
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text{*a more convenient form for Lemma 2*}
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lemma TFin_subsetD:
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     "[| n<=m;  m \<in> TFin(S,next);  n \<in> TFin(S,next);  next \<in> increasing(S) |]
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      ==> n=m | next`n <= m"
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by (blast dest: 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:
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     "[| m \<in> TFin(S,next);  n \<in> TFin(S,next);  next \<in> increasing(S) |]
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      ==> n <= m | m<=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 increasingD2 [THEN subset_trans] TFin_is_subset)
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done
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text{*Lemma 3 of section 3.3*}
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lemma equal_next_upper:
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     "[| n \<in> TFin(S,next);  m \<in> TFin(S,next);  m = next`m |] ==> n <= 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_next_Union:
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     "[| m \<in> TFin(S,next);  next \<in> increasing(S) |]
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      ==> m = next`m <-> m = Union(TFin(S,next))"
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apply (rule iffI)
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apply (rule Union_upper [THEN equalityI])
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apply (rule_tac [2] equal_next_upper [THEN Union_least])
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apply (assumption+)
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apply (erule ssubst)
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apply (rule increasingD2 [THEN equalityI], assumption)
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apply (blast del: subsetI
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             intro: subsetI TFin_UnionI TFin.nextI TFin_is_subset)+
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done
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subsection{*Hausdorff's Theorem: Every Set Contains a Maximal Chain*}
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text{*NOTE: We assume the partial ordering is @{text "\<subseteq>"}, the subset
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relation!*}
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text{** Defining the "next" operation for Hausdorff's Theorem **}
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lemma chain_subset_Pow: "chain(A) <= Pow(A)"
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apply (unfold chain_def)
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apply (rule Collect_subset)
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done
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lemma super_subset_chain: "super(A,c) <= chain(A)"
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apply (unfold super_def)
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apply (rule Collect_subset)
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done
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lemma maxchain_subset_chain: "maxchain(A) <= chain(A)"
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apply (unfold maxchain_def)
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apply (rule Collect_subset)
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done
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lemma choice_super:
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     "[| ch \<in> (\<Pi> X \<in> Pow(chain(S)) - {0}. X); X \<in> chain(S);  X \<notin> maxchain(S) |]
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      ==> ch ` super(S,X) \<in> super(S,X)"
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apply (erule apply_type)
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apply (unfold super_def maxchain_def, blast)
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done
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lemma choice_not_equals:
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     "[| ch \<in> (\<Pi> X \<in> Pow(chain(S)) - {0}. X); X \<in> chain(S);  X \<notin> maxchain(S) |]
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      ==> ch ` super(S,X) \<noteq> X"
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apply (rule notI)
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apply (drule choice_super, assumption, assumption)
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apply (simp add: super_def)
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done
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text{*This justifies Definition 4.4*}
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lemma Hausdorff_next_exists:
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     "ch \<in> (\<Pi> X \<in> Pow(chain(S))-{0}. X) ==>
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      \<exists>next \<in> increasing(S). \<forall>X \<in> Pow(S).
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                   next`X = if(X \<in> chain(S)-maxchain(S), ch`super(S,X), X)"
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apply (rule_tac x="\<lambda>X\<in>Pow(S).
