src/ZF/Ordinal.thy
 author paulson Thu May 16 09:16:22 2002 +0200 (2002-05-16) changeset 13155 dcbf6cb95534 parent 12114 a8e860c86252 child 13162 660a71e712af permissions -rw-r--r--
converting Ordinal.ML to Isar format
1 (*  Title:      ZF/Ordinal.thy
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
4     Copyright   1994  University of Cambridge
6 Ordinals in Zermelo-Fraenkel Set Theory
7 *)
9 theory Ordinal = WF + Bool + equalities:
11 constdefs
13   Memrel        :: "i=>i"
14     "Memrel(A)   == {z: A*A . EX x y. z=<x,y> & x:y }"
16   Transset  :: "i=>o"
17     "Transset(i) == ALL x:i. x<=i"
19   Ord  :: "i=>o"
20     "Ord(i)      == Transset(i) & (ALL x:i. Transset(x))"
22   lt        :: "[i,i] => o"  (infixl "<" 50)   (*less-than on ordinals*)
23     "i<j         == i:j & Ord(j)"
25   Limit         :: "i=>o"
26     "Limit(i)    == Ord(i) & 0<i & (ALL y. y<i --> succ(y)<i)"
28 syntax
29   "le"          :: "[i,i] => o"  (infixl 50)   (*less-than or equals*)
31 translations
32   "x le y"      == "x < succ(y)"
34 syntax (xsymbols)
35   "op le"       :: "[i,i] => o"  (infixl "\<le>" 50)  (*less-than or equals*)
38 (*** Rules for Transset ***)
40 (** Three neat characterisations of Transset **)
42 lemma Transset_iff_Pow: "Transset(A) <-> A<=Pow(A)"
43 by (unfold Transset_def, blast)
45 lemma Transset_iff_Union_succ: "Transset(A) <-> Union(succ(A)) = A"
46 apply (unfold Transset_def)
47 apply (blast elim!: equalityE)
48 done
50 lemma Transset_iff_Union_subset: "Transset(A) <-> Union(A) <= A"
51 by (unfold Transset_def, blast)
53 (** Consequences of downwards closure **)
55 lemma Transset_doubleton_D:
56     "[| Transset(C); {a,b}: C |] ==> a:C & b: C"
57 by (unfold Transset_def, blast)
59 lemma Transset_Pair_D:
60     "[| Transset(C); <a,b>: C |] ==> a:C & b: C"
62 apply (blast dest: Transset_doubleton_D)
63 done
65 lemma Transset_includes_domain:
66     "[| Transset(C); A*B <= C; b: B |] ==> A <= C"
67 by (blast dest: Transset_Pair_D)
69 lemma Transset_includes_range:
70     "[| Transset(C); A*B <= C; a: A |] ==> B <= C"
71 by (blast dest: Transset_Pair_D)
73 (** Closure properties **)
75 lemma Transset_0: "Transset(0)"
76 by (unfold Transset_def, blast)
78 lemma Transset_Un:
79     "[| Transset(i);  Transset(j) |] ==> Transset(i Un j)"
80 by (unfold Transset_def, blast)
82 lemma Transset_Int:
83     "[| Transset(i);  Transset(j) |] ==> Transset(i Int j)"
84 by (unfold Transset_def, blast)
86 lemma Transset_succ: "Transset(i) ==> Transset(succ(i))"
87 by (unfold Transset_def, blast)
89 lemma Transset_Pow: "Transset(i) ==> Transset(Pow(i))"
90 by (unfold Transset_def, blast)
92 lemma Transset_Union: "Transset(A) ==> Transset(Union(A))"
93 by (unfold Transset_def, blast)
95 lemma Transset_Union_family:
96     "[| !!i. i:A ==> Transset(i) |] ==> Transset(Union(A))"
97 by (unfold Transset_def, blast)
99 lemma Transset_Inter_family:
100     "[| j:A;  !!i. i:A ==> Transset(i) |] ==> Transset(Inter(A))"
101 by (unfold Transset_def, blast)
