src/ZF/Ordinal.thy
 author wenzelm Sat Mar 13 16:44:12 2010 +0100 (2010-03-13) changeset 35762 af3ff2ba4c54 parent 24893 b8ef7afe3a6b child 46820 c656222c4dc1 permissions -rw-r--r--
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1 (*  Title:      ZF/Ordinal.thy
2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
3     Copyright   1994  University of Cambridge
4 *)
8 theory Ordinal imports WF Bool equalities begin
10 definition
11   Memrel        :: "i=>i"  where
12     "Memrel(A)   == {z: A*A . EX x y. z=<x,y> & x:y }"
14 definition
15   Transset  :: "i=>o"  where
16     "Transset(i) == ALL x:i. x<=i"
18 definition
19   Ord  :: "i=>o"  where
20     "Ord(i)      == Transset(i) & (ALL x:i. Transset(x))"
22 definition
23   lt        :: "[i,i] => o"  (infixl "<" 50)   (*less-than on ordinals*)  where
24     "i<j         == i:j & Ord(j)"
26 definition
27   Limit         :: "i=>o"  where
28     "Limit(i)    == Ord(i) & 0<i & (ALL y. y<i --> succ(y)<i)"
30 abbreviation
31   le  (infixl "le" 50) where
32   "x le y == x < succ(y)"
34 notation (xsymbols)
35   le  (infixl "\<le>" 50)
37 notation (HTML output)
38   le  (infixl "\<le>" 50)
41 subsection{*Rules for Transset*}
43 subsubsection{*Three Neat Characterisations of Transset*}
45 lemma Transset_iff_Pow: "Transset(A) <-> A<=Pow(A)"
46 by (unfold Transset_def, blast)
48 lemma Transset_iff_Union_succ: "Transset(A) <-> Union(succ(A)) = A"
49 apply (unfold Transset_def)
50 apply (blast elim!: equalityE)
51 done
53 lemma Transset_iff_Union_subset: "Transset(A) <-> Union(A) <= A"
54 by (unfold Transset_def, blast)
56 subsubsection{*Consequences of Downwards Closure*}
58 lemma Transset_doubleton_D:
59     "[| Transset(C); {a,b}: C |] ==> a:C & b: C"
60 by (unfold Transset_def, blast)
62 lemma Transset_Pair_D:
63     "[| Transset(C); <a,b>: C |] ==> a:C & b: C"
65 apply (blast dest: Transset_doubleton_D)
66 done
68 lemma Transset_includes_domain:
69     "[| Transset(C); A*B <= C; b: B |] ==> A <= C"
70 by (blast dest: Transset_Pair_D)
72 lemma Transset_includes_range:
73     "[| Transset(C); A*B <= C; a: A |] ==> B <= C"
74 by (blast dest: Transset_Pair_D)
76 subsubsection{*Closure Properties*}
78 lemma Transset_0: "Transset(0)"
79 by (unfold Transset_def, blast)
81 lemma Transset_Un:
82     "[| Transset(i);  Transset(j) |] ==> Transset(i Un j)"
83 by (unfold Transset_def, blast)
85 lemma Transset_Int:
86     "[| Transset(i);  Transset(j) |] ==> Transset(i Int j)"
87 by (unfold Transset_def, blast)
89 lemma Transset_succ: "Transset(i) ==> Transset(succ(i))"
90 by (unfold Transset_def, blast)
92 lemma Transset_Pow: "Transset(i) ==> Transset(Pow(i))"
93 by (unfold Transset_def, blast)
95 lemma Transset_Union: "Transset(A) ==> Transset(Union(A))"
96 by (unfold Transset_def, blast)
98 lemma Transset_Union_family:
99     "[| !!i. i:A ==> Transset(i) |] ==> Transset(Union(A))"
100 by (unfold Transset_def, blast)
102 lemma Transset_Inter_family:
103     "[| !!i. i:A ==> Transset(i) |] ==> Transset(Inter(A))"
104 by (unfold Inter_def Transset_def, blast)