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                   if X \<in> chain(S) - maxchain(S) then ch ` super(S, X) else X"
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       in bexI)
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apply force
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apply (unfold increasing_def)
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apply (rule CollectI)
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apply (rule lam_type)
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apply (simp (no_asm_simp))
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apply (blast dest: super_subset_chain [THEN subsetD] 
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                   chain_subset_Pow [THEN subsetD] choice_super)
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txt{*Now, verify that it increases*}
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apply (simp (no_asm_simp) add: Pow_iff subset_refl)
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apply safe
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apply (drule choice_super)
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apply (assumption+)
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apply (simp add: super_def, blast)
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done
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text{*Lemma 4*}
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lemma TFin_chain_lemma4:
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     "[| c \<in> TFin(S,next);
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         ch \<in> (\<Pi> X \<in> Pow(chain(S))-{0}. X);
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         next \<in> increasing(S);
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         \<forall>X \<in> Pow(S). next`X =
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                          if(X \<in> chain(S)-maxchain(S), ch`super(S,X), X) |]
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     ==> c \<in> chain(S)"
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apply (erule TFin_induct)
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apply (simp (no_asm_simp) add: chain_subset_Pow [THEN subsetD, THEN PowD]
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            choice_super [THEN super_subset_chain [THEN subsetD]])
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apply (unfold chain_def)
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apply (rule CollectI, blast, safe)
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apply (rule_tac m1=B and n1=Ba in TFin_subset_linear [THEN disjE], fast+)
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      txt{*@{text "Blast_tac's"} slow*}
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done
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theorem Hausdorff: "\<exists>c. c \<in> maxchain(S)"
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apply (rule AC_Pi_Pow [THEN exE])
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apply (rule Hausdorff_next_exists [THEN bexE], assumption)
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apply (rename_tac ch "next")
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apply (subgoal_tac "Union (TFin (S,next)) \<in> chain (S) ")
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prefer 2
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 apply (blast intro!: TFin_chain_lemma4 subset_refl [THEN TFin_UnionI])
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apply (rule_tac x = "Union (TFin (S,next))" in exI)
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apply (rule classical)
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apply (subgoal_tac "next ` Union (TFin (S,next)) = Union (TFin (S,next))")
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apply (rule_tac [2] equal_next_Union [THEN iffD2, symmetric])
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apply (rule_tac [2] subset_refl [THEN TFin_UnionI])
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prefer 2 apply assumption
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apply (rule_tac [2] refl)
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apply (simp add: subset_refl [THEN TFin_UnionI,
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                              THEN TFin.dom_subset [THEN subsetD, THEN PowD]])
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apply (erule choice_not_equals [THEN notE])
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apply (assumption+)
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done
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subsection{*Zorn's Lemma: If All Chains in S Have Upper Bounds In S,
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       then S contains a Maximal Element*}
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text{*Used in the proof of Zorn's Lemma*}
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lemma chain_extend:
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    "[| c \<in> chain(A);  z \<in> A;  \<forall>x \<in> c. x<=z |] ==> cons(z,c) \<in> chain(A)"
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by (unfold chain_def, blast)
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lemma Zorn: "\<forall>c \<in> chain(S). Union(c) \<in> S ==> \<exists>y \<in> S. \<forall>z \<in> S. y<=z --> y=z"
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apply (rule Hausdorff [THEN exE])
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apply (simp add: maxchain_def)
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apply (rename_tac c)
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apply (rule_tac x = "Union (c)" in bexI)
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prefer 2 apply blast
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apply safe
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apply (rename_tac z)
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apply (rule classical)
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apply (subgoal_tac "cons (z,c) \<in> super (S,c) ")
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apply (blast elim: equalityE)
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apply (unfold super_def, safe)
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apply (fast elim: chain_extend)
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apply (fast elim: equalityE)
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done
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text {* Alternative version of Zorn's Lemma *}
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theorem Zorn2:
