src/HOL/Library/Zorn.thy
 author wenzelm Thu, 01 Aug 2013 00:18:45 +0200 changeset 52821 05eb2d77b195 parent 52199 d25fc4c0ff62 child 53374 a14d2a854c02 permissions -rw-r--r--
tuned proof;

(*  Title:      HOL/Library/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.
The extension of any well-founded relation to a well-order.
*)

theory Zorn
imports Order_Union
begin

subsection {* Zorn's Lemma for the Subset Relation *}

subsubsection {* Results that do not require an order *}

text {*Let @{text P} be a binary predicate on the set @{text A}.*}
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 {*A chain is a totally ordered subset of @{term A}.*}
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 {*We call a chain that is a proper superset of some set @{term X},
but not necessarily a chain itself, a superchain of @{term X}.*}
abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infix "<c" 50) where
"X <c C \<equiv> chain C \<and> X \<subset> C"

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

text {*We define the successor of a set to be an arbitrary
superchain, if such exists, or the set itself, otherwise.*}
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?]:
"\<lbrakk>C \<subseteq> A; \<And>x y. \<lbrakk>x \<in> C; y \<in> C\<rbrakk> \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x\<rbrakk> \<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"
by (auto simp: suc_def) (metis less_irrefl not_maxchain_Some)

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

text {*We build a set @{term \<C>} that is closed under applications
of @{term suc} and contains the union of all its subsets.*}
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 {*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>"}.*}
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+
text {*Thus closure under @{term suc} will hit a maximal chain
eventually, as is shown below.*}

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

text {*On chains, @{term suc} yields a chain.*}
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 chain X have "chain (suc X)" by (rule chain_suc)
moreover then have "suc X \<subseteq> A" unfolding chain_def by blast
ultimately 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 X \<in> \<C>
proof (induct)
case (suc X)
with * show ?case by (blast del: subsetI intro: subset_suc)
qed blast

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-, 1)
proof (induct arbitrary: Y)
case (suc X)
note * = \<And>Y. \<lbrakk>Y \<in> \<C>; Y \<subseteq> X\<rbrakk> \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X
with suc_Union_closed_total' [OF Y \<in> \<C> X \<in> \<C>]
have "Y \<subseteq> X \<or> suc X \<subseteq> Y" by blast
then show ?case
proof
assume "Y \<subseteq> X"
with * and Y \<in> \<C> 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 Y \<subseteq> suc X show ?thesis by blast
qed
next
case (Union X)
show ?case
proof (rule ccontr)
assume "\<not> ?thesis"
with Y \<subseteq> \<Union>X 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 X \<subseteq> \<C> have "x \<in> \<C>" by blast
from Union and x \<in> X
have *: "\<And>y. \<lbrakk>y \<in> \<C>; y \<subseteq> x\<rbrakk> \<Longrightarrow> x = y \<or> suc y \<subseteq> x" by blast
with suc_Union_closed_total' [OF Y \<in> \<C> x \<in> \<C>]
have "Y \<subseteq> x \<or> suc x \<subseteq> Y" by blast
then show False
proof
assume "Y \<subseteq> x"
with * [OF Y \<in> \<C>] have "x = Y \<or> suc Y \<subseteq> x" by blast
then show False
proof
assume "x = Y" with y \<in> x and y \<notin> Y show False by blast
next
assume "suc Y \<subseteq> x"
with x \<in> X have "suc Y \<subseteq> \<Union>X" by blast
with \<not> suc Y \<subseteq> \<Union>X show False by contradiction
qed
next
assume "suc x \<subseteq> Y"
moreover from suc_subset and y \<in> x have "y \<in> suc x" by blast
ultimately show False using y \<notin> Y by blast
qed
qed
qed

text {*The elements of @{term \<C>} are totally ordered by the subset relation.*}
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
then show ?thesis using suc_subset [of Y] 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 Y \<in> \<C> show ?thesis by blast
qed

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

lemma eq_suc_Union:
assumes "X \<in> \<C>"
shows "suc X = X \<longleftrightarrow> X = \<Union>\<C>"
proof
assume "suc X = X"
with suc_Union_closed_suc [OF suc_Union_closed_Union X \<in> \<C>]
have "\<Union>\<C> \<subseteq> X" .
with X \<in> \<C> show "X = \<Union>\<C>" by blast
next
from X \<in> \<C> have "suc X \<in> \<C>" by (rule suc)
then have "suc X \<subseteq> \<Union>\<C>" by blast
moreover assume "X = \<Union>\<C>"
ultimately have "suc X \<subseteq> X" by simp
moreover have "X \<subseteq> suc X" by (rule suc_subset)
ultimately show "suc X = X" ..
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 {*All elements of @{term \<C>} are chains.*}
lemma suc_Union_closed_chain:
assumes "X \<in> \<C>"
shows "chain X"
using assms
proof (induct)
case (suc X) then show ?case by (simp add: suc_def) (metis not_maxchain_Some)
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 chain v show ?thesis
proof (rule chain_total)
show "y \<in> v" by fact
show "x \<in> v" using u \<subseteq> v and x \<in> u by blast
qed
next
assume "v \<subseteq> u"
from chain u show ?thesis
proof (rule chain_total)
show "x \<in> u" by fact
show "y \<in> u" using v \<subseteq> u and y \<in> v by blast
qed
qed
qed
ultimately show ?case unfolding chain_def ..
qed

