src/HOL/Zorn.thy
 author nipkow Wed, 10 Jan 2018 15:25:09 +0100 changeset 67399 eab6ce8368fa parent 63572 c0cbfd2b5a45 child 67443 3abf6a722518 permissions -rw-r--r--
ran isabelle update_op on all sources
```
(*  Title:       HOL/Zorn.thy
Author:      Jacques D. Fleuriot
Author:      Tobias Nipkow, TUM
Author:      Christian Sternagel, JAIST

Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF).
The well-ordering theorem.
*)

section \<open>Zorn's Lemma\<close>

theory Zorn
imports Order_Relation Hilbert_Choice
begin

subsection \<open>Zorn's Lemma for the Subset Relation\<close>

subsubsection \<open>Results that do not require an order\<close>

text \<open>Let \<open>P\<close> be a binary predicate on the set \<open>A\<close>.\<close>
locale pred_on =
fixes A :: "'a set"
and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubset>" 50)
begin

abbreviation Peq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubseteq>" 50)
where "x \<sqsubseteq> y \<equiv> P\<^sup>=\<^sup>= x y"

text \<open>A chain is a totally ordered subset of \<open>A\<close>.\<close>
definition chain :: "'a set \<Rightarrow> bool"
where "chain C \<longleftrightarrow> C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x)"

text \<open>
We call a chain that is a proper superset of some set \<open>X\<close>,
but not necessarily a chain itself, a superchain of \<open>X\<close>.
\<close>
abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"  (infix "<c" 50)
where "X <c C \<equiv> chain C \<and> X \<subset> C"

text \<open>A maximal chain is a chain that does not have a superchain.\<close>
definition maxchain :: "'a set \<Rightarrow> bool"
where "maxchain C \<longleftrightarrow> chain C \<and> (\<nexists>S. C <c S)"

text \<open>
We define the successor of a set to be an arbitrary
superchain, if such exists, or the set itself, otherwise.
\<close>
definition suc :: "'a set \<Rightarrow> 'a set"
where "suc C = (if \<not> chain C \<or> maxchain C then C else (SOME D. C <c D))"

lemma chainI [Pure.intro?]: "C \<subseteq> A \<Longrightarrow> (\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x) \<Longrightarrow> chain C"
unfolding chain_def by blast

lemma chain_total: "chain C \<Longrightarrow> x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x"

lemma not_chain_suc [simp]: "\<not> chain X \<Longrightarrow> suc X = X"

lemma maxchain_suc [simp]: "maxchain X \<Longrightarrow> suc X = X"

lemma suc_subset: "X \<subseteq> suc X"
by (auto simp: suc_def maxchain_def intro: someI2)

lemma chain_empty [simp]: "chain {}"
by (auto simp: chain_def)

lemma not_maxchain_Some: "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> C <c (SOME D. C <c D)"
by (rule someI_ex) (auto simp: maxchain_def)

lemma suc_not_equals: "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> suc C \<noteq> C"
using not_maxchain_Some by (auto simp: suc_def)

lemma subset_suc:
assumes "X \<subseteq> Y"
shows "X \<subseteq> suc Y"
using assms by (rule subset_trans) (rule suc_subset)

text \<open>
We build a set @{term \<C>} that is closed under applications
of @{term suc} and contains the union of all its subsets.
\<close>
inductive_set suc_Union_closed ("\<C>")
where
suc: "X \<in> \<C> \<Longrightarrow> suc X \<in> \<C>"
| Union [unfolded Pow_iff]: "X \<in> Pow \<C> \<Longrightarrow> \<Union>X \<in> \<C>"

text \<open>
Since the empty set as well as the set itself is a subset of
every set, @{term \<C>} contains at least @{term "{} \<in> \<C>"} and
@{term "\<Union>\<C> \<in> \<C>"}.