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  "\<forall>c \<in> chain(S). \<exists>y \<in> S. \<forall>x \<in> c. x <= y ==> \<exists>y \<in> S. \<forall>z \<in> S. y<=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, 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 apply assumption
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apply rule
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apply (rule ccontr)
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apply (frule_tac z=z in chain_extend)
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  apply (assumption, blast)
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apply (unfold maxchain_def super_def)
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apply (blast elim!: equalityCE)
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done
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subsection{*Zermelo's Theorem: Every Set can be Well-Ordered*}
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text{*Lemma 5*}
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lemma TFin_well_lemma5:
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   324
     "[| n \<in> TFin(S,next);  Z <= TFin(S,next);  z:Z;  ~ Inter(Z) \<in> Z |]
paulson@13558
   325
      ==> \<forall>m \<in> Z. n <= m"
paulson@13134
   326
apply (erule TFin_induct)
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   327
prefer 2 apply blast txt{*second induction step is easy*}
paulson@13134
   328
apply (rule ballI)
paulson@13558
   329
apply (rule bspec [THEN TFin_subsetD, THEN disjE], auto)
paulson@13134
   330
apply (subgoal_tac "m = Inter (Z) ")
paulson@13134
   331
apply blast+
paulson@13134
   332
done
paulson@13134
   333
paulson@13558
   334
text{*Well-ordering of @{term "TFin(S,next)"} *}
paulson@13558
   335
lemma well_ord_TFin_lemma: "[| Z <= TFin(S,next);  z \<in> Z |] ==> Inter(Z) \<in> Z"
paulson@13134
   336
apply (rule classical)
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   337
apply (subgoal_tac "Z = {Union (TFin (S,next))}")
paulson@13134
   338
apply (simp (no_asm_simp) add: Inter_singleton)
paulson@13134
   339
apply (erule equal_singleton)
paulson@13134
   340
apply (rule Union_upper [THEN equalityI])
paulson@13269
   341
apply (rule_tac [2] subset_refl [THEN TFin_UnionI, THEN TFin_well_lemma5, THEN bspec], blast+)
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   342
done
paulson@13134
   343
paulson@13558
   344
text{*This theorem just packages the previous result*}
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   345
lemma well_ord_TFin:
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   346
     "next \<in> increasing(S) 
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   347
      ==> well_ord(TFin(S,next), Subset_rel(TFin(S,next)))"
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   348
apply (rule well_ordI)
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   349
apply (unfold Subset_rel_def linear_def)
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   350
txt{*Prove the well-foundedness goal*}
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   351
apply (rule wf_onI)
paulson@13269
   352
apply (frule well_ord_TFin_lemma, assumption)
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   353
apply (drule_tac x = "Inter (Z) " in bspec, assumption)
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   354
apply blast
paulson@13558
   355
txt{*Now prove the linearity goal*}
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   356
apply (intro ballI)
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   357
apply (case_tac "x=y")
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   358
 apply blast
paulson@13558
   359
txt{*The @{term "x\<noteq>y"} case remains*}
paulson@13134
   360
apply (rule_tac n1=x and m1=y in TFin_subset_linear [THEN disjE],
paulson@13269
   361
       assumption+, blast+)
paulson@13134
   362
done
paulson@13134
   363
paulson@13558
   364
text{** Defining the "next" operation for Zermelo's Theorem **}
paulson@13134
   365
paulson@13134
   366
lemma choice_Diff:
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   367
     "[| ch \<in> (\<Pi> X \<in> Pow(S) - {0}. X);  X \<subseteq> S;  X\<noteq>S |] ==> ch ` (S-X) \<in> S-X"
paulson@13134
   368
apply (erule apply_type)
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   369
apply (blast elim!: equalityE)
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   370
done
paulson@13134
   371
paulson@13558
   372
text{*This justifies Definition 6.1*}
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   373
lemma Zermelo_next_exists:
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   374
     "ch \<in> (\<Pi> X \<in> Pow(S)-{0}. X) ==>
paulson@13558
   375
           \<exists>next \<in> increasing(S). \<forall>X \<in> Pow(S).
paulson@13175
   376
                      next`X = (if X=S then S else cons(ch`(S-X), X))"
paulson@13175
   377
apply (rule_tac x="\<lambda>X\<in>Pow(S). if X=S then S else cons(ch`(S-X), X)"
paulson@13175
   378
       in bexI)
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   379
apply force
paulson@13134
   380
apply (unfold increasing_def)
paulson@13134
   381
apply (rule CollectI)
paulson@13134
   382
apply (rule lam_type)
paulson@13558
   383
txt{*Type checking is surprisingly hard!*}
paulson@13134
   384
apply (simp (no_asm_simp) add: Pow_iff cons_subset_iff subset_refl)
paulson@13134
   385
apply (blast intro!: choice_Diff [THEN DiffD1])
paulson@13558
   386
txt{*Verify that it increases*}
paulson@13558
   387
apply (intro allI impI)
paulson@13134
   388
apply (simp add: Pow_iff subset_consI subset_refl)
paulson@13134
   389
done
paulson@13134
   390
paulson@13134
   391
paulson@13558
   392
text{*The construction of the injection*}
paulson@13134
   393
lemma choice_imp_injection:
skalberg@14171
   394
     "[| ch \<in> (\<Pi> X \<in> Pow(S)-{0}. X);
paulson@13558
   395
         next \<in> increasing(S);
paulson@13558
   396
         \<forall>X \<in> Pow(S). next`X = if(X=S, S, cons(ch`(S-X), X)) |]
paulson@13558
   397
      ==> (\<lambda> x \<in> S. Union({y \<in> TFin(S,next). x \<notin> y}))
paulson@13558
   398
               \<in> inj(S, TFin(S,next) - {S})"
paulson@13134
   399
apply (rule_tac d = "%y. ch` (S-y) " in lam_injective)
paulson@13134
   400
apply (rule DiffI)
paulson@13134
   401
apply (rule Collect_subset [THEN TFin_UnionI])
paulson@13134
   402
apply (blast intro!: Collect_subset [THEN TFin_UnionI] elim: equalityE)
paulson@13558
   403
apply (subgoal_tac "x \<notin> Union ({y \<in> TFin (S,next) . x \<notin> y}) ")
paulson@13134
   404
prefer 2 apply (blast elim: equalityE)
paulson@13558
   405
apply (subgoal_tac "Union ({y \<in> TFin (S,next) . x \<notin> y}) \<noteq> S")
paulson@13134
   406
prefer 2 apply (blast elim: equalityE)
paulson@13558
   407
txt{*For proving @{text "x \<in> next`Union(...)"}.
