
wenzelm@32960: (*  Title:      HOL/Library/Zorn.thy
wenzelm@32960:     Author:     Jacques D. Fleuriot, Tobias Nipkow
wenzelm@32960: 
wenzelm@32960: Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF).
wenzelm@32960: The well-ordering theorem.
wenzelm@14706: *)
paulson@13551: 
wenzelm@14706: header {* Zorn's Lemma *}
paulson@13551: 
nipkow@15131: theory Zorn
blanchet@48750: imports Order_Relation
nipkow@15131: begin
paulson@13551: 
nipkow@26272: (* Define globally? In Set.thy? *)
haftmann@46980: definition chain_subset :: "'a set set \<Rightarrow> bool" ("chain\<^bsub>\<subseteq>\<^esub>")
haftmann@46980: where
haftmann@46980:   "chain\<^bsub>\<subseteq>\<^esub> C \<equiv> \<forall>A\<in>C.\<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A"
nipkow@26272: 
wenzelm@14706: text{*
wenzelm@14706:   The lemma and section numbers refer to an unpublished article
wenzelm@14706:   \cite{Abrial-Laffitte}.
wenzelm@14706: *}
paulson@13551: 
haftmann@46980: definition chain :: "'a set set \<Rightarrow> 'a set set set"
haftmann@46980: where
haftmann@46980:   "chain S = {F. F \<subseteq> S \<and> chain\<^bsub>\<subseteq>\<^esub> F}"
paulson@13551: 
haftmann@46980: definition super :: "'a set set \<Rightarrow> 'a set set \<Rightarrow> 'a set set set"
haftmann@46980: where
haftmann@46980:   "super S c = {d. d \<in> chain S \<and> c \<subset> d}"
paulson@13551: 
haftmann@46980: definition maxchain  ::  "'a set set \<Rightarrow> 'a set set set"
haftmann@46980: where
haftmann@46980:   "maxchain S = {c. c \<in> chain S \<and> super S c = {}}"
paulson@13551: 
haftmann@46980: definition succ :: "'a set set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
haftmann@46980: where
haftmann@46980:   "succ S c = (if c \<notin> chain S \<or> c \<in> maxchain S then c else SOME c'. c' \<in> super S c)"
haftmann@46980: 
haftmann@46980: inductive_set TFin :: "'a set set \<Rightarrow> 'a set set set"
haftmann@46980: for S :: "'a set set"
haftmann@46980: where
haftmann@46980:   succI:      "x \<in> TFin S \<Longrightarrow> succ S x \<in> TFin S"
haftmann@46980: | Pow_UnionI: "Y \<in> Pow (TFin S) \<Longrightarrow> \<Union>Y \<in> TFin S"
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"
berghofe@26806:   apply (auto simp add: succ_def super_def maxchain_def)
wenzelm@18585:   apply (rule contrapos_np, assumption)
wenzelm@46008:   apply (rule someI2)
wenzelm@46008:   apply blast+
wenzelm@17200:   done
paulson@13551: 
paulson@13551: lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]
paulson@13551: 
wenzelm@14706: lemma TFin_induct:
wenzelm@46008:   assumes H: "n \<in> TFin S" and
wenzelm@46008:     I: "!!x. x \<in> TFin S ==> P x ==> P (succ S x)"
wenzelm@46008:       "!!Y. Y \<subseteq> TFin S ==> Ball Y P ==> P (Union Y)"
wenzelm@46008:   shows "P n"
wenzelm@46008:   using H by induct (blast intro: I)+
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"
blanchet@48750:   by (unfold chain_def chain_subset_def) simp
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: 
nipkow@26191: lemma mem_super_Ex: "c \<in> chain S - maxchain S ==> EX d. d \<in> super S c"
blanchet@48750:   by (unfold super_def maxchain_def) simp
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@46008:   apply (rule someI2)
blanchet@48750:   apply simp+
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)
berghofe@26806:   apply (simp add: super_def less_le)
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]])
nipkow@26272:   apply (unfold chain_def chain_subset_def)
wenzelm@17200:   apply (rule CollectI, safe)
wenzelm@17200:    apply (drule bspec, assumption)
