(* Title: HOL/Cardinals/Wellorder_Extension.thy
Author: Christian Sternagel, JAIST
*)
section \<open>Extending Well-founded Relations to Wellorders\<close>
theory Wellorder_Extension
imports Main Order_Union
begin
subsection \<open>Extending Well-founded Relations to Wellorders\<close>
text \<open>A \emph{downset} (also lower set, decreasing set, initial segment, or
downward closed set) is closed w.r.t.\ smaller elements.\<close>
definition downset_on where
"downset_on A r = (\x y. (x, y) \ r \ y \ A \ x \ 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:
"(\x y. (x, y) \ r \ y \ A \ x \ A) \ downset_on A r"
by (auto simp: downset_on_def)
lemma downset_onD:
"downset_on A r \ (x, y) \ r \ y \ A \ x \ A"
unfolding downset_on_def by blast
text \<open>Extensions of relations w.r.t.\ a given set.\<close>
definition extension_on where
"extension_on A r s = (\x\A. \y\A. (x, y) \ s \ (x, y) \ r)"
lemma extension_onI:
"(\x y. \x \ A; y \ A; (x, y) \ s\ \ (x, y) \ r) \ extension_on A r s"
by (auto simp: extension_on_def)
lemma extension_onD:
"extension_on A r s \ x \ A \ y \ A \ (x, y) \ s \ (x, y) \ r"
by (auto simp: extension_on_def)
lemma downset_on_Union:
assumes "\r. r \ R \ downset_on (Field r) p"
shows "downset_on (Field (\R)) p"
using assms by (auto intro: downset_onI dest: downset_onD)
lemma chain_subset_extension_on_Union:
assumes "chain\<^sub>\ R" and "\r. r \ R \ extension_on (Field r) r p"
shows "extension_on (Field (\R)) (\R) p"
using assms
by (simp add: chain_subset_def extension_on_def)
(metis (no_types) mono_Field subsetD)
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 \<open>Every well-founded relation can be extended to a wellorder.\<close>
theorem well_order_extension:
assumes "wf p"
shows "\w. p \ w \ Well_order w"
proof -
let ?K = "{r. Well_order r \ downset_on (Field r) p \ extension_on (Field r) r p}"
define I where "I = init_seg_of \ ?K \ ?K"
have I_init: "I \ init_seg_of" by (simp add: I_def)
then have subch: "\R. R \ Chains I \ chain\<^sub>\ R"
by (auto simp: init_seg_of_def chain_subset_def Chains_def)
have Chains_wo: "\R r. R \ Chains I \ r \ R \
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)
have "\R \ ?K \ (\r\R. (r,\R) \ I)" if "R \ Chains I" for R
proof -
from that have Ris: "R \ Chains init_seg_of" using mono_Chains [OF I_init] by blast
have subch: "chain\<^sub>\ R" using \R \ Chains I\ I_init
by (auto simp: init_seg_of_def chain_subset_def Chains_def)
have "\r\R. Refl r" and "\r\R. trans r" and "\r\R. antisym r" and
"\r\R. Total r" and "\r\R. wf (r - Id)" and
"\r. r \ R \ downset_on (Field r) p" and
"\r. r \ R \ extension_on (Field r) r p"
using Chains_wo [OF \<open>R \<in> Chains I\<close>] by (simp_all add: order_on_defs)
have "Refl (\R)" using \\r\R. Refl r\ unfolding refl_on_def by fastforce
moreover have "trans (\R)"
by (rule chain_subset_trans_Union [OF subch \<open>\<forall>r\<in>R. trans r\<close>])
moreover have "antisym (\R)"
by (rule chain_subset_antisym_Union [OF subch \<open>\<forall>r\<in>R. antisym r\<close>])
moreover have "Total (\R)"
by (rule chain_subset_Total_Union [OF subch \<open>\<forall>r\<in>R. Total r\<close>])
moreover have "wf ((\R) - Id)"
proof -
have "(\R) - Id = \{r - Id | r. r \ R}" by blast
with \<open>\<forall>r\<in>R. wf (r - Id)\<close> wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
show ?thesis by fastforce
qed
ultimately have "Well_order (\R)" by (simp add: order_on_defs)
moreover have "\r\R. r initial_segment_of \R" using Ris
by (simp add: Chains_init_seg_of_Union)
moreover have "downset_on (Field (\R)) p"
by (rule downset_on_Union [OF \<open>\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p\<close>])
moreover have "extension_on (Field (\R)) (\R) p"
by (rule chain_subset_extension_on_Union [OF subch \<open>\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p\<close>])
ultimately show ?thesis
using mono_Chains [OF I_init] and \<open>R \<in> Chains I\<close>
by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo)
qed
then have 1: "\u\Field I. \r\R. (r, u) \ I" if "R\Chains I" for R
using that by (subst FI) blast
txt \<open>Zorn's Lemma yields a maximal wellorder m.\<close>
from Zorns_po_lemma [OF 0 1] obtain m :: "('a \ 'a) set"
where "Well_order m" and "downset_on (Field m) p" and "extension_on (Field m) m p" and
