src/HOL/Library/Zorn.thy
 author Andreas Lochbihler Wed Feb 27 10:33:30 2013 +0100 (2013-02-27) changeset 51288 be7e9a675ec9 parent 48750 a151db85a62b child 51500 01fe31f05aa8 permissions -rw-r--r--
add wellorder instance for Numeral_Type (suggested by Jesus Aransay)
1 (*  Title:      HOL/Library/Zorn.thy
2     Author:     Jacques D. Fleuriot, Tobias Nipkow
4 Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF).
5 The well-ordering theorem.
6 *)
8 header {* Zorn's Lemma *}
10 theory Zorn
11 imports Order_Relation
12 begin
14 (* Define globally? In Set.thy? *)
15 definition chain_subset :: "'a set set \<Rightarrow> bool" ("chain\<^bsub>\<subseteq>\<^esub>")
16 where
17   "chain\<^bsub>\<subseteq>\<^esub> C \<equiv> \<forall>A\<in>C.\<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A"
19 text{*
20   The lemma and section numbers refer to an unpublished article
21   \cite{Abrial-Laffitte}.
22 *}
24 definition chain :: "'a set set \<Rightarrow> 'a set set set"
25 where
26   "chain S = {F. F \<subseteq> S \<and> chain\<^bsub>\<subseteq>\<^esub> F}"
28 definition super :: "'a set set \<Rightarrow> 'a set set \<Rightarrow> 'a set set set"
29 where
30   "super S c = {d. d \<in> chain S \<and> c \<subset> d}"
32 definition maxchain  ::  "'a set set \<Rightarrow> 'a set set set"
33 where
34   "maxchain S = {c. c \<in> chain S \<and> super S c = {}}"
36 definition succ :: "'a set set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
37 where
38   "succ S c = (if c \<notin> chain S \<or> c \<in> maxchain S then c else SOME c'. c' \<in> super S c)"
40 inductive_set TFin :: "'a set set \<Rightarrow> 'a set set set"
41 for S :: "'a set set"
42 where
43   succI:      "x \<in> TFin S \<Longrightarrow> succ S x \<in> TFin S"
44 | Pow_UnionI: "Y \<in> Pow (TFin S) \<Longrightarrow> \<Union>Y \<in> TFin S"
47 subsection{*Mathematical Preamble*}
49 lemma Union_lemma0:
50     "(\<forall>x \<in> C. x \<subseteq> A | B \<subseteq> x) ==> Union(C) \<subseteq> A | B \<subseteq> Union(C)"
51   by blast
54 text{*This is theorem @{text increasingD2} of ZF/Zorn.thy*}
56 lemma Abrial_axiom1: "x \<subseteq> succ S x"
57   apply (auto simp add: succ_def super_def maxchain_def)
58   apply (rule contrapos_np, assumption)
59   apply (rule someI2)
60   apply blast+
61   done
63 lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]
65 lemma TFin_induct:
66   assumes H: "n \<in> TFin S" and
67     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"
70   using H by induct (blast intro: I)+
72 lemma succ_trans: "x \<subseteq> y ==> x \<subseteq> succ S y"
73   apply (erule subset_trans)
74   apply (rule Abrial_axiom1)
75   done
77 text{*Lemma 1 of section 3.1*}
78 lemma TFin_linear_lemma1:
79      "[| n \<in> TFin S;  m \<in> TFin S;
80          \<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m
81       |] ==> n \<subseteq> m | succ S m \<subseteq> n"
82   apply (erule TFin_induct)
83    apply (erule_tac [2] Union_lemma0)
84   apply (blast del: subsetI intro: succ_trans)
85   done
87 text{* Lemma 2 of section 3.2 *}
88 lemma TFin_linear_lemma2:
89      "m \<in> TFin S ==> \<forall>n \<in> TFin S. n \<subseteq> m --> n=m | succ S n \<subseteq> m"
90   apply (erule TFin_induct)
91    apply (rule impI [THEN ballI])
92    txt{*case split using @{text TFin_linear_lemma1}*}
93    apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
94      assumption+)
95     apply (drule_tac x = n in bspec, assumption)
96     apply (blast del: subsetI intro: succ_trans, blast)
97   txt{*second induction step*}
98   apply (rule impI [THEN ballI])
99   apply (rule Union_lemma0 [THEN disjE])
100     apply (rule_tac [3] disjI2)
101     prefer 2 apply blast
102    apply (rule ballI)
103    apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