103 (*** Natural Deduction rules for Ord ***)
105 lemma OrdI:
106     "[| Transset(i);  !!x. x:i ==> Transset(x) |]  ==>  Ord(i)"
109 lemma Ord_is_Transset: "Ord(i) ==> Transset(i)"
112 lemma Ord_contains_Transset:
113     "[| Ord(i);  j:i |] ==> Transset(j) "
114 by (unfold Ord_def, blast)
116 (*** Lemmas for ordinals ***)
118 lemma Ord_in_Ord: "[| Ord(i);  j:i |] ==> Ord(j)"
119 by (unfold Ord_def Transset_def, blast)
121 (* Ord(succ(j)) ==> Ord(j) *)
122 lemmas Ord_succD = Ord_in_Ord [OF _ succI1]
124 lemma Ord_subset_Ord: "[| Ord(i);  Transset(j);  j<=i |] ==> Ord(j)"
125 by (simp add: Ord_def Transset_def, blast)
127 lemma OrdmemD: "[| j:i;  Ord(i) |] ==> j<=i"
128 by (unfold Ord_def Transset_def, blast)
130 lemma Ord_trans: "[| i:j;  j:k;  Ord(k) |] ==> i:k"
131 by (blast dest: OrdmemD)
133 lemma Ord_succ_subsetI: "[| i:j;  Ord(j) |] ==> succ(i) <= j"
134 by (blast dest: OrdmemD)
137 (*** The construction of ordinals: 0, succ, Union ***)
139 lemma Ord_0 [iff,TC]: "Ord(0)"
140 by (blast intro: OrdI Transset_0)
142 lemma Ord_succ [TC]: "Ord(i) ==> Ord(succ(i))"
143 by (blast intro: OrdI Transset_succ Ord_is_Transset Ord_contains_Transset)
145 lemmas Ord_1 = Ord_0 [THEN Ord_succ]
147 lemma Ord_succ_iff [iff]: "Ord(succ(i)) <-> Ord(i)"
148 by (blast intro: Ord_succ dest!: Ord_succD)
150 lemma Ord_Un [TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Un j)"
151 apply (unfold Ord_def)
152 apply (blast intro!: Transset_Un)
153 done
155 lemma Ord_Int [TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Int j)"
156 apply (unfold Ord_def)
157 apply (blast intro!: Transset_Int)
158 done
161 lemma Ord_Inter:
162     "[| j:A;  !!i. i:A ==> Ord(i) |] ==> Ord(Inter(A))"
163 apply (rule Transset_Inter_family [THEN OrdI], assumption)
164 apply (blast intro: Ord_is_Transset)
165 apply (blast intro: Ord_contains_Transset)
166 done
168 lemma Ord_INT:
169     "[| j:A;  !!x. x:A ==> Ord(B(x)) |] ==> Ord(INT x:A. B(x))"
170 by (rule RepFunI [THEN Ord_Inter], assumption, blast)
172 (*There is no set of all ordinals, for then it would contain itself*)
173 lemma ON_class: "~ (ALL i. i:X <-> Ord(i))"
174 apply (rule notI)
175 apply (frule_tac x = "X" in spec)
176 apply (safe elim!: mem_irrefl)
177 apply (erule swap, rule OrdI [OF _ Ord_is_Transset])
179 apply (blast intro: Ord_in_Ord)+
180 done
182 (*** < is 'less than' for ordinals ***)
184 lemma ltI: "[| i:j;  Ord(j) |] ==> i<j"
185 by (unfold lt_def, blast)
187 lemma ltE:
188     "[| i<j;  [| i:j;  Ord(i);  Ord(j) |] ==> P |] ==> P"
189 apply (unfold lt_def)
190 apply (blast intro: Ord_in_Ord)
191 done
193 lemma ltD: "i<j ==> i:j"
194 by (erule ltE, assumption)
196 lemma not_lt0 [simp]: "~ i<0"
197 by (unfold lt_def, blast)
199 lemma lt_Ord: "j<i ==> Ord(j)"
200 by (erule ltE, assumption)
202 lemma lt_Ord2: "j<i ==> Ord(i)"
203 by (erule ltE, assumption)
205 (* "ja le j ==> Ord(j)" *)
206 lemmas le_Ord2 = lt_Ord2 [THEN Ord_succD]
208 (* i<0 ==> R *)
209 lemmas lt0E = not_lt0 [THEN notE, elim!]
