src/HOL/Library/Zorn.thy
 author haftmann Wed Dec 03 15:58:44 2008 +0100 (2008-12-03) changeset 28952 15a4b2cf8c34 parent 27476 964766deef47 child 30198 922f944f03b2 permissions -rw-r--r--
made repository layout more coherent with logical distribution structure; stripped some \$Id\$s
1 (*  Title       : HOL/Library/Zorn.thy
2     Author      : Jacques D. Fleuriot, Tobias Nipkow
3     Description : Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF)
4                   The well-ordering theorem
5 *)
7 header {* Zorn's Lemma *}
9 theory Zorn
10 imports "~~/src/HOL/Order_Relation"
11 begin
13 (* Define globally? In Set.thy? *)
14 definition chain_subset :: "'a set set \<Rightarrow> bool" ("chain\<^bsub>\<subseteq>\<^esub>") where
15 "chain\<^bsub>\<subseteq>\<^esub> C \<equiv> \<forall>A\<in>C.\<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A"
17 text{*
18   The lemma and section numbers refer to an unpublished article
19   \cite{Abrial-Laffitte}.
20 *}
22 definition
23   chain     ::  "'a set set => 'a set set set" where
24   "chain S  = {F. F \<subseteq> S & chain\<^bsub>\<subseteq>\<^esub> F}"
26 definition
27   super     ::  "['a set set,'a set set] => 'a set set set" where
28   "super S c = {d. d \<in> chain S & c \<subset> d}"
30 definition
31   maxchain  ::  "'a set set => 'a set set set" where
32   "maxchain S = {c. c \<in> chain S & super S c = {}}"
34 definition
35   succ      ::  "['a set set,'a set set] => 'a set set" where
36   "succ S c =
37     (if c \<notin> chain S | c \<in> maxchain S
38     then c else SOME c'. c' \<in> super S c)"
40 inductive_set
41   TFin :: "'a set set => 'a set set set"
42   for S :: "'a set set"
43   where
44     succI:        "x \<in> TFin S ==> succ S x \<in> TFin S"
45   | Pow_UnionI:   "Y \<in> Pow(TFin S) ==> Union(Y) \<in> TFin S"
48 subsection{*Mathematical Preamble*}
50 lemma Union_lemma0:
51     "(\<forall>x \<in> C. x \<subseteq> A | B \<subseteq> x) ==> Union(C) \<subseteq> A | B \<subseteq> Union(C)"
52   by blast
55 text{*This is theorem @{text increasingD2} of ZF/Zorn.thy*}
57 lemma Abrial_axiom1: "x \<subseteq> succ S x"
58   apply (auto simp add: succ_def super_def maxchain_def)
59   apply (rule contrapos_np, assumption)
60   apply (rule_tac Q="\<lambda>S. xa \<in> S" in someI2, blast+)
61   done
63 lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]
65 lemma TFin_induct:
66   assumes H: "n \<in> TFin S"
67   and I: "!!x. x \<in> TFin S ==> P x ==> P (succ S x)"
68     "!!Y. Y \<subseteq> TFin S ==> Ball Y P ==> P(Union Y)"
69   shows "P n" using H
70   apply (induct rule: TFin.induct [where P=P])
71    apply (blast intro: I)+
72   done
74 lemma succ_trans: "x \<subseteq> y ==> x \<subseteq> succ S y"
75   apply (erule subset_trans)
76   apply (rule Abrial_axiom1)
77   done
79 text{*Lemma 1 of section 3.1*}
80 lemma TFin_linear_lemma1:
81      "[| n \<in> TFin S;  m \<in> TFin S;
82          \<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m
83       |] ==> n \<subseteq> m | succ S m \<subseteq> n"
84   apply (erule TFin_induct)
85    apply (erule_tac [2] Union_lemma0)
86   apply (blast del: subsetI intro: succ_trans)
87   done
89 text{* Lemma 2 of section 3.2 *}
90 lemma TFin_linear_lemma2:
91      "m \<in> TFin S ==> \<forall>n \<in> TFin S. n \<subseteq> m --> n=m | succ S n \<subseteq> m"
92   apply (erule TFin_induct)
93    apply (rule impI [THEN ballI])
94    txt{*case split using @{text TFin_linear_lemma1}*}
95    apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
96      assumption+)
97     apply (drule_tac x = n in bspec, assumption)
98     apply (blast del: subsetI intro: succ_trans, blast)
99   txt{*second induction step*}