subsubsection {* Hausdorff's Maximum Principle *}

text {*There exists a maximal totally ordered subset of @{term A}. (Note that we do not
require @{term A} to be partially ordered.)*}

theorem Hausdorff: "\<exists>C. maxchain C"
proof -
let ?M = "\<Union>\<C>"
have "maxchain ?M"
proof (rule ccontr)
assume "\<not> maxchain ?M"
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 {*Make notation @{term \<C>} available again.*}
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 {*Hide constant @{const pred_on.suc_Union_closed}, which was just needed
for the proof of Hausforff's maximum principle.*}
hide_const pred_on.suc_Union_closed

lemma chain_mono:
assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<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 {* Results for the proper subset relation *}

interpretation subset: pred_on "A" "op \<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 subset.maxchain A C 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 \<Union>C \<subseteq> X by auto
ultimately have "subset.chain A ?C"
using subset.chain_extend [of A C X] and X \<in> A by auto
moreover assume "\<Union>C \<noteq> X"
moreover then have "C \<subset> ?C" using \<Union>C \<subseteq> X by auto
ultimately show False using * by blast
qed

subsubsection {* Zorn's lemma *}

text {*If every chain has an upper bound, then there is a maximal set.*}
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 "Y \<noteq> X"
with Y \<subseteq> X have "\<not> X \<subseteq> Y" by blast
from subset.chain_extend [OF subset.chain A M X \<in> A] and \<forall>X\<in>M. X \<subseteq> Y
have "subset.chain A ({X} \<union> M)" using Y \<subseteq> X by auto
moreover have "M \<subset> {X} \<union> M" using \<forall>X\<in>M. X \<subseteq> Y and \<not> X \<subseteq> Y by auto
ultimately show False
using subset.maxchain A M by (auto simp: subset.maxchain_def)
qed
qed
ultimately show ?thesis by blast
qed

text{*Alternative version of Zorn's lemma for the subset relation.*}
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 subset.maxchain A M]
show "\<Union>M = Z" .
qed
ultimately show ?thesis by blast
qed

subsection {* Zorn's Lemma for Partial Orders *}

text {*Relate old to new definitions.*}

(* Define globally? In Set.thy? *)
definition chain_subset :: "'a set set \<Rightarrow> bool" ("chain\<^sub>\<subseteq>") 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}"

(* Define globally? In Relation.thy? *)
definition Chains :: "('a \<times> 'a) set \<Rightarrow> 'a set set" 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; z \<in> S; \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chains S"
by (unfold chains_def chain_subset_def) 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"
by (auto simp add: chain_subset_def subset.chain_def)

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
by (metis Chains_subset Chains_subset' subset_antisym)

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{*Various other lemmas*}

lemma chainsD: "[| c \<in> chains S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
by (unfold chains_def chain_subset_def) blast

lemma chainsD2: "!!(c :: 'a set set). c \<in> chains S ==> c \<subseteq> S"
by (unfold chains_def) 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)
--{* Mirror r in the set of subsets below (wrt r) elements of A*}
let ?B = "%x. r\<inverse>  {x}" let ?S = "?B  Field r"
{
fix C assume 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A"
let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
have "C = ?B  ?A" using 1 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"
hence "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" using 2 by auto
thus "(a, b) \<in> r \<or> (b, a) \<in> r"
using Preorder r and a \<in> Field r and b \<in> Field r
qed
then obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" using u by auto
have "\<forall>A\<in>C. A \<subseteq> r\<inverse>  {u}" (is "?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
thus "(a, u) \<in> r" using uA and aB and Preorder r
by (auto simp add: preorder_on_def refl_on_def) (metis transD)
qed
then have "\<exists>u\<in>Field r. ?P u" using u \<in> Field r by blast
}
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
hence "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
using po and Preorder r and m \<in> Field r
by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
thus ?thesis using m \<in> Field r by blast
qed

subsection {* The Well Ordering Theorem *}

(* 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
initialSegmentOf :: "('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"
(metis UnCI Un_absorb2 subset_trans)