211 lemma lt_trans: "[| i<j;  j<k |] ==> i<k"
212 by (blast intro!: ltI elim!: ltE intro: Ord_trans)
214 lemma lt_not_sym: "i<j ==> ~ (j<i)"
215 apply (unfold lt_def)
216 apply (blast elim: mem_asym)
217 done
219 (* [| i<j;  ~P ==> j<i |] ==> P *)
220 lemmas lt_asym = lt_not_sym [THEN swap]
222 lemma lt_irrefl [elim!]: "i<i ==> P"
223 by (blast intro: lt_asym)
225 lemma lt_not_refl: "~ i<i"
226 apply (rule notI)
227 apply (erule lt_irrefl)
228 done
231 (** le is less than or equals;  recall  i le j  abbrevs  i<succ(j) !! **)
233 lemma le_iff: "i le j <-> i<j | (i=j & Ord(j))"
234 by (unfold lt_def, blast)
236 (*Equivalently, i<j ==> i < succ(j)*)
237 lemma leI: "i<j ==> i le j"
238 by (simp (no_asm_simp) add: le_iff)
240 lemma le_eqI: "[| i=j;  Ord(j) |] ==> i le j"
241 by (simp (no_asm_simp) add: le_iff)
243 lemmas le_refl = refl [THEN le_eqI]
245 lemma le_refl_iff [iff]: "i le i <-> Ord(i)"
246 by (simp (no_asm_simp) add: lt_not_refl le_iff)
248 lemma leCI: "(~ (i=j & Ord(j)) ==> i<j) ==> i le j"
249 by (simp add: le_iff, blast)
251 lemma leE:
252     "[| i le j;  i<j ==> P;  [| i=j;  Ord(j) |] ==> P |] ==> P"
253 by (simp add: le_iff, blast)
255 lemma le_anti_sym: "[| i le j;  j le i |] ==> i=j"
257 apply (blast elim: lt_asym)
258 done
260 lemma le0_iff [simp]: "i le 0 <-> i=0"
261 by (blast elim!: leE)
263 lemmas le0D = le0_iff [THEN iffD1, dest!]
265 (*** Natural Deduction rules for Memrel ***)
267 (*The lemmas MemrelI/E give better speed than [iff] here*)
268 lemma Memrel_iff [simp]: "<a,b> : Memrel(A) <-> a:b & a:A & b:A"
269 by (unfold Memrel_def, blast)
271 lemma MemrelI [intro!]: "[| a: b;  a: A;  b: A |] ==> <a,b> : Memrel(A)"
272 by auto
274 lemma MemrelE [elim!]:
275     "[| <a,b> : Memrel(A);
276         [| a: A;  b: A;  a:b |]  ==> P |]
277      ==> P"
278 by auto
280 lemma Memrel_type: "Memrel(A) <= A*A"
281 by (unfold Memrel_def, blast)
283 lemma Memrel_mono: "A<=B ==> Memrel(A) <= Memrel(B)"
284 by (unfold Memrel_def, blast)
286 lemma Memrel_0 [simp]: "Memrel(0) = 0"
287 by (unfold Memrel_def, blast)
289 lemma Memrel_1 [simp]: "Memrel(1) = 0"
290 by (unfold Memrel_def, blast)
292 (*The membership relation (as a set) is well-founded.