wenzelm@46008:    apply (rule_tac [2] m1 = Xa and n1 = X in TFin_subset_linear [THEN disjE])
wenzelm@46008:       apply 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:
nipkow@26272:   "[| c \<in> chain S; z \<in> S; \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chain S"
nipkow@26272: by (unfold chain_def chain_subset_def) blast
paulson@13551: 
paulson@13551: lemma chain_Union_upper: "[| c \<in> chain S; x \<in> c |] ==> x \<subseteq> Union(c)"
nipkow@26272: by auto
paulson@13551: 
paulson@13551: lemma chain_ball_Union_upper: "c \<in> chain S ==> \<forall>x \<in> c. x \<subseteq> Union(c)"
nipkow@26272: by auto
paulson@13551: 
paulson@13551: lemma maxchain_Zorn:
nipkow@26272:   "[| c \<in> maxchain S; u \<in> S; Union(c) \<subseteq> u |] ==> Union(c) = u"
nipkow@26272: apply (rule ccontr)
nipkow@26272: apply (simp add: maxchain_def)
nipkow@26272: apply (erule conjE)
nipkow@26272: apply (subgoal_tac "({u} Un c) \<in> super S c")
nipkow@26272:  apply simp
berghofe@26806: apply (unfold super_def less_le)
nipkow@26272: apply (blast intro: chain_extend dest: chain_Union_upper)
nipkow@26272: done
paulson@13551: 
paulson@13551: theorem Zorn_Lemma:
nipkow@26272:   "\<forall>c \<in> chain S. Union(c): S ==> \<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z --> y = z"
nipkow@26272: apply (cut_tac Hausdorff maxchain_subset_chain)
nipkow@26272: apply (erule exE)
nipkow@26272: apply (drule subsetD, assumption)
nipkow@26272: apply (drule bspec, assumption)
nipkow@26272: apply (rule_tac x = "Union(c)" in bexI)
nipkow@26272:  apply (rule ballI, rule impI)
nipkow@26272:  apply (blast dest!: maxchain_Zorn, assumption)
nipkow@26272: 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"
nipkow@26272: apply (cut_tac Hausdorff maxchain_subset_chain)
nipkow@26272: apply (erule exE)
nipkow@26272: apply (drule subsetD, assumption)
nipkow@26272: apply (drule bspec, assumption, erule bexE)
nipkow@26272: apply (rule_tac x = y in bexI)
nipkow@26272:  prefer 2 apply assumption
nipkow@26272: apply clarify
nipkow@26272: apply (rule ccontr)
nipkow@26272: apply (frule_tac z = x in chain_extend)
nipkow@26272:   apply (assumption, blast)
berghofe@26806: apply (unfold maxchain_def super_def less_le)
nipkow@26272: apply (blast elim!: equalityCE)
nipkow@26272: 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"
nipkow@26272: by (unfold chain_def chain_subset_def) blast
paulson@13551: 
paulson@13551: lemma chainD2: "!!(c :: 'a set set). c \<in> chain S ==> c \<subseteq> S"
nipkow@26272: by (unfold chain_def) blast
nipkow@26191: 
nipkow@26191: 
nipkow@26191: (* Define globally? In Relation.thy? *)
nipkow@26191: definition Chain :: "('a*'a)set \<Rightarrow> 'a set set" where
nipkow@26191: "Chain r \<equiv> {A. \<forall>a\<in>A.\<forall>b\<in>A. (a,b) : r \<or> (b,a) \<in> r}"
nipkow@26191: 
nipkow@26191: lemma mono_Chain: "r \<subseteq> s \<Longrightarrow> Chain r \<subseteq> Chain s"
nipkow@26191: unfolding Chain_def by blast
nipkow@26191: 
nipkow@26191: text{* Zorn's lemma for partial orders: *}
nipkow@26191: 
nipkow@26191: lemma Zorns_po_lemma:
nipkow@26191: assumes po: "Partial_order r" and u: "\<forall>C\<in>Chain r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a,u):r"
nipkow@26191: shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m,a):r \<longrightarrow> a=m"
nipkow@26191: proof-
nipkow@26295:   have "Preorder r" using po by(simp add:partial_order_on_def)
nipkow@26191: --{* Mirror r in the set of subsets below (wrt r) elements of A*}
nipkow@26191:   let ?B = "%x. r^-1 `` {x}" let ?S = "?B ` Field r"