104      assumption+, auto)
105   apply (blast intro!: Abrial_axiom1 [THEN subsetD])
106   done
108 text{*Re-ordering the premises of Lemma 2*}
109 lemma TFin_subsetD:
110      "[| n \<subseteq> m;  m \<in> TFin S;  n \<in> TFin S |] ==> n=m | succ S n \<subseteq> m"
111   by (rule TFin_linear_lemma2 [rule_format])
113 text{*Consequences from section 3.3 -- Property 3.2, the ordering is total*}
114 lemma TFin_subset_linear: "[| m \<in> TFin S;  n \<in> TFin S|] ==> n \<subseteq> m | m \<subseteq> n"
115   apply (rule disjE)
116     apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2])
117       apply (assumption+, erule disjI2)
118   apply (blast del: subsetI
119     intro: subsetI Abrial_axiom1 [THEN subset_trans])
120   done
122 text{*Lemma 3 of section 3.3*}
123 lemma eq_succ_upper: "[| n \<in> TFin S;  m \<in> TFin S;  m = succ S m |] ==> n \<subseteq> m"
124   apply (erule TFin_induct)
125    apply (drule TFin_subsetD)
126      apply (assumption+, force, blast)
127   done
129 text{*Property 3.3 of section 3.3*}
130 lemma equal_succ_Union: "m \<in> TFin S ==> (m = succ S m) = (m = Union(TFin S))"
131   apply (rule iffI)
132    apply (rule Union_upper [THEN equalityI])
133     apply assumption
134    apply (rule eq_succ_upper [THEN Union_least], assumption+)
135   apply (erule ssubst)
136   apply (rule Abrial_axiom1 [THEN equalityI])
137   apply (blast del: subsetI intro: subsetI TFin_UnionI TFin.succI)
138   done
140 subsection{*Hausdorff's Theorem: Every Set Contains a Maximal Chain.*}
142 text{*NB: We assume the partial ordering is @{text "\<subseteq>"},
143  the subset relation!*}
145 lemma empty_set_mem_chain: "({} :: 'a set set) \<in> chain S"
146   by (unfold chain_def chain_subset_def) simp
148 lemma super_subset_chain: "super S c \<subseteq> chain S"
149   by (unfold super_def) blast
151 lemma maxchain_subset_chain: "maxchain S \<subseteq> chain S"
152   by (unfold maxchain_def) blast
154 lemma mem_super_Ex: "c \<in> chain S - maxchain S ==> EX d. d \<in> super S c"
155   by (unfold super_def maxchain_def) simp
157 lemma select_super:
158      "c \<in> chain S - maxchain S ==> (\<some>c'. c': super S c): super S c"
159   apply (erule mem_super_Ex [THEN exE])
160   apply (rule someI2)
161   apply simp+
162   done
164 lemma select_not_equals:
165      "c \<in> chain S - maxchain S ==> (\<some>c'. c': super S c) \<noteq> c"
166   apply (rule notI)
167   apply (drule select_super)
168   apply (simp add: super_def less_le)
169   done
171 lemma succI3: "c \<in> chain S - maxchain S ==> succ S c = (\<some>c'. c': super S c)"
172   by (unfold succ_def) (blast intro!: if_not_P)
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 chain_subset_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       apply 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; \<forall>x \<in> c. x \<subseteq> (z:: 'a set) |] ==> {z} Un c \<in> chain S"
206 by (unfold chain_def chain_subset_def) blast
208 lemma chain_Union_upper: "[| c \<in> chain S; x \<in> c |] ==> x \<subseteq> Union(c)"
209 by auto
211 lemma chain_ball_Union_upper: "c \<in> chain S ==> \<forall>x \<in> c. x \<subseteq> Union(c)"
212 by auto
214 lemma maxchain_Zorn:
215   "[| c \<in> maxchain S; u \<in> S; Union(c) \<subseteq> u |] ==> Union(c) = u"
216 apply (rule ccontr)
218 apply (erule conjE)
219 apply (subgoal_tac "({u} Un c) \<in> super S c")
220  apply simp
221 apply (unfold super_def less_le)
222 apply (blast intro: chain_extend dest: chain_Union_upper)
223 done
225 theorem Zorn_Lemma:
226   "\<forall>c \<in> chain S. Union(c): S ==> \<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z --> y = z"
227 apply (cut_tac Hausdorff maxchain_subset_chain)
228 apply (erule exE)
