src/HOL/Zorn.thy
 author paulson Tue Apr 25 16:39:54 2017 +0100 (2017-04-25) changeset 65578 e4997c181cce parent 63572 c0cbfd2b5a45 child 67399 eab6ce8368fa permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
1 (*  Title:       HOL/Zorn.thy
2     Author:      Jacques D. Fleuriot
3     Author:      Tobias Nipkow, TUM
4     Author:      Christian Sternagel, JAIST
6 Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF).
7 The well-ordering theorem.
8 *)
10 section \<open>Zorn's Lemma\<close>
12 theory Zorn
13   imports Order_Relation Hilbert_Choice
14 begin
16 subsection \<open>Zorn's Lemma for the Subset Relation\<close>
18 subsubsection \<open>Results that do not require an order\<close>
20 text \<open>Let \<open>P\<close> be a binary predicate on the set \<open>A\<close>.\<close>
21 locale pred_on =
22   fixes A :: "'a set"
23     and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubset>" 50)
24 begin
26 abbreviation Peq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubseteq>" 50)
27   where "x \<sqsubseteq> y \<equiv> P\<^sup>=\<^sup>= x y"
29 text \<open>A chain is a totally ordered subset of \<open>A\<close>.\<close>
30 definition chain :: "'a set \<Rightarrow> bool"
31   where "chain C \<longleftrightarrow> C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. x \<sqsubseteq> y \<or> y \<sqsubseteq> x)"
33 text \<open>
34   We call a chain that is a proper superset of some set \<open>X\<close>,
35   but not necessarily a chain itself, a superchain of \<open>X\<close>.
36 \<close>
37 abbreviation superchain :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"  (infix "<c" 50)
38   where "X <c C \<equiv> chain C \<and> X \<subset> C"
40 text \<open>A maximal chain is a chain that does not have a superchain.\<close>
41 definition maxchain :: "'a set \<Rightarrow> bool"
42   where "maxchain C \<longleftrightarrow> chain C \<and> (\<nexists>S. C <c S)"
44 text \<open>
45   We define the successor of a set to be an arbitrary
46   superchain, if such exists, or the set itself, otherwise.
47 \<close>
48 definition suc :: "'a set \<Rightarrow> 'a set"
49   where "suc C = (if \<not> chain C \<or> maxchain C then C else (SOME D. C <c D))"
51 lemma chainI [Pure.intro?]: "C \<subseteq> A \<Longrightarrow> (\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x) \<Longrightarrow> chain C"
52   unfolding chain_def by blast
54 lemma chain_total: "chain C \<Longrightarrow> x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
55   by (simp add: chain_def)
57 lemma not_chain_suc [simp]: "\<not> chain X \<Longrightarrow> suc X = X"
58   by (simp add: suc_def)
60 lemma maxchain_suc [simp]: "maxchain X \<Longrightarrow> suc X = X"
61   by (simp add: suc_def)
63 lemma suc_subset: "X \<subseteq> suc X"
64   by (auto simp: suc_def maxchain_def intro: someI2)
66 lemma chain_empty [simp]: "chain {}"
67   by (auto simp: chain_def)
69 lemma not_maxchain_Some: "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> C <c (SOME D. C <c D)"
70   by (rule someI_ex) (auto simp: maxchain_def)
72 lemma suc_not_equals: "chain C \<Longrightarrow> \<not> maxchain C \<Longrightarrow> suc C \<noteq> C"
73   using not_maxchain_Some by (auto simp: suc_def)
75 lemma subset_suc:
76   assumes "X \<subseteq> Y"
77   shows "X \<subseteq> suc Y"
78   using assms by (rule subset_trans) (rule suc_subset)
80 text \<open>
81   We build a set @{term \<C>} that is closed under applications
82   of @{term suc} and contains the union of all its subsets.
83 \<close>
84 inductive_set suc_Union_closed ("\<C>")
85   where
86     suc: "X \<in> \<C> \<Longrightarrow> suc X \<in> \<C>"
87   | Union [unfolded Pow_iff]: "X \<in> Pow \<C> \<Longrightarrow> \<Union>X \<in> \<C>"
89 text \<open>
90   Since the empty set as well as the set itself is a subset of
91   every set, @{term \<C>} contains at least @{term "{} \<in> \<C>"} and
92   @{term "\<Union>\<C> \<in> \<C>"}.
