(*  Title       : HOL/Library/Zorn.thy

    ID          : $Id$

    Author      : Jacques D. Fleuriot

    Description : Zorn's Lemma -- see Larry Paulson's Zorn.thy in ZF

*)

header {* Zorn's Lemma *}

theory Zorn

imports Main

begin

text{*

  The lemma and section numbers refer to an unpublished article

  \cite{Abrial-Laffitte}.

*}

definition

  chain     ::  "'a set set => 'a set set set" where

  "chain S  = {F. F \<subseteq> S & (\<forall>x \<in> F. \<forall>y \<in> F. x \<subseteq> y | y \<subseteq> x)}"

definition

  super     ::  "['a set set,'a set set] => 'a set set set" where

  "super S c = {d. d \<in> chain S & c \<subset> d}"

definition

  maxchain  ::  "'a set set => 'a set set set" where

  "maxchain S = {c. c \<in> chain S & super S c = {}}"

definition

  succ      ::  "['a set set,'a set set] => 'a set set" where

  "succ S c =

    (if c \<notin> chain S | c \<in> maxchain S

    then c else SOME c'. c' \<in> super S c)"

consts

  TFin :: "'a set set => 'a set set set"

inductive "TFin S"

  intros

    succI:        "x \<in> TFin S ==> succ S x \<in> TFin S"

    Pow_UnionI:   "Y \<in> Pow(TFin S) ==> Union(Y) \<in> TFin S"

  monos          Pow_mono

subsection{*Mathematical Preamble*}

lemma Union_lemma0:

    "(\<forall>x \<in> C. x \<subseteq> A | B \<subseteq> x) ==> Union(C) \<subseteq> A | B \<subseteq> Union(C)"

  by blast

text{*This is theorem @{text increasingD2} of ZF/Zorn.thy*}

lemma Abrial_axiom1: "x \<subseteq> succ S x"

  apply (unfold succ_def)

  apply (rule split_if [THEN iffD2])

  apply (auto simp add: super_def maxchain_def psubset_def)

  apply (rule contrapos_np, assumption)

  apply (rule someI2, blast+)

  done

lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]

lemma TFin_induct:

          "[| n \<in> TFin S;

             !!x. [| x \<in> TFin S; P(x) |] ==> P(succ S x);

             !!Y. [| Y \<subseteq> TFin S; Ball Y P |] ==> P(Union Y) |]

          ==> P(n)"

  apply (induct set: TFin)

   apply blast+

  done

lemma succ_trans: "x \<subseteq> y ==> x \<subseteq> succ S y"

  apply (erule subset_trans)

  apply (rule Abrial_axiom1)

  done

text{*Lemma 1 of section 3.1*}

lemma TFin_linear_lemma1:

     "[| n \<in> TFin S;  m \<in> TFin S;

         \<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m

      |] ==> n \<subseteq> m | succ S m \<subseteq> n"

  apply (erule TFin_induct)

   apply (erule_tac [2] Union_lemma0)

  apply (blast del: subsetI intro: succ_trans)

  done

text{* Lemma 2 of section 3.2 *}

lemma TFin_linear_lemma2:

     "m \<in> TFin S ==> \<forall>n \<in> TFin S. n \<subseteq> m --> n=m | succ S n \<subseteq> m"

  apply (erule TFin_induct)

   apply (rule impI [THEN ballI])

   txt{*case split using @{text TFin_linear_lemma1}*}

   apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],

     assumption+)

    apply (drule_tac x = n in bspec, assumption)

    apply (blast del: subsetI intro: succ_trans, blast)

  txt{*second induction step*}

  apply (rule impI [THEN ballI])

  apply (rule Union_lemma0 [THEN disjE])

    apply (rule_tac [3] disjI2)

    prefer 2 apply blast

   apply (rule ballI)

   apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],

     assumption+, auto)

  apply (blast intro!: Abrial_axiom1 [THEN subsetD])

  done

text{*Re-ordering the premises of Lemma 2*}

lemma TFin_subsetD:

     "[| n \<subseteq> m;  m \<in> TFin S;  n \<in> TFin S |] ==> n=m | succ S n \<subseteq> m"

  by (rule TFin_linear_lemma2 [rule_format])

text{*Consequences from section 3.3 -- Property 3.2, the ordering is total*}

lemma TFin_subset_linear: "[| m \<in> TFin S;  n \<in> TFin S|] ==> n \<subseteq> m | m \<subseteq> n"

  apply (rule disjE)

    apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2])

      apply (assumption+, erule disjI2)

  apply (blast del: subsetI

    intro: subsetI Abrial_axiom1 [THEN subset_trans])

  done

text{*Lemma 3 of section 3.3*}

lemma eq_succ_upper: "[| n \<in> TFin S;  m \<in> TFin S;  m = succ S m |] ==> n \<subseteq> m"

  apply (erule TFin_induct)

   apply (drule TFin_subsetD)

     apply (assumption+, force, blast)

  done

text{*Property 3.3 of section 3.3*}

lemma equal_succ_Union: "m \<in> TFin S ==> (m = succ S m) = (m = Union(TFin S))"

  apply (rule iffI)

   apply (rule Union_upper [THEN equalityI])

    apply assumption

   apply (rule eq_succ_upper [THEN Union_least], assumption+)

  apply (erule ssubst)