293   Proof idea: show A<=B by applying the foundation axiom to A-B *)
294 lemma wf_Memrel: "wf(Memrel(A))"
295 apply (unfold wf_def)
296 apply (rule foundation [THEN disjE, THEN allI], erule disjI1, blast)
297 done
299 (*Transset(i) does not suffice, though ALL j:i.Transset(j) does*)
300 lemma trans_Memrel:
301     "Ord(i) ==> trans(Memrel(i))"
302 by (unfold Ord_def Transset_def trans_def, blast)
304 (*If Transset(A) then Memrel(A) internalizes the membership relation below A*)
305 lemma Transset_Memrel_iff:
306     "Transset(A) ==> <a,b> : Memrel(A) <-> a:b & b:A"
307 by (unfold Transset_def, blast)
310 (*** Transfinite induction ***)
312 (*Epsilon induction over a transitive set*)
313 lemma Transset_induct:
314     "[| i: k;  Transset(k);
315         !!x.[| x: k;  ALL y:x. P(y) |] ==> P(x) |]
316      ==>  P(i)"
318 apply (erule wf_Memrel [THEN wf_induct2], blast)
319 apply blast
320 done
322 (*Induction over an ordinal*)
323 lemmas Ord_induct = Transset_induct [OF _ Ord_is_Transset]
325 (*Induction over the class of ordinals -- a useful corollary of Ord_induct*)
327 lemma trans_induct:
328     "[| Ord(i);
329         !!x.[| Ord(x);  ALL y:x. P(y) |] ==> P(x) |]
330      ==>  P(i)"
331 apply (rule Ord_succ [THEN succI1 [THEN Ord_induct]], assumption)
332 apply (blast intro: Ord_succ [THEN Ord_in_Ord])
333 done
336 (*** Fundamental properties of the epsilon ordering (< on ordinals) ***)
339 (** Proving that < is a linear ordering on the ordinals **)
341 lemma Ord_linear [rule_format]:
342      "Ord(i) ==> (ALL j. Ord(j) --> i:j | i=j | j:i)"
343 apply (erule trans_induct)
344 apply (rule impI [THEN allI])
345 apply (erule_tac i=j in trans_induct)
346 apply (blast dest: Ord_trans)
347 done
349 (*The trichotomy law for ordinals!*)
350 lemma Ord_linear_lt:
351     "[| Ord(i);  Ord(j);  i<j ==> P;  i=j ==> P;  j<i ==> P |] ==> P"
353 apply (rule_tac i1=i and j1=j in Ord_linear [THEN disjE], blast+)
354 done
356 lemma Ord_linear2:
357     "[| Ord(i);  Ord(j);  i<j ==> P;  j le i ==> P |]  ==> P"
358 apply (rule_tac i = "i" and j = "j" in Ord_linear_lt)
359 apply (blast intro: leI le_eqI sym ) +
360 done
362 lemma Ord_linear_le:
363     "[| Ord(i);  Ord(j);  i le j ==> P;  j le i ==> P |]  ==> P"
364 apply (rule_tac i = "i" and j = "j" in Ord_linear_lt)
365 apply (blast intro: leI le_eqI ) +
366 done
368 lemma le_imp_not_lt: "j le i ==> ~ i<j"
369 by (blast elim!: leE elim: lt_asym)
371 lemma not_lt_imp_le: "[| ~ i<j;  Ord(i);  Ord(j) |] ==> j le i"
372 by (rule_tac i = "i" and j = "j" in Ord_linear2, auto)
374 (** Some rewrite rules for <, le **)
376 lemma Ord_mem_iff_lt: "Ord(j) ==> i:j <-> i<j"
377 by (unfold lt_def, blast)
379 lemma not_lt_iff_le: "[| Ord(i);  Ord(j) |] ==> ~ i<j <-> j le i"
380 by (blast dest: le_imp_not_lt not_lt_imp_le)
382 lemma not_le_iff_lt: "[| Ord(i);  Ord(j) |] ==> ~ i le j <-> j<i"
383 by (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym])
385 (*This is identical to 0<succ(i) *)
386 lemma Ord_0_le: "Ord(i) ==> 0 le i"
