Theory Zorn (Isabelle2005: October 2005)

# Theory Zorn

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theory Zorn
imports Main
begin

    (*  Title       : HOL/Library/Zorn.thy
        ID          : $Id: Zorn.thy,v 1.6 2005/08/31 13:46:37 wenzelm Exp $
        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      The lemma and section numbers refer to an unpublished article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}.
    *}

    constdefs
      chain     ::  "'a set set => 'a set set set"
      "chain S  == {F. F ⊆ S & (∀x ∈ F. ∀y ∈ F. x ⊆ y | y ⊆ x)}"

      super     ::  "['a set set,'a set set] => 'a set set set"
      "super S c == {d. d ∈ chain S & c ⊂ d}"

      maxchain  ::  "'a set set => 'a set set set"
      "maxchain S == {c. c ∈ chain S & super S c = {}}"

      succ      ::  "['a set set,'a set set] => 'a set set"
      "succ S c ==
        if c ∉ chain S | c ∈ maxchain S
        then c else SOME c'. c' ∈ super S c"

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

    inductive "TFin S"
      intros
        succI:        "x ∈ TFin S ==> succ S x ∈ TFin S"
        Pow_UnionI:   "Y ∈ Pow(TFin S) ==> Union(Y) ∈ TFin S"
      monos          Pow_mono

    subsection{*Mathematical Preamble*}

    lemma Union_lemma0:
        "(∀x ∈ C. x ⊆ A | B ⊆ x) ==> Union(C)<=A | B ⊆ Union(C)"
      by blast

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

    lemma Abrial_axiom1: "x ⊆ 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 swap, assumption)
      apply (rule someI2, blast+)
      done

    lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]

    lemma TFin_induct:
              "[| n ∈ TFin S;
                 !!x. [| x ∈ TFin S; P(x) |] ==> P(succ S x);
                 !!Y. [| Y ⊆ TFin S; Ball Y P |] ==> P(Union Y) |]
              ==> P(n)"
      apply (erule TFin.induct)
       apply blast+
      done

    lemma succ_trans: "x ⊆ y ==> x ⊆ succ S y"
      apply (erule subset_trans)
      apply (rule Abrial_axiom1)
      done

    text{*Lemma 1 of section 3.1*}
    lemma TFin_linear_lemma1:
         "[| n ∈ TFin S;  m ∈ TFin S;
             ∀x ∈ TFin S. x ⊆ m --> x = m | succ S x ⊆ m
          |] ==> n ⊆ m | succ S m ⊆ 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 ∈ TFin S ==> ∀n ∈ TFin S. n ⊆ m --> n=m | succ S n ⊆ 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 ⊆ m;  m ∈ TFin S;  n ∈ TFin S |] ==> n=m | succ S n ⊆ 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 ∈ TFin S;  n ∈ TFin S|] ==> n ⊆ m | m ⊆ 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 ∈ TFin S;  m ∈ TFin S;  m = succ S m |] ==> n ⊆ 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 ∈ TFin S ==> (m = succ S m) = (m = Union(TFin S))"
      apply (rule iffI)
       apply (rule Union_upper [THEN equalityI])
        apply (rule_tac [2] eq_succ_upper [THEN Union_least])
          apply (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 "⊆"},
     the subset relation!*}

    lemma empty_set_mem_chain: "({} :: 'a set set) ∈ chain S"
      by (unfold chain_def) auto

    lemma super_subset_chain: "super S c ⊆ chain S"
      by (unfold super_def) blast

    lemma maxchain_subset_chain: "maxchain S ⊆ chain S"
      by (unfold maxchain_def) blast

    lemma mem_super_Ex: "c ∈ chain S - maxchain S ==> ? d. d ∈ super S c"
      by (unfold super_def maxchain_def) auto

    lemma select_super: "c ∈ 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 ∈ chain S - maxchain S ==>
                              (\<some>c'. c': super S c) ≠ c"
      apply (rule notI)
      apply (drule select_super)
      apply (simp add: super_def psubset_def)
      done

    lemma succI3: "c ∈ 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 ∈ chain S - maxchain S ==> succ S c ≠ c"
      apply (frule succI3)
      apply (simp (no_asm_simp))
      apply (rule select_not_equals, assumption)
      done

    lemma TFin_chain_lemma4: "c ∈ 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: "∃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 ∈ chain S; z ∈ S;
            ∀x ∈ c. x<=(z:: 'a set) |] ==> {z} Un c ∈ chain S"
      by (unfold chain_def) blast

    lemma chain_Union_upper: "[| c ∈ chain S; x ∈ c |] ==> x ⊆ Union(c)"
      by (unfold chain_def) auto

    lemma chain_ball_Union_upper: "c ∈ chain S ==> ∀x ∈ c. x ⊆ Union(c)"
      by (unfold chain_def) auto

