author | wenzelm |
Sat, 27 May 2006 17:42:02 +0200 | |
changeset 19736 | d8d0f8f51d69 |
parent 18585 | 5d379fe2eb74 |
child 21404 | eb85850d3eb7 |
permissions | -rw-r--r-- |
14706 | 1 |
(* Title : HOL/Library/Zorn.thy |
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ID : $Id$ |
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Author : Jacques D. Fleuriot |
|
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Description : Zorn's Lemma -- see Larry Paulson's Zorn.thy in ZF |
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*) |
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header {* Zorn's Lemma *} |
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|
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theory Zorn |
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imports Main |
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begin |
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text{* |
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The lemma and section numbers refer to an unpublished article |
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\cite{Abrial-Laffitte}. |
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*} |
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|
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definition |
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chain :: "'a set set => 'a set set set" |
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"chain S = {F. F \<subseteq> S & (\<forall>x \<in> F. \<forall>y \<in> F. x \<subseteq> y | y \<subseteq> x)}" |
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super :: "['a set set,'a set set] => 'a set set set" |
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"super S c = {d. d \<in> chain S & c \<subset> d}" |
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maxchain :: "'a set set => 'a set set set" |
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"maxchain S = {c. c \<in> chain S & super S c = {}}" |
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succ :: "['a set set,'a set set] => 'a set set" |
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"succ S c = |
30 |
(if c \<notin> chain S | c \<in> maxchain S |
|
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then c else SOME c'. c' \<in> super S c)" |
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consts |
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TFin :: "'a set set => 'a set set set" |
|
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inductive "TFin S" |
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intros |
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succI: "x \<in> TFin S ==> succ S x \<in> TFin S" |
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Pow_UnionI: "Y \<in> Pow(TFin S) ==> Union(Y) \<in> TFin S" |
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monos Pow_mono |
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subsection{*Mathematical Preamble*} |
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lemma Union_lemma0: |
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"(\<forall>x \<in> C. x \<subseteq> A | B \<subseteq> x) ==> Union(C) \<subseteq> A | B \<subseteq> Union(C)" |
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by blast |
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text{*This is theorem @{text increasingD2} of ZF/Zorn.thy*} |
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lemma Abrial_axiom1: "x \<subseteq> succ S x" |
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apply (unfold succ_def) |
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apply (rule split_if [THEN iffD2]) |
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apply (auto simp add: super_def maxchain_def psubset_def) |
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apply (rule contrapos_np, assumption) |
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apply (rule someI2, blast+) |
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done |
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lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI] |
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lemma TFin_induct: |
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"[| n \<in> TFin S; |
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!!x. [| x \<in> TFin S; P(x) |] ==> P(succ S x); |
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!!Y. [| Y \<subseteq> TFin S; Ball Y P |] ==> P(Union Y) |] |
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==> P(n)" |
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apply (induct set: TFin) |
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apply blast+ |
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done |
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69 |
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lemma succ_trans: "x \<subseteq> y ==> x \<subseteq> succ S y" |
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apply (erule subset_trans) |
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apply (rule Abrial_axiom1) |
|
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done |
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text{*Lemma 1 of section 3.1*} |
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lemma TFin_linear_lemma1: |
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"[| n \<in> TFin S; m \<in> TFin S; |
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\<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m |
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|] ==> n \<subseteq> m | succ S m \<subseteq> n" |
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apply (erule TFin_induct) |
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apply (erule_tac [2] Union_lemma0) |
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apply (blast del: subsetI intro: succ_trans) |
|
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done |
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84 |
|
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text{* Lemma 2 of section 3.2 *} |
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lemma TFin_linear_lemma2: |
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"m \<in> TFin S ==> \<forall>n \<in> TFin S. n \<subseteq> m --> n=m | succ S n \<subseteq> m" |
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apply (erule TFin_induct) |
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apply (rule impI [THEN ballI]) |
|
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txt{*case split using @{text TFin_linear_lemma1}*} |
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apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE], |
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assumption+) |
