(* 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The lemma and section numbers refer to an unpublished article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)"
inductive_set
TFin :: "'a set set => 'a set set set"
for S :: "'a set set"
where
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