--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Wellorder_Extension.thy Wed Nov 20 23:14:06 2013 +0100
@@ -0,0 +1,213 @@
+(* Title: HOL/Cardinals/Wellorder_Extension.thy
+ Author: Christian Sternagel, JAIST
+*)
+
+header {* Extending Well-founded Relations to Wellorders *}
+
+theory Wellorder_Extension
+imports "~~/src/HOL/Library/Zorn" Order_Union
+begin
+
+subsection {* Extending Well-founded Relations to Wellorders *}
+
+text {*A \emph{downset} (also lower set, decreasing set, initial segment, or
+downward closed set) is closed w.r.t.\ smaller elements.*}
+definition downset_on where
+ "downset_on A r = (\<forall>x y. (x, y) \<in> r \<and> y \<in> A \<longrightarrow> x \<in> A)"
+
+(*
+text {*Connection to order filters of the @{theory Cardinals} theory.*}
+lemma (in wo_rel) ofilter_downset_on_conv:
+ "ofilter A \<longleftrightarrow> downset_on A r \<and> A \<subseteq> Field r"
+ by (auto simp: downset_on_def ofilter_def under_def)
+*)
+
+lemma downset_onI:
+ "(\<And>x y. (x, y) \<in> r \<Longrightarrow> y \<in> A \<Longrightarrow> x \<in> A) \<Longrightarrow> downset_on A r"
+ by (auto simp: downset_on_def)
+
+lemma downset_onD:
+ "downset_on A r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> y \<in> A \<Longrightarrow> x \<in> A"
+ unfolding downset_on_def by blast
+
+text {*Extensions of relations w.r.t.\ a given set.*}
+definition extension_on where
+ "extension_on A r s = (\<forall>x\<in>A. \<forall>y\<in>A. (x, y) \<in> s \<longrightarrow> (x, y) \<in> r)"
+
+lemma extension_onI:
+ "(\<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"
+ by (auto simp: extension_on_def)
+
+lemma extension_onD:
+ "extension_on A r s \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> (x, y) \<in> s \<Longrightarrow> (x, y) \<in> r"
+ by (auto simp: extension_on_def)
+
+lemma downset_on_Union:
+ assumes "\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p"
+ shows "downset_on (Field (\<Union>R)) p"
+ using assms by (auto intro: downset_onI dest: downset_onD)
+
+lemma chain_subset_extension_on_Union:
+ assumes "chain\<^sub>\<subseteq> R" and "\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p"
+ shows "extension_on (Field (\<Union>R)) (\<Union>R) p"
+ using assms
+ by (simp add: chain_subset_def extension_on_def)
+ (metis (no_types) mono_Field set_mp)
+
+lemma downset_on_empty [simp]: "downset_on {} p"
+ by (auto simp: downset_on_def)
+
+lemma extension_on_empty [simp]: "extension_on {} p q"
+ by (auto simp: extension_on_def)
+
+text {*Every well-founded relation can be extended to a wellorder.*}
+theorem well_order_extension:
+ assumes "wf p"
+ shows "\<exists>w. p \<subseteq> w \<and> Well_order w"
+proof -
+ let ?K = "{r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p}"
+ def I \<equiv> "init_seg_of \<inter> ?K \<times> ?K"
+ have I_init: "I \<subseteq> init_seg_of" by (simp add: I_def)
+ then have subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
+ by (auto simp: init_seg_of_def chain_subset_def Chains_def)
+ have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow>
+ Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p"
+ by (simp add: Chains_def I_def) blast
+ have FI: "Field I = ?K" by (auto simp: I_def init_seg_of_def Field_def)
+ then have 0: "Partial_order I"
+ 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)
+ { fix R assume "R \<in> Chains I"
+ then have Ris: "R \<in> Chains init_seg_of" using mono_Chains [OF I_init] by blast
