src/ZF/Ordinal.thy
 author wenzelm Sun Oct 07 21:19:31 2007 +0200 (2007-10-07) changeset 24893 b8ef7afe3a6b parent 22808 a7daa74e2980 child 35762 af3ff2ba4c54 permissions -rw-r--r--
modernized specifications;
removed legacy ML bindings;
```     1 (*  Title:      ZF/Ordinal.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1994  University of Cambridge
```
```     5
```
```     6 *)
```
```     7
```
```     8 header{*Transitive Sets and Ordinals*}
```
```     9
```
```    10 theory Ordinal imports WF Bool equalities begin
```
```    11
```
```    12 definition
```
```    13   Memrel        :: "i=>i"  where
```
```    14     "Memrel(A)   == {z: A*A . EX x y. z=<x,y> & x:y }"
```
```    15
```
```    16 definition
```
```    17   Transset  :: "i=>o"  where
```
```    18     "Transset(i) == ALL x:i. x<=i"
```
```    19
```
```    20 definition
```
```    21   Ord  :: "i=>o"  where
```
```    22     "Ord(i)      == Transset(i) & (ALL x:i. Transset(x))"
```
```    23
```
```    24 definition
```
```    25   lt        :: "[i,i] => o"  (infixl "<" 50)   (*less-than on ordinals*)  where
```
```    26     "i<j         == i:j & Ord(j)"
```
```    27
```
```    28 definition
```
```    29   Limit         :: "i=>o"  where
```
```    30     "Limit(i)    == Ord(i) & 0<i & (ALL y. y<i --> succ(y)<i)"
```
```    31
```
```    32 abbreviation
```
```    33   le  (infixl "le" 50) where
```
```    34   "x le y == x < succ(y)"
```
```    35
```
```    36 notation (xsymbols)
```
```    37   le  (infixl "\<le>" 50)
```
```    38
```
```    39 notation (HTML output)
```
```    40   le  (infixl "\<le>" 50)
```
```    41
```
```    42
```
```    43 subsection{*Rules for Transset*}
```
```    44
```
```    45 subsubsection{*Three Neat Characterisations of Transset*}
```
```    46
```
```    47 lemma Transset_iff_Pow: "Transset(A) <-> A<=Pow(A)"
```
```    48 by (unfold Transset_def, blast)
```
```    49
```
```    50 lemma Transset_iff_Union_succ: "Transset(A) <-> Union(succ(A)) = A"
```
```    51 apply (unfold Transset_def)
```
```    52 apply (blast elim!: equalityE)
```
```    53 done
```
```    54
```
```    55 lemma Transset_iff_Union_subset: "Transset(A) <-> Union(A) <= A"
```
```    56 by (unfold Transset_def, blast)
```
```    57
```
```    58 subsubsection{*Consequences of Downwards Closure*}
```
```    59
```
```    60 lemma Transset_doubleton_D:
```
```    61     "[| Transset(C); {a,b}: C |] ==> a:C & b: C"
```
```    62 by (unfold Transset_def, blast)
```
```    63
```
```    64 lemma Transset_Pair_D:
```
```    65     "[| Transset(C); <a,b>: C |] ==> a:C & b: C"
```
```    66 apply (simp add: Pair_def)
```
```    67 apply (blast dest: Transset_doubleton_D)
```
```    68 done
```
```    69
```
```    70 lemma Transset_includes_domain:
```
```    71     "[| Transset(C); A*B <= C; b: B |] ==> A <= C"
```
```    72 by (blast dest: Transset_Pair_D)
```
```    73
```
```    74 lemma Transset_includes_range:
```
```    75     "[| Transset(C); A*B <= C; a: A |] ==> B <= C"