93 \<close>
94 lemma suc_Union_closed_empty: "{} \<in> \<C>"
95   and suc_Union_closed_Union: "\<Union>\<C> \<in> \<C>"
96   using Union [of "{}"] and Union [of "\<C>"] by simp_all
98 text \<open>Thus closure under @{term suc} will hit a maximal chain
99   eventually, as is shown below.\<close>
101 lemma suc_Union_closed_induct [consumes 1, case_names suc Union, induct pred: suc_Union_closed]:
102   assumes "X \<in> \<C>"
103     and "\<And>X. X \<in> \<C> \<Longrightarrow> Q X \<Longrightarrow> Q (suc X)"
104     and "\<And>X. X \<subseteq> \<C> \<Longrightarrow> \<forall>x\<in>X. Q x \<Longrightarrow> Q (\<Union>X)"
105   shows "Q X"
106   using assms by induct blast+
108 lemma suc_Union_closed_cases [consumes 1, case_names suc Union, cases pred: suc_Union_closed]:
109   assumes "X \<in> \<C>"
110     and "\<And>Y. X = suc Y \<Longrightarrow> Y \<in> \<C> \<Longrightarrow> Q"
111     and "\<And>Y. X = \<Union>Y \<Longrightarrow> Y \<subseteq> \<C> \<Longrightarrow> Q"
112   shows "Q"
113   using assms by cases simp_all
115 text \<open>On chains, @{term suc} yields a chain.\<close>
116 lemma chain_suc:
117   assumes "chain X"
118   shows "chain (suc X)"
119   using assms
120   by (cases "\<not> chain X \<or> maxchain X") (force simp: suc_def dest: not_maxchain_Some)+
122 lemma chain_sucD:
123   assumes "chain X"
124   shows "suc X \<subseteq> A \<and> chain (suc X)"
125 proof -
126   from \<open>chain X\<close> have *: "chain (suc X)"
127     by (rule chain_suc)
128   then have "suc X \<subseteq> A"
129     unfolding chain_def by blast
130   with * show ?thesis by blast
131 qed
133 lemma suc_Union_closed_total':
134   assumes "X \<in> \<C>" and "Y \<in> \<C>"
135     and *: "\<And>Z. Z \<in> \<C> \<Longrightarrow> Z \<subseteq> Y \<Longrightarrow> Z = Y \<or> suc Z \<subseteq> Y"
136   shows "X \<subseteq> Y \<or> suc Y \<subseteq> X"
137   using \<open>X \<in> \<C>\<close>
138 proof induct
139   case (suc X)
140   with * show ?case by (blast del: subsetI intro: subset_suc)
141 next
142   case Union
143   then show ?case by blast
144 qed
146 lemma suc_Union_closed_subsetD:
147   assumes "Y \<subseteq> X" and "X \<in> \<C>" and "Y \<in> \<C>"
148   shows "X = Y \<or> suc Y \<subseteq> X"
149   using assms(2,3,1)
150 proof (induct arbitrary: Y)
151   case (suc X)
152   note * = \<open>\<And>Y. Y \<in> \<C> \<Longrightarrow> Y \<subseteq> X \<Longrightarrow> X = Y \<or> suc Y \<subseteq> X\<close>
153   with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>X \<in> \<C>\<close>]
154   have "Y \<subseteq> X \<or> suc X \<subseteq> Y" by blast
155   then show ?case
156   proof
157     assume "Y \<subseteq> X"
158     with * and \<open>Y \<in> \<C>\<close> have "X = Y \<or> suc Y \<subseteq> X" by blast
159     then show ?thesis
160     proof
161       assume "X = Y"
162       then show ?thesis by simp
163     next
164       assume "suc Y \<subseteq> X"
165       then have "suc Y \<subseteq> suc X" by (rule subset_suc)
166       then show ?thesis by simp
167     qed
168   next
169     assume "suc X \<subseteq> Y"
170     with \<open>Y \<subseteq> suc X\<close> show ?thesis by blast
171   qed
172 next
173   case (Union X)
174   show ?case
175   proof (rule ccontr)
176     assume "\<not> ?thesis"
177     with \<open>Y \<subseteq> \<Union>X\<close> obtain x y z
178       where "\<not> suc Y \<subseteq> \<Union>X"
179         and "x \<in> X" and "y \<in> x" and "y \<notin> Y"
180         and "z \<in> suc Y" and "\<forall>x\<in>X. z \<notin> x" by blast
181     with \<open>X \<subseteq> \<C>\<close> have "x \<in> \<C>" by blast
182     from Union and \<open>x \<in> X\<close> have *: "\<And>y. y \<in> \<C> \<Longrightarrow> y \<subseteq> x \<Longrightarrow> x = y \<or> suc y \<subseteq> x"
183       by blast
184     with suc_Union_closed_total' [OF \<open>Y \<in> \<C>\<close> \<open>x \<in> \<C>\<close>] have "Y \<subseteq> x \<or> suc x \<subseteq> Y"
185       by blast
186     then show False
187     proof
188       assume "Y \<subseteq> x"
189       with * [OF \<open>Y \<in> \<C>\<close>] have "x = Y \<or> suc Y \<subseteq> x" by blast
190       then show False
191       proof
192         assume "x = Y"
193         with \<open>y \<in> x\<close> and \<open>y \<notin> Y\<close> show False by blast
194       next
195         assume "suc Y \<subseteq> x"
196         with \<open>x \<in> X\<close> have "suc Y \<subseteq> \<Union>X" by blast
