
wenzelm@14706: (*  Title       : HOL/Library/Zorn.thy
paulson@13652:     ID          : $Id$
paulson@13652:     Author      : Jacques D. Fleuriot
wenzelm@14706:     Description : Zorn's Lemma -- see Larry Paulson's Zorn.thy in ZF
wenzelm@14706: *)
paulson@13551: 
wenzelm@14706: header {* Zorn's Lemma *}
paulson@13551: 
nipkow@15131: theory Zorn
nipkow@15140: imports Main
nipkow@15131: begin
paulson@13551: 
wenzelm@14706: text{*
wenzelm@14706:   The lemma and section numbers refer to an unpublished article
wenzelm@14706:   \cite{Abrial-Laffitte}.
wenzelm@14706: *}
paulson@13551: 
wenzelm@19736: definition
wenzelm@21404:   chain     ::  "'a set set => 'a set set set" where
wenzelm@19736:   "chain S  = {F. F \<subseteq> S & (\<forall>x \<in> F. \<forall>y \<in> F. x \<subseteq> y | y \<subseteq> x)}"
paulson@13551: 
wenzelm@21404: definition
wenzelm@21404:   super     ::  "['a set set,'a set set] => 'a set set set" where
wenzelm@19736:   "super S c = {d. d \<in> chain S & c \<subset> d}"
paulson@13551: 
wenzelm@21404: definition
wenzelm@21404:   maxchain  ::  "'a set set => 'a set set set" where
wenzelm@19736:   "maxchain S = {c. c \<in> chain S & super S c = {}}"
paulson@13551: 
wenzelm@21404: definition
wenzelm@21404:   succ      ::  "['a set set,'a set set] => 'a set set" where
wenzelm@19736:   "succ S c =
wenzelm@19736:     (if c \<notin> chain S | c \<in> maxchain S
wenzelm@19736:     then c else SOME c'. c' \<in> super S c)"
paulson@13551: 
berghofe@23755: inductive_set
wenzelm@14706:   TFin :: "'a set set => 'a set set set"
berghofe@23755:   for S :: "'a set set"
berghofe@23755:   where
paulson@13551:     succI:        "x \<in> TFin S ==> succ S x \<in> TFin S"
berghofe@23755:   | Pow_UnionI:   "Y \<in> Pow(TFin S) ==> Union(Y) \<in> TFin S"
paulson@13551:   monos          Pow_mono
paulson@13551: 
paulson@13551: 
paulson@13551: subsection{*Mathematical Preamble*}
paulson@13551: 
wenzelm@17200: lemma Union_lemma0:
paulson@18143:     "(\<forall>x \<in> C. x \<subseteq> A | B \<subseteq> x) ==> Union(C) \<subseteq> A | B \<subseteq> Union(C)"
wenzelm@17200:   by blast
paulson@13551: 
paulson@13551: 
paulson@13551: text{*This is theorem @{text increasingD2} of ZF/Zorn.thy*}
wenzelm@17200: 
paulson@13551: lemma Abrial_axiom1: "x \<subseteq> succ S x"
wenzelm@17200:   apply (unfold succ_def)
wenzelm@17200:   apply (rule split_if [THEN iffD2])
wenzelm@17200:   apply (auto simp add: super_def maxchain_def psubset_def)
wenzelm@18585:   apply (rule contrapos_np, assumption)
wenzelm@17200:   apply (rule someI2, blast+)
wenzelm@17200:   done
paulson@13551: 
paulson@13551: lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]
paulson@13551: 
wenzelm@14706: lemma TFin_induct:
wenzelm@14706:           "[| n \<in> TFin S;
wenzelm@14706:              !!x. [| x \<in> TFin S; P(x) |] ==> P(succ S x);
wenzelm@14706:              !!Y. [| Y \<subseteq> TFin S; Ball Y P |] ==> P(Union Y) |]
paulson@13551:           ==> P(n)"
wenzelm@19736:   apply (induct set: TFin)
wenzelm@17200:    apply blast+
wenzelm@17200:   done
paulson@13551: 
paulson@13551: lemma succ_trans: "x \<subseteq> y ==> x \<subseteq> succ S y"
wenzelm@17200:   apply (erule subset_trans)
