Theory Zorn

theory Zorn
imports Order_Union
(*  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.
*)


header {* Zorn's Lemma *}

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 => 'a => bool" (infix "\<sqsubset>" 50)
begin

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

text {*A chain is a totally ordered subset of @{term A}.*}
definition chain :: "'a set => bool" where
"chain C <-> C ⊆ A ∧ (∀x∈C. ∀y∈C. x \<sqsubseteq> y ∨ 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 => 'a set => bool" (infix "<c" 50) where
"X <c C ≡ chain C ∧ X ⊂ C"

text {*A maximal chain is a chain that does not have a superchain.*}
definition maxchain :: "'a set => bool" where
"maxchain C <-> chain C ∧ ¬ (∃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 => 'a set" where
"suc C = (if ¬ chain C ∨ maxchain C then C else (SOME D. C <c D))"

lemma chainI [Pure.intro?]:
"[|C ⊆ A; !!x y. [|x ∈ C; y ∈ C|] ==> x \<sqsubseteq> y ∨ y \<sqsubseteq> x|] ==> chain C"
unfolding chain_def by blast

lemma chain_total:
"chain C ==> x ∈ C ==> y ∈ C ==> x \<sqsubseteq> y ∨ y \<sqsubseteq> x"
by (simp add: chain_def)

lemma not_chain_suc [simp]: "¬ chain X ==> suc X = X"
by (simp add: suc_def)

lemma maxchain_suc [simp]: "maxchain X ==> suc X = X"
by (simp add: suc_def)

lemma suc_subset: "X ⊆ 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 ==> ¬ maxchain C ==> C <c (SOME D. C <c D)"
by (rule someI_ex) (auto simp: maxchain_def)

lemma suc_not_equals:
"chain C ==> ¬ maxchain C ==> suc C ≠ C"
by (auto simp: suc_def) (metis less_irrefl not_maxchain_Some)

lemma subset_suc:
assumes "X ⊆ Y" shows "X ⊆ 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 ∈ \<C> ==> suc X ∈ \<C>" |
Union [unfolded Pow_iff]: "X ∈ Pow \<C> ==> \<Union>X ∈ \<C>"

text {*Since the empty set as well as the set itself is a subset of
every set, @{term \<C>} contains at least @{term "{} ∈ \<C>"} and
@{term "\<Union>\<C> ∈ \<C>"}.*}

lemma
suc_Union_closed_empty: "{} ∈ \<C>" and
suc_Union_closed_Union: "\<Union>\<C> ∈ \<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 ∈ \<C>"
and "!!X. [|X ∈ \<C>; Q X|] ==> Q (suc X)"
and "!!X. [|X ⊆ \<C>; ∀x∈X. Q x|] ==> 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 ∈ \<C>"
and "!!Y. [|X = suc Y; Y ∈ \<C>|] ==> Q"
and "!!Y. [|X = \<Union>Y; Y ⊆ \<C>|] ==> 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 "¬ chain X ∨ maxchain X")
(force simp: suc_def dest: not_maxchain_Some)+

lemma chain_sucD:
assumes "chain X" shows "suc X ⊆ A ∧ chain (suc X)"
proof -
from `chain X` have *: "chain (suc X)" by (rule chain_suc)
then have "suc X ⊆ A" unfolding chain_def by blast
with * show ?thesis by blast
qed