\<close>
lemma suc_Union_closed_empty: "{} \<in> \<C>"
and suc_Union_closed_Union: "\<Union>\<C> \<in> \<C>"
using Union [of "{}"] and Union [of "\<C>"] by simp_all

text \<open>Thus closure under @{term suc} will hit a maximal chain
eventually, as is shown below.\<close>

lemma suc_Union_closed_induct [consumes 1, case_names suc Union, induct pred: suc_Union_closed]:
assumes "X \<in> \<C>"
and "\<And>X. X \<in> \<C> \<Longrightarrow> Q X \<Longrightarrow> Q (suc X)"
and "\<And>X. X \<subseteq> \<C> \<Longrightarrow> \<forall>x\<in>X. Q x \<Longrightarrow> Q (\<Union>X)"
shows "Q X"
using assms by induct blast+

lemma suc_Union_closed_cases [consumes 1, case_names suc Union, cases pred: suc_Union_closed]:
assumes "X \<in> \<C>"
and "\<And>Y. X = suc Y \<Longrightarrow> Y \<in> \<C> \<Longrightarrow> Q"
and "\<And>Y. X = \<Union>Y \<Longrightarrow> Y \<subseteq> \<C> \<Longrightarrow> Q"
shows "Q"
using assms by cases simp_all

text \<open>On chains, @{term suc} yields a chain.\<close>
lemma chain_suc:
assumes "chain X"
shows "chain (suc X)"
using assms
by (cases "\<not> chain X \<or> maxchain X") (force simp: suc_def dest: not_maxchain_Some)+

lemma chain_sucD:
assumes "chain X"
shows "suc X \<subseteq> A \<and> chain (suc X)"
proof -
from \<open>chain X\<close> have *: "chain (suc X)"
by (rule chain_suc)
then have "suc X \<subseteq> A"
unfolding chain_def by blast
with * show ?thesis by blast
qed

lemma suc_Union_closed_total':
assumes "X \<in> \<C>" and "Y \<in> \<C>"
and *: "\<And>Z. Z \<in> \<C> \<Longrightarrow> Z \<subseteq> Y \<Longrightarrow> Z = Y \<or> suc Z \<subseteq> Y"
shows "X \<subseteq> Y \<or> suc Y \<subseteq> X"
using \<open>X \<in> \<C>\<close>
proof induct
case (suc X)
with * show ?case by (blast del: subsetI intro: subset_suc)
next
case Union
then show ?case by blast
qed

lemma suc_Union_closed_subsetD:
assumes "Y \<subseteq> X" and "X \<in> \<C>" and "Y \<in> \<C>"
shows "X = Y \<or> suc Y \<subseteq> X"
using assms(2,3,1)
proof (induct arbitrary: Y)
case (suc X)
note * = \<open>\<And>Y. Y \<in> \<C> \<Longrightarrow> Y \<subseteq> X \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X\<close>
with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>X \<in> \<C>\<close>]
have "Y \<subseteq> X \<or> suc X \<subseteq> Y" by blast
then show ?case
proof
assume "Y \<subseteq> X"
with * and \<open>Y \<in> \<C>\<close> have "X = Y \<or> suc Y \<subseteq> X" by blast
then show ?thesis
proof
assume "X = Y"
then show ?thesis by simp
next
assume "suc Y \<subseteq> X"
then have "suc Y \<subseteq> suc X" by (rule subset_suc)
then show ?thesis by simp
qed
next
assume "suc X \<subseteq> Y"
with \<open>Y \<subseteq> suc X\<close> show ?thesis by blast
qed
next
case (Union X)
show ?case
proof (rule ccontr)
assume "\<not> ?thesis"
with \<open>Y \<subseteq> \<Union>X\<close> obtain x y z
where "\<not> suc Y \<subseteq> \<Union>X"
and "x \<in> X" and "y \<in> x" and "y \<notin> Y"
and "z \<in> suc Y" and "\<forall>x\<in>X. z \<notin> x" by blast
with \<open>X \<subseteq> \<C>\<close> have "x \<in> \<C>" by blast
from Union and \<open>x \<in> X\<close> have *: "\<And>y. y \<in> \<C> \<Longrightarrow> y \<subseteq> x \<Longrightarrow> x = y \<or> suc y \<subseteq> x"
by blast
with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>x \<in> \<C>\<close>] have "Y \<subseteq> x \<or> suc x \<subseteq> Y"
by blast
then show False
proof
assume "Y \<subseteq> x"
with * [OF \<open>Y \<in> \<C>\<close>] have "x = Y \<or> suc Y \<subseteq> x" by blast
then show False
proof
assume "x = Y"
with \<open>y \<in> x\<close> and \<open>y \<notin> Y\<close> show False by blast
next
assume "suc Y \<subseteq> x"
with \<open>x \<in> X\<close> have "suc Y \<subseteq> \<Union>X" by blast