106 lemma Transset_UN:
107      "(!!x. x \<in> A ==> Transset(B(x))) ==> Transset (\<Union>x\<in>A. B(x))"
108 by (rule Transset_Union_family, auto)
110 lemma Transset_INT:
111      "(!!x. x \<in> A ==> Transset(B(x))) ==> Transset (\<Inter>x\<in>A. B(x))"
112 by (rule Transset_Inter_family, auto)
115 subsection{*Lemmas for Ordinals*}
117 lemma OrdI:
118     "[| Transset(i);  !!x. x:i ==> Transset(x) |]  ==>  Ord(i)"
121 lemma Ord_is_Transset: "Ord(i) ==> Transset(i)"
124 lemma Ord_contains_Transset:
125     "[| Ord(i);  j:i |] ==> Transset(j) "
126 by (unfold Ord_def, blast)
129 lemma Ord_in_Ord: "[| Ord(i);  j:i |] ==> Ord(j)"
130 by (unfold Ord_def Transset_def, blast)
132 (*suitable for rewriting PROVIDED i has been fixed*)
133 lemma Ord_in_Ord': "[| j:i; Ord(i) |] ==> Ord(j)"
134 by (blast intro: Ord_in_Ord)
136 (* Ord(succ(j)) ==> Ord(j) *)
137 lemmas Ord_succD = Ord_in_Ord [OF _ succI1]
139 lemma Ord_subset_Ord: "[| Ord(i);  Transset(j);  j<=i |] ==> Ord(j)"
140 by (simp add: Ord_def Transset_def, blast)
142 lemma OrdmemD: "[| j:i;  Ord(i) |] ==> j<=i"
143 by (unfold Ord_def Transset_def, blast)
145 lemma Ord_trans: "[| i:j;  j:k;  Ord(k) |] ==> i:k"
146 by (blast dest: OrdmemD)
148 lemma Ord_succ_subsetI: "[| i:j;  Ord(j) |] ==> succ(i) <= j"
149 by (blast dest: OrdmemD)
152 subsection{*The Construction of Ordinals: 0, succ, Union*}
154 lemma Ord_0 [iff,TC]: "Ord(0)"
155 by (blast intro: OrdI Transset_0)
157 lemma Ord_succ [TC]: "Ord(i) ==> Ord(succ(i))"
158 by (blast intro: OrdI Transset_succ Ord_is_Transset Ord_contains_Transset)
160 lemmas Ord_1 = Ord_0 [THEN Ord_succ]
162 lemma Ord_succ_iff [iff]: "Ord(succ(i)) <-> Ord(i)"
163 by (blast intro: Ord_succ dest!: Ord_succD)
165 lemma Ord_Un [intro,simp,TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Un j)"
166 apply (unfold Ord_def)
167 apply (blast intro!: Transset_Un)
168 done
170 lemma Ord_Int [TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Int j)"
171 apply (unfold Ord_def)
172 apply (blast intro!: Transset_Int)
173 done
175 (*There is no set of all ordinals, for then it would contain itself*)
176 lemma ON_class: "~ (ALL i. i:X <-> Ord(i))"
177 apply (rule notI)
178 apply (frule_tac x = X in spec)
179 apply (safe elim!: mem_irrefl)
180 apply (erule swap, rule OrdI [OF _ Ord_is_Transset])
182 apply (blast intro: Ord_in_Ord)+
183 done
185 subsection{*< is 'less Than' for Ordinals*}
187 lemma ltI: "[| i:j;  Ord(j) |] ==> i<j"
188 by (unfold lt_def, blast)
190 lemma ltE:
191     "[| i<j;  [| i:j;  Ord(i);  Ord(j) |] ==> P |] ==> P"
192 apply (unfold lt_def)
193 apply (blast intro: Ord_in_Ord)
194 done
196 lemma ltD: "i<j ==> i:j"
197 by (erule ltE, assumption)
199 lemma not_lt0 [simp]: "~ i<0"
200 by (unfold lt_def, blast)
202 lemma lt_Ord: "j<i ==> Ord(j)"
203 by (erule ltE, assumption)
205 lemma lt_Ord2: "j<i ==> Ord(i)"
206 by (erule ltE, assumption)
208 (* "ja le j ==> Ord(j)" *)
209 lemmas le_Ord2 = lt_Ord2 [THEN Ord_succD]
211 (* i<0 ==> R *)
212 lemmas lt0E = not_lt0 [THEN notE, elim!]