lemma suc_Union_closed_total':
assumes "X ∈ \<C>" and "Y ∈ \<C>"
and *: "!!Z. Z ∈ \<C> ==> Z ⊆ Y ==> Z = Y ∨ suc Z ⊆ Y"
shows "X ⊆ Y ∨ suc Y ⊆ X"
using `X ∈ \<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 ⊆ X" and "X ∈ \<C>" and "Y ∈ \<C>"
shows "X = Y ∨ suc Y ⊆ X"
using assms(2-, 1)
proof (induct arbitrary: Y)
case (suc X)
note * = `!!Y. [|Y ∈ \<C>; Y ⊆ X|] ==> X = Y ∨ suc Y ⊆ X`
with suc_Union_closed_total' [OF `Y ∈ \<C>` `X ∈ \<C>`]
have "Y ⊆ X ∨ suc X ⊆ Y" by blast
then show ?case
proof
assume "Y ⊆ X"
with * and `Y ∈ \<C>` have "X = Y ∨ suc Y ⊆ X" by blast
then show ?thesis
proof
assume "X = Y" then show ?thesis by simp
next
assume "suc Y ⊆ X"
then have "suc Y ⊆ suc X" by (rule subset_suc)
then show ?thesis by simp
qed
next
assume "suc X ⊆ Y"
with `Y ⊆ suc X` show ?thesis by blast
qed
next
case (Union X)
show ?case
proof (rule ccontr)
assume "¬ ?thesis"
with `Y ⊆ \<Union>X` obtain x y z
where "¬ suc Y ⊆ \<Union>X"
and "x ∈ X" and "y ∈ x" and "y ∉ Y"
and "z ∈ suc Y" and "∀x∈X. z ∉ x" by blast
with `X ⊆ \<C>` have "x ∈ \<C>" by blast
from Union and `x ∈ X`
have *: "!!y. [|y ∈ \<C>; y ⊆ x|] ==> x = y ∨ suc y ⊆ x" by blast
with suc_Union_closed_total' [OF `Y ∈ \<C>` `x ∈ \<C>`]
have "Y ⊆ x ∨ suc x ⊆ Y" by blast
then show False
proof
assume "Y ⊆ x"
with * [OF `Y ∈ \<C>`] have "x = Y ∨ suc Y ⊆ x" by blast
then show False
proof
assume "x = Y" with `y ∈ x` and `y ∉ Y` show False by blast
next
assume "suc Y ⊆ x"
with `x ∈ X` have "suc Y ⊆ \<Union>X" by blast
with `¬ suc Y ⊆ \<Union>X` show False by contradiction
qed
next
assume "suc x ⊆ Y"
moreover from suc_subset and `y ∈ x` have "y ∈ suc x" by blast
ultimately show False using `y ∉ 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 ∈ \<C>" and "Y ∈ \<C>"
shows "X ⊆ Y ∨ Y ⊆ X"
proof (cases "∀Z∈\<C>. Z ⊆ Y --> Z = Y ∨ suc Z ⊆ Y")
case True
with suc_Union_closed_total' [OF assms]
have "X ⊆ Y ∨ suc Y ⊆ X" by blast
then show ?thesis using suc_subset [of Y] by blast
next
case False
then obtain Z
where "Z ∈ \<C>" and "Z ⊆ Y" and "Z ≠ Y" and "¬ suc Z ⊆ Y" by blast
with suc_Union_closed_subsetD and `Y ∈ \<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 ∈ \<C>" and "Y ∈ \<C>" and "suc Y = Y"
shows "X ⊆ Y"
using `X ∈ \<C>`
proof (induct)
case (suc X)
with `Y ∈ \<C>` and suc_Union_closed_subsetD
have "X = Y ∨ suc X ⊆ Y" by blast
then show ?case by (auto simp: `suc Y = Y`)
qed blast

lemma eq_suc_Union:
assumes "X ∈ \<C>"
shows "suc X = X <-> X = \<Union>\<C>"
proof
assume "suc X = X"
with suc_Union_closed_suc [OF suc_Union_closed_Union `X ∈ \<C>`]
have "\<Union>\<C> ⊆ X" .
with `X ∈ \<C>` show "X = \<Union>\<C>" by blast
next
from `X ∈ \<C>` have "suc X ∈ \<C>" by (rule suc)
then have "suc X ⊆ \<Union>\<C>" by blast
moreover assume "X = \<Union>\<C>"
ultimately have "suc X ⊆ X" by simp
moreover have "X ⊆ suc X" by (rule suc_subset)
ultimately show "suc X = X" ..
qed

lemma suc_in_carrier:
assumes "X ⊆ A"
shows "suc X ⊆ A"
using assms
by (cases "¬ chain X ∨ maxchain X")
(auto dest: chain_sucD)