with \<open>\<not> suc Y \<subseteq> \<Union>X\<close> show False by contradiction
qed
next
assume "suc x \<subseteq> Y"
moreover from suc_subset and \<open>y \<in> x\<close> have "y \<in> suc x" by blast
ultimately show False using \<open>y \<notin> Y\<close> by blast
qed
qed
qed

text \<open>The elements of @{term \<C>} are totally ordered by the subset relation.\<close>
lemma suc_Union_closed_total:
assumes "X \<in> \<C>" and "Y \<in> \<C>"
shows "X \<subseteq> Y \<or> Y \<subseteq> X"
proof (cases "\<forall>Z\<in>\<C>. Z \<subseteq> Y \<longrightarrow> Z = Y \<or> suc Z \<subseteq> Y")
case True
with suc_Union_closed_total' [OF assms]
have "X \<subseteq> Y \<or> suc Y \<subseteq> X" by blast
with suc_subset [of Y] show ?thesis by blast
next
case False
then obtain Z where "Z \<in> \<C>" and "Z \<subseteq> Y" and "Z \<noteq> Y" and "\<not> suc Z \<subseteq> Y"
by blast
with suc_Union_closed_subsetD and \<open>Y \<in> \<C>\<close> show ?thesis
by blast
qed

text \<open>Once we hit a fixed point w.r.t. @{term suc}, all other elements
of @{term \<C>} are subsets of this fixed point.\<close>
lemma suc_Union_closed_suc:
assumes "X \<in> \<C>" and "Y \<in> \<C>" and "suc Y = Y"
shows "X \<subseteq> Y"
using \<open>X \<in> \<C>\<close>
proof induct
case (suc X)
with \<open>Y \<in> \<C>\<close> and suc_Union_closed_subsetD have "X = Y \<or> suc X \<subseteq> Y"
by blast
then show ?case
by (auto simp: \<open>suc Y = Y\<close>)
next
case Union
then show ?case by blast
qed

lemma eq_suc_Union:
assumes "X \<in> \<C>"
shows "suc X = X \<longleftrightarrow> X = \<Union>\<C>"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?lhs
then have "\<Union>\<C> \<subseteq> X"
by (rule suc_Union_closed_suc [OF suc_Union_closed_Union \<open>X \<in> \<C>\<close>])
with \<open>X \<in> \<C>\<close> show ?rhs
by blast
next
from \<open>X \<in> \<C>\<close> have "suc X \<in> \<C>" by (rule suc)
then have "suc X \<subseteq> \<Union>\<C>" by blast
moreover assume ?rhs
ultimately have "suc X \<subseteq> X" by simp
moreover have "X \<subseteq> suc X" by (rule suc_subset)
ultimately show ?lhs ..
qed

lemma suc_in_carrier:
assumes "X \<subseteq> A"
shows "suc X \<subseteq> A"
using assms
by (cases "\<not> chain X \<or> maxchain X") (auto dest: chain_sucD)

lemma suc_Union_closed_in_carrier:
assumes "X \<in> \<C>"
shows "X \<subseteq> A"
using assms
by induct (auto dest: suc_in_carrier)

text \<open>All elements of @{term \<C>} are chains.\<close>
lemma suc_Union_closed_chain:
assumes "X \<in> \<C>"
shows "chain X"
using assms
proof induct
case (suc X)
then show ?case
using not_maxchain_Some by (simp add: suc_def)
next
case (Union X)
then have "\<Union>X \<subseteq> A"
by (auto dest: suc_Union_closed_in_carrier)
moreover have "\<forall>x\<in>\<Union>X. \<forall>y\<in>\<Union>X. x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
proof (intro ballI)
fix x y
assume "x \<in> \<Union>X" and "y \<in> \<Union>X"
then obtain u v where "x \<in> u" and "u \<in> X" and "y \<in> v" and "v \<in> X"
by blast
with Union have "u \<in> \<C>" and "v \<in> \<C>" and "chain u" and "chain v"
by blast+
with suc_Union_closed_total have "u \<subseteq> v \<or> v \<subseteq> u"
by blast
then show "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
proof
assume "u \<subseteq> v"
from \<open>chain v\<close> show ?thesis
proof (rule chain_total)
show "y \<in> v" by fact
show "x \<in> v" using \<open>u \<subseteq> v\<close> and \<open>x \<in> u\<close> by blast
qed
next
assume "v \<subseteq> u"
from \<open>chain u\<close> show ?thesis
proof (rule chain_total)
show "x \<in> u" by fact
show "y \<in> u" using \<open>v \<subseteq> u\<close> and \<open>y \<in> v\<close> by blast
qed
qed
qed
ultimately show ?case unfolding chain_def ..