+ have subch: "chain\<^sub>\<subseteq> R" using `R \<in> Chains I` I_init
+ by (auto simp: init_seg_of_def chain_subset_def Chains_def)
+ have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r" and
+ "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)" and
+ "\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p" and
+ "\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p"
+ using Chains_wo [OF `R \<in> Chains I`] by (simp_all add: order_on_defs)
+ have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` unfolding refl_on_def by fastforce
+ moreover have "trans (\<Union>R)"
+ by (rule chain_subset_trans_Union [OF subch `\<forall>r\<in>R. trans r`])
+ moreover have "antisym (\<Union>R)"
+ by (rule chain_subset_antisym_Union [OF subch `\<forall>r\<in>R. antisym r`])
+ moreover have "Total (\<Union>R)"
+ by (rule chain_subset_Total_Union [OF subch `\<forall>r\<in>R. Total r`])
+ moreover have "wf ((\<Union>R) - Id)"
+ proof -
+ have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
+ with `\<forall>r\<in>R. wf (r - Id)` wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
+ show ?thesis by fastforce
+ qed
+ ultimately have "Well_order (\<Union>R)" by (simp add: order_on_defs)
+ moreover have "\<forall>r\<in>R. r initial_segment_of \<Union>R" using Ris
+ by (simp add: Chains_init_seg_of_Union)
+ moreover have "downset_on (Field (\<Union>R)) p"
+ by (rule downset_on_Union [OF `\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p`])
+ moreover have "extension_on (Field (\<Union>R)) (\<Union>R) p"
+ by (rule chain_subset_extension_on_Union [OF subch `\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p`])
+ ultimately have "\<Union>R \<in> ?K \<and> (\<forall>r\<in>R. (r,\<Union>R) \<in> I)"
+ using mono_Chains [OF I_init] and `R \<in> Chains I`
+ by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo)
+ }
+ 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
+ txt {*Zorn's Lemma yields a maximal wellorder m.*}
+ from Zorns_po_lemma [OF 0 1] obtain m :: "('a \<times> 'a) set"
+ where "Well_order m" and "downset_on (Field m) p" and "extension_on (Field m) m p" and
+ max: "\<forall>r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p \<and>
+ (m, r) \<in> I \<longrightarrow> r = m"
+ by (auto simp: FI)
+ have "Field p \<subseteq> Field m"
+ proof (rule ccontr)
+ let ?Q = "Field p - Field m"
+ assume "\<not> (Field p \<subseteq> Field m)"
+ with assms [unfolded wf_eq_minimal, THEN spec, of ?Q]
+ obtain x where "x \<in> Field p" and "x \<notin> Field m" and
+ min: "\<forall>y. (y, x) \<in> p \<longrightarrow> y \<notin> ?Q" by blast
+ txt {*Add @{term x} as topmost element to @{term m}.*}
+ let ?s = "{(y, x) | y. y \<in> Field m}"
+ let ?m = "insert (x, x) m \<union> ?s"
+ have Fm: "Field ?m = insert x (Field m)" by (auto simp: Field_def)
+ have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
+ using `Well_order m` by (simp_all add: order_on_defs)
+ txt {*We show that the extension is a wellorder.*}
+ have "Refl ?m" using `Refl m` Fm by (auto simp: refl_on_def)
+ moreover have "trans ?m" using `trans m` `x \<notin> Field m`
+ unfolding trans_def Field_def Domain_unfold Domain_converse [symmetric] by blast
+ moreover have "antisym ?m" using `antisym m` `x \<notin> Field m`
+ unfolding antisym_def Field_def Domain_unfold Domain_converse [symmetric] by blast
+ moreover have "Total ?m" using `Total m` Fm by (auto simp: Relation.total_on_def)