100   apply (rule impI [THEN ballI])
101   apply (rule Union_lemma0 [THEN disjE])
102     apply (rule_tac [3] disjI2)
103     prefer 2 apply blast
104    apply (rule ballI)
105    apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
106      assumption+, auto)
107   apply (blast intro!: Abrial_axiom1 [THEN subsetD])
108   done
110 text{*Re-ordering the premises of Lemma 2*}
111 lemma TFin_subsetD:
112      "[| n \<subseteq> m;  m \<in> TFin S;  n \<in> TFin S |] ==> n=m | succ S n \<subseteq> m"
113   by (rule TFin_linear_lemma2 [rule_format])
115 text{*Consequences from section 3.3 -- Property 3.2, the ordering is total*}
116 lemma TFin_subset_linear: "[| m \<in> TFin S;  n \<in> TFin S|] ==> n \<subseteq> m | m \<subseteq> n"
117   apply (rule disjE)
118     apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2])
119       apply (assumption+, erule disjI2)
120   apply (blast del: subsetI
121     intro: subsetI Abrial_axiom1 [THEN subset_trans])
122   done
124 text{*Lemma 3 of section 3.3*}
125 lemma eq_succ_upper: "[| n \<in> TFin S;  m \<in> TFin S;  m = succ S m |] ==> n \<subseteq> m"
126   apply (erule TFin_induct)
127    apply (drule TFin_subsetD)
128      apply (assumption+, force, blast)
129   done
131 text{*Property 3.3 of section 3.3*}
132 lemma equal_succ_Union: "m \<in> TFin S ==> (m = succ S m) = (m = Union(TFin S))"
133   apply (rule iffI)
134    apply (rule Union_upper [THEN equalityI])
135     apply assumption
136    apply (rule eq_succ_upper [THEN Union_least], assumption+)
137   apply (erule ssubst)
138   apply (rule Abrial_axiom1 [THEN equalityI])
139   apply (blast del: subsetI intro: subsetI TFin_UnionI TFin.succI)
140   done
142 subsection{*Hausdorff's Theorem: Every Set Contains a Maximal Chain.*}
144 text{*NB: We assume the partial ordering is @{text "\<subseteq>"},
145  the subset relation!*}
147 lemma empty_set_mem_chain: "({} :: 'a set set) \<in> chain S"
148 by (unfold chain_def chain_subset_def) auto
150 lemma super_subset_chain: "super S c \<subseteq> chain S"
151   by (unfold super_def) blast
153 lemma maxchain_subset_chain: "maxchain S \<subseteq> chain S"
154   by (unfold maxchain_def) blast
156 lemma mem_super_Ex: "c \<in> chain S - maxchain S ==> EX d. d \<in> super S c"
157   by (unfold super_def maxchain_def) auto
159 lemma select_super:
160      "c \<in> chain S - maxchain S ==> (\<some>c'. c': super S c): super S c"
161   apply (erule mem_super_Ex [THEN exE])
162   apply (rule someI2 [where Q="%X. X : super S c"], auto)
163   done
165 lemma select_not_equals:
166      "c \<in> chain S - maxchain S ==> (\<some>c'. c': super S c) \<noteq> c"
167   apply (rule notI)
168   apply (drule select_super)
169   apply (simp add: super_def less_le)
170   done
172 lemma succI3: "c \<in> chain S - maxchain S ==> succ S c = (\<some>c'. c': super S c)"
173   by (unfold succ_def) (blast intro!: if_not_P)
175 lemma succ_not_equals: "c \<in> chain S - maxchain S ==> succ S c \<noteq> c"
176   apply (frule succI3)
177   apply (simp (no_asm_simp))
178   apply (rule select_not_equals, assumption)
179   done
181 lemma TFin_chain_lemma4: "c \<in> TFin S ==> (c :: 'a set set): chain S"
182   apply (erule TFin_induct)
183    apply (simp add: succ_def select_super [THEN super_subset_chain[THEN subsetD]])
184   apply (unfold chain_def chain_subset_def)
185   apply (rule CollectI, safe)
186    apply (drule bspec, assumption)
187    apply (rule_tac [2] m1 = Xa and n1 = X in TFin_subset_linear [THEN disjE],
188      best+)
189   done
191 theorem Hausdorff: "\<exists>c. (c :: 'a set set): maxchain S"
192   apply (rule_tac x = "Union (TFin S)" in exI)
193   apply (rule classical)