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apply (drule_tac x = n in bspec, assumption) |
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apply (blast del: subsetI intro: succ_trans, blast) |
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txt{*second induction step*} |
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apply (rule impI [THEN ballI]) |
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apply (rule Union_lemma0 [THEN disjE]) |
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apply (rule_tac [3] disjI2) |
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prefer 2 apply blast |
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apply (rule ballI) |
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apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE], |
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assumption+, auto) |
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apply (blast intro!: Abrial_axiom1 [THEN subsetD]) |
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done |
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105 |
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text{*Re-ordering the premises of Lemma 2*} |
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lemma TFin_subsetD: |
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"[| n \<subseteq> m; m \<in> TFin S; n \<in> TFin S |] ==> n=m | succ S n \<subseteq> m" |
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by (rule TFin_linear_lemma2 [rule_format]) |
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110 |
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text{*Consequences from section 3.3 -- Property 3.2, the ordering is total*} |
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lemma TFin_subset_linear: "[| m \<in> TFin S; n \<in> TFin S|] ==> n \<subseteq> m | m \<subseteq> n" |
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apply (rule disjE) |
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apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2]) |
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apply (assumption+, erule disjI2) |
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apply (blast del: subsetI |
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intro: subsetI Abrial_axiom1 [THEN subset_trans]) |
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done |
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119 |
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text{*Lemma 3 of section 3.3*} |
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lemma eq_succ_upper: "[| n \<in> TFin S; m \<in> TFin S; m = succ S m |] ==> n \<subseteq> m" |
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apply (erule TFin_induct) |
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apply (drule TFin_subsetD) |
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apply (assumption+, force, blast) |
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done |
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126 |
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text{*Property 3.3 of section 3.3*} |
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lemma equal_succ_Union: "m \<in> TFin S ==> (m = succ S m) = (m = Union(TFin S))" |
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apply (rule iffI) |
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apply (rule Union_upper [THEN equalityI]) |
|
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apply assumption |
132 |
apply (rule eq_succ_upper [THEN Union_least], assumption+) |
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apply (erule ssubst) |
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apply (rule Abrial_axiom1 [THEN equalityI]) |
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apply (blast del: subsetI intro: subsetI TFin_UnionI TFin.succI) |
|
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done |
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137 |
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subsection{*Hausdorff's Theorem: Every Set Contains a Maximal Chain.*} |
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|
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text{*NB: We assume the partial ordering is @{text "\<subseteq>"}, |
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|
141 |
the subset relation!*} |
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142 |
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lemma empty_set_mem_chain: "({} :: 'a set set) \<in> chain S" |
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by (unfold chain_def) auto |
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145 |
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lemma super_subset_chain: "super S c \<subseteq> chain S" |
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by (unfold super_def) blast |
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148 |
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lemma maxchain_subset_chain: "maxchain S \<subseteq> chain S" |
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by (unfold maxchain_def) blast |
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151 |
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lemma mem_super_Ex: "c \<in> chain S - maxchain S ==> ? d. d \<in> super S c" |
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by (unfold super_def maxchain_def) auto |
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154 |
|
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lemma select_super: |
156 |
"c \<in> chain S - maxchain S ==> (\<some>c'. c': super S c): super S c" |
|
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apply (erule mem_super_Ex [THEN exE]) |
158 |
apply (rule someI2, auto) |
|
159 |
done |
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|
160 |
|
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lemma select_not_equals: |
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"c \<in> chain S - maxchain S ==> (\<some>c'. c': super S c) \<noteq> c" |
|
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apply (rule notI) |
164 |
apply (drule select_super) |
|
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apply (simp add: super_def psubset_def) |
|
166 |
done |
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|
167 |
|
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lemma succI3: "c \<in> chain S - maxchain S ==> succ S c = (\<some>c'. c': super S c)" |
169 |
by (unfold succ_def) (blast intro!: if_not_P) |
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170 |
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171 |
lemma succ_not_equals: "c \<in> chain S - maxchain S ==> succ S c \<noteq> c" |