```
```    76 by (blast dest: Transset_Pair_D)
```
```    77
```
```    78 subsubsection{*Closure Properties*}
```
```    79
```
```    80 lemma Transset_0: "Transset(0)"
```
```    81 by (unfold Transset_def, blast)
```
```    82
```
```    83 lemma Transset_Un:
```
```    84     "[| Transset(i);  Transset(j) |] ==> Transset(i Un j)"
```
```    85 by (unfold Transset_def, blast)
```
```    86
```
```    87 lemma Transset_Int:
```
```    88     "[| Transset(i);  Transset(j) |] ==> Transset(i Int j)"
```
```    89 by (unfold Transset_def, blast)
```
```    90
```
```    91 lemma Transset_succ: "Transset(i) ==> Transset(succ(i))"
```
```    92 by (unfold Transset_def, blast)
```
```    93
```
```    94 lemma Transset_Pow: "Transset(i) ==> Transset(Pow(i))"
```
```    95 by (unfold Transset_def, blast)
```
```    96
```
```    97 lemma Transset_Union: "Transset(A) ==> Transset(Union(A))"
```
```    98 by (unfold Transset_def, blast)
```
```    99
```
```   100 lemma Transset_Union_family:
```
```   101     "[| !!i. i:A ==> Transset(i) |] ==> Transset(Union(A))"
```
```   102 by (unfold Transset_def, blast)
```
```   103
```
```   104 lemma Transset_Inter_family:
```
```   105     "[| !!i. i:A ==> Transset(i) |] ==> Transset(Inter(A))"
```
```   106 by (unfold Inter_def Transset_def, blast)
```
```   107
```
```   108 lemma Transset_UN:
```
```   109      "(!!x. x \<in> A ==> Transset(B(x))) ==> Transset (\<Union>x\<in>A. B(x))"
```
```   110 by (rule Transset_Union_family, auto)
```
```   111
```
```   112 lemma Transset_INT:
```
```   113      "(!!x. x \<in> A ==> Transset(B(x))) ==> Transset (\<Inter>x\<in>A. B(x))"
```
```   114 by (rule Transset_Inter_family, auto)
```
```   115
```
```   116
```
```   117 subsection{*Lemmas for Ordinals*}
```
```   118
```
```   119 lemma OrdI:
```
```   120     "[| Transset(i);  !!x. x:i ==> Transset(x) |]  ==>  Ord(i)"
```
```   121 by (simp add: Ord_def)
```
```   122
```
```   123 lemma Ord_is_Transset: "Ord(i) ==> Transset(i)"
```
```   124 by (simp add: Ord_def)
```
```   125
```
```   126 lemma Ord_contains_Transset:
```
```   127     "[| Ord(i);  j:i |] ==> Transset(j) "
```
```   128 by (unfold Ord_def, blast)
```
```   129
```
```   130
```
```   131 lemma Ord_in_Ord: "[| Ord(i);  j:i |] ==> Ord(j)"
```
```   132 by (unfold Ord_def Transset_def, blast)
```
```   133
```
```   134 (*suitable for rewriting PROVIDED i has been fixed*)
```
```   135 lemma Ord_in_Ord': "[| j:i; Ord(i) |] ==> Ord(j)"
```
```   136 by (blast intro: Ord_in_Ord)
```
```   137
```
```   138 (* Ord(succ(j)) ==> Ord(j) *)
```
```   139 lemmas Ord_succD = Ord_in_Ord [OF _ succI1]
```
```   140
```
```   141 lemma Ord_subset_Ord: "[| Ord(i);  Transset(j);  j<=i |] ==> Ord(j)"
```
```   142 by (simp add: Ord_def Transset_def, blast)
```
```   143
```
```   144 lemma OrdmemD: "[| j:i;  Ord(i) |] ==> j<=i"
```
```   145 by (unfold Ord_def Transset_def, blast)
```
```   146
```