nipkow@26191:   have "\<forall>C \<in> chain ?S. EX U:?S. ALL A:C. A\<subseteq>U"
nipkow@26272:   proof (auto simp:chain_def chain_subset_def)
nipkow@26191:     fix C assume 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C.\<forall>B\<in>C. A\<subseteq>B | B\<subseteq>A"
nipkow@26191:     let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
nipkow@26191:     have "C = ?B ` ?A" using 1 by(auto simp: image_def)
nipkow@26191:     have "?A\<in>Chain r"
nipkow@26191:     proof (simp add:Chain_def, intro allI impI, elim conjE)
nipkow@26191:       fix a b
nipkow@26191:       assume "a \<in> Field r" "?B a \<in> C" "b \<in> Field r" "?B b \<in> C"
blanchet@48750:       hence "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" using 2 by simp
nipkow@26191:       thus "(a, b) \<in> r \<or> (b, a) \<in> r" using `Preorder r` `a:Field r` `b:Field r`
wenzelm@32960:         by (simp add:subset_Image1_Image1_iff)
nipkow@26191:     qed
nipkow@26191:     then obtain u where uA: "u:Field r" "\<forall>a\<in>?A. (a,u) : r" using u by auto
nipkow@26191:     have "\<forall>A\<in>C. A \<subseteq> r^-1 `` {u}" (is "?P u")
nipkow@26191:     proof auto
nipkow@26191:       fix a B assume aB: "B:C" "a:B"
nipkow@26191:       with 1 obtain x where "x:Field r" "B = r^-1 `` {x}" by auto
nipkow@26191:       thus "(a,u) : r" using uA aB `Preorder r`
blanchet@48750:         by (simp add: preorder_on_def refl_on_def) (rule transD, blast+)
nipkow@26191:     qed
nipkow@26191:     thus "EX u:Field r. ?P u" using `u:Field r` by blast
nipkow@26191:   qed
nipkow@26191:   from Zorn_Lemma2[OF this]
nipkow@26191:   obtain m B where "m:Field r" "B = r^-1 `` {m}"
nipkow@26191:     "\<forall>x\<in>Field r. B \<subseteq> r^-1 `` {x} \<longrightarrow> B = r^-1 `` {x}"
ballarin@27064:     by auto
nipkow@26191:   hence "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m" using po `Preorder r` `m:Field r`
nipkow@26191:     by(auto simp:subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
nipkow@26191:   thus ?thesis using `m:Field r` by blast
nipkow@26191: qed
nipkow@26191: 
nipkow@26191: (* The initial segment of a relation appears generally useful.
nipkow@26191:    Move to Relation.thy?
nipkow@26191:    Definition correct/most general?
nipkow@26191:    Naming?
nipkow@26191: *)
nipkow@26191: definition init_seg_of :: "(('a*'a)set * ('a*'a)set)set" where
nipkow@26191: "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: 
nipkow@26191: abbreviation initialSegmentOf :: "('a*'a)set \<Rightarrow> ('a*'a)set \<Rightarrow> bool"
nipkow@26191:              (infix "initial'_segment'_of" 55) where
nipkow@26191: "r initial_segment_of s == (r,s):init_seg_of"
nipkow@26191: 
nipkow@30198: lemma refl_on_init_seg_of[simp]: "r initial_segment_of r"
nipkow@26191: by(simp add:init_seg_of_def)
nipkow@26191: 
nipkow@26191: lemma trans_init_seg_of:
nipkow@26191:   "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
blanchet@48750: by (simp (no_asm_use) add: init_seg_of_def) (metis (no_types) in_mono order_trans)
nipkow@26191: 
nipkow@26191: lemma antisym_init_seg_of:
nipkow@26191:   "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r=s"
huffman@35175: unfolding init_seg_of_def by safe
nipkow@26191: 
nipkow@26191: lemma Chain_init_seg_of_Union:
nipkow@26191:   "R \<in> Chain init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
blanchet@48750: by(simp add: init_seg_of_def Chain_def Ball_def) blast
nipkow@26191: 
nipkow@26272: lemma chain_subset_trans_Union:
nipkow@26272:   "chain\<^bsub>\<subseteq>\<^esub> R \<Longrightarrow> \<forall>r\<in>R. trans r \<Longrightarrow> trans(\<Union>R)"