214 lemma lt_trans: "[| i<j;  j<k |] ==> i<k"
215 by (blast intro!: ltI elim!: ltE intro: Ord_trans)
217 lemma lt_not_sym: "i<j ==> ~ (j<i)"
218 apply (unfold lt_def)
219 apply (blast elim: mem_asym)
220 done
222 (* [| i<j;  ~P ==> j<i |] ==> P *)
223 lemmas lt_asym = lt_not_sym [THEN swap]
225 lemma lt_irrefl [elim!]: "i<i ==> P"
226 by (blast intro: lt_asym)
228 lemma lt_not_refl: "~ i<i"
229 apply (rule notI)
230 apply (erule lt_irrefl)
231 done
234 (** le is less than or equals;  recall  i le j  abbrevs  i<succ(j) !! **)
236 lemma le_iff: "i le j <-> i<j | (i=j & Ord(j))"
237 by (unfold lt_def, blast)
239 (*Equivalently, i<j ==> i < succ(j)*)
240 lemma leI: "i<j ==> i le j"
241 by (simp (no_asm_simp) add: le_iff)
243 lemma le_eqI: "[| i=j;  Ord(j) |] ==> i le j"
244 by (simp (no_asm_simp) add: le_iff)
246 lemmas le_refl = refl [THEN le_eqI]
248 lemma le_refl_iff [iff]: "i le i <-> Ord(i)"
249 by (simp (no_asm_simp) add: lt_not_refl le_iff)
251 lemma leCI: "(~ (i=j & Ord(j)) ==> i<j) ==> i le j"
252 by (simp add: le_iff, blast)
254 lemma leE:
255     "[| i le j;  i<j ==> P;  [| i=j;  Ord(j) |] ==> P |] ==> P"
256 by (simp add: le_iff, blast)
258 lemma le_anti_sym: "[| i le j;  j le i |] ==> i=j"
260 apply (blast elim: lt_asym)
261 done
263 lemma le0_iff [simp]: "i le 0 <-> i=0"
264 by (blast elim!: leE)
266 lemmas le0D = le0_iff [THEN iffD1, dest!]
268 subsection{*Natural Deduction Rules for Memrel*}
270 (*The lemmas MemrelI/E give better speed than [iff] here*)
271 lemma Memrel_iff [simp]: "<a,b> : Memrel(A) <-> a:b & a:A & b:A"
272 by (unfold Memrel_def, blast)
274 lemma MemrelI [intro!]: "[| a: b;  a: A;  b: A |] ==> <a,b> : Memrel(A)"
275 by auto
277 lemma MemrelE [elim!]:
278     "[| <a,b> : Memrel(A);
279         [| a: A;  b: A;  a:b |]  ==> P |]
280      ==> P"
281 by auto
283 lemma Memrel_type: "Memrel(A) <= A*A"
284 by (unfold Memrel_def, blast)
286 lemma Memrel_mono: "A<=B ==> Memrel(A) <= Memrel(B)"
287 by (unfold Memrel_def, blast)
289 lemma Memrel_0 [simp]: "Memrel(0) = 0"
290 by (unfold Memrel_def, blast)
292 lemma Memrel_1 [simp]: "Memrel(1) = 0"
293 by (unfold Memrel_def, blast)
295 lemma relation_Memrel: "relation(Memrel(A))"
296 by (simp add: relation_def Memrel_def)
298 (*The membership relation (as a set) is well-founded.
