src/HOL/Library/Zorn.thy
 author nipkow Mon Aug 16 14:22:27 2004 +0200 (2004-08-16) changeset 15131 c69542757a4d parent 14706 71590b7733b7 child 15140 322485b816ac permissions -rw-r--r--
1 (*  Title       : HOL/Library/Zorn.thy
2     ID          : \$Id\$
3     Author      : Jacques D. Fleuriot
4     Description : Zorn's Lemma -- see Larry Paulson's Zorn.thy in ZF
5 *)
7 header {* Zorn's Lemma *}
9 theory Zorn
10 import Main
11 begin
13 text{*
14   The lemma and section numbers refer to an unpublished article
15   \cite{Abrial-Laffitte}.
16 *}
18 constdefs
19   chain     ::  "'a set set => 'a set set set"
20   "chain S  == {F. F \<subseteq> S & (\<forall>x \<in> F. \<forall>y \<in> F. x \<subseteq> y | y \<subseteq> x)}"
22   super     ::  "['a set set,'a set set] => 'a set set set"
23   "super S c == {d. d \<in> chain S & c \<subset> d}"
25   maxchain  ::  "'a set set => 'a set set set"
26   "maxchain S == {c. c \<in> chain S & super S c = {}}"
28   succ      ::  "['a set set,'a set set] => 'a set set"
29   "succ S c ==
30     if c \<notin> chain S | c \<in> maxchain S
31     then c else SOME c'. c' \<in> super S c"
33 consts
34   TFin :: "'a set set => 'a set set set"
36 inductive "TFin S"
37   intros
38     succI:        "x \<in> TFin S ==> succ S x \<in> TFin S"
39     Pow_UnionI:   "Y \<in> Pow(TFin S) ==> Union(Y) \<in> TFin S"
40   monos          Pow_mono
43 subsection{*Mathematical Preamble*}
45 lemma Union_lemma0: "(\<forall>x \<in> C. x \<subseteq> A | B \<subseteq> x) ==> Union(C)<=A | B \<subseteq> Union(C)"
46 by blast
49 text{*This is theorem @{text increasingD2} of ZF/Zorn.thy*}
50 lemma Abrial_axiom1: "x \<subseteq> succ S x"
51 apply (unfold succ_def)
52 apply (rule split_if [THEN iffD2])
53 apply (auto simp add: super_def maxchain_def psubset_def)
54 apply (rule swap, assumption)
55 apply (rule someI2, blast+)
56 done
58 lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]
60 lemma TFin_induct:
61           "[| n \<in> TFin S;
62              !!x. [| x \<in> TFin S; P(x) |] ==> P(succ S x);
63              !!Y. [| Y \<subseteq> TFin S; Ball Y P |] ==> P(Union Y) |]
64           ==> P(n)"
65 apply (erule TFin.induct, blast+)
66 done
68 lemma succ_trans: "x \<subseteq> y ==> x \<subseteq> succ S y"
69 apply (erule subset_trans)
70 apply (rule Abrial_axiom1)
71 done
73 text{*Lemma 1 of section 3.1*}
74 lemma TFin_linear_lemma1:
75      "[| n \<in> TFin S;  m \<in> TFin S;
76          \<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m
77       |] ==> n \<subseteq> m | succ S m \<subseteq> n"
78 apply (erule TFin_induct)
79 apply (erule_tac [2] Union_lemma0) (*or just blast*)
80 apply (blast del: subsetI intro: succ_trans)
81 done
83 text{* Lemma 2 of section 3.2 *}
84 lemma TFin_linear_lemma2:
85      "m \<in> TFin S ==> \<forall>n \<in> TFin S. n \<subseteq> m --> n=m | succ S n \<subseteq> m"
86 apply (erule TFin_induct)
87 apply (rule impI [THEN ballI])
88 txt{*case split using @{text TFin_linear_lemma1}*}
89 apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
90        assumption+)
91 apply (drule_tac x = n in bspec, assumption)