197         with \<open>\<not> suc Y \<subseteq> \<Union>X\<close> show False by contradiction
198       qed
199     next
200       assume "suc x \<subseteq> Y"
201       moreover from suc_subset and \<open>y \<in> x\<close> have "y \<in> suc x" by blast
202       ultimately show False using \<open>y \<notin> Y\<close> by blast
203     qed
204   qed
205 qed
207 text \<open>The elements of @{term \<C>} are totally ordered by the subset relation.\<close>
208 lemma suc_Union_closed_total:
209   assumes "X \<in> \<C>" and "Y \<in> \<C>"
210   shows "X \<subseteq> Y \<or> Y \<subseteq> X"
211 proof (cases "\<forall>Z\<in>\<C>. Z \<subseteq> Y \<longrightarrow> Z = Y \<or> suc Z \<subseteq> Y")
212   case True
213   with suc_Union_closed_total' [OF assms]
214   have "X \<subseteq> Y \<or> suc Y \<subseteq> X" by blast
215   with suc_subset [of Y] show ?thesis by blast
216 next
217   case False
218   then obtain Z where "Z \<in> \<C>" and "Z \<subseteq> Y" and "Z \<noteq> Y" and "\<not> suc Z \<subseteq> Y"
219     by blast
220   with suc_Union_closed_subsetD and \<open>Y \<in> \<C>\<close> show ?thesis
221     by blast
222 qed
224 text \<open>Once we hit a fixed point w.r.t. @{term suc}, all other elements
225   of @{term \<C>} are subsets of this fixed point.\<close>
226 lemma suc_Union_closed_suc:
227   assumes "X \<in> \<C>" and "Y \<in> \<C>" and "suc Y = Y"
228   shows "X \<subseteq> Y"
229   using \<open>X \<in> \<C>\<close>
230 proof induct
231   case (suc X)
232   with \<open>Y \<in> \<C>\<close> and suc_Union_closed_subsetD have "X = Y \<or> suc X \<subseteq> Y"
233     by blast
234   then show ?case
235     by (auto simp: \<open>suc Y = Y\<close>)
236 next
237   case Union
238   then show ?case by blast
239 qed
241 lemma eq_suc_Union:
242   assumes "X \<in> \<C>"
243   shows "suc X = X \<longleftrightarrow> X = \<Union>\<C>"
244     (is "?lhs \<longleftrightarrow> ?rhs")
245 proof
246   assume ?lhs
247   then have "\<Union>\<C> \<subseteq> X"
248     by (rule suc_Union_closed_suc [OF suc_Union_closed_Union \<open>X \<in> \<C>\<close>])
249   with \<open>X \<in> \<C>\<close> show ?rhs
250     by blast
251 next
252   from \<open>X \<in> \<C>\<close> have "suc X \<in> \<C>" by (rule suc)
253   then have "suc X \<subseteq> \<Union>\<C>" by blast
254   moreover assume ?rhs
255   ultimately have "suc X \<subseteq> X" by simp
256   moreover have "X \<subseteq> suc X" by (rule suc_subset)
257   ultimately show ?lhs ..
258 qed
260 lemma suc_in_carrier:
261   assumes "X \<subseteq> A"
262   shows "suc X \<subseteq> A"
263   using assms
264   by (cases "\<not> chain X \<or> maxchain X") (auto dest: chain_sucD)
266 lemma suc_Union_closed_in_carrier:
267   assumes "X \<in> \<C>"
268   shows "X \<subseteq> A"
269   using assms
270   by induct (auto dest: suc_in_carrier)
272 text \<open>All elements of @{term \<C>} are chains.\<close>
273 lemma suc_Union_closed_chain:
274   assumes "X \<in> \<C>"
275   shows "chain X"
276   using assms
277 proof induct
278   case (suc X)
279   then show ?case
280     using not_maxchain_Some by (simp add: suc_def)
281 next
282   case (Union X)
283   then have "\<Union>X \<subseteq> A"
284     by (auto dest: suc_Union_closed_in_carrier)
285   moreover have "\<forall>x\<in>\<Union>X. \<forall>y\<in>\<Union>X. x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
286   proof (intro ballI)
287     fix x y
288     assume "x \<in> \<Union>X" and "y \<in> \<Union>X"
289     then obtain u v where "x \<in> u" and "u \<in> X" and "y \<in> v" and "v \<in> X"
290       by blast
291     with Union have "u \<in> \<C>" and "v \<in> \<C>" and "chain u" and "chain v"
292       by blast+
293     with suc_Union_closed_total have "u \<subseteq> v \<or> v \<subseteq> u"
294       by blast
295     then show "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
296     proof
297       assume "u \<subseteq> v"
298       from \<open>chain v\<close> show ?thesis
299       proof (rule chain_total)
300         show "y \<in> v" by fact
301         show "x \<in> v" using \<open>u \<subseteq> v\<close> and \<open>x \<in> u\<close> by blast
302       qed
303     next
304       assume "v \<subseteq> u"
305       from \<open>chain u\<close> show ?thesis
306       proof (rule chain_total)
307         show "x \<in> u" by fact
308         show "y \<in> u" using \<open>v \<subseteq> u\<close> and \<open>y \<in> v\<close> by blast
309       qed
310     qed
311   qed
312   ultimately show ?case unfolding chain_def ..