blanchet@48750: apply(simp add:chain_subset_def)
nipkow@26191: apply(simp (no_asm_use) add:trans_def)
blanchet@48750: by (metis subsetD)
nipkow@26191: 
nipkow@26272: lemma chain_subset_antisym_Union:
nipkow@26272:   "chain\<^bsub>\<subseteq>\<^esub> R \<Longrightarrow> \<forall>r\<in>R. antisym r \<Longrightarrow> antisym(\<Union>R)"
blanchet@48750: by (simp add:chain_subset_def antisym_def) (metis subsetD)
nipkow@26191: 
nipkow@26272: lemma chain_subset_Total_Union:
nipkow@26272: assumes "chain\<^bsub>\<subseteq>\<^esub> R" "\<forall>r\<in>R. Total r"
nipkow@26191: shows "Total (\<Union>R)"
nipkow@26295: proof (simp add: total_on_def Ball_def, auto del:disjCI)
nipkow@26191:   fix r s a b assume A: "r:R" "s:R" "a:Field r" "b:Field s" "a\<noteq>b"
nipkow@26272:   from `chain\<^bsub>\<subseteq>\<^esub> R` `r:R` `s:R` have "r\<subseteq>s \<or> s\<subseteq>r"
nipkow@26272:     by(simp add:chain_subset_def)
nipkow@26191:   thus "(\<exists>r\<in>R. (a,b) \<in> r) \<or> (\<exists>r\<in>R. (b,a) \<in> r)"
nipkow@26191:   proof
nipkow@26191:     assume "r\<subseteq>s" hence "(a,b):s \<or> (b,a):s" using assms(2) A
nipkow@26295:       by(simp add:total_on_def)(metis mono_Field subsetD)
nipkow@26191:     thus ?thesis using `s:R` by blast
nipkow@26191:   next
nipkow@26191:     assume "s\<subseteq>r" hence "(a,b):r \<or> (b,a):r" using assms(2) A
nipkow@26295:       by(simp add:total_on_def)(metis mono_Field subsetD)
nipkow@26191:     thus ?thesis using `r:R` by blast
nipkow@26191:   qed
nipkow@26191: qed
nipkow@26191: 
nipkow@26191: lemma wf_Union_wf_init_segs:
nipkow@26191: assumes "R \<in> Chain init_seg_of" and "\<forall>r\<in>R. wf r" shows "wf(\<Union>R)"
nipkow@26191: proof(simp add:wf_iff_no_infinite_down_chain, rule ccontr, auto)
nipkow@26191:   fix f assume 1: "\<forall>i. \<exists>r\<in>R. (f(Suc i), f i) \<in> r"
nipkow@26191:   then obtain r where "r:R" and "(f(Suc 0), f 0) : r" by auto
nipkow@26191:   { fix i have "(f(Suc i), f i) \<in> r"
nipkow@26191:     proof(induct i)
nipkow@26191:       case 0 show ?case by fact
nipkow@26191:     next
nipkow@26191:       case (Suc i)
nipkow@26191:       moreover obtain s where "s\<in>R" and "(f(Suc(Suc i)), f(Suc i)) \<in> s"
wenzelm@32960:         using 1 by auto
nipkow@26191:       moreover hence "s initial_segment_of r \<or> r initial_segment_of s"
wenzelm@32960:         using assms(1) `r:R` by(simp add: Chain_def)
nipkow@26191:       ultimately show ?case by(simp add:init_seg_of_def) blast
nipkow@26191:     qed
nipkow@26191:   }
nipkow@26191:   thus False using assms(2) `r:R`
nipkow@26191:     by(simp add:wf_iff_no_infinite_down_chain) blast
nipkow@26191: qed
nipkow@26191: 
huffman@27476: lemma initial_segment_of_Diff:
huffman@27476:   "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s"
huffman@27476: unfolding init_seg_of_def by blast
huffman@27476: 
nipkow@26191: lemma Chain_inits_DiffI:
nipkow@26191:   "R \<in> Chain init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chain init_seg_of"
huffman@27476: unfolding Chain_def by (blast intro: initial_segment_of_Diff)
nipkow@26191: 
nipkow@26272: theorem well_ordering: "\<exists>r::('a*'a)set. Well_order r \<and> Field r = UNIV"
nipkow@26191: proof-
nipkow@26191: -- {*The initial segment relation on well-orders: *}
nipkow@26191:   let ?WO = "{r::('a*'a)set. Well_order r}"
nipkow@26191:   def I \<equiv> "init_seg_of \<inter> ?WO \<times> ?WO"
blanchet@48750:   have I_init: "I \<subseteq> init_seg_of" by(simp add: I_def)
nipkow@26272:   hence subch: "!!R. R : Chain I \<Longrightarrow> chain\<^bsub>\<subseteq>\<^esub> R"
