Theory Zorn

(*  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).
*)

section Zorn's Lemma and the Well-ordering Theorem

theory Zorn
  imports Order_Relation Hilbert_Choice
begin

subsection Zorn's Lemma for the Subset Relation

subsubsection Results that do not require an order

text Let P› be a binary predicate on the set A›.
locale pred_on =
  fixes A :: "'a set"
    and P :: "'a  'a  bool"  (infix "" 50)
begin

abbreviation Peq :: "'a  'a  bool"  (infix "" 50)
  where "x  y  P== x y"

text A chain is a totally ordered subset of A›.
definition chain :: "'a set  bool"
  where "chain C  C  A  (xC. yC. x  y  y  x)"

text 
  We call a chain that is a proper superset of some set X›,
  but not necessarily a chain itself, a superchain of 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  y  y  x)  chain C"
  unfolding chain_def by blast

lemma chain_total: "chain C  x  C  y  C  x  y  y  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"
  using not_maxchain_Some by (auto simp: suc_def)

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

text 
  We build a set term𝒞 that is closed under applications
  of termsuc 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  X  𝒞"

text 
  Since the empty set as well as the set itself is a subset of
  every set, term𝒞 contains at least term{}  𝒞 and
  term𝒞  𝒞.

lemma suc_Union_closed_empty: "{}  𝒞"
  and suc_Union_closed_Union: "𝒞  𝒞"
  using Union [of "{}"] and Union [of "𝒞"] by simp_all

text Thus closure under termsuc 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  xX. Q x  Q (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 = Y  Y  𝒞  Q"
  shows "Q"
  using assms by cases simp_all

text On chains, termsuc 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)
next
  case Union
  then show ?case by blast
qed

lemma suc_Union_closed_subsetD:
  assumes "Y  X" and "X  𝒞" and "Y  𝒞"
  shows "X = Y  suc Y  X"
  using assms(2,3,1)
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 subset_suc show ?thesis
      by fastforce
  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  X obtain x y z
      where "¬ suc Y  X"
        and "x  X" and "y  x" and "y  Y"
        and "z  suc Y" and "xX. 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] y  x y  Y x  X ¬ suc Y  X show False
        by blast
    next
      assume "suc x  Y"
      with y  Y suc_subset y  x show False by blast
    qed
  qed
qed

text The elements of term𝒞 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 "Z𝒞. Z  Y  Z = Y  suc Z  Y")
  case True
  with suc_Union_closed_total' [OF assms]
  have "X  Y  suc Y  X" by blast
  with suc_subset [of Y] show ?thesis 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. termsuc, all other elements
  of term𝒞 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)
next
  case Union
  then show ?case by blast
qed

lemma eq_suc_Union:
  assumes "X  𝒞"
  shows "suc X = X  X = 𝒞"
    (is "?lhs  ?rhs")
proof
  assume ?lhs
  then have "𝒞  X"
    by (rule suc_Union_closed_suc [OF suc_Union_closed_Union X  𝒞])
  with X  𝒞 show ?rhs
    by blast
next
  from X  𝒞 have "suc X  𝒞" by (rule suc)
  then have "suc X  𝒞" by blast
  moreover assume ?rhs
  ultimately have "suc X  X" by simp
  moreover have "X  suc X" by (rule suc_subset)
  ultimately show ?lhs ..
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 term𝒞 are chains.
lemma suc_Union_closed_chain:
  assumes "X  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 "X  A"
    by (auto dest: suc_Union_closed_in_carrier)
  moreover have "xX. yX. x  y  y  x"
  proof (intro ballI)
    fix x y
    assume "x  X" and "y  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  y  y  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 A›. (Note that we do not
  require A› to be partially ordered.)

theorem Hausdorff: "C. maxchain C"
proof -
  let ?M = "𝒞"
  have "maxchain ?M"
  proof (rule ccontr)
    assume "¬ ?thesis"
    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 term𝒞 available again.
no_notation suc_Union_closed  ("𝒞")

lemma chain_extend: "chain C  z  A  xC. x  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 constpred_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" "(⊂)" for A .