299   Proof idea: show A<=B by applying the foundation axiom to A-B *)
300 lemma wf_Memrel: "wf(Memrel(A))"
301 apply (unfold wf_def)
302 apply (rule foundation [THEN disjE, THEN allI], erule disjI1, blast)
303 done
305 text{*The premise @{term "Ord(i)"} does not suffice.*}
306 lemma trans_Memrel:
307     "Ord(i) ==> trans(Memrel(i))"
308 by (unfold Ord_def Transset_def trans_def, blast)
310 text{*However, the following premise is strong enough.*}
311 lemma Transset_trans_Memrel:
312     "\<forall>j\<in>i. Transset(j) ==> trans(Memrel(i))"
313 by (unfold Transset_def trans_def, blast)
315 (*If Transset(A) then Memrel(A) internalizes the membership relation below A*)
316 lemma Transset_Memrel_iff:
317     "Transset(A) ==> <a,b> : Memrel(A) <-> a:b & b:A"
318 by (unfold Transset_def, blast)
321 subsection{*Transfinite Induction*}
323 (*Epsilon induction over a transitive set*)
324 lemma Transset_induct:
325     "[| i: k;  Transset(k);
326         !!x.[| x: k;  ALL y:x. P(y) |] ==> P(x) |]
327      ==>  P(i)"
329 apply (erule wf_Memrel [THEN wf_induct2], blast+)
330 done
332 (*Induction over an ordinal*)
333 lemmas Ord_induct [consumes 2] = Transset_induct [OF _ Ord_is_Transset]
334 lemmas Ord_induct_rule = Ord_induct [rule_format, consumes 2]
336 (*Induction over the class of ordinals -- a useful corollary of Ord_induct*)
338 lemma trans_induct [consumes 1]:
339     "[| Ord(i);
340         !!x.[| Ord(x);  ALL y:x. P(y) |] ==> P(x) |]
341      ==>  P(i)"
342 apply (rule Ord_succ [THEN succI1 [THEN Ord_induct]], assumption)
343 apply (blast intro: Ord_succ [THEN Ord_in_Ord])
344 done
346 lemmas trans_induct_rule = trans_induct [rule_format, consumes 1]
349 (*** Fundamental properties of the epsilon ordering (< on ordinals) ***)
352 subsubsection{*Proving That < is a Linear Ordering on the Ordinals*}
354 lemma Ord_linear [rule_format]:
355      "Ord(i) ==> (ALL j. Ord(j) --> i:j | i=j | j:i)"
356 apply (erule trans_induct)
357 apply (rule impI [THEN allI])
358 apply (erule_tac i=j in trans_induct)
359 apply (blast dest: Ord_trans)
360 done
362 (*The trichotomy law for ordinals!*)
363 lemma Ord_linear_lt:
364     "[| Ord(i);  Ord(j);  i<j ==> P;  i=j ==> P;  j<i ==> P |] ==> P"
366 apply (rule_tac i1=i and j1=j in Ord_linear [THEN disjE], blast+)
367 done
369 lemma Ord_linear2:
370     "[| Ord(i);  Ord(j);  i<j ==> P;  j le i ==> P |]  ==> P"
371 apply (rule_tac i = i and j = j in Ord_linear_lt)
372 apply (blast intro: leI le_eqI sym ) +
373 done
375 lemma Ord_linear_le:
376     "[| Ord(i);  Ord(j);  i le j ==> P;  j le i ==> P |]  ==> P"
377 apply (rule_tac i = i and j = j in Ord_linear_lt)
378 apply (blast intro: leI le_eqI ) +
379 done
381 lemma le_imp_not_lt: "j le i ==> ~ i<j"
382 by (blast elim!: leE elim: lt_asym)
