src/HOL/Library/Zorn.thy
 changeset 52199 d25fc4c0ff62 parent 52183 667961fa6a60 child 52821 05eb2d77b195
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52197:20071aef2a3b 52199:d25fc4c0ff62
     3     Author:     Tobias Nipkow, TUM
     3     Author:     Tobias Nipkow, TUM
     4     Author:     Christian Sternagel, JAIST
     4     Author:     Christian Sternagel, JAIST
     5
     5
     6 Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF).
     6 Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF).
     7 The well-ordering theorem.
     7 The well-ordering theorem.

     8 The extension of any well-founded relation to a well-order.
     8 *)
     9 *)
     9
    10
    10 header {* Zorn's Lemma *}
    11 header {* Zorn's Lemma *}
    11
    12
    12 theory Zorn
    13 theory Zorn
    13 imports Order_Relation
    14 imports Order_Union
    14 begin
    15 begin
    15
    16
    16 subsection {* Zorn's Lemma for the Subset Relation *}
    17 subsection {* Zorn's Lemma for the Subset Relation *}
    17
    18
    18 subsubsection {* Results that do not require an order *}
    19 subsubsection {* Results that do not require an order *}
   710   moreover have "wf (?r - Id)" by (rule wf_subset [OF wf (r - Id)]) blast
   711   moreover have "wf (?r - Id)" by (rule wf_subset [OF wf (r - Id)]) blast
   711   ultimately have "Well_order ?r" by (simp add: order_on_defs)
   712   ultimately have "Well_order ?r" by (simp add: order_on_defs)
   712   with 1 show ?thesis by metis
   713   with 1 show ?thesis by metis
   713 qed
   714 qed
   714
   715

   716 subsection {* Extending Well-founded Relations to Well-Orders *}

   717

   718 text {*A \emph{downset} (also lower set, decreasing set, initial segment, or

   719 downward closed set) is closed w.r.t.\ smaller elements.*}

   720 definition downset_on where

   721   "downset_on A r = (\<forall>x y. (x, y) \<in> r \<and> y \<in> A \<longrightarrow> x \<in> A)"

   722

   723 (*

   724 text {*Connection to order filters of the @{theory Cardinals} theory.*}

   725 lemma (in wo_rel) ofilter_downset_on_conv:

   726   "ofilter A \<longleftrightarrow> downset_on A r \<and> A \<subseteq> Field r"

   727   by (auto simp: downset_on_def ofilter_def under_def)

   728 *)

   729

   730 lemma downset_onI:

   731   "(\<And>x y. (x, y) \<in> r \<Longrightarrow> y \<in> A \<Longrightarrow> x \<in> A) \<Longrightarrow> downset_on A r"

   732   by (auto simp: downset_on_def)

   733

   734 lemma downset_onD:

   735   "downset_on A r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> y \<in> A \<Longrightarrow> x \<in> A"

   736   by (auto simp: downset_on_def)

   737

   738 text {*Extensions of relations w.r.t.\ a given set.*}

   739 definition extension_on where

   740   "extension_on A r s = (\<forall>x\<in>A. \<forall>y\<in>A. (x, y) \<in> s \<longrightarrow> (x, y) \<in> r)"

   741

   742 lemma extension_onI:

   743   "(\<And>x y. \<lbrakk>x \<in> A; y \<in> A; (x, y) \<in> s\<rbrakk> \<Longrightarrow> (x, y) \<in> r) \<Longrightarrow> extension_on A r s"

   744   by (auto simp: extension_on_def)

   745

   746 lemma extension_onD:

   747   "extension_on A r s \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> (x, y) \<in> s \<Longrightarrow> (x, y) \<in> r"

   748   by (auto simp: extension_on_def)

   749

   750 lemma downset_on_Union:

   751   assumes "\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p"

   752   shows "downset_on (Field (\<Union>R)) p"

   753   using assms by (auto intro: downset_onI dest: downset_onD)

   754

   755 lemma chain_subset_extension_on_Union:

   756   assumes "chain\<^sub>\<subseteq> R" and "\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p"

   757   shows "extension_on (Field (\<Union>R)) (\<Union>R) p"

   758   using assms

   759   by (simp add: chain_subset_def extension_on_def)

   760      (metis Field_def mono_Field set_mp)

   761

   762 lemma downset_on_empty [simp]: "downset_on {} p"

   763   by (auto simp: downset_on_def)

   764

   765 lemma extension_on_empty [simp]: "extension_on {} p q"

   766   by (auto simp: extension_on_def)

   767

   768 text {*Every well-founded relation can be extended to a well-order.*}

   769 theorem well_order_extension:

   770   assumes "wf p"

   771   shows "\<exists>w. p \<subseteq> w \<and> Well_order w"

   772 proof -

   773   let ?K = "{r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p}"

   774   def I \<equiv> "init_seg_of \<inter> ?K \<times> ?K"

   775   have I_init: "I \<subseteq> init_seg_of" by (simp add: I_def)

   776   then have subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"

   777     by (auto simp: init_seg_of_def chain_subset_def Chains_def)

   778   have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow>

   779       Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p"

   780     by (simp add: Chains_def I_def) blast

   781   have FI: "Field I = ?K" by (auto simp: I_def init_seg_of_def Field_def)

   782   then have 0: "Partial_order I"

   783     by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def

   784       trans_def I_def elim: trans_init_seg_of)

   785   { fix R assume "R \<in> Chains I"

   786     then have Ris: "R \<in> Chains init_seg_of" using mono_Chains [OF I_init] by blast

   787     have subch: "chain\<^sub>\<subseteq> R" using R \<in> Chains I I_init

   788       by (auto simp: init_seg_of_def chain_subset_def Chains_def)

   789     have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r" and

   790       "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)" and

   791       "\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p" and

   792       "\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p"

   793       using Chains_wo [OF R \<in> Chains I] by (simp_all add: order_on_defs)

   794     have "Refl (\<Union>R)" using \<forall>r\<in>R. Refl r by (auto simp: refl_on_def)

   795     moreover have "trans (\<Union>R)"

   796       by (rule chain_subset_trans_Union [OF subch \<forall>r\<in>R. trans r])

   797     moreover have "antisym (\<Union>R)"

   798       by (rule chain_subset_antisym_Union [OF subch \<forall>r\<in>R. antisym r])

   799     moreover have "Total (\<Union>R)"

   800       by (rule chain_subset_Total_Union [OF subch \<forall>r\<in>R. Total r])

   801     moreover have "wf ((\<Union>R) - Id)"

   802     proof -

   803       have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast

   804       with \<forall>r\<in>R. wf (r - Id) wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]

   805       show ?thesis by (simp (no_asm_simp)) blast

   806     qed

   807     ultimately have "Well_order (\<Union>R)" by (simp add: order_on_defs)

   808     moreover have "\<forall>r\<in>R. r initial_segment_of \<Union>R" using Ris

   809       by (simp add: Chains_init_seg_of_Union)

   810     moreover have "downset_on (Field (\<Union>R)) p"

   811       by (rule downset_on_Union [OF \<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p])

   812     moreover have "extension_on (Field (\<Union>R)) (\<Union>R) p"

   813       by (rule chain_subset_extension_on_Union [OF subch \<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p])

   814     ultimately have "\<Union>R \<in> ?K \<and> (\<forall>r\<in>R. (r,\<Union>R) \<in> I)"

   815       using mono_Chains [OF I_init] and R \<in> Chains I

   816       by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo)

   817   }

   818   then have 1: "\<forall>R\<in>Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" by (subst FI) blast

   819   txt {*Zorn's Lemma yields a maximal well-order m.*}

   820   from Zorns_po_lemma [OF 0 1] obtain m :: "('a \<times> 'a) set"

   821     where "Well_order m" and "downset_on (Field m) p" and "extension_on (Field m) m p" and

   822     max: "\<forall>r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p \<and>

   823       (m, r) \<in> I \<longrightarrow> r = m"

   824     by (auto simp: FI)

   825   have "Field p \<subseteq> Field m"

   826   proof (rule ccontr)

   827     let ?Q = "Field p - Field m"

   828     assume "\<not> (Field p \<subseteq> Field m)"

   829     with assms [unfolded wf_eq_minimal, THEN spec, of ?Q]

   830       obtain x where "x \<in> Field p" and "x \<notin> Field m" and

   831       min: "\<forall>y. (y, x) \<in> p \<longrightarrow> y \<notin> ?Q" by blast

   832     txt {*Add @{term x} as topmost element to @{term m}.*}

   833     let ?s = "{(y, x) | y. y \<in> Field m}"

   834     let ?m = "insert (x, x) m \<union> ?s"

   835     have Fm: "Field ?m = insert x (Field m)" by (auto simp: Field_def)

   836     have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"

   837       using Well_order m by (simp_all add: order_on_defs)

   838     txt {*We show that the extension is a well-order.*}

   839     have "Refl ?m" using Refl m Fm by (auto simp: refl_on_def)

   840     moreover have "trans ?m" using trans m x \<notin> Field m

   841       unfolding trans_def Field_def Domain_unfold Domain_converse [symmetric] by blast