wenzelm@17200:   apply (rule Abrial_axiom1)
wenzelm@17200:   done
paulson@13551: 
paulson@13551: text{*Lemma 1 of section 3.1*}
paulson@13551: lemma TFin_linear_lemma1:
wenzelm@14706:      "[| n \<in> TFin S;  m \<in> TFin S;
wenzelm@14706:          \<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m
paulson@13551:       |] ==> n \<subseteq> m | succ S m \<subseteq> n"
wenzelm@17200:   apply (erule TFin_induct)
wenzelm@17200:    apply (erule_tac [2] Union_lemma0)
wenzelm@17200:   apply (blast del: subsetI intro: succ_trans)
wenzelm@17200:   done
paulson@13551: 
paulson@13551: text{* Lemma 2 of section 3.2 *}
paulson@13551: lemma TFin_linear_lemma2:
paulson@13551:      "m \<in> TFin S ==> \<forall>n \<in> TFin S. n \<subseteq> m --> n=m | succ S n \<subseteq> m"
wenzelm@17200:   apply (erule TFin_induct)
wenzelm@17200:    apply (rule impI [THEN ballI])
wenzelm@17200:    txt{*case split using @{text TFin_linear_lemma1}*}
wenzelm@17200:    apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
wenzelm@17200:      assumption+)
wenzelm@17200:     apply (drule_tac x = n in bspec, assumption)
wenzelm@17200:     apply (blast del: subsetI intro: succ_trans, blast)
wenzelm@17200:   txt{*second induction step*}
wenzelm@17200:   apply (rule impI [THEN ballI])
wenzelm@17200:   apply (rule Union_lemma0 [THEN disjE])
wenzelm@17200:     apply (rule_tac [3] disjI2)
wenzelm@17200:     prefer 2 apply blast
wenzelm@17200:    apply (rule ballI)
wenzelm@17200:    apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
wenzelm@17200:      assumption+, auto)
wenzelm@17200:   apply (blast intro!: Abrial_axiom1 [THEN subsetD])
wenzelm@17200:   done
paulson@13551: 
paulson@13551: text{*Re-ordering the premises of Lemma 2*}
paulson@13551: lemma TFin_subsetD:
paulson@13551:      "[| n \<subseteq> m;  m \<in> TFin S;  n \<in> TFin S |] ==> n=m | succ S n \<subseteq> m"
wenzelm@17200:   by (rule TFin_linear_lemma2 [rule_format])
paulson@13551: 
paulson@13551: text{*Consequences from section 3.3 -- Property 3.2, the ordering is total*}
paulson@13551: lemma TFin_subset_linear: "[| m \<in> TFin S;  n \<in> TFin S|] ==> n \<subseteq> m | m \<subseteq> n"
wenzelm@17200:   apply (rule disjE)
wenzelm@17200:     apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2])
wenzelm@17200:       apply (assumption+, erule disjI2)
wenzelm@17200:   apply (blast del: subsetI
wenzelm@17200:     intro: subsetI Abrial_axiom1 [THEN subset_trans])
wenzelm@17200:   done
paulson@13551: 
paulson@13551: text{*Lemma 3 of section 3.3*}
paulson@13551: lemma eq_succ_upper: "[| n \<in> TFin S;  m \<in> TFin S;  m = succ S m |] ==> n \<subseteq> m"
wenzelm@17200:   apply (erule TFin_induct)
wenzelm@17200:    apply (drule TFin_subsetD)
wenzelm@17200:      apply (assumption+, force, blast)
wenzelm@17200:   done
paulson@13551: 
paulson@13551: text{*Property 3.3 of section 3.3*}
paulson@13551: lemma equal_succ_Union: "m \<in> TFin S ==> (m = succ S m) = (m = Union(TFin S))"
wenzelm@17200:   apply (rule iffI)
wenzelm@17200:    apply (rule Union_upper [THEN equalityI])
paulson@18143:     apply assumption
paulson@18143:    apply (rule eq_succ_upper [THEN Union_least], assumption+)