194   apply (subgoal_tac "succ S (Union (TFin S)) = Union (TFin S) ")
195    prefer 2
196    apply (blast intro!: TFin_UnionI equal_succ_Union [THEN iffD2, symmetric])
197   apply (cut_tac subset_refl [THEN TFin_UnionI, THEN TFin_chain_lemma4])
198   apply (drule DiffI [THEN succ_not_equals], blast+)
199   done
202 subsection{*Zorn's Lemma: If All Chains Have Upper Bounds Then
203                                There Is  a Maximal Element*}
205 lemma chain_extend:
206   "[| c \<in> chain S; z \<in> S; \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chain S"
207 by (unfold chain_def chain_subset_def) blast
209 lemma chain_Union_upper: "[| c \<in> chain S; x \<in> c |] ==> x \<subseteq> Union(c)"
210 by auto
212 lemma chain_ball_Union_upper: "c \<in> chain S ==> \<forall>x \<in> c. x \<subseteq> Union(c)"
213 by 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 less_le)
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 less_le)
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 chain_subset_def) blast
261 lemma chainD2: "!!(c :: 'a set set). c \<in> chain S ==> c \<subseteq> S"
262 by (unfold chain_def) blast
265 (* Define globally? In Relation.thy? *)
266 definition Chain :: "('a*'a)set \<Rightarrow> 'a set set" where
267 "Chain r \<equiv> {A. \<forall>a\<in>A.\<forall>b\<in>A. (a,b) : r \<or> (b,a) \<in> r}"
269 lemma mono_Chain: "r \<subseteq> s \<Longrightarrow> Chain r \<subseteq> Chain s"
270 unfolding Chain_def by blast
272 text{* Zorn's lemma for partial orders: *}
274 lemma Zorns_po_lemma:
275 assumes po: "Partial_order r" and u: "\<forall>C\<in>Chain r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a,u):r"
276 shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m,a):r \<longrightarrow> a=m"
277 proof-
278   have "Preorder r" using po by(simp add:partial_order_on_def)
279 --{* Mirror r in the set of subsets below (wrt r) elements of A*}
280   let ?B = "%x. r^-1 `` {x}" let ?S = "?B ` Field r"
281   have "\<forall>C \<in> chain ?S. EX U:?S. ALL A:C. A\<subseteq>U"
282   proof (auto simp:chain_def chain_subset_def)
283     fix C assume 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C.\<forall>B\<in>C. A\<subseteq>B | B\<subseteq>A"
284     let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
285     have "C = ?B ` ?A" using 1 by(auto simp: image_def)
286     have "?A\<in>Chain r"
287     proof (simp add:Chain_def, intro allI impI, elim conjE)
288       fix a b
289       assume "a \<in> Field r" "?B a \<in> C" "b \<in> Field r" "?B b \<in> C"
290       hence "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" using 2 by auto
291       thus "(a, b) \<in> r \<or> (b, a) \<in> r" using `Preorder r` `a:Field r` `b:Field r`
293     qed
294     then obtain u where uA: "u:Field r" "\<forall>a\<in>?A. (a,u) : r" using u by auto
295     have "\<forall>A\<in>C. A \<subseteq> r^-1 `` {u}" (is "?P u")
296     proof auto
297       fix a B assume aB: "B:C" "a:B"
298       with 1 obtain x where "x:Field r" "B = r^-1 `` {x}" by auto
299       thus "(a,u) : r" using uA aB `Preorder r`
300 	by (auto simp add: preorder_on_def refl_def) (metis transD)
301     qed
302     thus "EX u:Field r. ?P u" using `u:Field r` by blast
303   qed
304   from Zorn_Lemma2[OF this]
305   obtain m B where "m:Field r" "B = r^-1 `` {m}"
306     "\<forall>x\<in>Field r. B \<subseteq> r^-1 `` {x} \<longrightarrow> B = r^-1 `` {x}"
307     by auto
308   hence "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m" using po `Preorder r` `m:Field r`
309     by(auto simp:subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
310   thus ?thesis using `m:Field r` by blast
311 qed
313 (* The initial segment of a relation appears generally useful.