313 qed
315 subsubsection \<open>Hausdorff's Maximum Principle\<close>
317 text \<open>There exists a maximal totally ordered subset of \<open>A\<close>. (Note that we do not
318   require \<open>A\<close> to be partially ordered.)\<close>
320 theorem Hausdorff: "\<exists>C. maxchain C"
321 proof -
322   let ?M = "\<Union>\<C>"
323   have "maxchain ?M"
324   proof (rule ccontr)
325     assume "\<not> ?thesis"
326     then have "suc ?M \<noteq> ?M"
327       using suc_not_equals and suc_Union_closed_chain [OF suc_Union_closed_Union] by simp
328     moreover have "suc ?M = ?M"
329       using eq_suc_Union [OF suc_Union_closed_Union] by simp
330     ultimately show False by contradiction
331   qed
332   then show ?thesis by blast
333 qed
335 text \<open>Make notation @{term \<C>} available again.\<close>
336 no_notation suc_Union_closed  ("\<C>")
338 lemma chain_extend: "chain C \<Longrightarrow> z \<in> A \<Longrightarrow> \<forall>x\<in>C. x \<sqsubseteq> z \<Longrightarrow> chain ({z} \<union> C)"
339   unfolding chain_def by blast
341 lemma maxchain_imp_chain: "maxchain C \<Longrightarrow> chain C"
342   by (simp add: maxchain_def)
344 end
346 text \<open>Hide constant @{const pred_on.suc_Union_closed}, which was just needed
347   for the proof of Hausforff's maximum principle.\<close>
348 hide_const pred_on.suc_Union_closed
350 lemma chain_mono:
351   assumes "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> P x y \<Longrightarrow> Q x y"
352     and "pred_on.chain A P C"
353   shows "pred_on.chain A Q C"
354   using assms unfolding pred_on.chain_def by blast
357 subsubsection \<open>Results for the proper subset relation\<close>
359 interpretation subset: pred_on "A" "op \<subset>" for A .
361 lemma subset_maxchain_max:
362   assumes "subset.maxchain A C"
363     and "X \<in> A"
364     and "\<Union>C \<subseteq> X"
365   shows "\<Union>C = X"
366 proof (rule ccontr)
367   let ?C = "{X} \<union> C"
368   from \<open>subset.maxchain A C\<close> have "subset.chain A C"
369     and *: "\<And>S. subset.chain A S \<Longrightarrow> \<not> C \<subset> S"
370     by (auto simp: subset.maxchain_def)
371   moreover have "\<forall>x\<in>C. x \<subseteq> X" using \<open>\<Union>C \<subseteq> X\<close> by auto
372   ultimately have "subset.chain A ?C"
373     using subset.chain_extend [of A C X] and \<open>X \<in> A\<close> by auto
374   moreover assume **: "\<Union>C \<noteq> X"
375   moreover from ** have "C \<subset> ?C" using \<open>\<Union>C \<subseteq> X\<close> by auto
376   ultimately show False using * by blast
377 qed
380 subsubsection \<open>Zorn's lemma\<close>
382 text \<open>If every chain has an upper bound, then there is a maximal set.\<close>
383 lemma subset_Zorn:
384   assumes "\<And>C. subset.chain A C \<Longrightarrow> \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U"
385   shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
386 proof -
387   from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
388   then have "subset.chain A M"
389     by (rule subset.maxchain_imp_chain)
390   with assms obtain Y where "Y \<in> A" and "\<forall>X\<in>M. X \<subseteq> Y"
391     by blast
392   moreover have "\<forall>X\<in>A. Y \<subseteq> X \<longrightarrow> Y = X"
393   proof (intro ballI impI)
394     fix X
395     assume "X \<in> A" and "Y \<subseteq> X"
396     show "Y = X"
397     proof (rule ccontr)
398       assume "\<not> ?thesis"
399       with \<open>Y \<subseteq> X\<close> have "\<not> X \<subseteq> Y" by blast
400       from subset.chain_extend [OF \<open>subset.chain A M\<close> \<open>X \<in> A\<close>] and \<open>\<forall>X\<in>M. X \<subseteq> Y\<close>
401       have "subset.chain A ({X} \<union> M)"
402         using \<open>Y \<subseteq> X\<close> by auto
403       moreover have "M \<subset> {X} \<union> M"
404         using \<open>\<forall>X\<in>M. X \<subseteq> Y\<close> and \<open>\<not> X \<subseteq> Y\<close> by auto
405       ultimately show False
406         using \<open>subset.maxchain A M\<close> by (auto simp: subset.maxchain_def)
407     qed
408   qed
409   ultimately show ?thesis by blast
410 qed
412 text \<open>Alternative version of Zorn's lemma for the subset relation.\<close>
413 lemma subset_Zorn':
414   assumes "\<And>C. subset.chain A C \<Longrightarrow> \<Union>C \<in> A"
415   shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
416 proof -
417   from subset.Hausdorff [of A] obtain M where "subset.maxchain A M" ..
418   then have "subset.chain A M"
419     by (rule subset.maxchain_imp_chain)
420   with assms have "\<Union>M \<in> A" .
421   moreover have "\<forall>Z\<in>A. \<Union>M \<subseteq> Z \<longrightarrow> \<Union>M = Z"
422   proof (intro ballI impI)
423     fix Z
424     assume "Z \<in> A" and "\<Union>M \<subseteq> Z"
425     with subset_maxchain_max [OF \<open>subset.maxchain A M\<close>]
426       show "\<Union>M = Z" .