nipkow@26272:     by(auto simp:init_seg_of_def chain_subset_def Chain_def)
nipkow@26191:   have Chain_wo: "!!R r. R \<in> Chain I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
nipkow@26191:     by(simp add:Chain_def I_def) blast
nipkow@26191:   have FI: "Field I = ?WO" by(auto simp add:I_def init_seg_of_def Field_def)
nipkow@26191:   hence 0: "Partial_order I"
nipkow@30198:     by(auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def trans_def I_def elim!: trans_init_seg_of)
nipkow@26191: -- {*I-chains have upper bounds in ?WO wrt I: their Union*}
nipkow@26191:   { fix R assume "R \<in> Chain I"
nipkow@26191:     hence Ris: "R \<in> Chain init_seg_of" using mono_Chain[OF I_init] by blast
nipkow@26272:     have subch: "chain\<^bsub>\<subseteq>\<^esub> R" using `R : Chain I` I_init
nipkow@26272:       by(auto simp:init_seg_of_def chain_subset_def Chain_def)
nipkow@26191:     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:          "\<forall>r\<in>R. wf(r-Id)"
nipkow@26295:       using Chain_wo[OF `R \<in> Chain I`] by(simp_all add:order_on_defs)
nipkow@30198:     have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` by(auto simp:refl_on_def)
nipkow@26191:     moreover have "trans (\<Union>R)"
nipkow@26272:       by(rule chain_subset_trans_Union[OF subch `\<forall>r\<in>R. trans r`])
nipkow@26191:     moreover have "antisym(\<Union>R)"
nipkow@26272:       by(rule chain_subset_antisym_Union[OF subch `\<forall>r\<in>R. antisym r`])
nipkow@26191:     moreover have "Total (\<Union>R)"
nipkow@26272:       by(rule chain_subset_Total_Union[OF subch `\<forall>r\<in>R. Total r`])
nipkow@26191:     moreover have "wf((\<Union>R)-Id)"
nipkow@26191:     proof-
nipkow@26191:       have "(\<Union>R)-Id = \<Union>{r-Id|r. r \<in> R}" by blast
nipkow@26191:       with `\<forall>r\<in>R. wf(r-Id)` wf_Union_wf_init_segs[OF Chain_inits_DiffI[OF Ris]]
nipkow@26191:       show ?thesis by (simp (no_asm_simp)) blast
nipkow@26191:     qed
nipkow@26295:     ultimately have "Well_order (\<Union>R)" by(simp add:order_on_defs)
nipkow@26191:     moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R" using Ris
nipkow@26191:       by(simp add: Chain_init_seg_of_Union)
nipkow@26191:     ultimately have "\<Union>R : ?WO \<and> (\<forall>r\<in>R. (r,\<Union>R) : I)"
nipkow@26191:       using mono_Chain[OF I_init] `R \<in> Chain I`
blanchet@48750:       by(simp (no_asm) add:I_def del:Field_Union)(metis Chain_wo)
nipkow@26191:   }
nipkow@26191:   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: --{*Zorn's Lemma yields a maximal well-order m:*}
nipkow@26191:   then obtain m::"('a*'a)set" where "Well_order m" and
nipkow@26191:     max: "\<forall>r. Well_order r \<and> (m,r):I \<longrightarrow> r=m"
nipkow@26191:     using Zorns_po_lemma[OF 0 1] by (auto simp:FI)
nipkow@26191: --{*Now show by contradiction that m covers the whole type:*}
nipkow@26191:   { fix x::'a assume "x \<notin> Field m"
nipkow@26191: --{*We assume that x is not covered and extend m at the top with x*}
nipkow@26191:     have "m \<noteq> {}"
nipkow@26191:     proof
nipkow@26191:       assume "m={}"
nipkow@26191:       moreover have "Well_order {(x,x)}"
haftmann@46752:         by(simp add:order_on_defs refl_on_def trans_def antisym_def total_on_def Field_def Domain_unfold Domain_converse [symmetric])
nipkow@26191:       ultimately show False using max
wenzelm@32960:         by (auto simp:I_def init_seg_of_def simp del:Field_insert)
nipkow@26191:     qed
nipkow@26191:     hence "Field m \<noteq> {}" by(auto simp:Field_def)
nipkow@26295:     moreover have "wf(m-Id)" using `Well_order m`