387 by (erule not_lt_iff_le [THEN iffD1], auto)
389 lemma Ord_0_lt: "[| Ord(i);  i~=0 |] ==> 0<i"
390 apply (erule not_le_iff_lt [THEN iffD1])
391 apply (rule Ord_0, blast)
392 done
394 lemma Ord_0_lt_iff: "Ord(i) ==> i~=0 <-> 0<i"
395 by (blast intro: Ord_0_lt)
398 (*** Results about less-than or equals ***)
400 (** For ordinals, j<=i (subset) implies j le i (less-than or equals) **)
402 lemma zero_le_succ_iff [iff]: "0 le succ(x) <-> Ord(x)"
403 by (blast intro: Ord_0_le elim: ltE)
405 lemma subset_imp_le: "[| j<=i;  Ord(i);  Ord(j) |] ==> j le i"
406 apply (rule not_lt_iff_le [THEN iffD1], assumption)
407 apply assumption
408 apply (blast elim: ltE mem_irrefl)
409 done
411 lemma le_imp_subset: "i le j ==> i<=j"
412 by (blast dest: OrdmemD elim: ltE leE)
414 lemma le_subset_iff: "j le i <-> j<=i & Ord(i) & Ord(j)"
415 by (blast dest: subset_imp_le le_imp_subset elim: ltE)
417 lemma le_succ_iff: "i le succ(j) <-> i le j | i=succ(j) & Ord(i)"
418 apply (simp (no_asm) add: le_iff)
419 apply blast
420 done
422 (*Just a variant of subset_imp_le*)
423 lemma all_lt_imp_le: "[| Ord(i);  Ord(j);  !!x. x<j ==> x<i |] ==> j le i"
424 by (blast intro: not_lt_imp_le dest: lt_irrefl)
426 (** Transitive laws **)
428 lemma lt_trans1: "[| i le j;  j<k |] ==> i<k"
429 by (blast elim!: leE intro: lt_trans)
431 lemma lt_trans2: "[| i<j;  j le k |] ==> i<k"
432 by (blast elim!: leE intro: lt_trans)
434 lemma le_trans: "[| i le j;  j le k |] ==> i le k"
435 by (blast intro: lt_trans1)
437 lemma succ_leI: "i<j ==> succ(i) le j"
438 apply (rule not_lt_iff_le [THEN iffD1])
439 apply (blast elim: ltE leE lt_asym)+
440 done
442 (*Identical to  succ(i) < succ(j) ==> i<j  *)
443 lemma succ_leE: "succ(i) le j ==> i<j"
444 apply (rule not_le_iff_lt [THEN iffD1])
445 apply (blast elim: ltE leE lt_asym)+
446 done
448 lemma succ_le_iff [iff]: "succ(i) le j <-> i<j"
449 by (blast intro: succ_leI succ_leE)
451 lemma succ_le_imp_le: "succ(i) le succ(j) ==> i le j"
452 by (blast dest!: succ_leE)
454 lemma lt_subset_trans: "[| i <= j;  j<k;  Ord(i) |] ==> i<k"
455 apply (rule subset_imp_le [THEN lt_trans1])
456 apply (blast intro: elim: ltE) +
457 done
459 (** Union and Intersection **)
461 lemma Un_upper1_le: "[| Ord(i); Ord(j) |] ==> i le i Un j"
462 by (rule Un_upper1 [THEN subset_imp_le], auto)
464 lemma Un_upper2_le: "[| Ord(i); Ord(j) |] ==> j le i Un j"
465 by (rule Un_upper2 [THEN subset_imp_le], auto)
467 (*Replacing k by succ(k') yields the similar rule for le!*)
468 lemma Un_least_lt: "[| i<k;  j<k |] ==> i Un j < k"
469 apply (rule_tac i = "i" and j = "j" in Ord_linear_le)
470 apply (auto simp add: Un_commute le_subset_iff subset_Un_iff lt_Ord)
471 done
473 lemma Un_least_lt_iff: "[| Ord(i); Ord(j) |] ==> i Un j < k  <->  i<k & j<k"
474 apply (safe intro!: Un_least_lt)
475 apply (rule_tac [2] Un_upper2_le [THEN lt_trans1])
476 apply (rule Un_upper1_le [THEN lt_trans1], auto)
477 done
479 lemma Un_least_mem_iff:
480     "[| Ord(i); Ord(j); Ord(k) |] ==> i Un j : k  <->  i:k & j:k"