  apply (rule Abrial_axiom1 [THEN equalityI])

  apply (blast del: subsetI intro: subsetI TFin_UnionI TFin.succI)

  done

subsection{*Hausdorff's Theorem: Every Set Contains a Maximal Chain.*}

text{*NB: We assume the partial ordering is @{text "\<subseteq>"},

 the subset relation!*}

lemma empty_set_mem_chain: "({} :: 'a set set) \<in> chain S"

  by (unfold chain_def) auto

lemma super_subset_chain: "super S c \<subseteq> chain S"

  by (unfold super_def) blast

lemma maxchain_subset_chain: "maxchain S \<subseteq> chain S"

  by (unfold maxchain_def) blast

lemma mem_super_Ex: "c \<in> chain S - maxchain S ==> ? d. d \<in> super S c"

  by (unfold super_def maxchain_def) auto

lemma select_super:

     "c \<in> chain S - maxchain S ==> (\<some>c'. c': super S c): super S c"

  apply (erule mem_super_Ex [THEN exE])

  apply (rule someI2, auto)

  done

lemma select_not_equals:

     "c \<in> chain S - maxchain S ==> (\<some>c'. c': super S c) \<noteq> c"

  apply (rule notI)

  apply (drule select_super)

  apply (simp add: super_def psubset_def)

  done

lemma succI3: "c \<in> chain S - maxchain S ==> succ S c = (\<some>c'. c': super S c)"

  by (unfold succ_def) (blast intro!: if_not_P)

lemma succ_not_equals: "c \<in> chain S - maxchain S ==> succ S c \<noteq> c"

  apply (frule succI3)

  apply (simp (no_asm_simp))

  apply (rule select_not_equals, assumption)

  done

lemma TFin_chain_lemma4: "c \<in> TFin S ==> (c :: 'a set set): chain S"

  apply (erule TFin_induct)

   apply (simp add: succ_def select_super [THEN super_subset_chain[THEN subsetD]])

  apply (unfold chain_def)

  apply (rule CollectI, safe)

   apply (drule bspec, assumption)

   apply (rule_tac [2] m1 = Xa and n1 = X in TFin_subset_linear [THEN disjE],

     blast+)

  done

theorem Hausdorff: "\<exists>c. (c :: 'a set set): maxchain S"

  apply (rule_tac x = "Union (TFin S)" in exI)

  apply (rule classical)

  apply (subgoal_tac "succ S (Union (TFin S)) = Union (TFin S) ")

   prefer 2

   apply (blast intro!: TFin_UnionI equal_succ_Union [THEN iffD2, symmetric])

  apply (cut_tac subset_refl [THEN TFin_UnionI, THEN TFin_chain_lemma4])

  apply (drule DiffI [THEN succ_not_equals], blast+)

  done

subsection{*Zorn's Lemma: If All Chains Have Upper Bounds Then

                               There Is  a Maximal Element*}

lemma chain_extend:

    "[| c \<in> chain S; z \<in> S;

        \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chain S"

  by (unfold chain_def) blast

lemma chain_Union_upper: "[| c \<in> chain S; x \<in> c |] ==> x \<subseteq> Union(c)"

  by (unfold chain_def) auto

lemma chain_ball_Union_upper: "c \<in> chain S ==> \<forall>x \<in> c. x \<subseteq> Union(c)"

  by (unfold chain_def) auto

lemma maxchain_Zorn:

     "[| c \<in> maxchain S; u \<in> S; Union(c) \<subseteq> u |] ==> Union(c) = u"

  apply (rule ccontr)

  apply (simp add: maxchain_def)

  apply (erule conjE)

  apply (subgoal_tac "({u} Un c) \<in> super S c")

   apply simp

  apply (unfold super_def psubset_def)

  apply (blast intro: chain_extend dest: chain_Union_upper)

  done

theorem Zorn_Lemma:

    "\<forall>c \<in> chain S. Union(c): S ==> \<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z --> y = z"

  apply (cut_tac Hausdorff maxchain_subset_chain)

  apply (erule exE)

  apply (drule subsetD, assumption)

  apply (drule bspec, assumption)

  apply (rule_tac x = "Union(c)" in bexI)

   apply (rule ballI, rule impI)

   apply (blast dest!: maxchain_Zorn, assumption)

  done

subsection{*Alternative version of Zorn's Lemma*}

lemma Zorn_Lemma2:

  "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>x \<in> c. x \<subseteq> y

    ==> \<exists>y \<in> S. \<forall>x \<in> S. (y :: 'a set) \<subseteq> x --> y = x"

  apply (cut_tac Hausdorff maxchain_subset_chain)

  apply (erule exE)

  apply (drule subsetD, assumption)

  apply (drule bspec, assumption, erule bexE)

  apply (rule_tac x = y in bexI)

   prefer 2 apply assumption

  apply clarify

  apply (rule ccontr)

  apply (frule_tac z = x in chain_extend)

    apply (assumption, blast)

  apply (unfold maxchain_def super_def psubset_def)

  apply (blast elim!: equalityCE)

  done

text{*Various other lemmas*}

lemma chainD: "[| c \<in> chain S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"

  by (unfold chain_def) blast

lemma chainD2: "!!(c :: 'a set set). c \<in> chain S ==> c \<subseteq> S"

  by (unfold chain_def) blast

end