qed

subsubsection \<open>Hausdorff's Maximum Principle\<close>

text \<open>There exists a maximal totally ordered subset of \<open>A\<close>. (Note that we do not
require \<open>A\<close> to be partially ordered.)\<close>

theorem Hausdorff: "\<exists>C. maxchain C"
proof -
let ?M = "\<Union>\<C>"
have "maxchain ?M"
proof (rule ccontr)
assume "\<not> ?thesis"
then have "suc ?M \<noteq> ?M"
using suc_not_equals and suc_Union_closed_chain [OF suc_Union_closed_Union] by simp
moreover have "suc ?M = ?M"
using eq_suc_Union [OF suc_Union_closed_Union] by simp
qed
then show ?thesis by blast
qed

text \<open>Make notation @{term \<C>} available again.\<close>
no_notation suc_Union_closed  ("\<C>")

lemma chain_extend: "chain C \<Longrightarrow> z \<in> A \<Longrightarrow> \<forall>x\<in>C. x \<sqsubseteq> z \<Longrightarrow> chain ({z} \<union> C)"
unfolding chain_def by blast

lemma maxchain_imp_chain: "maxchain C \<Longrightarrow> chain C"

end

text \<open>Hide constant @{const pred_on.suc_Union_closed}, which was just needed
for the proof of Hausforff's maximum principle.\<close>
hide_const pred_on.suc_Union_closed

lemma chain_mono:
assumes "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> P x y \<Longrightarrow> Q x y"
and "pred_on.chain A P C"
shows "pred_on.chain A Q C"
using assms unfolding pred_on.chain_def by blast

subsubsection \<open>Results for the proper subset relation\<close>

interpretation subset: pred_on "A" "(\<subset>)" for A .

lemma subset_maxchain_max:
assumes "subset.maxchain A C"
and "X \<in> A"
and "\<Union>C \<subseteq> X"
shows "\<Union>C = X"
proof (rule ccontr)
let ?C = "{X} \<union> C"
from \<open>subset.maxchain A C\<close> have "subset.chain A C"
and *: "\<And>S. subset.chain A S \<Longrightarrow> \<not> C \<subset> S"
by (auto simp: subset.maxchain_def)
moreover have "\<forall>x\<in>C. x \<subseteq> X" using \<open>\<Union>C \<subseteq> X\<close> by auto
ultimately have "subset.chain A ?C"
using subset.chain_extend [of A C X] and \<open>X \<in> A\<close> by auto
moreover assume **: "\<Union>C \<noteq> X"
moreover from ** have "C \<subset> ?C" using \<open>\<Union>C \<subseteq> X\<close> by auto
ultimately show False using * by blast
qed

subsubsection \<open>Zorn's lemma\<close>

text \<open>If every chain has an upper bound, then there is a maximal set.\<close>
lemma subset_Zorn:
assumes "\<And>C. subset.chain A C \<Longrightarrow> \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U"
shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
proof -
from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
then have "subset.chain A M"
by (rule subset.maxchain_imp_chain)
with assms obtain Y where "Y \<in> A" and "\<forall>X\<in>M. X \<subseteq> Y"
by blast
moreover have "\<forall>X\<in>A. Y \<subseteq> X \<longrightarrow> Y = X"
proof (intro ballI impI)
fix X
assume "X \<in> A" and "Y \<subseteq> X"
show "Y = X"
proof (rule ccontr)
assume "\<not> ?thesis"
with \<open>Y \<subseteq> X\<close> have "\<not> X \<subseteq> Y" by blast
from subset.chain_extend [OF \<open>subset.chain A M\<close> \<open>X \<in> A\<close>] and \<open>\<forall>X\<in>M. X \<subseteq> Y\<close>
have "subset.chain A ({X} \<union> M)"
using \<open>Y \<subseteq> X\<close> by auto
moreover have "M \<subset> {X} \<union> M"
using \<open>\<forall>X\<in>M. X \<subseteq> Y\<close> and \<open>\<not> X \<subseteq> Y\<close> by auto
ultimately show False
using \<open>subset.maxchain A M\<close> by (auto simp: subset.maxchain_def)
qed
qed
ultimately show ?thesis by blast
qed

text \<open>Alternative version of Zorn's lemma for the subset relation.\<close>
lemma subset_Zorn':
assumes "\<And>C. subset.chain A C \<Longrightarrow> \<Union>C \<in> A"
shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
proof -
from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
then have "subset.chain A M"
by (rule subset.maxchain_imp_chain)
with assms have "\<Union>M \<in> A" .