92 apply (blast del: subsetI intro: succ_trans, blast)
93 txt{*second induction step*}
94 apply (rule impI [THEN ballI])
95 apply (rule Union_lemma0 [THEN disjE])
96 apply (rule_tac [3] disjI2)
97  prefer 2 apply blast
98 apply (rule ballI)
99 apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
100        assumption+, auto)
101 apply (blast intro!: Abrial_axiom1 [THEN subsetD])
102 done
104 text{*Re-ordering the premises of Lemma 2*}
105 lemma TFin_subsetD:
106      "[| n \<subseteq> m;  m \<in> TFin S;  n \<in> TFin S |] ==> n=m | succ S n \<subseteq> m"
107 apply (rule TFin_linear_lemma2 [rule_format])
108 apply (assumption+)
109 done
111 text{*Consequences from section 3.3 -- Property 3.2, the ordering is total*}
112 lemma TFin_subset_linear: "[| m \<in> TFin S;  n \<in> TFin S|] ==> n \<subseteq> m | m \<subseteq> n"
113 apply (rule disjE)
114 apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2])
115 apply (assumption+, erule disjI2)
116 apply (blast del: subsetI
117              intro: subsetI Abrial_axiom1 [THEN subset_trans])
118 done
120 text{*Lemma 3 of section 3.3*}
121 lemma eq_succ_upper: "[| n \<in> TFin S;  m \<in> TFin S;  m = succ S m |] ==> n \<subseteq> m"
122 apply (erule TFin_induct)
123 apply (drule TFin_subsetD)
124 apply (assumption+, force, blast)
125 done
127 text{*Property 3.3 of section 3.3*}
128 lemma equal_succ_Union: "m \<in> TFin S ==> (m = succ S m) = (m = Union(TFin S))"
129 apply (rule iffI)
130 apply (rule Union_upper [THEN equalityI])
131 apply (rule_tac [2] eq_succ_upper [THEN Union_least])
132 apply (assumption+)
133 apply (erule ssubst)
134 apply (rule Abrial_axiom1 [THEN equalityI])
135 apply (blast del: subsetI
136              intro: subsetI TFin_UnionI TFin.succI)
137 done
139 subsection{*Hausdorff's Theorem: Every Set Contains a Maximal Chain.*}
141 text{*NB: We assume the partial ordering is @{text "\<subseteq>"},
142  the subset relation!*}
144 lemma empty_set_mem_chain: "({} :: 'a set set) \<in> chain S"
145 by (unfold chain_def, auto)
147 lemma super_subset_chain: "super S c \<subseteq> chain S"
148 by (unfold super_def, fast)
150 lemma maxchain_subset_chain: "maxchain S \<subseteq> chain S"
151 by (unfold maxchain_def, fast)
153 lemma mem_super_Ex: "c \<in> chain S - maxchain S ==> ? d. d \<in> super S c"
154 by (unfold super_def maxchain_def, auto)
156 lemma select_super: "c \<in> chain S - maxchain S ==>
157                           (@c'. c': super S c): super S c"
158 apply (erule mem_super_Ex [THEN exE])
159 apply (rule someI2, auto)
160 done
162 lemma select_not_equals: "c \<in> chain S - maxchain S ==>
163                           (@c'. c': super S c) \<noteq> c"
164 apply (rule notI)
165 apply (drule select_super)
166 apply (simp add: super_def psubset_def)
167 done
169 lemma succI3: "c \<in> chain S - maxchain S ==> succ S c = (@c'. c': super S c)"
170 apply (unfold succ_def)
171 apply (fast intro!: if_not_P)
172 done
174 lemma succ_not_equals: "c \<in> chain S - maxchain S ==> succ S c \<noteq> c"
175 apply (frule succI3)
176 apply (simp (no_asm_simp))
177 apply (rule select_not_equals, assumption)
178 done