427   qed
428   ultimately show ?thesis by blast
429 qed
432 subsection \<open>Zorn's Lemma for Partial Orders\<close>
434 text \<open>Relate old to new definitions.\<close>
436 definition chain_subset :: "'a set set \<Rightarrow> bool"  ("chain\<^sub>\<subseteq>")  (* Define globally? In Set.thy? *)
437   where "chain\<^sub>\<subseteq> C \<longleftrightarrow> (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)"
439 definition chains :: "'a set set \<Rightarrow> 'a set set set"
440   where "chains A = {C. C \<subseteq> A \<and> chain\<^sub>\<subseteq> C}"
442 definition Chains :: "('a \<times> 'a) set \<Rightarrow> 'a set set"  (* Define globally? In Relation.thy? *)
443   where "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"
445 lemma chains_extend: "c \<in> chains S \<Longrightarrow> z \<in> S \<Longrightarrow> \<forall>x \<in> c. x \<subseteq> z \<Longrightarrow> {z} \<union> c \<in> chains S"
446   for z :: "'a set"
447   unfolding chains_def chain_subset_def by blast
449 lemma mono_Chains: "r \<subseteq> s \<Longrightarrow> Chains r \<subseteq> Chains s"
450   unfolding Chains_def by blast
452 lemma chain_subset_alt_def: "chain\<^sub>\<subseteq> C = subset.chain UNIV C"
453   unfolding chain_subset_def subset.chain_def by fast
455 lemma chains_alt_def: "chains A = {C. subset.chain A C}"
456   by (simp add: chains_def chain_subset_alt_def subset.chain_def)
458 lemma Chains_subset: "Chains r \<subseteq> {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
459   by (force simp add: Chains_def pred_on.chain_def)
461 lemma Chains_subset':
462   assumes "refl r"
463   shows "{C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C} \<subseteq> Chains r"
464   using assms
465   by (auto simp add: Chains_def pred_on.chain_def refl_on_def)
467 lemma Chains_alt_def:
468   assumes "refl r"
469   shows "Chains r = {C. pred_on.chain UNIV (\<lambda>x y. (x, y) \<in> r) C}"
470   using assms Chains_subset Chains_subset' by blast
472 lemma Zorn_Lemma: "\<forall>C\<in>chains A. \<Union>C \<in> A \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
473   using subset_Zorn' [of A] by (force simp: chains_alt_def)
475 lemma Zorn_Lemma2: "\<forall>C\<in>chains A. \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U \<Longrightarrow> \<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
476   using subset_Zorn [of A] by (auto simp: chains_alt_def)
478 text \<open>Various other lemmas\<close>
480 lemma chainsD: "c \<in> chains S \<Longrightarrow> x \<in> c \<Longrightarrow> y \<in> c \<Longrightarrow> x \<subseteq> y \<or> y \<subseteq> x"
481   unfolding chains_def chain_subset_def by blast
483 lemma chainsD2: "c \<in> chains S \<Longrightarrow> c \<subseteq> S"
484   unfolding chains_def by blast
486 lemma Zorns_po_lemma:
487   assumes po: "Partial_order r"
488     and u: "\<forall>C\<in>Chains r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r"
489   shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
490 proof -
491   have "Preorder r"
492     using po by (simp add: partial_order_on_def)
493   txt \<open>Mirror \<open>r\<close> in the set of subsets below (wrt \<open>r\<close>) elements of \<open>A\<close>.\<close>
494   let ?B = "\<lambda>x. r\<inverse> `` {x}"
495   let ?S = "?B ` Field r"
496   have "\<exists>u\<in>Field r. \<forall>A\<in>C. A \<subseteq> r\<inverse> `` {u}"  (is "\<exists>u\<in>Field r. ?P u")
497     if 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A" for C
498   proof -
499     let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
500     from 1 have "C = ?B ` ?A" by (auto simp: image_def)
501     have "?A \<in> Chains r"
502     proof (simp add: Chains_def, intro allI impI, elim conjE)
503       fix a b
504       assume "a \<in> Field r" and "?B a \<in> C" and "b \<in> Field r" and "?B b \<in> C"
505       with 2 have "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" by auto
506       then show "(a, b) \<in> r \<or> (b, a) \<in> r"
507         using \<open>Preorder r\<close> and \<open>a \<in> Field r\<close> and \<open>b \<in> Field r\<close>
508         by (simp add:subset_Image1_Image1_iff)
509     qed
510     with u obtain u where uA: "u \<in> Field r" "\<forall>a\<in>?A. (a, u) \<in> r" by auto
511     have "?P u"
512     proof auto
513       fix a B assume aB: "B \<in> C" "a \<in> B"
514       with 1 obtain x where "x \<in> Field r" and "B = r\<inverse> `` {x}" by auto
515       then show "(a, u) \<in> r"
516         using uA and aB and \<open>Preorder r\<close>
517         unfolding preorder_on_def refl_on_def by simp (fast dest: transD)
518     qed
519     then show ?thesis
520       using \<open>u \<in> Field r\<close> by blast
521   qed
522   then have "\<forall>C\<in>chains ?S. \<exists>U\<in>?S. \<forall>A\<in>C. A \<subseteq> U"
523     by (auto simp: chains_def chain_subset_def)
524   from Zorn_Lemma2 [OF this] obtain m B
525     where "m \<in> Field r"
526       and "B = r\<inverse> `` {m}"
527       and "\<forall>x\<in>Field r. B \<subseteq> r\<inverse> `` {x} \<longrightarrow> r\<inverse> `` {x} = B"
528     by auto
529   then have "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
530     using po and \<open>Preorder r\<close> and \<open>m \<in> Field r\<close>
531     by (auto simp: subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
532   then show ?thesis
533     using \<open>m \<in> Field r\<close> by blast
534 qed
537 subsection \<open>The Well Ordering Theorem\<close>
539 (* The initial segment of a relation appears generally useful.