```   147 lemma Ord_trans: "[| i:j;  j:k;  Ord(k) |] ==> i:k"
```
```   148 by (blast dest: OrdmemD)
```
```   149
```
```   150 lemma Ord_succ_subsetI: "[| i:j;  Ord(j) |] ==> succ(i) <= j"
```
```   151 by (blast dest: OrdmemD)
```
```   152
```
```   153
```
```   154 subsection{*The Construction of Ordinals: 0, succ, Union*}
```
```   155
```
```   156 lemma Ord_0 [iff,TC]: "Ord(0)"
```
```   157 by (blast intro: OrdI Transset_0)
```
```   158
```
```   159 lemma Ord_succ [TC]: "Ord(i) ==> Ord(succ(i))"
```
```   160 by (blast intro: OrdI Transset_succ Ord_is_Transset Ord_contains_Transset)
```
```   161
```
```   162 lemmas Ord_1 = Ord_0 [THEN Ord_succ]
```
```   163
```
```   164 lemma Ord_succ_iff [iff]: "Ord(succ(i)) <-> Ord(i)"
```
```   165 by (blast intro: Ord_succ dest!: Ord_succD)
```
```   166
```
```   167 lemma Ord_Un [intro,simp,TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Un j)"
```
```   168 apply (unfold Ord_def)
```
```   169 apply (blast intro!: Transset_Un)
```
```   170 done
```
```   171
```
```   172 lemma Ord_Int [TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Int j)"
```
```   173 apply (unfold Ord_def)
```
```   174 apply (blast intro!: Transset_Int)
```
```   175 done
```
```   176
```
```   177 (*There is no set of all ordinals, for then it would contain itself*)
```
```   178 lemma ON_class: "~ (ALL i. i:X <-> Ord(i))"
```
```   179 apply (rule notI)
```
```   180 apply (frule_tac x = X in spec)
```
```   181 apply (safe elim!: mem_irrefl)
```
```   182 apply (erule swap, rule OrdI [OF _ Ord_is_Transset])
```
```   183 apply (simp add: Transset_def)
```
```   184 apply (blast intro: Ord_in_Ord)+
```
```   185 done
```
```   186
```
```   187 subsection{*< is 'less Than' for Ordinals*}
```
```   188
```
```   189 lemma ltI: "[| i:j;  Ord(j) |] ==> i<j"
```
```   190 by (unfold lt_def, blast)
```
```   191
```
```   192 lemma ltE:
```
```   193     "[| i<j;  [| i:j;  Ord(i);  Ord(j) |] ==> P |] ==> P"
```
```   194 apply (unfold lt_def)
```
```   195 apply (blast intro: Ord_in_Ord)
```
```   196 done
```
```   197
```
```   198 lemma ltD: "i<j ==> i:j"
```
```   199 by (erule ltE, assumption)
```
```   200
```
```   201 lemma not_lt0 [simp]: "~ i<0"
```
```   202 by (unfold lt_def, blast)
```
```   203
```
```   204 lemma lt_Ord: "j<i ==> Ord(j)"
```
```   205 by (erule ltE, assumption)
```
```   206
```
```   207 lemma lt_Ord2: "j<i ==> Ord(i)"
```
```   208 by (erule ltE, assumption)
```
```   209
```
```   210 (* "ja le j ==> Ord(j)" *)
```
```   211 lemmas le_Ord2 = lt_Ord2 [THEN Ord_succD]
```
```   212
```
```   213 (* i<0 ==> R *)
```
```   214 lemmas lt0E = not_lt0 [THEN notE, elim!]
```
```   215
```
```   216 lemma lt_trans: "[| i<j;  j<k |] ==> i<k"
```
```   217 by (blast intro!: ltI elim!: ltE intro: Ord_trans)
```
```   218
```
```   219 lemma lt_not_sym: "i<j ==> ~ (j<i)"
```
```   220 apply (unfold lt_def)
```
```   221 apply (blast elim: mem_asym)
```
```   222 done
```