lemma subset_maxchain_max:
  assumes "subset.maxchain A C"
    and "X  A"
    and "C  X"
  shows "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 "xC. x  X" using 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 **: "C  X"
  moreover from ** have "C  ?C" using C  X by auto
  ultimately show False using * by blast
qed

lemma subset_chain_def: "𝒜. subset.chain 𝒜 𝒞 = (𝒞  𝒜  (X𝒞. Y𝒞. X  Y  Y  X))"
  by (auto simp: subset.chain_def)

lemma subset_chain_insert:
  "subset.chain 𝒜 (insert B )  B  𝒜  (X. X  B  B  X)  subset.chain 𝒜 "
  by (fastforce simp add: subset_chain_def)

subsubsection Zorn's lemma

text If every chain has an upper bound, then there is a maximal set.
theorem subset_Zorn:
  assumes "C. subset.chain A C  UA. XC. X  U"
  shows "MA. XA. 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 "XM. X  Y"
    by blast
  moreover have "XA. Y  X  Y = X"
  proof (intro ballI impI)
    fix X
    assume "X  A" and "Y  X"
    show "Y = X"
    proof (rule ccontr)
      assume "¬ ?thesis"
      with Y  X have "¬ X  Y" by blast
      from subset.chain_extend [OF subset.chain A M X  A] and XM. X  Y
      have "subset.chain A ({X}  M)"
        using Y  X by auto
      moreover have "M  {X}  M"
        using XM. 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  C  A"
  shows "MA. XA. 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 "M  A" .
  moreover have "ZA. M  Z  M = Z"
  proof (intro ballI impI)
    fix Z
    assume "Z  A" and "M  Z"
    with subset_maxchain_max [OF subset.maxchain A M]
      show "M = Z" .
  qed
  ultimately show ?thesis by blast
qed


subsection Zorn's Lemma for Partial Orders

text Relate old to new definitions.

definition chain_subset :: "'a set set  bool"  ("chain")  (* Define globally? In Set.thy? *)
  where "chain C  (AC. BC. A  B  B  A)"

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

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

lemma chains_extend: "c  chains S  z  S  x  c. x  z  {z}  c  chains S"
  for z :: "'a set"
  unfolding chains_def chain_subset_def by 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"
  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  {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 Chains_subset Chains_subset' by blast

lemma Chains_relation_of:
  assumes "C  Chains (relation_of P A)" shows "C  A"
  using assms unfolding Chains_def relation_of_def by auto

lemma pairwise_chain_Union:
  assumes P: "S. S  𝒞  pairwise R S" and "chain 𝒞"
  shows "pairwise R (𝒞)"
  using chain 𝒞 unfolding pairwise_def chain_subset_def
  by (blast intro: P [unfolded pairwise_def, rule_format])

lemma Zorn_Lemma: "Cchains A. C  A  MA. XA. M  X  X = M"
  using subset_Zorn' [of A] by (force simp: chains_alt_def)

lemma Zorn_Lemma2: "Cchains A. UA. XC. X  U  MA. XA. M  X  X = M"
  using subset_Zorn [of A] by (auto simp: chains_alt_def)

subsection Other variants of Zorn's Lemma

lemma chainsD: "c  chains S  x  c  y  c  x  y  y  x"
  unfolding chains_def chain_subset_def by blast

lemma chainsD2: "c  chains S  c  S"
  unfolding chains_def by blast

lemma Zorns_po_lemma:
  assumes po: "Partial_order r"
    and u: "C. C  Chains r  uField r. aC. (a, u)  r"
  shows "mField r. aField r. (m, a)  r  a = m"
proof -
  have "Preorder r"
    using po by (simp add: partial_order_on_def)
  txt Mirror r› in the set of subsets below (wrt r›) elements of A›.
  let ?B = "λx. r¯ `` {x}"
  let ?S = "?B ` Field r"
  have "uField r. AC. A  r¯ `` {u}"  (is "uField r. ?P u")
    if 1: "C  ?S" and 2: "AC. BC. A  B  B  A" for C
  proof -
    let ?A = "{xField r. MC. M = ?B x}"
    from 1 have "C = ?B ` ?A" 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"
      with 2 have "?B a  ?B b  ?B b  ?B a" by auto
      then show "(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"
      by (auto simp: dest: u)
    have "?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
      then show "(a, u)  r"
        using uA and aB and Preorder r
        unfolding preorder_on_def refl_on_def by simp (fast dest: transD)
    qed
    then show ?thesis
      using u  Field r by blast
  qed
  then have "Cchains ?S. U?S. AC. 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 "xField r. B  r¯ `` {x}  r¯ `` {x} = B"
    by auto
  then have "aField 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)
  then show ?thesis
    using m  Field r by blast
qed