540    Move to Relation.thy?
541    Definition correct/most general?
542    Naming?
543 *)
544 definition init_seg_of :: "(('a \<times> 'a) set \<times> ('a \<times> 'a) set) set"
545   where "init_seg_of = {(r, s). r \<subseteq> s \<and> (\<forall>a b c. (a, b) \<in> s \<and> (b, c) \<in> r \<longrightarrow> (a, b) \<in> r)}"
547 abbreviation initial_segment_of_syntax :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"
548     (infix "initial'_segment'_of" 55)
549   where "r initial_segment_of s \<equiv> (r, s) \<in> init_seg_of"
551 lemma refl_on_init_seg_of [simp]: "r initial_segment_of r"
552   by (simp add: init_seg_of_def)
554 lemma trans_init_seg_of:
555   "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
556   by (simp (no_asm_use) add: init_seg_of_def) blast
558 lemma antisym_init_seg_of: "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r = s"
559   unfolding init_seg_of_def by safe
561 lemma Chains_init_seg_of_Union: "R \<in> Chains init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
562   by (auto simp: init_seg_of_def Ball_def Chains_def) blast
564 lemma chain_subset_trans_Union:
565   assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. trans r"
566   shows "trans (\<Union>R)"
567 proof (intro transI, elim UnionE)
568   fix S1 S2 :: "'a rel" and x y z :: 'a
569   assume "S1 \<in> R" "S2 \<in> R"
570   with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1"
571     unfolding chain_subset_def by blast
572   moreover assume "(x, y) \<in> S1" "(y, z) \<in> S2"
573   ultimately have "((x, y) \<in> S1 \<and> (y, z) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, z) \<in> S2)"
574     by blast
575   with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "(x, z) \<in> \<Union>R"
576     by (auto elim: transE)
577 qed
579 lemma chain_subset_antisym_Union:
580   assumes "chain\<^sub>\<subseteq> R" "\<forall>r\<in>R. antisym r"
581   shows "antisym (\<Union>R)"
582 proof (intro antisymI, elim UnionE)
583   fix S1 S2 :: "'a rel" and x y :: 'a
584   assume "S1 \<in> R" "S2 \<in> R"
585   with assms(1) have "S1 \<subseteq> S2 \<or> S2 \<subseteq> S1"
586     unfolding chain_subset_def by blast
587   moreover assume "(x, y) \<in> S1" "(y, x) \<in> S2"
588   ultimately have "((x, y) \<in> S1 \<and> (y, x) \<in> S1) \<or> ((x, y) \<in> S2 \<and> (y, x) \<in> S2)"
589     by blast
590   with \<open>S1 \<in> R\<close> \<open>S2 \<in> R\<close> assms(2) show "x = y"
591     unfolding antisym_def by auto
592 qed
594 lemma chain_subset_Total_Union:
595   assumes "chain\<^sub>\<subseteq> R" and "\<forall>r\<in>R. Total r"
596   shows "Total (\<Union>R)"
597 proof (simp add: total_on_def Ball_def, auto del: disjCI)
598   fix r s a b
599   assume A: "r \<in> R" "s \<in> R" "a \<in> Field r" "b \<in> Field s" "a \<noteq> b"
600   from \<open>chain\<^sub>\<subseteq> R\<close> and \<open>r \<in> R\<close> and \<open>s \<in> R\<close> have "r \<subseteq> s \<or> s \<subseteq> r"
601     by (auto simp add: chain_subset_def)
602   then show "(\<exists>r\<in>R. (a, b) \<in> r) \<or> (\<exists>r\<in>R. (b, a) \<in> r)"
603   proof
604     assume "r \<subseteq> s"
605     then have "(a, b) \<in> s \<or> (b, a) \<in> s"
606       using assms(2) A mono_Field[of r s]
607       by (auto simp add: total_on_def)
608     then show ?thesis
609       using \<open>s \<in> R\<close> by blast
610   next
611     assume "s \<subseteq> r"
612     then have "(a, b) \<in> r \<or> (b, a) \<in> r"
613       using assms(2) A mono_Field[of s r]
614       by (fastforce simp add: total_on_def)
615     then show ?thesis
616       using \<open>r \<in> R\<close> by blast
617   qed
618 qed
620 lemma wf_Union_wf_init_segs:
621   assumes "R \<in> Chains init_seg_of"
622     and "\<forall>r\<in>R. wf r"
623   shows "wf (\<Union>R)"
624 proof (simp add: wf_iff_no_infinite_down_chain, rule ccontr, auto)
625   fix f
626   assume 1: "\<forall>i. \<exists>r\<in>R. (f (Suc i), f i) \<in> r"
627   then obtain r where "r \<in> R" and "(f (Suc 0), f 0) \<in> r" by auto
628   have "(f (Suc i), f i) \<in> r" for i
629   proof (induct i)
630     case 0
631     show ?case by fact
632   next
633     case (Suc i)
634     then obtain s where s: "s \<in> R" "(f (Suc (Suc i)), f(Suc i)) \<in> s"
635       using 1 by auto
636     then have "s initial_segment_of r \<or> r initial_segment_of s"
637       using assms(1) \<open>r \<in> R\<close> by (simp add: Chains_def)
638     with Suc s show ?case by (simp add: init_seg_of_def) blast
639   qed
640   then show False
641     using assms(2) and \<open>r \<in> R\<close>
642     by (simp add: wf_iff_no_infinite_down_chain) blast