229 apply (drule subsetD, assumption)
230 apply (drule bspec, assumption)
231 apply (rule_tac x = "Union(c)" in bexI)
232  apply (rule ballI, rule impI)
233  apply (blast dest!: maxchain_Zorn, assumption)
234 done
236 subsection{*Alternative version of Zorn's Lemma*}
238 lemma Zorn_Lemma2:
239   "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>x \<in> c. x \<subseteq> y
240     ==> \<exists>y \<in> S. \<forall>x \<in> S. (y :: 'a set) \<subseteq> x --> y = x"
241 apply (cut_tac Hausdorff maxchain_subset_chain)
242 apply (erule exE)
243 apply (drule subsetD, assumption)
244 apply (drule bspec, assumption, erule bexE)
245 apply (rule_tac x = y in bexI)
246  prefer 2 apply assumption
247 apply clarify
248 apply (rule ccontr)
249 apply (frule_tac z = x in chain_extend)
250   apply (assumption, blast)
251 apply (unfold maxchain_def super_def less_le)
252 apply (blast elim!: equalityCE)
253 done
255 text{*Various other lemmas*}
257 lemma chainD: "[| c \<in> chain S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
258 by (unfold chain_def chain_subset_def) blast
260 lemma chainD2: "!!(c :: 'a set set). c \<in> chain S ==> c \<subseteq> S"
261 by (unfold chain_def) blast
264 (* Define globally? In Relation.thy? *)
265 definition Chain :: "('a*'a)set \<Rightarrow> 'a set set" where
266 "Chain r \<equiv> {A. \<forall>a\<in>A.\<forall>b\<in>A. (a,b) : r \<or> (b,a) \<in> r}"
268 lemma mono_Chain: "r \<subseteq> s \<Longrightarrow> Chain r \<subseteq> Chain s"
269 unfolding Chain_def by blast
271 text{* Zorn's lemma for partial orders: *}
273 lemma Zorns_po_lemma:
274 assumes po: "Partial_order r" and u: "\<forall>C\<in>Chain r. \<exists>u\<in>Field r. \<forall>a\<in>C. (a,u):r"
275 shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m,a):r \<longrightarrow> a=m"
276 proof-
277   have "Preorder r" using po by(simp add:partial_order_on_def)
278 --{* Mirror r in the set of subsets below (wrt r) elements of A*}
279   let ?B = "%x. r^-1 `` {x}" let ?S = "?B ` Field r"
280   have "\<forall>C \<in> chain ?S. EX U:?S. ALL A:C. A\<subseteq>U"
281   proof (auto simp:chain_def chain_subset_def)
282     fix C assume 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C.\<forall>B\<in>C. A\<subseteq>B | B\<subseteq>A"
283     let ?A = "{x\<in>Field r. \<exists>M\<in>C. M = ?B x}"
284     have "C = ?B ` ?A" using 1 by(auto simp: image_def)
285     have "?A\<in>Chain r"
286     proof (simp add:Chain_def, intro allI impI, elim conjE)
287       fix a b
288       assume "a \<in> Field r" "?B a \<in> C" "b \<in> Field r" "?B b \<in> C"
289       hence "?B a \<subseteq> ?B b \<or> ?B b \<subseteq> ?B a" using 2 by simp
290       thus "(a, b) \<in> r \<or> (b, a) \<in> r" using `Preorder r` `a:Field r` `b:Field r`
292     qed
293     then obtain u where uA: "u:Field r" "\<forall>a\<in>?A. (a,u) : r" using u by auto
294     have "\<forall>A\<in>C. A \<subseteq> r^-1 `` {u}" (is "?P u")
295     proof auto
296       fix a B assume aB: "B:C" "a:B"
297       with 1 obtain x where "x:Field r" "B = r^-1 `` {x}" by auto
298       thus "(a,u) : r" using uA aB `Preorder r`
299         by (simp add: preorder_on_def refl_on_def) (rule transD, blast+)
300     qed
301     thus "EX u:Field r. ?P u" using `u:Field r` by blast
302   qed
303   from Zorn_Lemma2[OF this]
304   obtain m B where "m:Field r" "B = r^-1 `` {m}"
305     "\<forall>x\<in>Field r. B \<subseteq> r^-1 `` {x} \<longrightarrow> B = r^-1 `` {x}"
306     by auto
307   hence "\<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m" using po `Preorder r` `m:Field r`
308     by(auto simp:subset_Image1_Image1_iff Partial_order_eq_Image1_Image1_iff)
309   thus ?thesis using `m:Field r` by blast
310 qed
312 (* The initial segment of a relation appears generally useful.