lemma predicate_Zorn:
  assumes po: "partial_order_on A (relation_of P A)"
    and ch: "C. C  Chains (relation_of P A)  u  A. a  C. P a u"
  shows "m  A. a  A. P m a  a = m"
proof -
  have "a  A" if "C  Chains (relation_of P A)" and "a  C" for C a
    using that unfolding Chains_def relation_of_def by auto
  moreover have "(a, u)  relation_of P A" if "a  A" and "u  A" and "P a u" for a u
    unfolding relation_of_def using that by auto
  ultimately have "mA. aA. (m, a)  relation_of P A  a = m"
    using Zorns_po_lemma[OF Partial_order_relation_ofI[OF po], rule_format] ch
    unfolding Field_relation_of[OF partial_order_onD(1)[OF po]] by blast
  then show ?thesis
    by (auto simp: relation_of_def)
qed

lemma Union_in_chain: "finite ;   {}; subset.chain 𝒜     "
proof (induction  rule: finite_induct)
  case (insert B )
  show ?case
  proof (cases " = {}")
    case False
    then show ?thesis
      using insert sup.absorb2 by (auto simp: subset_chain_insert dest!: bspec [where x=""])
  qed auto
qed simp

lemma Inter_in_chain: "finite ;   {}; subset.chain 𝒜     "
proof (induction  rule: finite_induct)
  case (insert B )
  show ?case
  proof (cases " = {}")
    case False
    then show ?thesis
      using insert inf.absorb2 by (auto simp: subset_chain_insert dest!: bspec [where x=""])
  qed auto
qed simp

lemma finite_subset_Union_chain:
  assumes "finite A" "A  " "  {}" and sub: "subset.chain 𝒜 "
  obtains B where "B  " "A  B"
proof -
  obtain  where : "finite " "  " "A  "
    using assms by (auto intro: finite_subset_Union)
  show thesis
  proof (cases " = {}")
    case True
    then show ?thesis
      using A     {} that by fastforce
  next
    case False
    show ?thesis
    proof
      show "  "
        using sub    finite 
        by (simp add: Union_in_chain False subset.chain_def subset_iff)
      show "A  "
        using A   by blast
    qed
  qed
qed

lemma subset_Zorn_nonempty:
  assumes "𝒜  {}" and ch: "𝒞. 𝒞{}; subset.chain 𝒜 𝒞  𝒞  𝒜"
  shows "M𝒜. X𝒜. M  X  X = M"
proof (rule subset_Zorn)
  show "U𝒜. X𝒞. X  U" if "subset.chain 𝒜 𝒞" for 𝒞
  proof (cases "𝒞 = {}")
    case True
    then show ?thesis
      using 𝒜  {} by blast
  next
    case False
    show ?thesis
      by (blast intro!: ch False that Union_upper)
  qed
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 initial_segment_of_syntax :: "('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) blast

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  rR  r initial_segment_of R"
  by (auto simp: init_seg_of_def Ball_def Chains_def) blast

lemma chain_subset_trans_Union:
  assumes "chain R" "rR. trans r"
  shows "trans (R)"
proof (intro transI, elim UnionE)
  fix S1 S2 :: "'a rel" and x y z :: 'a
  assume "S1  R" "S2  R"
  with assms(1) have "S1  S2  S2  S1"
    unfolding chain_subset_def by blast
  moreover assume "(x, y)  S1" "(y, z)  S2"
  ultimately have "((x, y)  S1  (y, z)  S1)  ((x, y)  S2  (y, z)  S2)"
    by blast
  with S1  R S2  R assms(2) show "(x, z)  R"
    by (auto elim: transE)
qed

lemma chain_subset_antisym_Union:
  assumes "chain R" "rR. antisym r"
  shows "antisym (R)"
proof (intro antisymI, elim UnionE)
  fix S1 S2 :: "'a rel" and x y :: 'a
  assume "S1  R" "S2  R"
  with assms(1) have "S1  S2  S2  S1"
    unfolding chain_subset_def by blast
  moreover assume "(x, y)  S1" "(y, x)  S2"
  ultimately have "((x, y)  S1  (y, x)  S1)  ((x, y)  S2  (y, x)  S2)"
    by blast
  with S1  R S2  R assms(2) show "x = y"
    unfolding antisym_def by auto
qed

lemma chain_subset_Total_Union:
  assumes "chain R" and "rR. Total r"
  shows "Total (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)
  then show "(rR. (a, b)  r)  (rR. (b, a)  r)"
  proof
    assume "r  s"
    then have "(a, b)  s  (b, a)  s"
      using assms(2) A mono_Field[of r s]
      by (auto simp add: total_on_def)
    then show ?thesis
      using s  R by blast
  next
    assume "s  r"
    then have "(a, b)  r  (b, a)  r"
      using assms(2) A mono_Field[of s r]
      by (fastforce simp add: total_on_def)
    then show ?thesis
      using r  R by blast
  qed
qed