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apply (frule succI3) |
173 |
apply (simp (no_asm_simp)) |
|
174 |
apply (rule select_not_equals, assumption) |
|
175 |
done |
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176 |
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|
177 |
lemma TFin_chain_lemma4: "c \<in> TFin S ==> (c :: 'a set set): chain S" |
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apply (erule TFin_induct) |
179 |
apply (simp add: succ_def select_super [THEN super_subset_chain[THEN subsetD]]) |
|
180 |
apply (unfold chain_def) |
|
181 |
apply (rule CollectI, safe) |
|
182 |
apply (drule bspec, assumption) |
|
183 |
apply (rule_tac [2] m1 = Xa and n1 = X in TFin_subset_linear [THEN disjE], |
|
184 |
blast+) |
|
185 |
done |
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14706 | 186 |
|
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187 |
theorem Hausdorff: "\<exists>c. (c :: 'a set set): maxchain S" |
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apply (rule_tac x = "Union (TFin S)" in exI) |
17200 | 189 |
apply (rule classical) |
190 |
apply (subgoal_tac "succ S (Union (TFin S)) = Union (TFin S) ") |
|
191 |
prefer 2 |
|
192 |
apply (blast intro!: TFin_UnionI equal_succ_Union [THEN iffD2, symmetric]) |
|
193 |
apply (cut_tac subset_refl [THEN TFin_UnionI, THEN TFin_chain_lemma4]) |
|
194 |
apply (drule DiffI [THEN succ_not_equals], blast+) |
|
195 |
done |
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changeset
|
196 |
|
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197 |
|
14706 | 198 |
subsection{*Zorn's Lemma: If All Chains Have Upper Bounds Then |
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|
199 |
There Is a Maximal Element*} |
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|
200 |
|
14706 | 201 |
lemma chain_extend: |
202 |
"[| c \<in> chain S; z \<in> S; |
|
18143 | 203 |
\<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chain S" |
17200 | 204 |
by (unfold chain_def) blast |
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|
205 |
|
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|
206 |
lemma chain_Union_upper: "[| c \<in> chain S; x \<in> c |] ==> x \<subseteq> Union(c)" |
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by (unfold chain_def) auto |
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changeset
|
208 |
|
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changeset
|
209 |
lemma chain_ball_Union_upper: "c \<in> chain S ==> \<forall>x \<in> c. x \<subseteq> Union(c)" |
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by (unfold chain_def) auto |
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changeset
|
211 |
|
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changeset
|
212 |
lemma maxchain_Zorn: |
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|
213 |
"[| c \<in> maxchain S; u \<in> S; Union(c) \<subseteq> u |] ==> Union(c) = u" |
17200 | 214 |
apply (rule ccontr) |
215 |
apply (simp add: maxchain_def) |
|
216 |
apply (erule conjE) |
|
18143 | 217 |
apply (subgoal_tac "({u} Un c) \<in> super S c") |
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apply simp |
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apply (unfold super_def psubset_def) |
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apply (blast intro: chain_extend dest: chain_Union_upper) |
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done |
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222 |
|
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theorem Zorn_Lemma: |
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"\<forall>c \<in> chain S. Union(c): S ==> \<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z --> y = z" |
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apply (cut_tac Hausdorff maxchain_subset_chain) |
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apply (erule exE) |
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apply (drule subsetD, assumption) |
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apply (drule bspec, assumption) |
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apply (rule_tac x = "Union(c)" in bexI) |
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apply (rule ballI, rule impI) |
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apply (blast dest!: maxchain_Zorn, assumption) |
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done |
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233 |
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subsection{*Alternative version of Zorn's Lemma*} |
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lemma Zorn_Lemma2: |
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"\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>x \<in> c. x \<subseteq> y |
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==> \<exists>y \<in> S. \<forall>x \<in> S. (y :: 'a set) \<subseteq> x --> y = x" |
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apply (cut_tac Hausdorff maxchain_subset_chain) |
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apply (erule exE) |
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apply (drule subsetD, assumption) |
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apply (drule bspec, assumption, erule bexE) |
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apply (rule_tac x = y in bexI) |
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prefer 2 apply assumption |
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apply clarify |
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apply (rule ccontr) |
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apply (frule_tac z = x in chain_extend) |
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apply (assumption, blast) |
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apply (unfold maxchain_def super_def psubset_def) |
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apply (blast elim!: equalityCE) |
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done |
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text{*Various other lemmas*} |
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|
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lemma chainD: "[| c \<in> chain S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x" |
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by (unfold chain_def) blast |
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257 |
|
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lemma chainD2: "!!(c :: 'a set set). c \<in> chain S ==> c \<subseteq> S" |
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by (unfold chain_def) blast |
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260 |
|
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end |