314    Move to Relation.thy?
315    Definition correct/most general?
316    Naming?
317 *)
318 definition init_seg_of :: "(('a*'a)set * ('a*'a)set)set" where
319 "init_seg_of == {(r,s). r \<subseteq> s \<and> (\<forall>a b c. (a,b):s \<and> (b,c):r \<longrightarrow> (a,b):r)}"
321 abbreviation initialSegmentOf :: "('a*'a)set \<Rightarrow> ('a*'a)set \<Rightarrow> bool"
322              (infix "initial'_segment'_of" 55) where
323 "r initial_segment_of s == (r,s):init_seg_of"
325 lemma refl_init_seg_of[simp]: "r initial_segment_of r"
328 lemma trans_init_seg_of:
329   "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
331   (metis Domain_iff UnCI Un_absorb2 subset_trans)
333 lemma antisym_init_seg_of:
334   "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r=s"
335 by(auto simp:init_seg_of_def)
337 lemma Chain_init_seg_of_Union:
338   "R \<in> Chain init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
339 by(auto simp add:init_seg_of_def Chain_def Ball_def) blast
341 lemma chain_subset_trans_Union:
342   "chain\<^bsub>\<subseteq>\<^esub> R \<Longrightarrow> \<forall>r\<in>R. trans r \<Longrightarrow> trans(\<Union>R)"
345 apply (metis subsetD)
346 done
348 lemma chain_subset_antisym_Union:
349   "chain\<^bsub>\<subseteq>\<^esub> R \<Longrightarrow> \<forall>r\<in>R. antisym r \<Longrightarrow> antisym(\<Union>R)"
351 apply (metis subsetD)
352 done
354 lemma chain_subset_Total_Union:
355 assumes "chain\<^bsub>\<subseteq>\<^esub> R" "\<forall>r\<in>R. Total r"
356 shows "Total (\<Union>R)"
357 proof (simp add: total_on_def Ball_def, auto del:disjCI)
358   fix r s a b assume A: "r:R" "s:R" "a:Field r" "b:Field s" "a\<noteq>b"
359   from `chain\<^bsub>\<subseteq>\<^esub> R` `r:R` `s:R` have "r\<subseteq>s \<or> s\<subseteq>r"
361   thus "(\<exists>r\<in>R. (a,b) \<in> r) \<or> (\<exists>r\<in>R. (b,a) \<in> r)"
362   proof
363     assume "r\<subseteq>s" hence "(a,b):s \<or> (b,a):s" using assms(2) A
365     thus ?thesis using `s:R` by blast
366   next
367     assume "s\<subseteq>r" hence "(a,b):r \<or> (b,a):r" using assms(2) A
369     thus ?thesis using `r:R` by blast
370   qed
371 qed
373 lemma wf_Union_wf_init_segs:
374 assumes "R \<in> Chain init_seg_of" and "\<forall>r\<in>R. wf r" shows "wf(\<Union>R)"
375 proof(simp add:wf_iff_no_infinite_down_chain, rule ccontr, auto)
376   fix f assume 1: "\<forall>i. \<exists>r\<in>R. (f(Suc i), f i) \<in> r"
377   then obtain r where "r:R" and "(f(Suc 0), f 0) : r" by auto
378   { fix i have "(f(Suc i), f i) \<in> r"
379     proof(induct i)
380       case 0 show ?case by fact
381     next
382       case (Suc i)
383       moreover obtain s where "s\<in>R" and "(f(Suc(Suc i)), f(Suc i)) \<in> s"
384 	using 1 by auto
385       moreover hence "s initial_segment_of r \<or> r initial_segment_of s"
386 	using assms(1) `r:R` by(simp add: Chain_def)
387       ultimately show ?case by(simp add:init_seg_of_def) blast
388     qed
389   }
390   thus False using assms(2) `r:R`
392 qed
394 lemma initial_segment_of_Diff:
395   "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s"
396 unfolding init_seg_of_def by blast
398 lemma Chain_inits_DiffI:
399   "R \<in> Chain init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chain init_seg_of"