384 lemma not_lt_imp_le: "[| ~ i<j;  Ord(i);  Ord(j) |] ==> j le i"
385 by (rule_tac i = i and j = j in Ord_linear2, auto)
387 subsubsection{*Some Rewrite Rules for <, le*}
389 lemma Ord_mem_iff_lt: "Ord(j) ==> i:j <-> i<j"
390 by (unfold lt_def, blast)
392 lemma not_lt_iff_le: "[| Ord(i);  Ord(j) |] ==> ~ i<j <-> j le i"
393 by (blast dest: le_imp_not_lt not_lt_imp_le)
395 lemma not_le_iff_lt: "[| Ord(i);  Ord(j) |] ==> ~ i le j <-> j<i"
396 by (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym])
398 (*This is identical to 0<succ(i) *)
399 lemma Ord_0_le: "Ord(i) ==> 0 le i"
400 by (erule not_lt_iff_le [THEN iffD1], auto)
402 lemma Ord_0_lt: "[| Ord(i);  i~=0 |] ==> 0<i"
403 apply (erule not_le_iff_lt [THEN iffD1])
404 apply (rule Ord_0, blast)
405 done
407 lemma Ord_0_lt_iff: "Ord(i) ==> i~=0 <-> 0<i"
408 by (blast intro: Ord_0_lt)
411 subsection{*Results about Less-Than or Equals*}
413 (** For ordinals, j<=i (subset) implies j le i (less-than or equals) **)
415 lemma zero_le_succ_iff [iff]: "0 le succ(x) <-> Ord(x)"
416 by (blast intro: Ord_0_le elim: ltE)
418 lemma subset_imp_le: "[| j<=i;  Ord(i);  Ord(j) |] ==> j le i"
419 apply (rule not_lt_iff_le [THEN iffD1], assumption+)
420 apply (blast elim: ltE mem_irrefl)
421 done
423 lemma le_imp_subset: "i le j ==> i<=j"
424 by (blast dest: OrdmemD elim: ltE leE)
426 lemma le_subset_iff: "j le i <-> j<=i & Ord(i) & Ord(j)"
427 by (blast dest: subset_imp_le le_imp_subset elim: ltE)
429 lemma le_succ_iff: "i le succ(j) <-> i le j | i=succ(j) & Ord(i)"
430 apply (simp (no_asm) add: le_iff)
431 apply blast
432 done
434 (*Just a variant of subset_imp_le*)
435 lemma all_lt_imp_le: "[| Ord(i);  Ord(j);  !!x. x<j ==> x<i |] ==> j le i"
436 by (blast intro: not_lt_imp_le dest: lt_irrefl)
438 subsubsection{*Transitivity Laws*}
440 lemma lt_trans1: "[| i le j;  j<k |] ==> i<k"
441 by (blast elim!: leE intro: lt_trans)
443 lemma lt_trans2: "[| i<j;  j le k |] ==> i<k"
444 by (blast elim!: leE intro: lt_trans)
446 lemma le_trans: "[| i le j;  j le k |] ==> i le k"
447 by (blast intro: lt_trans1)
449 lemma succ_leI: "i<j ==> succ(i) le j"
450 apply (rule not_lt_iff_le [THEN iffD1])
451 apply (blast elim: ltE leE lt_asym)+
452 done
454 (*Identical to  succ(i) < succ(j) ==> i<j  *)
455 lemma succ_leE: "succ(i) le j ==> i<j"
456 apply (rule not_le_iff_lt [THEN iffD1])
457 apply (blast elim: ltE leE lt_asym)+
458 done
460 lemma succ_le_iff [iff]: "succ(i) le j <-> i<j"
461 by (blast intro: succ_leI succ_leE)
463 lemma succ_le_imp_le: "succ(i) le succ(j) ==> i le j"
464 by (blast dest!: succ_leE)
466 lemma lt_subset_trans: "[| i <= j;  j<k;  Ord(i) |] ==> i<k"
467 apply (rule subset_imp_le [THEN lt_trans1])
468 apply (blast intro: elim: ltE) +
469 done
471 lemma lt_imp_0_lt: "j<i ==> 0<i"
472 by (blast intro: lt_trans1 Ord_0_le [OF lt_Ord])