   842     moreover have "antisym ?m" using antisym m x \<notin> Field m

   843       unfolding antisym_def Field_def Domain_unfold Domain_converse [symmetric] by blast

   844     moreover have "Total ?m" using Total m Fm by (auto simp: Relation.total_on_def)

   845     moreover have "wf (?m - Id)"

   846     proof -

   847       have "wf ?s" using x \<notin> Field m

   848         by (simp add: wf_eq_minimal Field_def Domain_unfold Domain_converse [symmetric]) metis

   849       thus ?thesis using wf (m - Id) x \<notin> Field m

   850         wf_subset [OF wf ?s Diff_subset]

   851         by (fastforce intro!: wf_Un simp add: Un_Diff Field_def)

   852     qed

   853     ultimately have "Well_order ?m" by (simp add: order_on_defs)

   854     moreover have "extension_on (Field ?m) ?m p"

   855       using extension_on (Field m) m p downset_on (Field m) p

   856       by (subst Fm) (auto simp: extension_on_def dest: downset_onD)

   857     moreover have "downset_on (Field ?m) p"

   858       using downset_on (Field m) p and min

   859       by (subst Fm, simp add: downset_on_def Field_def) (metis Domain_iff)

   860     moreover have "(m, ?m) \<in> I"

   861       using Well_order m and Well_order ?m and

   862       downset_on (Field m) p and downset_on (Field ?m) p and

   863       extension_on (Field m) m p and extension_on (Field ?m) ?m p and

   864       Refl m and x \<notin> Field m

   865       by (auto simp: I_def init_seg_of_def refl_on_def)

   866     ultimately

   867     --{*This contradicts maximality of m:*}

   868     show False using max and x \<notin> Field m unfolding Field_def by blast

   869   qed

   870   have "p \<subseteq> m"

   871     using Field p \<subseteq> Field m and extension_on (Field m) m p

   872     by (force simp: Field_def extension_on_def)

   873   with Well_order m show ?thesis by blast

   874 qed

   875

   876 text {*Every well-founded relation can be extended to a total well-order.*}

   877 corollary total_well_order_extension:

   878   assumes "wf p"

   879   shows "\<exists>w. p \<subseteq> w \<and> Well_order w \<and> Field w = UNIV"

   880 proof -

   881   from well_order_extension [OF assms] obtain w

   882     where "p \<subseteq> w" and wo: "Well_order w" by blast

   883   let ?A = "UNIV - Field w"

   884   from well_order_on [of ?A] obtain w' where wo': "well_order_on ?A w'" ..

   885   have [simp]: "Field w' = ?A" using rel.well_order_on_Well_order [OF wo'] by simp

   886   have *: "Field w \<inter> Field w' = {}" by simp

   887   let ?w = "w \<union>o w'"

   888   have "p \<subseteq> ?w" using p \<subseteq> w by (auto simp: Osum_def)

   889   moreover have "Well_order ?w" using Osum_Well_order [OF * wo] and wo' by simp

   890   moreover have "Field ?w = UNIV" by (simp add: Field_Osum)

   891   ultimately show ?thesis by blast

   892 qed

   893

   894 corollary well_order_on_extension:

   895   assumes "wf p" and "Field p \<subseteq> A"

   896   shows "\<exists>w. p \<subseteq> w \<and> well_order_on A w"

   897 proof -

   898   from total_well_order_extension [OF wf p] obtain r

   899     where "p \<subseteq> r" and wo: "Well_order r" and univ: "Field r = UNIV" by blast

   900   let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"

   901   from p \<subseteq> r have "p \<subseteq> ?r" using Field p \<subseteq> A by (auto simp: Field_def)

   902   have 1: "Field ?r = A" using wo univ

   903     by (fastforce simp: Field_def order_on_defs refl_on_def)

   904   have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"

   905     using Well_order r by (simp_all add: order_on_defs)

   906   have "refl_on A ?r" using Refl r by (auto simp: refl_on_def univ)

   907   moreover have "trans ?r" using trans r

   908     unfolding trans_def by blast

   909   moreover have "antisym ?r" using antisym r

   910     unfolding antisym_def by blast

   911   moreover have "total_on A ?r" using Total r by (simp add: total_on_def univ)

   912   moreover have "wf (?r - Id)" by (rule wf_subset [OF wf(r - Id)]) blast

   913   ultimately have "well_order_on A ?r" by (simp add: order_on_defs)

   914   with p \<subseteq> ?r show ?thesis by blast

   915 qed

   916
   715 end
   917 end
   716
   918