lemma suc_Union_closed_in_carrier:
assumes "X ∈ \<C>"
shows "X ⊆ 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 ∈ \<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 ⊆ A" by (auto dest: suc_Union_closed_in_carrier)
moreover have "∀x∈\<Union>X. ∀y∈\<Union>X. x \<sqsubseteq> y ∨ y \<sqsubseteq> x"
proof (intro ballI)
fix x y
assume "x ∈ \<Union>X" and "y ∈ \<Union>X"
then obtain u v where "x ∈ u" and "u ∈ X" and "y ∈ v" and "v ∈ X" by blast
with Union have "u ∈ \<C>" and "v ∈ \<C>" and "chain u" and "chain v" by blast+
with suc_Union_closed_total have "u ⊆ v ∨ v ⊆ u" by blast
then show "x \<sqsubseteq> y ∨ y \<sqsubseteq> x"
proof
assume "u ⊆ v"
from `chain v` show ?thesis
proof (rule chain_total)
show "y ∈ v" by fact
show "x ∈ v" using `u ⊆ v` and `x ∈ u` by blast
qed
next
assume "v ⊆ u"
from `chain u` show ?thesis
proof (rule chain_total)
show "x ∈ u" by fact
show "y ∈ u" using `v ⊆ u` and `y ∈ 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: "∃C. maxchain C"
proof -
let ?M = "\<Union>\<C>"
have "maxchain ?M"
proof (rule ccontr)
assume "¬ maxchain ?M"
then have "suc ?M ≠ ?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
ultimately show False by contradiction
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 ==> z ∈ A ==> ∀x∈C. x \<sqsubseteq> z ==> chain ({z} ∪ C)"
unfolding chain_def by blast

lemma maxchain_imp_chain:
"maxchain C ==> chain C"
by (simp add: maxchain_def)

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 "!!x y. [|x ∈ A; y ∈ A; P x y|] ==> 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 ⊂" for A .

lemma subset_maxchain_max:
assumes "subset.maxchain A C" and "X ∈ A" and "\<Union>C ⊆ X"
shows "\<Union>C = X"
proof (rule ccontr)
let ?C = "{X} ∪ C"
from `subset.maxchain A C` have "subset.chain A C"
and *: "!!S. subset.chain A S ==> ¬ C ⊂ S"
by (auto simp: subset.maxchain_def)
moreover have "∀x∈C. x ⊆ X" using `\<Union>C ⊆ X` by auto
ultimately have "subset.chain A ?C"
using subset.chain_extend [of A C X] and `X ∈ A` by auto
moreover assume **: "\<Union>C ≠ X"
moreover from ** have "C ⊂ ?C" using `\<Union>C ⊆ 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 "!!C. subset.chain A C ==> ∃U∈A. ∀X∈C. X ⊆ U"
shows "∃M∈A. ∀X∈A. M ⊆ X --> 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 ∈ A" and "∀X∈M. X ⊆ Y" by blast
moreover have "∀X∈A. Y ⊆ X --> Y = X"
proof (intro ballI impI)
fix X
assume "X ∈ A" and "Y ⊆ X"
show "Y = X"
proof (rule ccontr)
assume "Y ≠ X"
with `Y ⊆ X` have "¬ X ⊆ Y" by blast
from subset.chain_extend [OF `subset.chain A M` `X ∈ A`] and `∀X∈M. X ⊆ Y`
have "subset.chain A ({X} ∪ M)" using `Y ⊆ X` by auto
moreover have "M ⊂ {X} ∪ M" using `∀X∈M. X ⊆ Y` and `¬ X ⊆ 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 "!!C. subset.chain A C ==> \<Union>C ∈ A"
shows "∃M∈A. ∀X∈A. M ⊆ X --> 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 ∈ A" .
moreover have "∀Z∈A. \<Union>M ⊆ Z --> \<Union>M = Z"
proof (intro ballI impI)
fix Z
assume "Z ∈ A" and "\<Union>M ⊆ 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 => bool" ("chain") where
"chain C <-> (∀A∈C. ∀B∈C. A ⊆ B ∨ B ⊆ A)"

definition chains :: "'a set set => 'a set set set" where
"chains A = {C. C ⊆ A ∧ chain C}"

(* Define globally? In Relation.thy? *)
definition Chains :: "('a × 'a) set => 'a set set" where
"Chains r = {C. ∀a∈C. ∀b∈C. (a, b) ∈ r ∨ (b, a) ∈ r}"

lemma chains_extend:
"[| c ∈ chains S; z ∈ S; ∀x ∈ c. x ⊆ (z:: 'a set) |] ==> {z} Un c ∈ chains S"
by (unfold chains_def chain_subset_def) blast

lemma mono_Chains: "r ⊆ s ==> Chains r ⊆ Chains s"
unfolding Chains_def by blast

lemma chain_subset_alt_def: "chain 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 ⊆ {C. pred_on.chain UNIV (λx y. (x, y) ∈ r) C}"
by (force simp add: Chains_def pred_on.chain_def)