max: "\r. Well_order r \ downset_on (Field r) p \ extension_on (Field r) r p \
(m, r) \<in> I \<longrightarrow> r = m"
by (auto simp: FI)
have "Field p \ Field m"
proof (rule ccontr)
let ?Q = "Field p - Field m"
assume "\ (Field p \ Field m)"
with assms [unfolded wf_eq_minimal, THEN spec, of ?Q]
obtain x where "x \ Field p" and "x \ Field m" and
min: "\y. (y, x) \ p \ y \ ?Q" by blast
txt \<open>Add \<^term>\<open>x\<close> as topmost element to \<^term>\<open>m\<close>.\<close>
let ?s = "{(y, x) | y. y \ Field m}"
let ?m = "insert (x, x) m \ ?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 \<open>Well_order m\<close> by (simp_all add: order_on_defs)
txt \<open>We show that the extension is a wellorder.\<close>
have "Refl ?m" using \<open>Refl m\<close> Fm by (auto simp: refl_on_def)
moreover have "trans ?m" using \<open>trans m\<close> \<open>x \<notin> Field m\<close>
unfolding trans_def Field_def Domain_unfold Domain_converse [symmetric] by blast
moreover have "antisym ?m" using \<open>antisym m\<close> \<open>x \<notin> Field m\<close>
unfolding antisym_def Field_def Domain_unfold Domain_converse [symmetric] by blast
moreover have "Total ?m" using \<open>Total m\<close> Fm by (auto simp: Relation.total_on_def)
moreover have "wf (?m - Id)"
proof -
have "wf ?s" using \<open>x \<notin> Field m\<close>
by (simp add: wf_eq_minimal Field_def Domain_unfold Domain_converse [symmetric]) metis
thus ?thesis using \<open>wf (m - Id)\<close> \<open>x \<notin> Field m\<close>
wf_subset [OF \<open>wf ?s\<close> 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 \<open>extension_on (Field m) m p\<close> \<open>downset_on (Field m) p\<close>
by (subst Fm) (auto simp: extension_on_def dest: downset_onD)
moreover have "downset_on (Field ?m) p"
apply (subst Fm)
using \<open>downset_on (Field m) p\<close> and min
unfolding downset_on_def Field_def by blast
moreover have "(m, ?m) \ I"
using \<open>Well_order m\<close> and \<open>Well_order ?m\<close> and
\<open>downset_on (Field m) p\<close> and \<open>downset_on (Field ?m) p\<close> and
\<open>extension_on (Field m) m p\<close> and \<open>extension_on (Field ?m) ?m p\<close> and
\<open>Refl m\<close> and \<open>x \<notin> Field m\<close>
by (auto simp: I_def init_seg_of_def refl_on_def)
ultimately
\<comment> \<open>This contradicts maximality of m:\<close>
show False using max and \<open>x \<notin> Field m\<close> unfolding Field_def by blast
qed
have "p \ m"
using \<open>Field p \<subseteq> Field m\<close> and \<open>extension_on (Field m) m p\<close>
unfolding Field_def extension_on_def by auto fast
with \<open>Well_order m\<close> show ?thesis by blast
qed
text \<open>Every well-founded relation can be extended to a total wellorder.\<close>
corollary total_well_order_extension:
assumes "wf p"
shows "\w. p \ w \ Well_order w \ Field w = UNIV"
proof -
from well_order_extension [OF assms] obtain w
where "p \ 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 well_order_on_Well_order [OF wo'] by simp
have *: "Field w \ Field w' = {}" by simp
let ?w = "w \o w'"
have "p \ ?w" using \p \ 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 \ A"
shows "\w. p \ w \ well_order_on A w"
proof -
from total_well_order_extension [OF \<open>wf p\<close>] obtain r
where "p \ r" and wo: "Well_order r" and univ: "Field r = UNIV" by blast
let ?r = "{(x, y). x \ A \ y \ A \ (x, y) \ r}"
from \<open>p \<subseteq> r\<close> have "p \<subseteq> ?r" using \<open>Field p \<subseteq> A\<close> by (auto simp: Field_def)
have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"
using \<open>Well_order r\<close> by (simp_all add: order_on_defs)
have "refl_on A ?r" using \<open>Refl r\<close> by (auto simp: refl_on_def univ)
moreover have "trans ?r" using \<open>trans r\<close>
unfolding trans_def by blast
moreover have "antisym ?r" using \<open>antisym r\<close>
unfolding antisym_def by blast
moreover have "total_on A ?r" using \<open>Total r\<close> by (simp add: total_on_def univ)
moreover have "wf (?r - Id)" by (rule wf_subset [OF \<open>wf(r - Id)\<close>]) blast
ultimately have "well_order_on A ?r" by (simp add: order_on_defs)
with \<open>p \<subseteq> ?r\<close> show ?thesis by blast
qed
end
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