moreover have "\<forall>Z\<in>A. \<Union>M \<subseteq> Z \<longrightarrow> \<Union>M = Z"
proof (intro ballI impI)
fix Z
assume "Z \<in> A" and "\<Union>M \<subseteq> Z"
with subset_maxchain_max [OF \<open>subset.maxchain A M\<close>]
show "\<Union>M = Z" .
qed
ultimately show ?thesis by blast
qed

subsection \<open>Zorn's Lemma for Partial Orders\<close>

text \<open>Relate old to new definitions.\<close>

definition chain_subset :: "'a set set \<Rightarrow> bool"  ("chain\<^sub>\<subseteq>")  (* Define globally? In Set.thy? *)
where "chain\<^sub>\<subseteq> C \<longleftrightarrow> (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)"

definition chains :: "'a set set \<Rightarrow> 'a set set set"
where "chains A = {C. C \<subseteq> A \<and> chain\<^sub>\<subseteq> C}"

definition Chains :: "('a \<times> 'a) set \<Rightarrow> 'a set set"  (* Define globally? In Relation.thy? *)
where "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"

lemma chains_extend: "c \<in> chains S \<Longrightarrow> z \<in> S \<Longrightarrow> \<forall>x \<in> c. x \<subseteq> z \<Longrightarrow> {z} \<union> c \<in> chains S"
for z :: "'a set"
unfolding chains_def chain_subset_def by blast

lemma mono_Chains: "r \<subseteq> s \<Longrightarrow> Chains r \<subseteq> Chains s"
unfolding Chains_def by blast

lemma chain_subset_alt_def: "chain\<^sub>\<subseteq> C = subset.chain UNIV C"
unfolding chain_subset_def subset.chain_def by fast

lemma chains_alt_def: "chains A = {C. subset.chain A C}"
by (simp add: chains_def chain_subset_alt_def subset.chain_def)

lemma Chains_subset: "Chains r \<subseteq> {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
by (force simp add: Chains_def pred_on.chain_def)

lemma Chains_subset':
assumes "refl r"
shows "{C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C} \<subseteq> Chains r"
using assms
by (auto simp add: Chains_def pred_on.chain_def refl_on_def)

lemma Chains_alt_def:
assumes "refl r"
shows "Chains r = {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
using assms Chains_subset Chains_subset' by blast

lemma Zorn_Lemma: "\<forall>C\<in>chains A. \<Union>C \<in> A \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
using subset_Zorn' [of A] by (force simp: chains_alt_def)

lemma Zorn_Lemma2: "\<forall>C\<in>chains A. \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
using subset_Zorn [of A] by (auto simp: chains_alt_def)

text \<open>Various other lemmas\<close>

lemma chainsD: "c \<in> chains S \<Longrightarrow> x \<in> c \<Longrightarrow> y \<in> c \<Longrightarrow> x \<subseteq> y \<or> y \<subseteq> x"
unfolding chains_def chain_subset_def by blast

lemma chainsD2: "c \<in> chains S \<Longrightarrow> c \<subseteq> S"
unfolding chains_def by blast

lemma Zorns_po_lemma:
assumes po: "Partial_order r"
and u: "\<forall>C\<in>Chains r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r"
shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
proof -
have "Preorder r"
using po by (simp add: partial_order_on_def)
txt \<open>Mirror \<open>r\<close> in the set of subsets below (wrt \<open>r\<close>) elements of \<open>A\<close>.\<close>
let ?B = "\<lambda>x. r\<inverse> `` {x}"
let ?S = "?B ` Field r"
have "\<exists>u\<in>Field r. \<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}"  (is "\<exists>u\<in>Field r. ?P u")
if 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A" for C
proof -
let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
from 1 have "C = ?B ` ?A" by (auto simp: image_def)
have "?A \<in> Chains r"
proof (simp add: Chains_def, intro allI impI, elim conjE)
fix a b
assume "a \<in> Field r" and "?B a \<in> C" and "b \<in> Field r" and "?B b \<in> C"
with 2 have "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" by auto
then show "(a, b) \<in> r \<or> (b, a) \<in> r"
using \<open>Preorder r\<close> and \<open>a \<in> Field r\<close> and \<open>b \<in> Field r\<close>
qed
with u obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" by auto
have "?P u"
proof auto
fix a B assume aB: "B \<in> C" "a \<in> B"
with 1 obtain x where "x \<in> Field r" and "B = r\<inverse> `` {x}" by auto
then show "(a, u) \<in> r"
using uA and aB and \<open>Preorder r\<close>
unfolding preorder_on_def refl_on_def by simp (fast dest: transD)
qed
then show ?thesis
using \<open>u \<in> Field r\<close> by blast
qed
then have "\<forall>C\<in>chains ?S. \<exists>U\<in>?S. \<forall>A\<in>C. A \<subseteq> U"
by (auto simp: chains_def chain_subset_def)
from Zorn_Lemma2 [OF this] obtain m B
where "m \<in> Field r"
and "B = r\<inverse> `` {m}"
and "\<forall>x\<in>Field r. B \<subseteq> r\<inverse> `` {x} \<longrightarrow> r\<inverse> `` {x} = B"
by auto
then have "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
using po and \<open>Preorder r\<close> and \<open>m \<in> Field r\<close>
by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
then show ?thesis
using \<open>m \<in> Field r\<close> by blast
qed

subsection \<open>The Well Ordering Theorem\<close>

(* The initial segment of a relation appears generally useful.