paulson@13558
   408
  Abrial and Laffitte's justification appears to be faulty.*}
paulson@13558
   409
apply (subgoal_tac "~ next ` Union ({y \<in> TFin (S,next) . x \<notin> y}) 
paulson@13558
   410
                    <= Union ({y \<in> TFin (S,next) . x \<notin> y}) ")
paulson@13558
   411
 prefer 2
paulson@13558
   412
 apply (simp del: Union_iff
wenzelm@32960
   413
             add: Collect_subset [THEN TFin_UnionI, THEN TFin_is_subset]
wenzelm@32960
   414
             Pow_iff cons_subset_iff subset_refl choice_Diff [THEN DiffD2])
paulson@13558
   415
apply (subgoal_tac "x \<in> next ` Union ({y \<in> TFin (S,next) . x \<notin> y}) ")
paulson@13558
   416
 prefer 2
paulson@13558
   417
 apply (blast intro!: Collect_subset [THEN TFin_UnionI] TFin.nextI)
paulson@13558
   418
txt{*End of the lemmas!*}
paulson@13134
   419
apply (simp add: Collect_subset [THEN TFin_UnionI, THEN TFin_is_subset])
paulson@13134
   420
done
paulson@13134
   421
paulson@13558
   422
text{*The wellordering theorem*}
paulson@13558
   423
theorem AC_well_ord: "\<exists>r. well_ord(S,r)"
paulson@13134
   424
apply (rule AC_Pi_Pow [THEN exE])
paulson@13269
   425
apply (rule Zermelo_next_exists [THEN bexE], assumption)
paulson@13134
   426
apply (rule exI)
paulson@13134
   427
apply (rule well_ord_rvimage)
paulson@13134
   428
apply (erule_tac [2] well_ord_TFin)
paulson@13269
   429
apply (rule choice_imp_injection [THEN inj_weaken_type], blast+)
paulson@13134
   430
done
paulson@13558
   431
ballarin@27704
   432
ballarin@27704
   433
subsection {* Zorn's Lemma for Partial Orders *}
ballarin@27704
   434
ballarin@27704
   435
text {* Reimported from HOL by Clemens Ballarin. *}
ballarin@27704
   436
ballarin@27704
   437
ballarin@27704
   438
definition Chain :: "i => i" where
ballarin@27704
   439
  "Chain(r) = {A : Pow(field(r)). ALL a:A. ALL b:A. <a, b> : r | <b, a> : r}"
ballarin@27704
   440
ballarin@27704
   441
lemma mono_Chain:
ballarin@27704
   442
  "r \<subseteq> s ==> Chain(r) \<subseteq> Chain(s)"
ballarin@27704
   443
  unfolding Chain_def
ballarin@27704
   444
  by blast
ballarin@27704
   445
ballarin@27704
   446
theorem Zorn_po:
ballarin@27704
   447
  assumes po: "Partial_order(r)"
ballarin@27704
   448
    and u: "ALL C:Chain(r). EX u:field(r). ALL a:C. <a, u> : r"
ballarin@27704
   449
  shows "EX m:field(r). ALL a:field(r). <m, a> : r --> a = m"
ballarin@27704
   450
proof -
ballarin@27704
   451
  have "Preorder(r)" using po by (simp add: partial_order_on_def)
ballarin@27704
   452
  --{* Mirror r in the set of subsets below (wrt r) elements of A (?). *}
ballarin@27704
   453
  let ?B = "lam x:field(r). r -`` {x}" let ?S = "?B `` field(r)"
ballarin@27704
   454
  have "ALL C:chain(?S). EX U:?S. ALL A:C. A \<subseteq> U"
ballarin@27704
   455
  proof (clarsimp simp: chain_def Subset_rel_def bex_image_simp)
ballarin@27704
   456
    fix C
ballarin@27704
   457
    assume 1: "C \<subseteq> ?S" and 2: "ALL A:C. ALL B:C. A \<subseteq> B | B \<subseteq> A"
ballarin@27704
   458
    let ?A = "{x : field(r). EX M:C. M = ?B`x}"
ballarin@27704
   459
    have "C = ?B `` ?A" using 1
ballarin@27704
   460
      apply (auto simp: image_def)