+ moreover have "wf (?m - Id)"
+ proof -
+ have "wf ?s" using `x \<notin> Field m`
+ by (simp add: wf_eq_minimal Field_def Domain_unfold Domain_converse [symmetric]) metis
+ thus ?thesis using `wf (m - Id)` `x \<notin> Field m`
+ wf_subset [OF `wf ?s` Diff_subset]
+ by (fastforce intro!: wf_Un simp add: Un_Diff Field_def)
+ qed
+ ultimately have "Well_order ?m" by (simp add: order_on_defs)
+ moreover have "extension_on (Field ?m) ?m p"
+ using `extension_on (Field m) m p` `downset_on (Field m) p`
+ by (subst Fm) (auto simp: extension_on_def dest: downset_onD)
+ moreover have "downset_on (Field ?m) p"
+ apply (subst Fm)
+ using `downset_on (Field m) p` and min
+ unfolding downset_on_def Field_def by blast
+ moreover have "(m, ?m) \<in> I"
+ using `Well_order m` and `Well_order ?m` and
+ `downset_on (Field m) p` and `downset_on (Field ?m) p` and
+ `extension_on (Field m) m p` and `extension_on (Field ?m) ?m p` and
+ `Refl m` and `x \<notin> Field m`
+ by (auto simp: I_def init_seg_of_def refl_on_def)
+ ultimately
+ --{*This contradicts maximality of m:*}
+ show False using max and `x \<notin> Field m` unfolding Field_def by blast
+ qed
+ have "p \<subseteq> m"
+ using `Field p \<subseteq> Field m` and `extension_on (Field m) m p`
+ unfolding Field_def extension_on_def by auto fast
+ with `Well_order m` show ?thesis by blast
+qed
+
+text {*Every well-founded relation can be extended to a total wellorder.*}
+corollary total_well_order_extension:
+ assumes "wf p"
+ shows "\<exists>w. p \<subseteq> w \<and> Well_order w \<and> Field w = UNIV"
+proof -
+ from well_order_extension [OF assms] obtain w
+ where "p \<subseteq> w" and wo: "Well_order w" by blast
+ let ?A = "UNIV - Field w"
+ from well_order_on [of ?A] obtain w' where wo': "well_order_on ?A w'" ..
+ have [simp]: "Field w' = ?A" using rel.well_order_on_Well_order [OF wo'] by simp
+ have *: "Field w \<inter> Field w' = {}" by simp
+ let ?w = "w \<union>o w'"
+ have "p \<subseteq> ?w" using `p \<subseteq> w` by (auto simp: Osum_def)
+ moreover have "Well_order ?w" using Osum_Well_order [OF * wo] and wo' by simp
+ moreover have "Field ?w = UNIV" by (simp add: Field_Osum)
+ ultimately show ?thesis by blast
+qed
+
+corollary well_order_on_extension:
+ assumes "wf p" and "Field p \<subseteq> A"
+ shows "\<exists>w. p \<subseteq> w \<and> well_order_on A w"
+proof -
+ from total_well_order_extension [OF `wf p`] obtain r
+ where "p \<subseteq> r" and wo: "Well_order r" and univ: "Field r = UNIV" by blast
+ let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
+ from `p \<subseteq> r` have "p \<subseteq> ?r" using `Field p \<subseteq> A` by (auto simp: Field_def)
+ have 1: "Field ?r = A" using wo univ
+ by (fastforce simp: Field_def order_on_defs refl_on_def)
+ have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"
+ using `Well_order r` by (simp_all add: order_on_defs)
+ have "refl_on A ?r" using `Refl r` by (auto simp: refl_on_def univ)
+ moreover have "trans ?r" using `trans r`
+ unfolding trans_def by blast
+ moreover have "antisym ?r" using `antisym r`
+ unfolding antisym_def by blast
+ moreover have "total_on A ?r" using `Total r` by (simp add: total_on_def univ)
+ moreover have "wf (?r - Id)" by (rule wf_subset [OF `wf(r - Id)`]) blast
+ ultimately have "well_order_on A ?r" by (simp add: order_on_defs)
+ with `p \<subseteq> ?r` show ?thesis by blast
+qed
+
+end