nipkow@26295:       by(simp add:well_order_on_def)
nipkow@26191: --{*The extension of m by x:*}
nipkow@26191:     let ?s = "{(a,x)|a. a : Field m}" let ?m = "insert (x,x) m Un ?s"
nipkow@26191:     have Fm: "Field ?m = insert x (Field m)"
nipkow@26191:       apply(simp add:Field_insert Field_Un)
nipkow@26191:       unfolding Field_def by auto
nipkow@26191:     have "Refl m" "trans m" "antisym m" "Total m" "wf(m-Id)"
nipkow@26295:       using `Well_order m` by(simp_all add:order_on_defs)
nipkow@26191: --{*We show that the extension is a well-order*}
nipkow@30198:     have "Refl ?m" using `Refl m` Fm by(auto simp:refl_on_def)
nipkow@26191:     moreover have "trans ?m" using `trans m` `x \<notin> Field m`
haftmann@46752:       unfolding trans_def Field_def Domain_unfold Domain_converse [symmetric] by blast
nipkow@26191:     moreover have "antisym ?m" using `antisym m` `x \<notin> Field m`
haftmann@46752:       unfolding antisym_def Field_def Domain_unfold Domain_converse [symmetric] by blast
nipkow@26295:     moreover have "Total ?m" using `Total m` Fm by(auto simp: total_on_def)
nipkow@26191:     moreover have "wf(?m-Id)"
nipkow@26191:     proof-
nipkow@26191:       have "wf ?s" using `x \<notin> Field m`
blanchet@48750:         by(simp add:wf_eq_minimal Field_def Domain_unfold Domain_converse [symmetric]) metis
nipkow@26191:       thus ?thesis using `wf(m-Id)` `x \<notin> Field m`
wenzelm@32960:         wf_subset[OF `wf ?s` Diff_subset]
nipkow@44890:         by (fastforce intro!: wf_Un simp add: Un_Diff Field_def)
nipkow@26191:     qed
nipkow@26295:     ultimately have "Well_order ?m" by(simp add:order_on_defs)
nipkow@26191: --{*We show that the extension is above m*}
nipkow@26191:     moreover hence "(m,?m) : I" using `Well_order m` `x \<notin> Field m`
haftmann@46752:       by(fastforce simp:I_def init_seg_of_def Field_def Domain_unfold Domain_converse [symmetric])
nipkow@26191:     ultimately
nipkow@26191: --{*This contradicts maximality of m:*}
nipkow@26191:     have False using max `x \<notin> Field m` unfolding Field_def by blast
nipkow@26191:   }
nipkow@26191:   hence "Field m = UNIV" by auto
nipkow@26272:   moreover with `Well_order m` have "Well_order m" by simp
nipkow@26272:   ultimately show ?thesis by blast
nipkow@26272: qed
nipkow@26272: 
nipkow@26295: corollary well_order_on: "\<exists>r::('a*'a)set. well_order_on A r"
nipkow@26272: proof -
nipkow@26272:   obtain r::"('a*'a)set" where wo: "Well_order r" and univ: "Field r = UNIV"
nipkow@26272:     using well_ordering[where 'a = "'a"] by blast
nipkow@26272:   let ?r = "{(x,y). x:A & y:A & (x,y):r}"
nipkow@26272:   have 1: "Field ?r = A" using wo univ
haftmann@46752:     by(fastforce simp: Field_def Domain_unfold Domain_converse [symmetric] order_on_defs refl_on_def)
nipkow@26272:   have "Refl r" "trans r" "antisym r" "Total r" "wf(r-Id)"
nipkow@26295:     using `Well_order r` by(simp_all add:order_on_defs)
nipkow@30198:   have "Refl ?r" using `Refl r` by(auto simp:refl_on_def 1 univ)
nipkow@26272:   moreover have "trans ?r" using `trans r`
nipkow@26272:     unfolding trans_def by blast
nipkow@26272:   moreover have "antisym ?r" using `antisym r`
nipkow@26272:     unfolding antisym_def by blast
nipkow@26295:   moreover have "Total ?r" using `Total r` by(simp add:total_on_def 1 univ)
nipkow@26272:   moreover have "wf(?r - Id)" by(rule wf_subset[OF `wf(r-Id)`]) blast
nipkow@26295:   ultimately have "Well_order ?r" by(simp add:order_on_defs)
nipkow@26295:   with 1 show ?thesis by metis
nipkow@26191: qed
nipkow@26191: 
paulson@13551: end