643 qed
645 lemma initial_segment_of_Diff: "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s"
646   unfolding init_seg_of_def by blast
648 lemma Chains_inits_DiffI: "R \<in> Chains init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chains init_seg_of"
649   unfolding Chains_def by (blast intro: initial_segment_of_Diff)
651 theorem well_ordering: "\<exists>r::'a rel. Well_order r \<and> Field r = UNIV"
652 proof -
653 \<comment> \<open>The initial segment relation on well-orders:\<close>
654   let ?WO = "{r::'a rel. Well_order r}"
655   define I where "I = init_seg_of \<inter> ?WO \<times> ?WO"
656   then have I_init: "I \<subseteq> init_seg_of" by simp
657   then have subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
658     unfolding init_seg_of_def chain_subset_def Chains_def by blast
659   have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
660     by (simp add: Chains_def I_def) blast
661   have FI: "Field I = ?WO"
662     by (auto simp add: I_def init_seg_of_def Field_def)
663   then have 0: "Partial_order I"
664     by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def
665         trans_def I_def elim!: trans_init_seg_of)
666 \<comment> \<open>\<open>I\<close>-chains have upper bounds in \<open>?WO\<close> wrt \<open>I\<close>: their Union\<close>
667   have "\<Union>R \<in> ?WO \<and> (\<forall>r\<in>R. (r, \<Union>R) \<in> I)" if "R \<in> Chains I" for R
668   proof -
669     from that have Ris: "R \<in> Chains init_seg_of"
670       using mono_Chains [OF I_init] by blast
671     have subch: "chain\<^sub>\<subseteq> R"
672       using \<open>R : Chains I\<close> I_init by (auto simp: init_seg_of_def chain_subset_def Chains_def)
673     have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r"
674       and "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)"
675       using Chains_wo [OF \<open>R \<in> Chains I\<close>] by (simp_all add: order_on_defs)
676     have "Refl (\<Union>R)"
677       using \<open>\<forall>r\<in>R. Refl r\<close> unfolding refl_on_def by fastforce
678     moreover have "trans (\<Union>R)"
679       by (rule chain_subset_trans_Union [OF subch \<open>\<forall>r\<in>R. trans r\<close>])
680     moreover have "antisym (\<Union>R)"
681       by (rule chain_subset_antisym_Union [OF subch \<open>\<forall>r\<in>R. antisym r\<close>])
682     moreover have "Total (\<Union>R)"
683       by (rule chain_subset_Total_Union [OF subch \<open>\<forall>r\<in>R. Total r\<close>])
684     moreover have "wf ((\<Union>R) - Id)"
685     proof -
686       have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
687       with \<open>\<forall>r\<in>R. wf (r - Id)\<close> and wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
688       show ?thesis by fastforce
689     qed
690     ultimately have "Well_order (\<Union>R)"
691       by (simp add:order_on_defs)
692     moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R"
693       using Ris by (simp add: Chains_init_seg_of_Union)
694     ultimately show ?thesis
695       using mono_Chains [OF I_init] Chains_wo[of R] and \<open>R \<in> Chains I\<close>
696       unfolding I_def by blast
697   qed
698   then have 1: "\<forall>R \<in> Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I"
699     by (subst FI) blast
700 \<comment>\<open>Zorn's Lemma yields a maximal well-order \<open>m\<close>:\<close>
701   then obtain m :: "'a rel"
702     where "Well_order m"
703       and max: "\<forall>r. Well_order r \<and> (m, r) \<in> I \<longrightarrow> r = m"
704     using Zorns_po_lemma[OF 0 1] unfolding FI by fastforce
705 \<comment>\<open>Now show by contradiction that \<open>m\<close> covers the whole type:\<close>
706   have False if "x \<notin> Field m" for x :: 'a
707   proof -
708 \<comment>\<open>Assuming that \<open>x\<close> is not covered and extend \<open>m\<close> at the top with \<open>x\<close>\<close>
709     have "m \<noteq> {}"
710     proof
711       assume "m = {}"
712       moreover have "Well_order {(x, x)}"
713         by (simp add: order_on_defs refl_on_def trans_def antisym_def total_on_def Field_def)
714       ultimately show False using max
715         by (auto simp: I_def init_seg_of_def simp del: Field_insert)
716     qed
717     then have "Field m \<noteq> {}" by (auto simp: Field_def)
718     moreover have "wf (m - Id)"
719       using \<open>Well_order m\<close> by (simp add: well_order_on_def)
720 \<comment>\<open>The extension of \<open>m\<close> by \<open>x\<close>:\<close>
721     let ?s = "{(a, x) | a. a \<in> Field m}"
722     let ?m = "insert (x, x) m \<union> ?s"