Move to Relation.thy?
Definition correct/most general?
Naming?
*)
definition init_seg_of :: "(('a \<times> 'a) set \<times> ('a \<times> 'a) set) set"
where "init_seg_of = {(r, s). r \<subseteq> s \<and> (\<forall>a b c. (a, b) \<in> s \<and> (b, c) \<in> r \<longrightarrow> (a, b) \<in> r)}"

abbreviation initial_segment_of_syntax :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"
(infix "initial'_segment'_of" 55)
where "r initial_segment_of s \<equiv> (r, s) \<in> init_seg_of"

lemma refl_on_init_seg_of [simp]: "r initial_segment_of r"

lemma trans_init_seg_of:
"r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
by (simp (no_asm_use) add: init_seg_of_def) blast

lemma antisym_init_seg_of: "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s"
unfolding init_seg_of_def by safe

lemma Chains_init_seg_of_Union: "R \<in> Chains init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
by (auto simp: init_seg_of_def Ball_def Chains_def) blast

lemma chain_subset_trans_Union:
assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. trans r"
shows "trans (\<Union>R)"
proof (intro transI, elim UnionE)
fix S1 S2 :: "'a rel" and x y z :: 'a
assume "S1 \<in> R" "S2 \<in> R"
with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1"
unfolding chain_subset_def by blast
moreover assume "(x, y) \<in> S1" "(y, z) \<in> S2"
ultimately have "((x, y) \<in> S1 \<and> (y, z) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, z) \<in> S2)"
by blast
with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "(x, z) \<in> \<Union>R"
by (auto elim: transE)
qed

lemma chain_subset_antisym_Union:
assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. antisym r"
shows "antisym (\<Union>R)"
proof (intro antisymI, elim UnionE)
fix S1 S2 :: "'a rel" and x y :: 'a
assume "S1 \<in> R" "S2 \<in> R"
with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1"
unfolding chain_subset_def by blast
moreover assume "(x, y) \<in> S1" "(y, x) \<in> S2"
ultimately have "((x, y) \<in> S1 \<and> (y, x) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, x) \<in> S2)"
by blast
with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "x = y"
unfolding antisym_def by auto
qed

lemma chain_subset_Total_Union:
assumes "chain\<^sub>\<subseteq> R" and "\<forall>r\<in>R. Total r"
shows "Total (\<Union>R)"
proof (simp add: total_on_def Ball_def, auto del: disjCI)
fix r s a b
assume A: "r \<in> R" "s \<in> R" "a \<in> Field r" "b \<in> Field s" "a \<noteq> b"
from \<open>chain\<^sub>\<subseteq> R\<close> and \<open>r \<in> R\<close> and \<open>s \<in> R\<close> have "r \<subseteq> s \<or> s \<subseteq> r"
then show "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
proof
assume "r \<subseteq> s"
then have "(a, b) \<in> s \<or> (b, a) \<in> s"
using assms(2) A mono_Field[of r s]
then show ?thesis
using \<open>s \<in> R\<close> by blast
next
assume "s \<subseteq> r"
then have "(a, b) \<in> r \<or> (b, a) \<in> r"
using assms(2) A mono_Field[of s r]
then show ?thesis
using \<open>r \<in> R\<close> by blast
qed
qed

lemma wf_Union_wf_init_segs:
assumes "R \<in> Chains init_seg_of"
and "\<forall>r\<in>R. wf r"
shows "wf (\<Union>R)"
proof (simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto)
fix f
assume 1: "\<forall>i. \<exists>r\<in>R. (f (Suc i), f i) \<in> r"
then obtain r where "r \<in> R" and "(f (Suc 0), f 0) \<in> r" by auto
have "(f (Suc i), f i) \<in> r" for i
proof (induct i)
case 0
show ?case by fact
next
case (Suc i)
then obtain s where s: "s \<in> R" "(f (Suc (Suc i)), f(Suc i)) \<in> s"
using 1 by auto
then have "s initial_segment_of r \<or> r initial_segment_of s"
using assms(1) \<open>r \<in> R\<close> by (simp add: Chains_def)