481 apply (insert Un_least_lt_iff [of i j k])
483 done
485 (*Replacing k by succ(k') yields the similar rule for le!*)
486 lemma Int_greatest_lt: "[| i<k;  j<k |] ==> i Int j < k"
487 apply (rule_tac i = "i" and j = "j" in Ord_linear_le)
488 apply (auto simp add: Int_commute le_subset_iff subset_Int_iff lt_Ord)
489 done
491 (*FIXME: the Intersection duals are missing!*)
493 (*** Results about limits ***)
495 lemma Ord_Union: "[| !!i. i:A ==> Ord(i) |] ==> Ord(Union(A))"
496 apply (rule Ord_is_Transset [THEN Transset_Union_family, THEN OrdI])
497 apply (blast intro: Ord_contains_Transset)+
498 done
500 lemma Ord_UN: "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(UN x:A. B(x))"
501 by (rule Ord_Union, blast)
503 (* No < version; consider (UN i:nat.i)=nat *)
504 lemma UN_least_le:
505     "[| Ord(i);  !!x. x:A ==> b(x) le i |] ==> (UN x:A. b(x)) le i"
506 apply (rule le_imp_subset [THEN UN_least, THEN subset_imp_le])
507 apply (blast intro: Ord_UN elim: ltE)+
508 done
510 lemma UN_succ_least_lt:
511     "[| j<i;  !!x. x:A ==> b(x)<j |] ==> (UN x:A. succ(b(x))) < i"
512 apply (rule ltE, assumption)
513 apply (rule UN_least_le [THEN lt_trans2])
514 apply (blast intro: succ_leI)+
515 done
517 lemma UN_upper_le:
518      "[| a: A;  i le b(a);  Ord(UN x:A. b(x)) |] ==> i le (UN x:A. b(x))"
519 apply (frule ltD)
520 apply (rule le_imp_subset [THEN subset_trans, THEN subset_imp_le])
521 apply (blast intro: lt_Ord UN_upper)+
522 done
524 lemma le_implies_UN_le_UN:
525     "[| !!x. x:A ==> c(x) le d(x) |] ==> (UN x:A. c(x)) le (UN x:A. d(x))"
526 apply (rule UN_least_le)
527 apply (rule_tac [2] UN_upper_le)
528 apply (blast intro: Ord_UN le_Ord2)+
529 done
531 lemma Ord_equality: "Ord(i) ==> (UN y:i. succ(y)) = i"
532 by (blast intro: Ord_trans)
534 (*Holds for all transitive sets, not just ordinals*)
535 lemma Ord_Union_subset: "Ord(i) ==> Union(i) <= i"
536 by (blast intro: Ord_trans)
539 (*** Limit ordinals -- general properties ***)
541 lemma Limit_Union_eq: "Limit(i) ==> Union(i) = i"
542 apply (unfold Limit_def)
543 apply (fast intro!: ltI elim!: ltE elim: Ord_trans)
544 done
546 lemma Limit_is_Ord: "Limit(i) ==> Ord(i)"
547 apply (unfold Limit_def)
548 apply (erule conjunct1)
549 done
551 lemma Limit_has_0: "Limit(i) ==> 0 < i"
552 apply (unfold Limit_def)
553 apply (erule conjunct2 [THEN conjunct1])
554 done
556 lemma Limit_has_succ: "[| Limit(i);  j<i |] ==> succ(j) < i"
557 by (unfold Limit_def, blast)
559 lemma non_succ_LimitI:
560     "[| 0<i;  ALL y. succ(y) ~= i |] ==> Limit(i)"
561 apply (unfold Limit_def)
562 apply (safe del: subsetI)
563 apply (rule_tac [2] not_le_iff_lt [THEN iffD1])
564 apply (simp_all add: lt_Ord lt_Ord2)
565 apply (blast elim: leE lt_asym)
566 done
568 lemma succ_LimitE [elim!]: "Limit(succ(i)) ==> P"
569 apply (rule lt_irrefl)
570 apply (rule Limit_has_succ, assumption)
571 apply (erule Limit_is_Ord [THEN Ord_succD, THEN le_refl])
572 done
574 lemma not_succ_Limit [simp]: "~ Limit(succ(i))"