lemma wf_Union_wf_init_segs:
  assumes "R  Chains init_seg_of"
    and "rR. wf r"
  shows "wf (R)"
proof (simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto)
  fix f
  assume 1: "i. rR. (f (Suc i), f i)  r"
  then obtain r where "r  R" and "(f (Suc 0), f 0)  r" by auto
  have "(f (Suc i), f i)  r" for i
  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
  then show 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}"
  define I where "I = init_seg_of  ?WO × ?WO"
  then have I_init: "I  init_seg_of" by simp
  then have subch: "R. R  Chains I  chain R"
    unfolding init_seg_of_def chain_subset_def Chains_def by blast
  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)
  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)
― ‹I›-chains have upper bounds in ?WO› wrt I›: their Union
  have "R  ?WO  (rR. (r, R)  I)" if "R  Chains I" for R
  proof -
    from that have Ris: "R  Chains init_seg_of"
      using mono_Chains [OF I_init] by blast
    have subch: "chain R"
      using R  Chains I I_init by (auto simp: init_seg_of_def chain_subset_def Chains_def)
    have "rR. Refl r" and "rR. trans r" and "rR. antisym r"
      and "rR. Total r" and "rR. wf (r - Id)"
      using Chains_wo [OF R  Chains I] by (simp_all add: order_on_defs)
    have "Refl (R)"
      using rR. Refl r unfolding refl_on_def by fastforce
    moreover have "trans (R)"
      by (rule chain_subset_trans_Union [OF subch rR. trans r])
    moreover have "antisym (R)"
      by (rule chain_subset_antisym_Union [OF subch rR. antisym r])
    moreover have "Total (R)"
      by (rule chain_subset_Total_Union [OF subch rR. Total r])
    moreover have "wf ((R) - Id)"
    proof -
      have "(R) - Id = {r - Id | r. r  R}" by blast
      with rR. wf (r - Id) and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
      show ?thesis by fastforce
    qed
    ultimately have "Well_order (R)"
      by (simp add:order_on_defs)
    moreover have "r  R. r initial_segment_of R"
      using Ris by (simp add: Chains_init_seg_of_Union)
    ultimately show ?thesis
      using mono_Chains [OF I_init] Chains_wo[of R] and R  Chains I
      unfolding I_def by blast
  qed
  then have 1: "uField I. rR. (r, u)  I" if "R  Chains I" for R
    using that 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] unfolding FI by fastforce
― ‹Now show by contradiction that m› covers the whole type:
  have False if "x  Field m" for x :: 'a
  proof -
― ‹Assuming 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
    then have "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 unfolding refl_on_def by blast
    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: wf_eq_minimal Field_def Bex_def)
      then show ?thesis
        using wf (m - Id) and x  Field m wf_subset [OF wf ?s Diff_subset]
        by (auto simp: Un_Diff Field_def intro: wf_Un)
    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›:
    show False
      using max and x  Field m unfolding Field_def by blast
  qed
  then have "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)
  from Well_order r have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"
    by (simp_all add: order_on_defs)
  from Refl r have "Refl ?r"
    by (auto simp: refl_on_def 1 univ)
  moreover from trans r have "trans ?r"
    unfolding trans_def by blast
  moreover from antisym r have "antisym ?r"
    unfolding antisym_def by blast
  moreover from Total r have "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 auto
qed

lemma dependent_wf_choice:
  fixes P :: "('a  'b)  'a  'b  bool"
  assumes "wf R"
    and adm: "f g x r. (z. (z, x)  R  f z = g z)  P f x r = P g x r"
    and P: "x f. (y. (y, x)  R  P f y (f y))  r. P f x r"
  shows "f. x. P f x (f x)"
proof (intro exI allI)
  fix x
  define f where "f  wfrec R (λf x. SOME r. P f x r)"
  from wf R 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 wf R])
      show "adm_wf R (λf x. SOME r. P f x r)"
        by (auto simp: adm_wf_def intro!: arg_cong[where f=Eps] adm)
      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 "r f g x. (y. y < x  f y = g y)  P f x r = P g x r"
    and P: "x f. (y. y < x  P f y (f y))  r. P f x r"
  shows "f. x. P f x (f x)"
  using wf by (rule dependent_wf_choice) (auto intro!: assms)

end