wenzelm@17200:   apply (erule ssubst)
wenzelm@17200:   apply (rule Abrial_axiom1 [THEN equalityI])
wenzelm@17200:   apply (blast del: subsetI intro: subsetI TFin_UnionI TFin.succI)
wenzelm@17200:   done
paulson@13551: 
paulson@13551: subsection{*Hausdorff's Theorem: Every Set Contains a Maximal Chain.*}
paulson@13551: 
wenzelm@14706: text{*NB: We assume the partial ordering is @{text "\<subseteq>"},
paulson@13551:  the subset relation!*}
paulson@13551: 
paulson@13551: lemma empty_set_mem_chain: "({} :: 'a set set) \<in> chain S"
wenzelm@17200:   by (unfold chain_def) auto
paulson@13551: 
paulson@13551: lemma super_subset_chain: "super S c \<subseteq> chain S"
wenzelm@17200:   by (unfold super_def) blast
paulson@13551: 
paulson@13551: lemma maxchain_subset_chain: "maxchain S \<subseteq> chain S"
wenzelm@17200:   by (unfold maxchain_def) blast
paulson@13551: 
paulson@13551: lemma mem_super_Ex: "c \<in> chain S - maxchain S ==> ? d. d \<in> super S c"
wenzelm@17200:   by (unfold super_def maxchain_def) auto
paulson@13551: 
paulson@18143: lemma select_super:
paulson@18143:      "c \<in> chain S - maxchain S ==> (\<some>c'. c': super S c): super S c"
wenzelm@17200:   apply (erule mem_super_Ex [THEN exE])
wenzelm@17200:   apply (rule someI2, auto)
wenzelm@17200:   done
paulson@13551: 
paulson@18143: lemma select_not_equals:
paulson@18143:      "c \<in> chain S - maxchain S ==> (\<some>c'. c': super S c) \<noteq> c"
wenzelm@17200:   apply (rule notI)
wenzelm@17200:   apply (drule select_super)
wenzelm@17200:   apply (simp add: super_def psubset_def)
wenzelm@17200:   done
paulson@13551: 
wenzelm@17200: lemma succI3: "c \<in> chain S - maxchain S ==> succ S c = (\<some>c'. c': super S c)"
wenzelm@17200:   by (unfold succ_def) (blast intro!: if_not_P)
paulson@13551: 
paulson@13551: lemma succ_not_equals: "c \<in> chain S - maxchain S ==> succ S c \<noteq> c"
wenzelm@17200:   apply (frule succI3)
wenzelm@17200:   apply (simp (no_asm_simp))
wenzelm@17200:   apply (rule select_not_equals, assumption)
wenzelm@17200:   done
paulson@13551: 
paulson@13551: lemma TFin_chain_lemma4: "c \<in> TFin S ==> (c :: 'a set set): chain S"
wenzelm@17200:   apply (erule TFin_induct)
wenzelm@17200:    apply (simp add: succ_def select_super [THEN super_subset_chain[THEN subsetD]])
wenzelm@17200:   apply (unfold chain_def)
wenzelm@17200:   apply (rule CollectI, safe)
wenzelm@17200:    apply (drule bspec, assumption)
wenzelm@17200:    apply (rule_tac [2] m1 = Xa and n1 = X in TFin_subset_linear [THEN disjE],
wenzelm@17200:      blast+)
wenzelm@17200:   done
wenzelm@14706: 
paulson@13551: theorem Hausdorff: "\<exists>c. (c :: 'a set set): maxchain S"
paulson@18143:   apply (rule_tac x = "Union (TFin S)" in exI)
wenzelm@17200:   apply (rule classical)
wenzelm@17200:   apply (subgoal_tac "succ S (Union (TFin S)) = Union (TFin S) ")
wenzelm@17200:    prefer 2
wenzelm@17200:    apply (blast intro!: TFin_UnionI equal_succ_Union [THEN iffD2, symmetric])
wenzelm@17200:   apply (cut_tac subset_refl [THEN TFin_UnionI, THEN TFin_chain_lemma4])
wenzelm@17200:   apply (drule DiffI [THEN succ_not_equals], blast+)
wenzelm@17200:   done
paulson@13551: 