ballarin@27704
   461
      apply rule
ballarin@27704
   462
      apply rule
ballarin@27704
   463
      apply (drule subsetD) apply assumption
ballarin@27704
   464
      apply (erule CollectE)
ballarin@27704
   465
      apply rule apply assumption
ballarin@27704
   466
      apply (erule bexE)
ballarin@27704
   467
      apply rule prefer 2 apply assumption
ballarin@27704
   468
      apply rule
ballarin@27704
   469
      apply (erule lamE) apply simp
ballarin@27704
   470
      apply assumption
ballarin@27704
   471
ballarin@27704
   472
      apply (thin_tac "C \<subseteq> ?X")
ballarin@27704
   473
      apply (fast elim: lamE)
ballarin@27704
   474
      done
ballarin@27704
   475
    have "?A : Chain(r)"
ballarin@27704
   476
    proof (simp add: Chain_def subsetI, intro conjI ballI impI)
ballarin@27704
   477
      fix a b
ballarin@27704
   478
      assume "a : field(r)" "r -`` {a} : C" "b : field(r)" "r -`` {b} : C"
ballarin@27704
   479
      hence "r -`` {a} \<subseteq> r -`` {b} | r -`` {b} \<subseteq> r -`` {a}" using 2 by auto
ballarin@27704
   480
      then show "<a, b> : r | <b, a> : r"
wenzelm@32960
   481
        using `Preorder(r)` `a : field(r)` `b : field(r)`
wenzelm@32960
   482
        by (simp add: subset_vimage1_vimage1_iff)
ballarin@27704
   483
    qed
ballarin@27704
   484
    then obtain u where uA: "u : field(r)" "ALL a:?A. <a, u> : r"
ballarin@27704
   485
      using u
ballarin@27704
   486
      apply auto
ballarin@27704
   487
      apply (drule bspec) apply assumption
ballarin@27704
   488
      apply auto
ballarin@27704
   489
      done
ballarin@27704
   490
    have "ALL A:C. A \<subseteq> r -`` {u}"
ballarin@27704
   491
    proof (auto intro!: vimageI)
ballarin@27704
   492
      fix a B
ballarin@27704
   493
      assume aB: "B : C" "a : B"
ballarin@27704
   494
      with 1 obtain x where "x : field(r)" "B = r -`` {x}"
wenzelm@32960
   495
        apply -
wenzelm@32960
   496
        apply (drule subsetD) apply assumption
wenzelm@32960
   497
        apply (erule imageE)
wenzelm@32960
   498
        apply (erule lamE)
wenzelm@32960
   499
        apply simp
wenzelm@32960
   500
        done
ballarin@27704
   501
      then show "<a, u> : r" using uA aB `Preorder(r)`
wenzelm@32960
   502
        by (auto simp: preorder_on_def refl_def) (blast dest: trans_onD)+
ballarin@27704
   503
    qed
ballarin@27704
   504
    then show "EX U:field(r). ALL A:C. A \<subseteq> r -`` {U}"
ballarin@27704
   505
      using `u : field(r)` ..
ballarin@27704
   506
  qed
ballarin@27704
   507
  from Zorn2 [OF this]
ballarin@27704
   508
  obtain m B where "m : field(r)" "B = r -`` {m}"
ballarin@27704
   509
    "ALL x:field(r). B \<subseteq> r -`` {x} --> B = r -`` {x}"
ballarin@27704
   510
    by (auto elim!: lamE simp: ball_image_simp)
ballarin@27704
   511
  then have "ALL a:field(r). <m, a> : r --> a = m"
ballarin@27704
   512
    using po `Preorder(r)` `m : field(r)`
ballarin@27704
   513
    by (auto simp: subset_vimage1_vimage1_iff Partial_order_eq_vimage1_vimage1_iff)
ballarin@27704
   514
  then show ?thesis using `m : field(r)` by blast
ballarin@27704
   515
qed
ballarin@27704
   516
lcp@516
   517
end