```   223
```
```   224 (* [| i<j;  ~P ==> j<i |] ==> P *)
```
```   225 lemmas lt_asym = lt_not_sym [THEN swap]
```
```   226
```
```   227 lemma lt_irrefl [elim!]: "i<i ==> P"
```
```   228 by (blast intro: lt_asym)
```
```   229
```
```   230 lemma lt_not_refl: "~ i<i"
```
```   231 apply (rule notI)
```
```   232 apply (erule lt_irrefl)
```
```   233 done
```
```   234
```
```   235
```
```   236 (** le is less than or equals;  recall  i le j  abbrevs  i<succ(j) !! **)
```
```   237
```
```   238 lemma le_iff: "i le j <-> i<j | (i=j & Ord(j))"
```
```   239 by (unfold lt_def, blast)
```
```   240
```
```   241 (*Equivalently, i<j ==> i < succ(j)*)
```
```   242 lemma leI: "i<j ==> i le j"
```
```   243 by (simp (no_asm_simp) add: le_iff)
```
```   244
```
```   245 lemma le_eqI: "[| i=j;  Ord(j) |] ==> i le j"
```
```   246 by (simp (no_asm_simp) add: le_iff)
```
```   247
```
```   248 lemmas le_refl = refl [THEN le_eqI]
```
```   249
```
```   250 lemma le_refl_iff [iff]: "i le i <-> Ord(i)"
```
```   251 by (simp (no_asm_simp) add: lt_not_refl le_iff)
```
```   252
```
```   253 lemma leCI: "(~ (i=j & Ord(j)) ==> i<j) ==> i le j"
```
```   254 by (simp add: le_iff, blast)
```
```   255
```
```   256 lemma leE:
```
```   257     "[| i le j;  i<j ==> P;  [| i=j;  Ord(j) |] ==> P |] ==> P"
```
```   258 by (simp add: le_iff, blast)
```
```   259
```
```   260 lemma le_anti_sym: "[| i le j;  j le i |] ==> i=j"
```
```   261 apply (simp add: le_iff)
```
```   262 apply (blast elim: lt_asym)
```
```   263 done
```
```   264
```
```   265 lemma le0_iff [simp]: "i le 0 <-> i=0"
```
```   266 by (blast elim!: leE)
```
```   267
```
```   268 lemmas le0D = le0_iff [THEN iffD1, dest!]
```
```   269
```
```   270 subsection{*Natural Deduction Rules for Memrel*}
```
```   271
```
```   272 (*The lemmas MemrelI/E give better speed than [iff] here*)
```
```   273 lemma Memrel_iff [simp]: "<a,b> : Memrel(A) <-> a:b & a:A & b:A"
```
```   274 by (unfold Memrel_def, blast)
```
```   275
```
```   276 lemma MemrelI [intro!]: "[| a: b;  a: A;  b: A |] ==> <a,b> : Memrel(A)"
```
```   277 by auto
```
```   278
```
```   279 lemma MemrelE [elim!]:
```
```   280     "[| <a,b> : Memrel(A);
```
```   281         [| a: A;  b: A;  a:b |]  ==> P |]
```
```   282      ==> P"
```
```   283 by auto
```
```   284
```
```   285 lemma Memrel_type: "Memrel(A) <= A*A"
```
```   286 by (unfold Memrel_def, blast)
```
```   287
```
```   288 lemma Memrel_mono: "A<=B ==> Memrel(A) <= Memrel(B)"
```
```   289 by (unfold Memrel_def, blast)
```
```   290
```
```   291 lemma Memrel_0 [simp]: "Memrel(0) = 0"
```
```   292 by (unfold Memrel_def, blast)
```
```   293
```
```   294 lemma Memrel_1 [simp]: "Memrel(1) = 0"
```
```   295 by (unfold Memrel_def, blast)
```
```   296
```
```   297 lemma relation_Memrel: "relation(Memrel(A))"
```
```   298 by (simp add: relation_def Memrel_def)
```
```   299
```
```   300 (*The membership relation (as a set) is well-founded.