723     have Fm: "Field ?m = insert x (Field m)"
724       by (auto simp: Field_def)
725     have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
726       using \<open>Well_order m\<close> by (simp_all add: order_on_defs)
727 \<comment>\<open>We show that the extension is a well-order\<close>
728     have "Refl ?m"
729       using \<open>Refl m\<close> Fm unfolding refl_on_def by blast
730     moreover have "trans ?m" using \<open>trans m\<close> and \<open>x \<notin> Field m\<close>
731       unfolding trans_def Field_def by blast
732     moreover have "antisym ?m"
733       using \<open>antisym m\<close> and \<open>x \<notin> Field m\<close> unfolding antisym_def Field_def by blast
734     moreover have "Total ?m"
735       using \<open>Total m\<close> and Fm by (auto simp: total_on_def)
736     moreover have "wf (?m - Id)"
737     proof -
738       have "wf ?s"
739         using \<open>x \<notin> Field m\<close> by (auto simp: wf_eq_minimal Field_def Bex_def)
740       then show ?thesis
741         using \<open>wf (m - Id)\<close> and \<open>x \<notin> Field m\<close> wf_subset [OF \<open>wf ?s\<close> Diff_subset]
742         by (auto simp: Un_Diff Field_def intro: wf_Un)
743     qed
744     ultimately have "Well_order ?m"
745       by (simp add: order_on_defs)
746 \<comment>\<open>We show that the extension is above \<open>m\<close>\<close>
747     moreover have "(m, ?m) \<in> I"
748       using \<open>Well_order ?m\<close> and \<open>Well_order m\<close> and \<open>x \<notin> Field m\<close>
749       by (fastforce simp: I_def init_seg_of_def Field_def)
750     ultimately
751 \<comment>\<open>This contradicts maximality of \<open>m\<close>:\<close>
752     show False
753       using max and \<open>x \<notin> Field m\<close> unfolding Field_def by blast
754   qed
755   then have "Field m = UNIV" by auto
756   with \<open>Well_order m\<close> show ?thesis by blast
757 qed
759 corollary well_order_on: "\<exists>r::'a rel. well_order_on A r"
760 proof -
761   obtain r :: "'a rel" where wo: "Well_order r" and univ: "Field r = UNIV"
762     using well_ordering [where 'a = "'a"] by blast
763   let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
764   have 1: "Field ?r = A"
765     using wo univ by (fastforce simp: Field_def order_on_defs refl_on_def)
766   from \<open>Well_order r\<close> have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"
767     by (simp_all add: order_on_defs)
768   from \<open>Refl r\<close> have "Refl ?r"
769     by (auto simp: refl_on_def 1 univ)
770   moreover from \<open>trans r\<close> have "trans ?r"
771     unfolding trans_def by blast
772   moreover from \<open>antisym r\<close> have "antisym ?r"
773     unfolding antisym_def by blast
774   moreover from \<open>Total r\<close> have "Total ?r"
775     by (simp add:total_on_def 1 univ)
776   moreover have "wf (?r - Id)"
777     by (rule wf_subset [OF \<open>wf (r - Id)\<close>]) blast
778   ultimately have "Well_order ?r"
779     by (simp add: order_on_defs)
780   with 1 show ?thesis by auto
781 qed
783 (* Move this to Hilbert Choice and wfrec to Wellfounded*)
785 lemma wfrec_def_adm: "f \<equiv> wfrec R F \<Longrightarrow> wf R \<Longrightarrow> adm_wf R F \<Longrightarrow> f = F f"
786   using wfrec_fixpoint by simp
788 lemma dependent_wf_choice:
789   fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
790   assumes "wf R"
791     and adm: "\<And>f g x r. (\<And>z. (z, x) \<in> R \<Longrightarrow> f z = g z) \<Longrightarrow> P f x r = P g x r"
792     and P: "\<And>x f. (\<And>y. (y, x) \<in> R \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
793   shows "\<exists>f. \<forall>x. P f x (f x)"
794 proof (intro exI allI)
795   fix x
796   define f where "f \<equiv> wfrec R (\<lambda>f x. SOME r. P f x r)"
797   from \<open>wf R\<close> show "P f x (f x)"
798   proof (induct x)
799     case (less x)
800     show "P f x (f x)"
801     proof (subst (2) wfrec_def_adm[OF f_def \<open>wf R\<close>])
802       show "adm_wf R (\<lambda>f x. SOME r. P f x r)"
803         by (auto simp add: adm_wf_def intro!: arg_cong[where f=Eps] ext adm)
804       show "P f x (Eps (P f x))"
805         using P by (rule someI_ex) fact
806     qed
807   qed
808 qed
810 lemma (in wellorder) dependent_wellorder_choice:
811   assumes "\<And>r f g x. (\<And>y. y < x \<Longrightarrow> f y = g y) \<Longrightarrow> P f x r = P g x r"
812     and P: "\<And>x f. (\<And>y. y < x \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r"
813   shows "\<exists>f. \<forall>x. P f x (f x)"
814   using wf by (rule dependent_wf_choice) (auto intro!: assms)
816 end