575 by blast
577 lemma Limit_le_succD: "[| Limit(i);  i le succ(j) |] ==> i le j"
578 by (blast elim!: leE)
580 (** Traditional 3-way case analysis on ordinals **)
582 lemma Ord_cases_disj: "Ord(i) ==> i=0 | (EX j. Ord(j) & i=succ(j)) | Limit(i)"
583 by (blast intro!: non_succ_LimitI Ord_0_lt)
585 lemma Ord_cases:
586     "[| Ord(i);
587         i=0                          ==> P;
588         !!j. [| Ord(j); i=succ(j) |] ==> P;
589         Limit(i)                     ==> P
590      |] ==> P"
591 by (drule Ord_cases_disj, blast)
593 lemma trans_induct3:
594      "[| Ord(i);
595          P(0);
596          !!x. [| Ord(x);  P(x) |] ==> P(succ(x));
597          !!x. [| Limit(x);  ALL y:x. P(y) |] ==> P(x)
598       |] ==> P(i)"
599 apply (erule trans_induct)
600 apply (erule Ord_cases, blast+)
601 done
603 ML
604 {*
605 val Memrel_def = thm "Memrel_def";
606 val Transset_def = thm "Transset_def";
607 val Ord_def = thm "Ord_def";
608 val lt_def = thm "lt_def";
609 val Limit_def = thm "Limit_def";
611 val Transset_iff_Pow = thm "Transset_iff_Pow";
612 val Transset_iff_Union_succ = thm "Transset_iff_Union_succ";
613 val Transset_iff_Union_subset = thm "Transset_iff_Union_subset";
614 val Transset_doubleton_D = thm "Transset_doubleton_D";
615 val Transset_Pair_D = thm "Transset_Pair_D";
616 val Transset_includes_domain = thm "Transset_includes_domain";
617 val Transset_includes_range = thm "Transset_includes_range";
618 val Transset_0 = thm "Transset_0";
619 val Transset_Un = thm "Transset_Un";
620 val Transset_Int = thm "Transset_Int";
621 val Transset_succ = thm "Transset_succ";
622 val Transset_Pow = thm "Transset_Pow";
623 val Transset_Union = thm "Transset_Union";
624 val Transset_Union_family = thm "Transset_Union_family";
625 val Transset_Inter_family = thm "Transset_Inter_family";
626 val OrdI = thm "OrdI";
627 val Ord_is_Transset = thm "Ord_is_Transset";
628 val Ord_contains_Transset = thm "Ord_contains_Transset";
629 val Ord_in_Ord = thm "Ord_in_Ord";
630 val Ord_succD = thm "Ord_succD";
631 val Ord_subset_Ord = thm "Ord_subset_Ord";
632 val OrdmemD = thm "OrdmemD";
633 val Ord_trans = thm "Ord_trans";
634 val Ord_succ_subsetI = thm "Ord_succ_subsetI";
635 val Ord_0 = thm "Ord_0";
636 val Ord_succ = thm "Ord_succ";
637 val Ord_1 = thm "Ord_1";
638 val Ord_succ_iff = thm "Ord_succ_iff";
639 val Ord_Un = thm "Ord_Un";
640 val Ord_Int = thm "Ord_Int";
641 val Ord_Inter = thm "Ord_Inter";
642 val Ord_INT = thm "Ord_INT";
643 val ON_class = thm "ON_class";
644 val ltI = thm "ltI";
645 val ltE = thm "ltE";
646 val ltD = thm "ltD";
647 val not_lt0 = thm "not_lt0";
648 val lt_Ord = thm "lt_Ord";
649 val lt_Ord2 = thm "lt_Ord2";
650 val le_Ord2 = thm "le_Ord2";
651 val lt0E = thm "lt0E";
652 val lt_trans = thm "lt_trans";
653 val lt_not_sym = thm "lt_not_sym";
654 val lt_asym = thm "lt_asym";
655 val lt_irrefl = thm "lt_irrefl";
656 val lt_not_refl = thm "lt_not_refl";
657 val le_iff = thm "le_iff";
658 val leI = thm "leI";
659 val le_eqI = thm "le_eqI";
