src/HOL/Library/Zorn.thy
changeset 52199 d25fc4c0ff62
parent 52183 667961fa6a60
child 52821 05eb2d77b195
equal deleted inserted replaced
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