with Suc s show ?case by (simp add: init_seg_of_def) blast
qed
then show False
using assms(2) and \<open>r \<in> R\<close>
qed

lemma initial_segment_of_Diff: "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s"
unfolding init_seg_of_def by blast

lemma Chains_inits_DiffI: "R \<in> Chains init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chains init_seg_of"
unfolding Chains_def by (blast intro: initial_segment_of_Diff)

theorem well_ordering: "\<exists>r::'a rel. Well_order r \<and> Field r = UNIV"
proof -
\<comment> \<open>The initial segment relation on well-orders:\<close>
let ?WO = "{r::'a rel. Well_order r}"
define I where "I = init_seg_of \<inter> ?WO \<times> ?WO"
then have I_init: "I \<subseteq> init_seg_of" by simp
then have subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
unfolding init_seg_of_def chain_subset_def Chains_def by blast
have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
by (simp add: Chains_def I_def) blast
have FI: "Field I = ?WO"
by (auto simp add: 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)
\<comment> \<open>\<open>I\<close>-chains have upper bounds in \<open>?WO\<close> wrt \<open>I\<close>: their Union\<close>
have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)" if "R \<in> Chains I" for R
proof -
from that have Ris: "R \<in> Chains init_seg_of"
using mono_Chains [OF I_init] by blast
have subch: "chain\<^sub>\<subseteq> R"
using \<open>R : Chains I\<close> 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)"
using Chains_wo [OF \<open>R \<in> Chains I\<close>] by (simp_all add: order_on_defs)
have "Refl (\<Union>R)"
using \<open>\<forall>r\<in>R. Refl r\<close> unfolding refl_on_def by fastforce
moreover have "trans (\<Union>R)"
by (rule chain_subset_trans_Union [OF subch \<open>\<forall>r\<in>R. trans r\<close>])
moreover have "antisym (\<Union>R)"
by (rule chain_subset_antisym_Union [OF subch \<open>\<forall>r\<in>R. antisym r\<close>])
moreover have "Total (\<Union>R)"
by (rule chain_subset_Total_Union [OF subch \<open>\<forall>r\<in>R. Total r\<close>])
moreover have "wf ((\<Union>R) - Id)"
proof -
have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
with \<open>\<forall>r\<in>R. wf (r - Id)\<close> and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
show ?thesis by fastforce
qed
ultimately have "Well_order (\<Union>R)"
moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R"
using Ris by (simp add: Chains_init_seg_of_Union)
ultimately show ?thesis
using mono_Chains [OF I_init] Chains_wo[of R] and \<open>R \<in> Chains I\<close>
unfolding I_def by blast
qed
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
\<comment>\<open>Zorn's Lemma yields a maximal well-order \<open>m\<close>:\<close>
then obtain m :: "'a rel"
where "Well_order m"
and max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m"
using Zorns_po_lemma[OF 0 1] unfolding FI by fastforce
\<comment>\<open>Now show by contradiction that \<open>m\<close> covers the whole type:\<close>
have False if "x \<notin> Field m" for x :: 'a
proof -
\<comment>\<open>Assuming that \<open>x\<close> is not covered and extend \<open>m\<close> at the top with \<open>x\<close>\<close>
have "m \<noteq> {}"
proof
assume "m = {}"
moreover have "Well_order {(x, x)}"
by (simp add: order_on_defs refl_on_def trans_def antisym_def total_on_def Field_def)
ultimately show False using max
by (auto simp: I_def init_seg_of_def simp del: Field_insert)
qed
then have "Field m \<noteq> {}" by (auto simp: Field_def)
moreover have "wf (m - Id)"
using \<open>Well_order m\<close> by (simp add: well_order_on_def)
\<comment>\<open>The extension of \<open>m\<close> by \<open>x\<close>:\<close>