    lemma maxchain_Zorn:
         "[| c ∈ maxchain S; u ∈ S; Union(c) ⊆ u |] ==> Union(c) = u"
      apply (rule ccontr)
      apply (simp add: maxchain_def)
      apply (erule conjE)
      apply (subgoal_tac " ({u} Un c) ∈ super S c")
       apply simp
      apply (unfold super_def psubset_def)
      apply (blast intro: chain_extend dest: chain_Union_upper)
      done

    theorem Zorn_Lemma:
        "∀c ∈ chain S. Union(c): S ==> ∃y ∈ S. ∀z ∈ S. y ⊆ 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:
      "∀c ∈ chain S. ∃y ∈ S. ∀x ∈ c. x ⊆ y
        ==> ∃y ∈ S. ∀x ∈ S. (y :: 'a set) ⊆ 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 ∈ chain S; x ∈ c; y ∈ c |] ==> x ⊆ y | y ⊆ x"
      by (unfold chain_def) blast

    lemma chainD2: "!!(c :: 'a set set). c ∈ chain S ==> c ⊆ S"
      by (unfold chain_def) blast

    end

### Mathematical Preamble

lemma Union_lemma0:

      ∀x∈C. x ⊆ A ∨ B ⊆ x ==> Union C ⊆ A ∨ B ⊆ Union C

lemma Abrial_axiom1:

      x ⊆ succ S x

lemmas TFin_UnionI:

      Y ⊆ TFin S ==> Union Y ∈ TFin S

lemmas TFin_UnionI:

      Y ⊆ TFin S ==> Union Y ∈ TFin S

lemma TFin_induct:

      [| n ∈ TFin S; !!x. [| x ∈ TFin S; P x |] ==> P (succ S x);
         !!Y. [| Y ⊆ TFin S; Ball Y P |] ==> P (Union Y) |]
      ==> P n

lemma succ_trans:

      x ⊆ y ==> x ⊆ succ S y

lemma TFin_linear_lemma1:

      [| n ∈ TFin S; m ∈ TFin S; ∀x∈TFin S. x ⊆ m --> x = m ∨ succ S x ⊆ m |]
      ==> n ⊆ m ∨ succ S m ⊆ n

lemma TFin_linear_lemma2:

      m ∈ TFin S ==> ∀n∈TFin S. n ⊆ m --> n = m ∨ succ S n ⊆ m

lemma TFin_subsetD:

      [| n ⊆ m; m ∈ TFin S; n ∈ TFin S |] ==> n = m ∨ succ S n ⊆ m

lemma TFin_subset_linear:

      [| m ∈ TFin S; n ∈ TFin S |] ==> n ⊆ m ∨ m ⊆ n

lemma eq_succ_upper:

      [| n ∈ TFin S; m ∈ TFin S; m = succ S m |] ==> n ⊆ m

lemma equal_succ_Union:

      m ∈ TFin S ==> (m = succ S m) = (m = Union (TFin S))

### Hausdorff's Theorem: Every Set Contains a Maximal Chain.

lemma empty_set_mem_chain:

      {} ∈ chain S

lemma super_subset_chain:

      super S c ⊆ chain S

lemma maxchain_subset_chain:

      maxchain S ⊆ chain S

lemma mem_super_Ex:

      c ∈ chain S - maxchain S ==> ∃d. d ∈ super S c

lemma select_super:

      c ∈ chain S - maxchain S ==> (SOME c'. c' ∈ super S c) ∈ super S c

lemma select_not_equals:

      c ∈ chain S - maxchain S ==> (SOME c'. c' ∈ super S c) ≠ c

lemma succI3:

      c ∈ chain S - maxchain S ==> succ S c = (SOME c'. c' ∈ super S c)

lemma succ_not_equals:

      c ∈ chain S - maxchain S ==> succ S c ≠ c

lemma TFin_chain_lemma4:

      c ∈ TFin S ==> c ∈ chain S

theorem Hausdorff:

      ∃c. c ∈ maxchain S

### Zorn's Lemma: If All Chains Have Upper Bounds Then There Is a Maximal Element

lemma chain_extend:

      [| c ∈ chain S; z ∈ S; ∀x∈c. x ⊆ z |] ==> {z} ∪ c ∈ chain S

lemma chain_Union_upper:

      [| c ∈ chain S; x ∈ c |] ==> x ⊆ Union c

lemma chain_ball_Union_upper:

      c ∈ chain S ==> ∀x∈c. x ⊆ Union c

lemma maxchain_Zorn:

      [| c ∈ maxchain S; u ∈ S; Union c ⊆ u |] ==> Union c = u

theorem Zorn_Lemma:

      ∀c∈chain S. Union c ∈ S ==> ∃y∈S. ∀z∈S. y ⊆ z --> y = z

### Alternative version of Zorn's Lemma

lemma Zorn_Lemma2:

      ∀c∈chain S. ∃y∈S. ∀x∈c. x ⊆ y ==> ∃y∈S. ∀x∈S. y ⊆ x --> y = x

lemma chainD:

      [| c ∈ chain S; x ∈ c; y ∈ c |] ==> x ⊆ y ∨ y ⊆ x

lemma chainD2:

      c ∈ chain S ==> c ⊆ S