lemma Chains_subset':
assumes "refl r"
shows "{C. pred_on.chain UNIV (λx y. (x, y) ∈ r) C} ⊆ 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 (λx y. (x, y) ∈ r) C}"
using assms
by (metis Chains_subset Chains_subset' subset_antisym)

lemma Zorn_Lemma:
"∀C∈chains A. \<Union>C ∈ A ==> ∃M∈A. ∀X∈A. M ⊆ X --> X = M"
using subset_Zorn' [of A] by (force simp: chains_alt_def)

lemma Zorn_Lemma2:
"∀C∈chains A. ∃U∈A. ∀X∈C. X ⊆ U ==> ∃M∈A. ∀X∈A. M ⊆ X --> X = M"
using subset_Zorn [of A] by (auto simp: chains_alt_def)

text{*Various other lemmas*}

lemma chainsD: "[| c ∈ chains S; x ∈ c; y ∈ c |] ==> x ⊆ y | y ⊆ x"
by (unfold chains_def chain_subset_def) blast

lemma chainsD2: "!!(c :: 'a set set). c ∈ chains S ==> c ⊆ S"
by (unfold chains_def) blast

lemma Zorns_po_lemma:
assumes po: "Partial_order r"
and u: "∀C∈Chains r. ∃u∈Field r. ∀a∈C. (a, u) ∈ r"
shows "∃m∈Field r. ∀a∈Field r. (m, a) ∈ r --> 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¯ `` {x}" let ?S = "?B ` Field r"
{
fix C assume 1: "C ⊆ ?S" and 2: "∀A∈C. ∀B∈C. A ⊆ B ∨ B ⊆ A"
let ?A = "{x∈Field r. ∃M∈C. M = ?B x}"
have "C = ?B ` ?A" using 1 by (auto simp: image_def)
have "?A ∈ Chains r"
proof (simp add: Chains_def, intro allI impI, elim conjE)
fix a b
assume "a ∈ Field r" and "?B a ∈ C" and "b ∈ Field r" and "?B b ∈ C"
hence "?B a ⊆ ?B b ∨ ?B b ⊆ ?B a" using 2 by auto
thus "(a, b) ∈ r ∨ (b, a) ∈ r"
using `Preorder r` and `a ∈ Field r` and `b ∈ Field r`
by (simp add:subset_Image1_Image1_iff)
qed
then obtain u where uA: "u ∈ Field r" "∀a∈?A. (a, u) ∈ r" using u by auto
have "∀A∈C. A ⊆ r¯ `` {u}" (is "?P u")
proof auto
fix a B assume aB: "B ∈ C" "a ∈ B"
with 1 obtain x where "x ∈ Field r" and "B = r¯ `` {x}" by auto
thus "(a, u) ∈ r" using uA and aB and `Preorder r`
by (auto simp add: preorder_on_def refl_on_def) (metis transD)
qed
then have "∃u∈Field r. ?P u" using `u ∈ Field r` by blast
}
then have "∀C∈chains ?S. ∃U∈?S. ∀A∈C. A ⊆ U"
by (auto simp: chains_def chain_subset_def)
from Zorn_Lemma2 [OF this]
obtain m B where "m ∈ Field r" and "B = r¯ `` {m}"
and "∀x∈Field r. B ⊆ r¯ `` {x} --> r¯ `` {x} = B"
by auto
hence "∀a∈Field r. (m, a) ∈ r --> a = m"
using po and `Preorder r` and `m ∈ Field r`
by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
thus ?thesis using `m ∈ 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 × 'a) set × ('a × 'a) set) set" where
"init_seg_of = {(r, s). r ⊆ s ∧ (∀a b c. (a, b) ∈ s ∧ (b, c) ∈ r --> (a, b) ∈ r)}"

abbreviation
initialSegmentOf :: "('a × 'a) set => ('a × 'a) set => bool" (infix "initial'_segment'_of" 55)
where
"r initial_segment_of s ≡ (r, s) ∈ init_seg_of"

lemma refl_on_init_seg_of [simp]: "r initial_segment_of r"
by (simp add: init_seg_of_def)

lemma trans_init_seg_of:
"r initial_segment_of s ==> s initial_segment_of t ==> r initial_segment_of t"
by (simp (no_asm_use) add: init_seg_of_def)
(metis UnCI Un_absorb2 subset_trans)