660 val le_refl = thm "le_refl";
661 val le_refl_iff = thm "le_refl_iff";
662 val leCI = thm "leCI";
663 val leE = thm "leE";
664 val le_anti_sym = thm "le_anti_sym";
665 val le0_iff = thm "le0_iff";
666 val le0D = thm "le0D";
667 val Memrel_iff = thm "Memrel_iff";
668 val MemrelI = thm "MemrelI";
669 val MemrelE = thm "MemrelE";
670 val Memrel_type = thm "Memrel_type";
671 val Memrel_mono = thm "Memrel_mono";
672 val Memrel_0 = thm "Memrel_0";
673 val Memrel_1 = thm "Memrel_1";
674 val wf_Memrel = thm "wf_Memrel";
675 val trans_Memrel = thm "trans_Memrel";
676 val Transset_Memrel_iff = thm "Transset_Memrel_iff";
677 val Transset_induct = thm "Transset_induct";
678 val Ord_induct = thm "Ord_induct";
679 val trans_induct = thm "trans_induct";
680 val Ord_linear = thm "Ord_linear";
681 val Ord_linear_lt = thm "Ord_linear_lt";
682 val Ord_linear2 = thm "Ord_linear2";
683 val Ord_linear_le = thm "Ord_linear_le";
684 val le_imp_not_lt = thm "le_imp_not_lt";
685 val not_lt_imp_le = thm "not_lt_imp_le";
686 val Ord_mem_iff_lt = thm "Ord_mem_iff_lt";
687 val not_lt_iff_le = thm "not_lt_iff_le";
688 val not_le_iff_lt = thm "not_le_iff_lt";
689 val Ord_0_le = thm "Ord_0_le";
690 val Ord_0_lt = thm "Ord_0_lt";
691 val Ord_0_lt_iff = thm "Ord_0_lt_iff";
692 val zero_le_succ_iff = thm "zero_le_succ_iff";
693 val subset_imp_le = thm "subset_imp_le";
694 val le_imp_subset = thm "le_imp_subset";
695 val le_subset_iff = thm "le_subset_iff";
696 val le_succ_iff = thm "le_succ_iff";
697 val all_lt_imp_le = thm "all_lt_imp_le";
698 val lt_trans1 = thm "lt_trans1";
699 val lt_trans2 = thm "lt_trans2";
700 val le_trans = thm "le_trans";
701 val succ_leI = thm "succ_leI";
702 val succ_leE = thm "succ_leE";
703 val succ_le_iff = thm "succ_le_iff";
704 val succ_le_imp_le = thm "succ_le_imp_le";
705 val lt_subset_trans = thm "lt_subset_trans";
706 val Un_upper1_le = thm "Un_upper1_le";
707 val Un_upper2_le = thm "Un_upper2_le";
708 val Un_least_lt = thm "Un_least_lt";
709 val Un_least_lt_iff = thm "Un_least_lt_iff";
710 val Un_least_mem_iff = thm "Un_least_mem_iff";
711 val Int_greatest_lt = thm "Int_greatest_lt";
712 val Ord_Union = thm "Ord_Union";
713 val Ord_UN = thm "Ord_UN";
714 val UN_least_le = thm "UN_least_le";
715 val UN_succ_least_lt = thm "UN_succ_least_lt";
716 val UN_upper_le = thm "UN_upper_le";
717 val le_implies_UN_le_UN = thm "le_implies_UN_le_UN";
718 val Ord_equality = thm "Ord_equality";
719 val Ord_Union_subset = thm "Ord_Union_subset";
720 val Limit_Union_eq = thm "Limit_Union_eq";
721 val Limit_is_Ord = thm "Limit_is_Ord";
722 val Limit_has_0 = thm "Limit_has_0";
723 val Limit_has_succ = thm "Limit_has_succ";
724 val non_succ_LimitI = thm "non_succ_LimitI";
725 val succ_LimitE = thm "succ_LimitE";
726 val not_succ_Limit = thm "not_succ_Limit";
727 val Limit_le_succD = thm "Limit_le_succD";
728 val Ord_cases_disj = thm "Ord_cases_disj";
729 val Ord_cases = thm "Ord_cases";
730 val trans_induct3 = thm "trans_induct3";
731 *}
733 end