474 lemma succ_lt_iff: "succ(i) < j <-> i<j & succ(i) \<noteq> j"
475 apply auto
476 apply (blast intro: lt_trans le_refl dest: lt_Ord)
477 apply (frule lt_Ord)
478 apply (rule not_le_iff_lt [THEN iffD1])
479   apply (blast intro: lt_Ord2)
480  apply blast
481 apply (simp add: lt_Ord lt_Ord2 le_iff)
482 apply (blast dest: lt_asym)
483 done
485 lemma Ord_succ_mem_iff: "Ord(j) ==> succ(i) \<in> succ(j) <-> i\<in>j"
486 apply (insert succ_le_iff [of i j])
488 done
490 subsubsection{*Union and Intersection*}
492 lemma Un_upper1_le: "[| Ord(i); Ord(j) |] ==> i le i Un j"
493 by (rule Un_upper1 [THEN subset_imp_le], auto)
495 lemma Un_upper2_le: "[| Ord(i); Ord(j) |] ==> j le i Un j"
496 by (rule Un_upper2 [THEN subset_imp_le], auto)
498 (*Replacing k by succ(k') yields the similar rule for le!*)
499 lemma Un_least_lt: "[| i<k;  j<k |] ==> i Un j < k"
500 apply (rule_tac i = i and j = j in Ord_linear_le)
501 apply (auto simp add: Un_commute le_subset_iff subset_Un_iff lt_Ord)
502 done
504 lemma Un_least_lt_iff: "[| Ord(i); Ord(j) |] ==> i Un j < k  <->  i<k & j<k"
505 apply (safe intro!: Un_least_lt)
506 apply (rule_tac [2] Un_upper2_le [THEN lt_trans1])
507 apply (rule Un_upper1_le [THEN lt_trans1], auto)
508 done
510 lemma Un_least_mem_iff:
511     "[| Ord(i); Ord(j); Ord(k) |] ==> i Un j : k  <->  i:k & j:k"
512 apply (insert Un_least_lt_iff [of i j k])
514 done
516 (*Replacing k by succ(k') yields the similar rule for le!*)
517 lemma Int_greatest_lt: "[| i<k;  j<k |] ==> i Int j < k"
518 apply (rule_tac i = i and j = j in Ord_linear_le)
519 apply (auto simp add: Int_commute le_subset_iff subset_Int_iff lt_Ord)
520 done
522 lemma Ord_Un_if:
523      "[| Ord(i); Ord(j) |] ==> i \<union> j = (if j<i then i else j)"
524 by (simp add: not_lt_iff_le le_imp_subset leI
525               subset_Un_iff [symmetric]  subset_Un_iff2 [symmetric])
527 lemma succ_Un_distrib:
528      "[| Ord(i); Ord(j) |] ==> succ(i \<union> j) = succ(i) \<union> succ(j)"
529 by (simp add: Ord_Un_if lt_Ord le_Ord2)
531 lemma lt_Un_iff:
532      "[| Ord(i); Ord(j) |] ==> k < i \<union> j <-> k < i | k < j";
533 apply (simp add: Ord_Un_if not_lt_iff_le)
534 apply (blast intro: leI lt_trans2)+
535 done
537 lemma le_Un_iff:
538      "[| Ord(i); Ord(j) |] ==> k \<le> i \<union> j <-> k \<le> i | k \<le> j";
539 by (simp add: succ_Un_distrib lt_Un_iff [symmetric])
541 lemma Un_upper1_lt: "[|k < i; Ord(j)|] ==> k < i Un j"
542 by (simp add: lt_Un_iff lt_Ord2)
544 lemma Un_upper2_lt: "[|k < j; Ord(i)|] ==> k < i Un j"
545 by (simp add: lt_Un_iff lt_Ord2)
548 lemma Ord_Union_succ_eq: "Ord(i) ==> \<Union>(succ(i)) = i"
549 by (blast intro: Ord_trans)
554 lemma Ord_Union [intro,simp,TC]: "[| !!i. i:A ==> Ord(i) |] ==> Ord(Union(A))"
555 apply (rule Ord_is_Transset [THEN Transset_Union_family, THEN OrdI])