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

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

lemma chain_subset_trans_Union:
"chain R ==> ∀r∈R. trans r ==> trans (\<Union>R)"
apply (auto simp add: chain_subset_def)
apply (simp (no_asm_use) add: trans_def)
apply (metis subsetD)
done

lemma chain_subset_antisym_Union:
"chain R ==> ∀r∈R. antisym r ==> antisym (\<Union>R)"
apply (auto simp add: chain_subset_def antisym_def)
apply (metis subsetD)
done

lemma chain_subset_Total_Union:
assumes "chain R" and "∀r∈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 ∈ R" "s ∈ R" "a ∈ Field r" "b ∈ Field s" "a ≠ b"
from `chain R` and `r ∈ R` and `s ∈ R` have "r ⊆ s ∨ s ⊆ r"
by (auto simp add: chain_subset_def)
thus "(∃r∈R. (a, b) ∈ r) ∨ (∃r∈R. (b, a) ∈ r)"
proof
assume "r ⊆ s" hence "(a, b) ∈ s ∨ (b, a) ∈ s" using assms(2) A
by (simp add: total_on_def) (metis mono_Field subsetD)
thus ?thesis using `s ∈ R` by blast
next
assume "s ⊆ r" hence "(a, b) ∈ r ∨ (b, a) ∈ r" using assms(2) A
by (simp add: total_on_def) (metis mono_Field subsetD)
thus ?thesis using `r ∈ R` by blast
qed
qed

lemma wf_Union_wf_init_segs:
assumes "R ∈ Chains init_seg_of" and "∀r∈R. wf r"
shows "wf (\<Union>R)"
proof(simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto)
fix f assume 1: "∀i. ∃r∈R. (f (Suc i), f i) ∈ r"
then obtain r where "r ∈ R" and "(f (Suc 0), f 0) ∈ r" by auto
{ fix i have "(f (Suc i), f i) ∈ r"
proof (induct i)
case 0 show ?case by fact
next
case (Suc i)
then obtain s where s: "s ∈ R" "(f (Suc (Suc i)), f(Suc i)) ∈ s"
using 1 by auto
then have "s initial_segment_of r ∨ r initial_segment_of s"
using assms(1) `r ∈ R` by (simp add: Chains_def)
with Suc s show ?case by (simp add: init_seg_of_def) blast
qed
}
thus False using assms(2) and `r ∈ R`
by (simp add: wf_iff_no_infinite_down_chain) blast
qed

lemma initial_segment_of_Diff:
"p initial_segment_of q ==> p - s initial_segment_of q - s"
unfolding init_seg_of_def by blast

lemma Chains_inits_DiffI:
"R ∈ Chains init_seg_of ==> {r - s |r. r ∈ R} ∈ Chains init_seg_of"
unfolding Chains_def by (blast intro: initial_segment_of_Diff)

theorem well_ordering: "∃r::'a rel. Well_order r ∧ Field r = UNIV"
proof -
-- {*The initial segment relation on well-orders: *}
let ?WO = "{r::'a rel. Well_order r}"
def I "init_seg_of ∩ ?WO × ?WO"
have I_init: "I ⊆ init_seg_of" by (auto simp: I_def)
hence subch: "!!R. R ∈ Chains I ==> chain 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"
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 ∈ Chains I"
hence Ris: "R ∈ Chains init_seg_of" using mono_Chains [OF I_init] by blast
have subch: "chain 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)"
using Chains_wo [OF `R ∈ Chains I`] by (simp_all add: order_on_defs)
have "Refl (\<Union>R)" using `∀r∈R. Refl r` by (auto simp: refl_on_def)
moreover have "trans (\<Union>R)"
by (rule chain_subset_trans_Union [OF subch `∀r∈R. trans r`])
moreover have "antisym (\<Union>R)"
by (rule chain_subset_antisym_Union [OF subch `∀r∈R. antisym r`])
moreover have "Total (\<Union>R)"
by (rule chain_subset_Total_Union [OF subch `∀r∈R. Total r`])
moreover have "wf ((\<Union>R) - Id)"
proof -
have "(\<Union>R) - Id = \<Union>{r - Id | r. r ∈ R}" by blast
with `∀r∈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 "∀r ∈ R. r initial_segment_of \<Union>R" using Ris
by(simp add: Chains_init_seg_of_Union)
ultimately have "\<Union>R ∈ ?WO ∧ (∀r∈R. (r, \<Union>R) ∈ I)"
using mono_Chains [OF I_init] and `R ∈ Chains I`
by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo)
}
hence 1: "∀R ∈ Chains I. ∃u∈Field I. ∀r∈R. (r, u) ∈ 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: "∀r. Well_order r ∧ (m, r) ∈ I --> 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 ∉ Field m"
--{*We assume that x is not covered and extend m at the top with x*}
have "m ≠ {}"
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 ≠ {}" by(auto simp:Field_def)
moreover have "wf (m - Id)" using `Well_order m`
by (simp add: well_order_on_def)
--{*The extension of m by x:*}
let ?s = "{(a, x) | a. a ∈ 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 `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 ∉ Field m`
unfolding trans_def Field_def by blast
moreover have "antisym ?m" using `antisym m` and `x ∉ 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 ∉ Field m`
by (auto simp add: wf_eq_minimal Field_def) metis
thus ?thesis using `wf (m - Id)` and `x ∉ 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 have "(m, ?m) ∈ I" using `Well_order ?m` and `Well_order m` and `x ∉ Field m`
by (fastforce simp: I_def init_seg_of_def Field_def)
ultimately
--{*This contradicts maximality of m:*}
have False using max and `x ∉ Field m` unfolding Field_def by blast
}
hence "Field m = UNIV" by auto
with `Well_order m` show ?thesis by blast
qed