let ?s = "{(a, x) | a. a \<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 \<open>Well_order m\<close> by (simp_all add: order_on_defs)
\<comment>\<open>We show that the extension is a well-order\<close>
have "Refl ?m"
using \<open>Refl m\<close> Fm unfolding refl_on_def by blast
moreover have "trans ?m" using \<open>trans m\<close> and \<open>x \<notin> Field m\<close>
unfolding trans_def Field_def by blast
moreover have "antisym ?m"
using \<open>antisym m\<close> and \<open>x \<notin> Field m\<close> unfolding antisym_def Field_def by blast
moreover have "Total ?m"
using \<open>Total m\<close> and Fm by (auto simp: total_on_def)
moreover have "wf (?m - Id)"
proof -
have "wf ?s"
using \<open>x \<notin> Field m\<close> by (auto simp: wf_eq_minimal Field_def Bex_def)
then show ?thesis
using \<open>wf (m - Id)\<close> and \<open>x \<notin> Field m\<close> wf_subset [OF \<open>wf ?s\<close> Diff_subset]
by (auto simp: Un_Diff Field_def intro: wf_Un)
qed
ultimately have "Well_order ?m"
\<comment>\<open>We show that the extension is above \<open>m\<close>\<close>
moreover have "(m, ?m) \<in> I"
using \<open>Well_order ?m\<close> and \<open>Well_order m\<close> and \<open>x \<notin> Field m\<close>
by (fastforce simp: I_def init_seg_of_def Field_def)
ultimately
show False
using max and \<open>x \<notin> Field m\<close> unfolding Field_def by blast
qed
then have "Field m = UNIV" by auto
with \<open>Well_order m\<close> show ?thesis by blast
qed

corollary well_order_on: "\<exists>r::'a rel. well_order_on A r"
proof -
obtain r :: "'a rel" where wo: "Well_order r" and univ: "Field r = UNIV"
using well_ordering [where 'a = "'a"] by blast
let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
have 1: "Field ?r = A"
using wo univ by (fastforce simp: Field_def order_on_defs refl_on_def)
from \<open>Well_order r\<close> have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"
from \<open>Refl r\<close> have "Refl ?r"
by (auto simp: refl_on_def 1 univ)
moreover from \<open>trans r\<close> have "trans ?r"
unfolding trans_def by blast
moreover from \<open>antisym r\<close> have "antisym ?r"
unfolding antisym_def by blast
moreover from \<open>Total r\<close> have "Total ?r"
moreover have "wf (?r - Id)"
by (rule wf_subset [OF \<open>wf (r - Id)\<close>]) blast
ultimately have "Well_order ?r"
with 1 show ?thesis by auto
qed

(* Move this to Hilbert Choice and wfrec to Wellfounded*)

lemma wfrec_def_adm: "f \<equiv> wfrec R F \<Longrightarrow> wf R \<Longrightarrow> adm_wf R F \<Longrightarrow> f = F f"
using wfrec_fixpoint by simp

lemma dependent_wf_choice:
fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
assumes "wf R"
and adm: "\<And>f g x r. (\<And>z. (z, x) \<in> R \<Longrightarrow> f z = g z) \<Longrightarrow> P f x r = P g x r"
and P: "\<And>x f. (\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
shows "\<exists>f. \<forall>x. P f x (f x)"
proof (intro exI allI)
fix x
define f where "f \<equiv> wfrec R (\<lambda>f x. SOME r. P f x r)"
from \<open>wf R\<close> show "P f x (f x)"
proof (induct x)
case (less x)
show "P f x (f x)"
proof (subst (2) wfrec_def_adm[OF f_def \<open>wf R\<close>])
show "adm_wf R (\<lambda>f x. SOME r. P f x r)"
show "P f x (Eps (P f x))"
using P by (rule someI_ex) fact
qed
qed
qed

lemma (in wellorder) dependent_wellorder_choice:
assumes "\<And>r f g x. (\<And>y. y < x \<Longrightarrow> f y = g y) \<Longrightarrow> P f x r = P g x r"
and P: "\<And>x f. (\<And>y. y < x \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
shows "\<exists>f. \<forall>x. P f x (f x)"
using wf by (rule dependent_wf_choice) (auto intro!: assms)

end
```