paulson@13551: 
wenzelm@14706: subsection{*Zorn's Lemma: If All Chains Have Upper Bounds Then
paulson@13551:                                There Is  a Maximal Element*}
paulson@13551: 
wenzelm@14706: lemma chain_extend:
wenzelm@14706:     "[| c \<in> chain S; z \<in> S;
paulson@18143:         \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chain S"
wenzelm@17200:   by (unfold chain_def) blast
paulson@13551: 
paulson@13551: lemma chain_Union_upper: "[| c \<in> chain S; x \<in> c |] ==> x \<subseteq> Union(c)"
wenzelm@17200:   by (unfold chain_def) auto
paulson@13551: 
paulson@13551: lemma chain_ball_Union_upper: "c \<in> chain S ==> \<forall>x \<in> c. x \<subseteq> Union(c)"
wenzelm@17200:   by (unfold chain_def) auto
paulson@13551: 
paulson@13551: lemma maxchain_Zorn:
paulson@13551:      "[| c \<in> maxchain S; u \<in> S; Union(c) \<subseteq> u |] ==> Union(c) = u"
wenzelm@17200:   apply (rule ccontr)
wenzelm@17200:   apply (simp add: maxchain_def)
wenzelm@17200:   apply (erule conjE)
paulson@18143:   apply (subgoal_tac "({u} Un c) \<in> super S c")
wenzelm@17200:    apply simp
wenzelm@17200:   apply (unfold super_def psubset_def)
wenzelm@17200:   apply (blast intro: chain_extend dest: chain_Union_upper)
wenzelm@17200:   done
paulson@13551: 
paulson@13551: theorem Zorn_Lemma:
wenzelm@17200:     "\<forall>c \<in> chain S. Union(c): S ==> \<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z --> y = z"
wenzelm@17200:   apply (cut_tac Hausdorff maxchain_subset_chain)
wenzelm@17200:   apply (erule exE)
wenzelm@17200:   apply (drule subsetD, assumption)
wenzelm@17200:   apply (drule bspec, assumption)
paulson@18143:   apply (rule_tac x = "Union(c)" in bexI)
wenzelm@17200:    apply (rule ballI, rule impI)
wenzelm@17200:    apply (blast dest!: maxchain_Zorn, assumption)
wenzelm@17200:   done
paulson@13551: 
paulson@13551: subsection{*Alternative version of Zorn's Lemma*}
paulson@13551: 
paulson@13551: lemma Zorn_Lemma2:
wenzelm@17200:   "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>x \<in> c. x \<subseteq> y
wenzelm@17200:     ==> \<exists>y \<in> S. \<forall>x \<in> S. (y :: 'a set) \<subseteq> x --> y = x"
wenzelm@17200:   apply (cut_tac Hausdorff maxchain_subset_chain)
wenzelm@17200:   apply (erule exE)
wenzelm@17200:   apply (drule subsetD, assumption)
wenzelm@17200:   apply (drule bspec, assumption, erule bexE)
wenzelm@17200:   apply (rule_tac x = y in bexI)
wenzelm@17200:    prefer 2 apply assumption
wenzelm@17200:   apply clarify
wenzelm@17200:   apply (rule ccontr)
wenzelm@17200:   apply (frule_tac z = x in chain_extend)
wenzelm@17200:     apply (assumption, blast)
wenzelm@17200:   apply (unfold maxchain_def super_def psubset_def)
wenzelm@17200:   apply (blast elim!: equalityCE)
wenzelm@17200:   done
paulson@13551: 
paulson@13551: text{*Various other lemmas*}
paulson@13551: 
paulson@13551: lemma chainD: "[| c \<in> chain S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
wenzelm@17200:   by (unfold chain_def) blast
paulson@13551: 
paulson@13551: lemma chainD2: "!!(c :: 'a set set). c \<in> chain S ==> c \<subseteq> S"
wenzelm@17200:   by (unfold chain_def) blast
paulson@13551: 
paulson@13551: end