```
```   301   Proof idea: show A<=B by applying the foundation axiom to A-B *)
```
```   302 lemma wf_Memrel: "wf(Memrel(A))"
```
```   303 apply (unfold wf_def)
```
```   304 apply (rule foundation [THEN disjE, THEN allI], erule disjI1, blast)
```
```   305 done
```
```   306
```
```   307 text{*The premise @{term "Ord(i)"} does not suffice.*}
```
```   308 lemma trans_Memrel:
```
```   309     "Ord(i) ==> trans(Memrel(i))"
```
```   310 by (unfold Ord_def Transset_def trans_def, blast)
```
```   311
```
```   312 text{*However, the following premise is strong enough.*}
```
```   313 lemma Transset_trans_Memrel:
```
```   314     "\<forall>j\<in>i. Transset(j) ==> trans(Memrel(i))"
```
```   315 by (unfold Transset_def trans_def, blast)
```
```   316
```
```   317 (*If Transset(A) then Memrel(A) internalizes the membership relation below A*)
```
```   318 lemma Transset_Memrel_iff:
```
```   319     "Transset(A) ==> <a,b> : Memrel(A) <-> a:b & b:A"
```
```   320 by (unfold Transset_def, blast)
```
```   321
```
```   322
```
```   323 subsection{*Transfinite Induction*}
```
```   324
```
```   325 (*Epsilon induction over a transitive set*)
```
```   326 lemma Transset_induct:
```
```   327     "[| i: k;  Transset(k);
```
```   328         !!x.[| x: k;  ALL y:x. P(y) |] ==> P(x) |]
```
```   329      ==>  P(i)"
```
```   330 apply (simp add: Transset_def)
```
```   331 apply (erule wf_Memrel [THEN wf_induct2], blast+)
```
```   332 done
```
```   333
```
```   334 (*Induction over an ordinal*)
```
```   335 lemmas Ord_induct [consumes 2] = Transset_induct [OF _ Ord_is_Transset]
```
```   336 lemmas Ord_induct_rule = Ord_induct [rule_format, consumes 2]
```
```   337
```
```   338 (*Induction over the class of ordinals -- a useful corollary of Ord_induct*)
```
```   339
```
```   340 lemma trans_induct [consumes 1]:
```
```   341     "[| Ord(i);
```
```   342         !!x.[| Ord(x);  ALL y:x. P(y) |] ==> P(x) |]
```
```   343      ==>  P(i)"
```
```   344 apply (rule Ord_succ [THEN succI1 [THEN Ord_induct]], assumption)
```
```   345 apply (blast intro: Ord_succ [THEN Ord_in_Ord])
```
```   346 done
```
```   347
```
```   348 lemmas trans_induct_rule = trans_induct [rule_format, consumes 1]
```
```   349
```
```   350
```
```   351 (*** Fundamental properties of the epsilon ordering (< on ordinals) ***)
```
```   352
```
```   353
```
```   354 subsubsection{*Proving That < is a Linear Ordering on the Ordinals*}
```
```   355
```
```   356 lemma Ord_linear [rule_format]:
```
```   357      "Ord(i) ==> (ALL j. Ord(j) --> i:j | i=j | j:i)"
```
```   358 apply (erule trans_induct)
```
```   359 apply (rule impI [THEN allI])
```
```   360 apply (erule_tac i=j in trans_induct)
```
```   361 apply (blast dest: Ord_trans)
```
```   362 done
```
```   363
```
```   364 (*The trichotomy law for ordinals!*)
```
```   365 lemma Ord_linear_lt:
```
```   366     "[| Ord(i);  Ord(j);  i<j ==> P;  i=j ==> P;  j<i ==> P |] ==> P"
```
```   367 apply (simp add: lt_def)
```
```   368 apply (rule_tac i1=i and j1=j in Ord_linear [THEN disjE], blast+)
```
```   369 done
```
```   370
```
```   371 lemma Ord_linear2:
```
```   372     "[| Ord(i);  Ord(j);  i<j ==> P;  j le i ==> P |]  ==> P"
```
```   373 apply (rule_tac i = i and j = j in Ord_linear_lt)
```
```   374 apply (blast intro: leI le_eqI sym ) +
```
```   375 done
```
```   376
```