313    Move to Relation.thy?
314    Definition correct/most general?
315    Naming?
316 *)
317 definition init_seg_of :: "(('a*'a)set * ('a*'a)set)set" where
318 "init_seg_of == {(r,s). r \<subseteq> s \<and> (\<forall>a b c. (a,b):s \<and> (b,c):r \<longrightarrow> (a,b):r)}"
320 abbreviation initialSegmentOf :: "('a*'a)set \<Rightarrow> ('a*'a)set \<Rightarrow> bool"
321              (infix "initial'_segment'_of" 55) where
322 "r initial_segment_of s == (r,s):init_seg_of"
324 lemma refl_on_init_seg_of[simp]: "r initial_segment_of r"
327 lemma trans_init_seg_of:
328   "r initial_segment_of s \<Longrightarrow> s initial_segment_of t \<Longrightarrow> r initial_segment_of t"
329 by (simp (no_asm_use) add: init_seg_of_def) (metis (no_types) in_mono order_trans)
331 lemma antisym_init_seg_of:
332   "r initial_segment_of s \<Longrightarrow> s initial_segment_of r \<Longrightarrow> r=s"
333 unfolding init_seg_of_def by safe
335 lemma Chain_init_seg_of_Union:
336   "R \<in> Chain init_seg_of \<Longrightarrow> r\<in>R \<Longrightarrow> r initial_segment_of \<Union>R"
337 by(simp add: init_seg_of_def Chain_def Ball_def) blast
339 lemma chain_subset_trans_Union:
340   "chain\<^bsub>\<subseteq>\<^esub> R \<Longrightarrow> \<forall>r\<in>R. trans r \<Longrightarrow> trans(\<Union>R)"
343 by (metis subsetD)
345 lemma chain_subset_antisym_Union:
346   "chain\<^bsub>\<subseteq>\<^esub> R \<Longrightarrow> \<forall>r\<in>R. antisym r \<Longrightarrow> antisym(\<Union>R)"
347 by (simp add:chain_subset_def antisym_def) (metis subsetD)
349 lemma chain_subset_Total_Union:
350 assumes "chain\<^bsub>\<subseteq>\<^esub> R" "\<forall>r\<in>R. Total r"
351 shows "Total (\<Union>R)"
352 proof (simp add: total_on_def Ball_def, auto del:disjCI)
353   fix r s a b assume A: "r:R" "s:R" "a:Field r" "b:Field s" "a\<noteq>b"
354   from `chain\<^bsub>\<subseteq>\<^esub> R` `r:R` `s:R` have "r\<subseteq>s \<or> s\<subseteq>r"
356   thus "(\<exists>r\<in>R. (a,b) \<in> r) \<or> (\<exists>r\<in>R. (b,a) \<in> r)"
357   proof
358     assume "r\<subseteq>s" hence "(a,b):s \<or> (b,a):s" using assms(2) A
360     thus ?thesis using `s:R` by blast
361   next
362     assume "s\<subseteq>r" hence "(a,b):r \<or> (b,a):r" using assms(2) A
364     thus ?thesis using `r:R` by blast
365   qed
366 qed
368 lemma wf_Union_wf_init_segs:
369 assumes "R \<in> Chain init_seg_of" and "\<forall>r\<in>R. wf r" shows "wf(\<Union>R)"
370 proof(simp add:wf_iff_no_infinite_down_chain, rule ccontr, auto)
371   fix f assume 1: "\<forall>i. \<exists>r\<in>R. (f(Suc i), f i) \<in> r"
372   then obtain r where "r:R" and "(f(Suc 0), f 0) : r" by auto
373   { fix i have "(f(Suc i), f i) \<in> r"
374     proof(induct i)
375       case 0 show ?case by fact
376     next
377       case (Suc i)
378       moreover obtain s where "s\<in>R" and "(f(Suc(Suc i)), f(Suc i)) \<in> s"
379         using 1 by auto
380       moreover hence "s initial_segment_of r \<or> r initial_segment_of s"
381         using assms(1) `r:R` by(simp add: Chain_def)
382       ultimately show ?case by(simp add:init_seg_of_def) blast
383     qed
384   }
385   thus False using assms(2) `r:R`
387 qed
389 lemma initial_segment_of_Diff:
390   "p initial_segment_of q \<Longrightarrow> p - s initial_segment_of q - s"
391 unfolding init_seg_of_def by blast