400 unfolding Chain_def by (blast intro: initial_segment_of_Diff)
402 theorem well_ordering: "\<exists>r::('a*'a)set. Well_order r \<and> Field r = UNIV"
403 proof-
404 -- {*The initial segment relation on well-orders: *}
405   let ?WO = "{r::('a*'a)set. Well_order r}"
406   def I \<equiv> "init_seg_of \<inter> ?WO \<times> ?WO"
407   have I_init: "I \<subseteq> init_seg_of" by(auto simp:I_def)
408   hence subch: "!!R. R : Chain I \<Longrightarrow> chain\<^bsub>\<subseteq>\<^esub> R"
409     by(auto simp:init_seg_of_def chain_subset_def Chain_def)
410   have Chain_wo: "!!R r. R \<in> Chain I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
412   have FI: "Field I = ?WO" by(auto simp add:I_def init_seg_of_def Field_def)
413   hence 0: "Partial_order I"
414     by(auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_def trans_def I_def elim!: trans_init_seg_of)
415 -- {*I-chains have upper bounds in ?WO wrt I: their Union*}
416   { fix R assume "R \<in> Chain I"
417     hence Ris: "R \<in> Chain init_seg_of" using mono_Chain[OF I_init] by blast
418     have subch: "chain\<^bsub>\<subseteq>\<^esub> R" using `R : Chain I` I_init
419       by(auto simp:init_seg_of_def chain_subset_def Chain_def)
420     have "\<forall>r\<in>R. Refl r" "\<forall>r\<in>R. trans r" "\<forall>r\<in>R. antisym r" "\<forall>r\<in>R. Total r"
421          "\<forall>r\<in>R. wf(r-Id)"
422       using Chain_wo[OF `R \<in> Chain I`] by(simp_all add:order_on_defs)
423     have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` by(auto simp:refl_def)
424     moreover have "trans (\<Union>R)"
425       by(rule chain_subset_trans_Union[OF subch `\<forall>r\<in>R. trans r`])
426     moreover have "antisym(\<Union>R)"
427       by(rule chain_subset_antisym_Union[OF subch `\<forall>r\<in>R. antisym r`])
428     moreover have "Total (\<Union>R)"
429       by(rule chain_subset_Total_Union[OF subch `\<forall>r\<in>R. Total r`])
430     moreover have "wf((\<Union>R)-Id)"
431     proof-
432       have "(\<Union>R)-Id = \<Union>{r-Id|r. r \<in> R}" by blast
433       with `\<forall>r\<in>R. wf(r-Id)` wf_Union_wf_init_segs[OF Chain_inits_DiffI[OF Ris]]
434       show ?thesis by (simp (no_asm_simp)) blast
435     qed
436     ultimately have "Well_order (\<Union>R)" by(simp add:order_on_defs)
437     moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R" using Ris
439     ultimately have "\<Union>R : ?WO \<and> (\<forall>r\<in>R. (r,\<Union>R) : I)"
440       using mono_Chain[OF I_init] `R \<in> Chain I`
441       by(simp (no_asm) add:I_def del:Field_Union)(metis Chain_wo subsetD)
442   }
443   hence 1: "\<forall>R \<in> Chain I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r,u) : I" by (subst FI) blast
444 --{*Zorn's Lemma yields a maximal well-order m:*}
445   then obtain m::"('a*'a)set" where "Well_order m" and
446     max: "\<forall>r. Well_order r \<and> (m,r):I \<longrightarrow> r=m"
447     using Zorns_po_lemma[OF 0 1] by (auto simp:FI)
448 --{*Now show by contradiction that m covers the whole type:*}
449   { fix x::'a assume "x \<notin> Field m"
450 --{*We assume that x is not covered and extend m at the top with x*}