corollary well_order_on: "∃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 ∈ A ∧ y ∈ A ∧ (x, y) ∈ 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 = (∀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 <-> downset_on A r ∧ A ⊆ 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"
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 = (∀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 (\<Union>R)) p"
using assms by (auto intro: downset_onI dest: downset_onD)

lemma chain_subset_extension_on_Union:
assumes "chain R" and "!!r. r ∈ R ==> 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 "∃w. p ⊆ w ∧ Well_order w"
proof -
let ?K = "{r. Well_order r ∧ downset_on (Field r) p ∧ extension_on (Field r) r p}"
def 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 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 ∧ downset_on (Field r) p ∧ 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 ∈ Chains I"
then have Ris: "R ∈ Chains init_seg_of" using mono_Chains [OF I_init] by blast
have subch: "chain 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 `R ∈ Chains I`] by (simp_all add: order_on_defs)
have "Refl (\<Union>R)" using `∀r∈R. Refl r` by (auto simp: refl_on_def)
moreover have "trans (\<Union>R)"
by (rule chain_subset_trans_Union [OF subch `∀r∈R. trans r`])
moreover have "antisym (\<Union>R)"
by (rule chain_subset_antisym_Union [OF subch `∀r∈R. antisym r`])
moreover have "Total (\<Union>R)"
by (rule chain_subset_Total_Union [OF subch `∀r∈R. Total r`])
moreover have "wf ((\<Union>R) - Id)"
proof -
have "(\<Union>R) - Id = \<Union>{r - Id | r. r ∈ R}" by blast
with `∀r∈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 "∀r∈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 `!!r. r ∈ R ==> downset_on (Field r) p`])
moreover have "extension_on (Field (\<Union>R)) (\<Union>R) p"
by (rule chain_subset_extension_on_Union [OF subch `!!r. r ∈ R ==> extension_on (Field r) r p`])
ultimately have "\<Union>R ∈ ?K ∧ (∀r∈R. (r,\<Union>R) ∈ I)"
using mono_Chains [OF I_init] and `R ∈ Chains I`
by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo)
}
then have 1: "∀R∈Chains I. ∃u∈Field I. ∀r∈R. (r, u) ∈ 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 × '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) ∈ I --> 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 {*Add @{term x} as topmost element to @{term m}.*}
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 `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 ∉ Field m`
unfolding trans_def Field_def Domain_unfold Domain_converse [symmetric] by blast
moreover have "antisym ?m" using `antisym m` `x ∉ 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 ∉ Field m`
by (simp add: wf_eq_minimal Field_def Domain_unfold Domain_converse [symmetric]) metis
thus ?thesis using `wf (m - Id)` `x ∉ 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) ∈ 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 ∉ 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 ∉ Field m` unfolding Field_def by blast
qed
have "p ⊆ m"
using `Field p ⊆ 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 "∃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 rel.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 `wf p`] 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 `p ⊆ r` have "p ⊆ ?r" using `Field p ⊆ 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 ⊆ ?r` show ?thesis by blast
qed

end