393 lemma Chain_inits_DiffI:
394   "R \<in> Chain init_seg_of \<Longrightarrow> {r - s |r. r \<in> R} \<in> Chain init_seg_of"
395 unfolding Chain_def by (blast intro: initial_segment_of_Diff)
397 theorem well_ordering: "\<exists>r::('a*'a)set. Well_order r \<and> Field r = UNIV"
398 proof-
399 -- {*The initial segment relation on well-orders: *}
400   let ?WO = "{r::('a*'a)set. Well_order r}"
401   def I \<equiv> "init_seg_of \<inter> ?WO \<times> ?WO"
402   have I_init: "I \<subseteq> init_seg_of" by(simp add: I_def)
403   hence subch: "!!R. R : Chain I \<Longrightarrow> chain\<^bsub>\<subseteq>\<^esub> R"
404     by(auto simp:init_seg_of_def chain_subset_def Chain_def)
405   have Chain_wo: "!!R r. R \<in> Chain I \<Longrightarrow> r \<in> R \<Longrightarrow> Well_order r"
407   have FI: "Field I = ?WO" by(auto simp add:I_def init_seg_of_def Field_def)
408   hence 0: "Partial_order I"
409     by(auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def trans_def I_def elim!: trans_init_seg_of)
410 -- {*I-chains have upper bounds in ?WO wrt I: their Union*}
411   { fix R assume "R \<in> Chain I"
412     hence Ris: "R \<in> Chain init_seg_of" using mono_Chain[OF I_init] by blast
413     have subch: "chain\<^bsub>\<subseteq>\<^esub> R" using `R : Chain I` I_init
414       by(auto simp:init_seg_of_def chain_subset_def Chain_def)
415     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"
416          "\<forall>r\<in>R. wf(r-Id)"
417       using Chain_wo[OF `R \<in> Chain I`] by(simp_all add:order_on_defs)
418     have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r` by(auto simp:refl_on_def)
419     moreover have "trans (\<Union>R)"
420       by(rule chain_subset_trans_Union[OF subch `\<forall>r\<in>R. trans r`])
421     moreover have "antisym(\<Union>R)"
422       by(rule chain_subset_antisym_Union[OF subch `\<forall>r\<in>R. antisym r`])
423     moreover have "Total (\<Union>R)"
424       by(rule chain_subset_Total_Union[OF subch `\<forall>r\<in>R. Total r`])
425     moreover have "wf((\<Union>R)-Id)"
426     proof-
427       have "(\<Union>R)-Id = \<Union>{r-Id|r. r \<in> R}" by blast
428       with `\<forall>r\<in>R. wf(r-Id)` wf_Union_wf_init_segs[OF Chain_inits_DiffI[OF Ris]]
429       show ?thesis by (simp (no_asm_simp)) blast
430     qed
431     ultimately have "Well_order (\<Union>R)" by(simp add:order_on_defs)
432     moreover have "\<forall>r \<in> R. r initial_segment_of \<Union>R" using Ris
434     ultimately have "\<Union>R : ?WO \<and> (\<forall>r\<in>R. (r,\<Union>R) : I)"
435       using mono_Chain[OF I_init] `R \<in> Chain I`
436       by(simp (no_asm) add:I_def del:Field_Union)(metis Chain_wo)
437   }
438   hence 1: "\<forall>R \<in> Chain I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r,u) : I" by (subst FI) blast
439 --{*Zorn's Lemma yields a maximal well-order m:*}
440   then obtain m::"('a*'a)set" where "Well_order m" and
441     max: "\<forall>r. Well_order r \<and> (m,r):I \<longrightarrow> r=m"
442     using Zorns_po_lemma[OF 0 1] by (auto simp:FI)
443 --{*Now show by contradiction that m covers the whole type:*}
444   { fix x::'a assume "x \<notin> Field m"
445 --{*We assume that x is not covered and extend m at the top with x*}
446     have "m \<noteq> {}"
447     proof
448       assume "m={}"