556 apply (blast intro: Ord_contains_Transset)+
557 done
559 lemma Ord_UN [intro,simp,TC]:
560      "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(\<Union>x\<in>A. B(x))"
561 by (rule Ord_Union, blast)
563 lemma Ord_Inter [intro,simp,TC]:
564     "[| !!i. i:A ==> Ord(i) |] ==> Ord(Inter(A))"
565 apply (rule Transset_Inter_family [THEN OrdI])
566 apply (blast intro: Ord_is_Transset)
568 apply (blast intro: Ord_contains_Transset)
569 done
571 lemma Ord_INT [intro,simp,TC]:
572     "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(\<Inter>x\<in>A. B(x))"
573 by (rule Ord_Inter, blast)
576 (* No < version; consider (\<Union>i\<in>nat.i)=nat *)
577 lemma UN_least_le:
578     "[| Ord(i);  !!x. x:A ==> b(x) le i |] ==> (\<Union>x\<in>A. b(x)) le i"
579 apply (rule le_imp_subset [THEN UN_least, THEN subset_imp_le])
580 apply (blast intro: Ord_UN elim: ltE)+
581 done
583 lemma UN_succ_least_lt:
584     "[| j<i;  !!x. x:A ==> b(x)<j |] ==> (\<Union>x\<in>A. succ(b(x))) < i"
585 apply (rule ltE, assumption)
586 apply (rule UN_least_le [THEN lt_trans2])
587 apply (blast intro: succ_leI)+
588 done
590 lemma UN_upper_lt:
591      "[| a\<in>A;  i < b(a);  Ord(\<Union>x\<in>A. b(x)) |] ==> i < (\<Union>x\<in>A. b(x))"
592 by (unfold lt_def, blast)
594 lemma UN_upper_le:
595      "[| a: A;  i le b(a);  Ord(\<Union>x\<in>A. b(x)) |] ==> i le (\<Union>x\<in>A. b(x))"
596 apply (frule ltD)
597 apply (rule le_imp_subset [THEN subset_trans, THEN subset_imp_le])
598 apply (blast intro: lt_Ord UN_upper)+
599 done
601 lemma lt_Union_iff: "\<forall>i\<in>A. Ord(i) ==> (j < \<Union>(A)) <-> (\<exists>i\<in>A. j<i)"
602 by (auto simp: lt_def Ord_Union)
604 lemma Union_upper_le:
605      "[| j: J;  i\<le>j;  Ord(\<Union>(J)) |] ==> i \<le> \<Union>J"
606 apply (subst Union_eq_UN)
607 apply (rule UN_upper_le, auto)
608 done
610 lemma le_implies_UN_le_UN:
611     "[| !!x. x:A ==> c(x) le d(x) |] ==> (\<Union>x\<in>A. c(x)) le (\<Union>x\<in>A. d(x))"
612 apply (rule UN_least_le)
613 apply (rule_tac [2] UN_upper_le)
614 apply (blast intro: Ord_UN le_Ord2)+
615 done
617 lemma Ord_equality: "Ord(i) ==> (\<Union>y\<in>i. succ(y)) = i"
618 by (blast intro: Ord_trans)
620 (*Holds for all transitive sets, not just ordinals*)
621 lemma Ord_Union_subset: "Ord(i) ==> Union(i) <= i"
622 by (blast intro: Ord_trans)
625 subsection{*Limit Ordinals -- General Properties*}
627 lemma Limit_Union_eq: "Limit(i) ==> Union(i) = i"
628 apply (unfold Limit_def)
629 apply (fast intro!: ltI elim!: ltE elim: Ord_trans)
630 done
632 lemma Limit_is_Ord: "Limit(i) ==> Ord(i)"
633 apply (unfold Limit_def)
634 apply (erule conjunct1)
635 done
637 lemma Limit_has_0: "Limit(i) ==> 0 < i"
638 apply (unfold Limit_def)
639 apply (erule conjunct2 [THEN conjunct1])