180 lemma TFin_chain_lemma4: "c \<in> TFin S ==> (c :: 'a set set): chain S"
181 apply (erule TFin_induct)
182 apply (simp add: succ_def select_super [THEN super_subset_chain[THEN subsetD]])
183 apply (unfold chain_def)
184 apply (rule CollectI, safe)
185 apply (drule bspec, assumption)
186 apply (rule_tac [2] m1 = Xa and n1 = X in TFin_subset_linear [THEN disjE],
187        blast+)
188 done
190 theorem Hausdorff: "\<exists>c. (c :: 'a set set): maxchain S"
191 apply (rule_tac x = "Union (TFin S) " in exI)
192 apply (rule classical)
193 apply (subgoal_tac "succ S (Union (TFin S)) = Union (TFin S) ")
194  prefer 2
195  apply (blast intro!: TFin_UnionI equal_succ_Union [THEN iffD2, symmetric])
196 apply (cut_tac subset_refl [THEN TFin_UnionI, THEN TFin_chain_lemma4])
197 apply (drule DiffI [THEN succ_not_equals], blast+)
198 done
201 subsection{*Zorn's Lemma: If All Chains Have Upper Bounds Then
202                                There Is  a Maximal Element*}
204 lemma chain_extend:
205     "[| c \<in> chain S; z \<in> S;
206         \<forall>x \<in> c. x<=(z:: 'a set) |] ==> {z} Un c \<in> chain S"
207 by (unfold chain_def, blast)
209 lemma chain_Union_upper: "[| c \<in> chain S; x \<in> c |] ==> x \<subseteq> Union(c)"
210 by (unfold chain_def, auto)
212 lemma chain_ball_Union_upper: "c \<in> chain S ==> \<forall>x \<in> c. x \<subseteq> Union(c)"
213 by (unfold chain_def, auto)
215 lemma maxchain_Zorn:
216      "[| c \<in> maxchain S; u \<in> S; Union(c) \<subseteq> u |] ==> Union(c) = u"
217 apply (rule ccontr)
219 apply (erule conjE)
220 apply (subgoal_tac " ({u} Un c) \<in> super S c")
221 apply simp
222 apply (unfold super_def psubset_def)
223 apply (blast intro: chain_extend dest: chain_Union_upper)
224 done
226 theorem Zorn_Lemma:
227      "\<forall>c \<in> chain S. Union(c): S ==> \<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z --> y = z"
228 apply (cut_tac Hausdorff maxchain_subset_chain)
229 apply (erule exE)
230 apply (drule subsetD, assumption)
231 apply (drule bspec, assumption)
232 apply (rule_tac x = "Union (c) " in bexI)
233 apply (rule ballI, rule impI)
234 apply (blast dest!: maxchain_Zorn, assumption)
235 done
237 subsection{*Alternative version of Zorn's Lemma*}
239 lemma Zorn_Lemma2:
240      "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>x \<in> c. x \<subseteq> y
241       ==> \<exists>y \<in> S. \<forall>x \<in> S. (y :: 'a set) \<subseteq> x --> y = x"
242 apply (cut_tac Hausdorff maxchain_subset_chain)
243 apply (erule exE)
244 apply (drule subsetD, assumption)
245 apply (drule bspec, assumption, erule bexE)
246 apply (rule_tac x = y in bexI)
247  prefer 2 apply assumption
248 apply clarify
249 apply (rule ccontr)
250 apply (frule_tac z = x in chain_extend)
251 apply (assumption, blast)
252 apply (unfold maxchain_def super_def psubset_def)
253 apply (blast elim!: equalityCE)
254 done
256 text{*Various other lemmas*}
258 lemma chainD: "[| c \<in> chain S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
259 by (unfold chain_def, blast)
261 lemma chainD2: "!!(c :: 'a set set). c \<in> chain S ==> c \<subseteq> S"
262 by (unfold chain_def, blast)
264 end