449       moreover have "Well_order {(x,x)}"
450         by(simp add:order_on_defs refl_on_def trans_def antisym_def total_on_def Field_def Domain_unfold Domain_converse [symmetric])
451       ultimately show False using max
452         by (auto simp:I_def init_seg_of_def simp del:Field_insert)
453     qed
454     hence "Field m \<noteq> {}" by(auto simp:Field_def)
455     moreover have "wf(m-Id)" using `Well_order m`
457 --{*The extension of m by x:*}
458     let ?s = "{(a,x)|a. a : Field m}" let ?m = "insert (x,x) m Un ?s"
459     have Fm: "Field ?m = insert x (Field m)"
461       unfolding Field_def by auto
462     have "Refl m" "trans m" "antisym m" "Total m" "wf(m-Id)"
463       using `Well_order m` by(simp_all add:order_on_defs)
464 --{*We show that the extension is a well-order*}
465     have "Refl ?m" using `Refl m` Fm by(auto simp:refl_on_def)
466     moreover have "trans ?m" using `trans m` `x \<notin> Field m`
467       unfolding trans_def Field_def Domain_unfold Domain_converse [symmetric] by blast
468     moreover have "antisym ?m" using `antisym m` `x \<notin> Field m`
469       unfolding antisym_def Field_def Domain_unfold Domain_converse [symmetric] by blast
470     moreover have "Total ?m" using `Total m` Fm by(auto simp: total_on_def)
471     moreover have "wf(?m-Id)"
472     proof-
473       have "wf ?s" using `x \<notin> Field m`
474         by(simp add:wf_eq_minimal Field_def Domain_unfold Domain_converse [symmetric]) metis
475       thus ?thesis using `wf(m-Id)` `x \<notin> Field m`
476         wf_subset[OF `wf ?s` Diff_subset]
477         by (fastforce intro!: wf_Un simp add: Un_Diff Field_def)
478     qed
479     ultimately have "Well_order ?m" by(simp add:order_on_defs)
480 --{*We show that the extension is above m*}
481     moreover hence "(m,?m) : I" using `Well_order m` `x \<notin> Field m`
482       by(fastforce simp:I_def init_seg_of_def Field_def Domain_unfold Domain_converse [symmetric])
483     ultimately
484 --{*This contradicts maximality of m:*}
485     have False using max `x \<notin> Field m` unfolding Field_def by blast
486   }
487   hence "Field m = UNIV" by auto
488   moreover with `Well_order m` have "Well_order m" by simp
489   ultimately show ?thesis by blast
490 qed
492 corollary well_order_on: "\<exists>r::('a*'a)set. well_order_on A r"
493 proof -
494   obtain r::"('a*'a)set" where wo: "Well_order r" and univ: "Field r = UNIV"
495     using well_ordering[where 'a = "'a"] by blast
496   let ?r = "{(x,y). x:A & y:A & (x,y):r}"
497   have 1: "Field ?r = A" using wo univ
498     by(fastforce simp: Field_def Domain_unfold Domain_converse [symmetric] order_on_defs refl_on_def)
499   have "Refl r" "trans r" "antisym r" "Total r" "wf(r-Id)"
500     using `Well_order r` by(simp_all add:order_on_defs)
501   have "Refl ?r" using `Refl r` by(auto simp:refl_on_def 1 univ)
502   moreover have "trans ?r" using `trans r`
503     unfolding trans_def by blast
504   moreover have "antisym ?r" using `antisym r`
505     unfolding antisym_def by blast
506   moreover have "Total ?r" using `Total r` by(simp add:total_on_def 1 univ)
507   moreover have "wf(?r - Id)" by(rule wf_subset[OF `wf(r-Id)`]) blast
508   ultimately have "Well_order ?r" by(simp add:order_on_defs)
509   with 1 show ?thesis by metis
510 qed
512 end