451     have "m \<noteq> {}"
452     proof
453       assume "m={}"
454       moreover have "Well_order {(x,x)}"
455 	by(simp add:order_on_defs refl_def trans_def antisym_def total_on_def Field_def Domain_def Range_def)
456       ultimately show False using max
457 	by (auto simp:I_def init_seg_of_def simp del:Field_insert)
458     qed
459     hence "Field m \<noteq> {}" by(auto simp:Field_def)
460     moreover have "wf(m-Id)" using `Well_order m`
462 --{*The extension of m by x:*}
463     let ?s = "{(a,x)|a. a : Field m}" let ?m = "insert (x,x) m Un ?s"
464     have Fm: "Field ?m = insert x (Field m)"
466       unfolding Field_def by auto
467     have "Refl m" "trans m" "antisym m" "Total m" "wf(m-Id)"
468       using `Well_order m` by(simp_all add:order_on_defs)
469 --{*We show that the extension is a well-order*}
470     have "Refl ?m" using `Refl m` Fm by(auto simp:refl_def)
471     moreover have "trans ?m" using `trans m` `x \<notin> Field m`
472       unfolding trans_def Field_def Domain_def Range_def by blast
473     moreover have "antisym ?m" using `antisym m` `x \<notin> Field m`
474       unfolding antisym_def Field_def Domain_def Range_def by blast
475     moreover have "Total ?m" using `Total m` Fm by(auto simp: total_on_def)
476     moreover have "wf(?m-Id)"
477     proof-
478       have "wf ?s" using `x \<notin> Field m`
479 	by(auto simp add:wf_eq_minimal Field_def Domain_def Range_def) metis
480       thus ?thesis using `wf(m-Id)` `x \<notin> Field m`
481 	wf_subset[OF `wf ?s` Diff_subset]
482 	by (fastsimp intro!: wf_Un simp add: Un_Diff Field_def)
483     qed
484     ultimately have "Well_order ?m" by(simp add:order_on_defs)
485 --{*We show that the extension is above m*}
486     moreover hence "(m,?m) : I" using `Well_order m` `x \<notin> Field m`
487       by(fastsimp simp:I_def init_seg_of_def Field_def Domain_def Range_def)
488     ultimately
489 --{*This contradicts maximality of m:*}
490     have False using max `x \<notin> Field m` unfolding Field_def by blast
491   }
492   hence "Field m = UNIV" by auto
493   moreover with `Well_order m` have "Well_order m" by simp
494   ultimately show ?thesis by blast
495 qed
497 corollary well_order_on: "\<exists>r::('a*'a)set. well_order_on A r"
498 proof -
499   obtain r::"('a*'a)set" where wo: "Well_order r" and univ: "Field r = UNIV"
500     using well_ordering[where 'a = "'a"] by blast
501   let ?r = "{(x,y). x:A & y:A & (x,y):r}"
502   have 1: "Field ?r = A" using wo univ
503     by(fastsimp simp: Field_def Domain_def Range_def order_on_defs refl_def)
504   have "Refl r" "trans r" "antisym r" "Total r" "wf(r-Id)"
505     using `Well_order r` by(simp_all add:order_on_defs)
506   have "Refl ?r" using `Refl r` by(auto simp:refl_def 1 univ)
507   moreover have "trans ?r" using `trans r`
508     unfolding trans_def by blast
509   moreover have "antisym ?r" using `antisym r`
510     unfolding antisym_def by blast
511   moreover have "Total ?r" using `Total r` by(simp add:total_on_def 1 univ)
512   moreover have "wf(?r - Id)" by(rule wf_subset[OF `wf(r-Id)`]) blast
513   ultimately have "Well_order ?r" by(simp add:order_on_defs)
514   with 1 show ?thesis by metis
515 qed
517 end