```   377 lemma Ord_linear_le:
```
```   378     "[| Ord(i);  Ord(j);  i le j ==> P;  j le i ==> P |]  ==> P"
```
```   379 apply (rule_tac i = i and j = j in Ord_linear_lt)
```
```   380 apply (blast intro: leI le_eqI ) +
```
```   381 done
```
```   382
```
```   383 lemma le_imp_not_lt: "j le i ==> ~ i<j"
```
```   384 by (blast elim!: leE elim: lt_asym)
```
```   385
```
```   386 lemma not_lt_imp_le: "[| ~ i<j;  Ord(i);  Ord(j) |] ==> j le i"
```
```   387 by (rule_tac i = i and j = j in Ord_linear2, auto)
```
```   388
```
```   389 subsubsection{*Some Rewrite Rules for <, le*}
```
```   390
```
```   391 lemma Ord_mem_iff_lt: "Ord(j) ==> i:j <-> i<j"
```
```   392 by (unfold lt_def, blast)
```
```   393
```
```   394 lemma not_lt_iff_le: "[| Ord(i);  Ord(j) |] ==> ~ i<j <-> j le i"
```
```   395 by (blast dest: le_imp_not_lt not_lt_imp_le)
```
```   396
```
```   397 lemma not_le_iff_lt: "[| Ord(i);  Ord(j) |] ==> ~ i le j <-> j<i"
```
```   398 by (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym])
```
```   399
```
```   400 (*This is identical to 0<succ(i) *)
```
```   401 lemma Ord_0_le: "Ord(i) ==> 0 le i"
```
```   402 by (erule not_lt_iff_le [THEN iffD1], auto)
```
```   403
```
```   404 lemma Ord_0_lt: "[| Ord(i);  i~=0 |] ==> 0<i"
```
```   405 apply (erule not_le_iff_lt [THEN iffD1])
```
```   406 apply (rule Ord_0, blast)
```
```   407 done
```
```   408
```
```   409 lemma Ord_0_lt_iff: "Ord(i) ==> i~=0 <-> 0<i"
```
```   410 by (blast intro: Ord_0_lt)
```
```   411
```
```   412
```
```   413 subsection{*Results about Less-Than or Equals*}
```
```   414
```
```   415 (** For ordinals, j<=i (subset) implies j le i (less-than or equals) **)
```
```   416
```
```   417 lemma zero_le_succ_iff [iff]: "0 le succ(x) <-> Ord(x)"
```
```   418 by (blast intro: Ord_0_le elim: ltE)
```
```   419
```
```   420 lemma subset_imp_le: "[| j<=i;  Ord(i);  Ord(j) |] ==> j le i"
```
```   421 apply (rule not_lt_iff_le [THEN iffD1], assumption+)
```
```   422 apply (blast elim: ltE mem_irrefl)
```
```   423 done
```
```   424
```
```   425 lemma le_imp_subset: "i le j ==> i<=j"
```
```   426 by (blast dest: OrdmemD elim: ltE leE)
```
```   427
```
```   428 lemma le_subset_iff: "j le i <-> j<=i & Ord(i) & Ord(j)"
```
```   429 by (blast dest: subset_imp_le le_imp_subset elim: ltE)
```
```   430
```
```   431 lemma le_succ_iff: "i le succ(j) <-> i le j | i=succ(j) & Ord(i)"
```
```   432 apply (simp (no_asm) add: le_iff)
```
```   433 apply blast
```
```   434 done
```
```   435
```
```   436 (*Just a variant of subset_imp_le*)
```
```   437 lemma all_lt_imp_le: "[| Ord(i);  Ord(j);  !!x. x<j ==> x<i |] ==> j le i"
```
```   438 by (blast intro: not_lt_imp_le dest: lt_irrefl)
```
```   439
```
```   440 subsubsection{*Transitivity Laws*}
```
```   441
```
```   442 lemma lt_trans1: "[| i le j;  j<k |] ==> i<k"
```
```   443 by (blast elim!: leE intro: lt_trans)
```
```   444
```
```   445 lemma lt_trans2: "[| i<j;  j le k |] ==> i<k"
```
```   446 by (blast elim!: leE intro: lt_trans)
```
```   447
```
```   448 lemma le_trans: "[| i le j;  j le k |] ==> i le k"
```
```   449 by (blast intro: lt_trans1)
```
```   450
```
```   451 lemma succ_leI: "i<j ==> succ(i) le j"
```
```   452 apply (rule not_lt_iff_le [THEN iffD1])
```
```   453 apply (blast elim: ltE leE lt_asym)+
```
```   454 done
```
```   455
```