640 done
642 lemma Limit_nonzero: "Limit(i) ==> i ~= 0"
643 by (drule Limit_has_0, blast)
645 lemma Limit_has_succ: "[| Limit(i);  j<i |] ==> succ(j) < i"
646 by (unfold Limit_def, blast)
648 lemma Limit_succ_lt_iff [simp]: "Limit(i) ==> succ(j) < i <-> (j<i)"
649 apply (safe intro!: Limit_has_succ)
650 apply (frule lt_Ord)
651 apply (blast intro: lt_trans)
652 done
654 lemma zero_not_Limit [iff]: "~ Limit(0)"
657 lemma Limit_has_1: "Limit(i) ==> 1 < i"
658 by (blast intro: Limit_has_0 Limit_has_succ)
660 lemma increasing_LimitI: "[| 0<l; \<forall>x\<in>l. \<exists>y\<in>l. x<y |] ==> Limit(l)"
661 apply (unfold Limit_def, simp add: lt_Ord2, clarify)
662 apply (drule_tac i=y in ltD)
663 apply (blast intro: lt_trans1 [OF _ ltI] lt_Ord2)
664 done
666 lemma non_succ_LimitI:
667     "[| 0<i;  ALL y. succ(y) ~= i |] ==> Limit(i)"
668 apply (unfold Limit_def)
669 apply (safe del: subsetI)
670 apply (rule_tac [2] not_le_iff_lt [THEN iffD1])
671 apply (simp_all add: lt_Ord lt_Ord2)
672 apply (blast elim: leE lt_asym)
673 done
675 lemma succ_LimitE [elim!]: "Limit(succ(i)) ==> P"
676 apply (rule lt_irrefl)
677 apply (rule Limit_has_succ, assumption)
678 apply (erule Limit_is_Ord [THEN Ord_succD, THEN le_refl])
679 done
681 lemma not_succ_Limit [simp]: "~ Limit(succ(i))"
682 by blast
684 lemma Limit_le_succD: "[| Limit(i);  i le succ(j) |] ==> i le j"
685 by (blast elim!: leE)
688 subsubsection{*Traditional 3-Way Case Analysis on Ordinals*}
690 lemma Ord_cases_disj: "Ord(i) ==> i=0 | (EX j. Ord(j) & i=succ(j)) | Limit(i)"
691 by (blast intro!: non_succ_LimitI Ord_0_lt)
693 lemma Ord_cases:
694     "[| Ord(i);
695         i=0                          ==> P;
696         !!j. [| Ord(j); i=succ(j) |] ==> P;
697         Limit(i)                     ==> P
698      |] ==> P"
699 by (drule Ord_cases_disj, blast)
701 lemma trans_induct3 [case_names 0 succ limit, consumes 1]:
702      "[| Ord(i);
703          P(0);
704          !!x. [| Ord(x);  P(x) |] ==> P(succ(x));
705          !!x. [| Limit(x);  ALL y:x. P(y) |] ==> P(x)
706       |] ==> P(i)"
707 apply (erule trans_induct)
708 apply (erule Ord_cases, blast+)
709 done
711 lemmas trans_induct3_rule = trans_induct3 [rule_format, case_names 0 succ limit, consumes 1]
713 text{*A set of ordinals is either empty, contains its own union, or its
714 union is a limit ordinal.*}
715 lemma Ord_set_cases:
716    "\<forall>i\<in>I. Ord(i) ==> I=0 \<or> \<Union>(I) \<in> I \<or> (\<Union>(I) \<notin> I \<and> Limit(\<Union>(I)))"
717 apply (clarify elim!: not_emptyE)
718 apply (cases "\<Union>(I)" rule: Ord_cases)
719    apply (blast intro: Ord_Union)
720   apply (blast intro: subst_elem)
721  apply auto
722 apply (clarify elim!: equalityE succ_subsetE)
724 apply (subgoal_tac "B = succ(j)", blast)
725 apply (rule le_anti_sym)