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:
"chain\<^sub>\<subseteq> R \<Longrightarrow> \<forall>r\<in>R. trans r \<Longrightarrow> trans (\<Union>R)"
apply (metis subsetD)
done

lemma chain_subset_antisym_Union:
"chain\<^sub>\<subseteq> R \<Longrightarrow> \<forall>r\<in>R. antisym r \<Longrightarrow> antisym (\<Union>R)"
apply (auto simp add: chain_subset_def antisym_def)
apply (metis subsetD)
done

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 chain\<^sub>\<subseteq> R and r \<in> R and s \<in> R have "r \<subseteq> s \<or> s \<subseteq> r"
thus "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
proof
assume "r \<subseteq> s" hence "(a, b) \<in> s \<or> (b, a) \<in> s" using assms(2) A
by (simp add: total_on_def) (metis mono_Field subsetD)
thus ?thesis using s \<in> R by blast
next
assume "s \<subseteq> r" hence "(a, b) \<in> r \<or> (b, a) \<in> r" using assms(2) A
by (simp add: total_on_def) (metis mono_Field subsetD)
thus ?thesis using r \<in> R 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
{ fix i have "(f (Suc i), f i) \<in> r"
proof (induct i)
case 0 show ?case by fact
next
case (Suc i)
moreover obtain s where "s \<in> R" and "(f (Suc (Suc i)), f(Suc i)) \<in> s"
using 1 by auto
moreover hence "s initial_segment_of r \<or> r initial_segment_of s"
using assms(1) r \<in> R by (simp add: Chains_def)
ultimately show ?case by (simp add: init_seg_of_def) blast
qed
}
thus False using assms(2) and r \<in> R
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 -
-- {*The initial segment relation on well-orders: *}
let ?WO = "{r::'a rel. Well_order r}"
def I \<equiv> "init_seg_of \<inter> ?WO \<times> ?WO"
have I_init: "I \<subseteq> init_seg_of" by (auto simp: I_def)
hence 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"
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)
hence 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)
-- {*I-chains have upper bounds in ?WO wrt I: their Union*}
{ fix R assume "R \<in> Chains I"
hence Ris: "R \<in> Chains init_seg_of" using mono_Chains [OF I_init] by blast
have subch: "chain\<^sub>\<subseteq> R" using R : 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)"
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 by (auto simp: refl_on_def)
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) and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
show ?thesis by (simp (no_asm_simp)) blast
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
ultimately have "\<Union>R \<in> ?WO \<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)
}
hence 1: "\<forall>R \<in> Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" by (subst FI) blast
--{*Zorn's Lemma yields a maximal well-order m:*}
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] by (auto simp:FI)
--{*Now show by contradiction that m covers the whole type:*}
{ fix x::'a assume "x \<notin> Field m"
--{*We assume that x is not covered and extend m at the top with x*}
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
hence "Field m \<noteq> {}" by(auto simp:Field_def)
moreover have "wf (m - Id)" using Well_order m
--{*The extension of m by x:*}
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 Well_order m by (simp_all add: order_on_defs)
--{*We show that the extension is a well-order*}
have "Refl ?m" using Refl m Fm by (auto simp: refl_on_def)
moreover have "trans ?m" using trans m and x \<notin> Field m
unfolding trans_def Field_def by blast
moreover have "antisym ?m" using antisym m and x \<notin> Field m
unfolding antisym_def Field_def by blast
moreover have "Total ?m" using Total m and Fm by (auto simp: total_on_def)
moreover have "wf (?m - Id)"
proof -
have "wf ?s" using x \<notin> Field m
by (auto simp add: wf_eq_minimal Field_def) metis
thus ?thesis using wf (m - Id) and 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)
--{*We show that the extension is above m*}
moreover hence "(m, ?m) \<in> I" using Well_order m and x \<notin> Field m
by (fastforce simp: I_def init_seg_of_def Field_def)
ultimately
have False using max and x \<notin> Field m unfolding Field_def by blast
}
hence "Field m = UNIV" by auto
moreover with Well_order m have "Well_order m" by simp
ultimately 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)
have "Refl r" and "trans r" and "antisym r" and "Total r" and "wf (r - Id)"
using Well_order r by (simp_all add: order_on_defs)
have "Refl ?r" using Refl r by (auto simp: refl_on_def 1 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 ?r" using Total r by (simp add:total_on_def 1 univ)
moreover have "wf (?r - Id)" by (rule wf_subset [OF wf (r - Id)]) blast
ultimately have "Well_order ?r" by (simp add: order_on_defs)
with 1 show ?thesis by metis
qed

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

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"
by (auto simp: downset_on_def)

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 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 well-order.*}
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 by (auto simp: refl_on_def)
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 (simp (no_asm_simp)) blast
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
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 well-order 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 well-order.*}
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"
using downset_on (Field m) p and min
by (subst Fm, simp add: downset_on_def Field_def) (metis Domain_iff)
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
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
by (force simp: Field_def extension_on_def)
with Well_order m show ?thesis by blast
qed

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