```   456 (*Identical to  succ(i) < succ(j) ==> i<j  *)
```
```   457 lemma succ_leE: "succ(i) le j ==> i<j"
```
```   458 apply (rule not_le_iff_lt [THEN iffD1])
```
```   459 apply (blast elim: ltE leE lt_asym)+
```
```   460 done
```
```   461
```
```   462 lemma succ_le_iff [iff]: "succ(i) le j <-> i<j"
```
```   463 by (blast intro: succ_leI succ_leE)
```
```   464
```
```   465 lemma succ_le_imp_le: "succ(i) le succ(j) ==> i le j"
```
```   466 by (blast dest!: succ_leE)
```
```   467
```
```   468 lemma lt_subset_trans: "[| i <= j;  j<k;  Ord(i) |] ==> i<k"
```
```   469 apply (rule subset_imp_le [THEN lt_trans1])
```
```   470 apply (blast intro: elim: ltE) +
```
```   471 done
```
```   472
```
```   473 lemma lt_imp_0_lt: "j<i ==> 0<i"
```
```   474 by (blast intro: lt_trans1 Ord_0_le [OF lt_Ord])
```
```   475
```
```   476 lemma succ_lt_iff: "succ(i) < j <-> i<j & succ(i) \<noteq> j"
```
```   477 apply auto
```
```   478 apply (blast intro: lt_trans le_refl dest: lt_Ord)
```
```   479 apply (frule lt_Ord)
```
```   480 apply (rule not_le_iff_lt [THEN iffD1])
```
```   481   apply (blast intro: lt_Ord2)
```
```   482  apply blast
```
```   483 apply (simp add: lt_Ord lt_Ord2 le_iff)
```
```   484 apply (blast dest: lt_asym)
```
```   485 done
```
```   486
```
```   487 lemma Ord_succ_mem_iff: "Ord(j) ==> succ(i) \<in> succ(j) <-> i\<in>j"
```
```   488 apply (insert succ_le_iff [of i j])
```
```   489 apply (simp add: lt_def)
```
```   490 done
```
```   491
```
```   492 subsubsection{*Union and Intersection*}
```
```   493
```
```   494 lemma Un_upper1_le: "[| Ord(i); Ord(j) |] ==> i le i Un j"
```
```   495 by (rule Un_upper1 [THEN subset_imp_le], auto)
```
```   496
```
```   497 lemma Un_upper2_le: "[| Ord(i); Ord(j) |] ==> j le i Un j"
```
```   498 by (rule Un_upper2 [THEN subset_imp_le], auto)
```
```   499
```
```   500 (*Replacing k by succ(k') yields the similar rule for le!*)
```
```   501 lemma Un_least_lt: "[| i<k;  j<k |] ==> i Un j < k"
```
```   502 apply (rule_tac i = i and j = j in Ord_linear_le)
```
```   503 apply (auto simp add: Un_commute le_subset_iff subset_Un_iff lt_Ord)
```
```   504 done
```
```   505
```
```   506 lemma Un_least_lt_iff: "[| Ord(i); Ord(j) |] ==> i Un j < k  <->  i<k & j<k"
```
```   507 apply (safe intro!: Un_least_lt)
```
```   508 apply (rule_tac [2] Un_upper2_le [THEN lt_trans1])
```
```   509 apply (rule Un_upper1_le [THEN lt_trans1], auto)
```
```   510 done
```
```   511
```
```   512 lemma Un_least_mem_iff:
```
```   513     "[| Ord(i); Ord(j); Ord(k) |] ==> i Un j : k  <->  i:k & j:k"
```
```   514 apply (insert Un_least_lt_iff [of i j k])
```
```   515 apply (simp add: lt_def)
```
```   516 done
```
```   517
```
```   518 (*Replacing k by succ(k') yields the similar rule for le!*)
```
```   519 lemma Int_greatest_lt: "[| i<k;  j<k |] ==> i Int j < k"
```
```   520 apply (rule_tac i = i and j = j in Ord_linear_le)
```
```   521 apply (auto simp add: Int_commute le_subset_iff subset_Int_iff lt_Ord)
```
```   522 done
```
```   523
```
```   524 lemma Ord_Un_if:
```
```   525      "[| Ord(i); Ord(j) |] ==> i \<union> j = (if j<i then i else j)"
```
```   526 by (simp add: not_lt_iff_le le_imp_subset leI
```
```   527               subset_Un_iff [symmetric]  subset_Un_iff2 [symmetric])
```
```   528
```
```   529 lemma succ_Un_distrib:
```
```   530      "[| Ord(i); Ord(j) |] ==> succ(i \<union> j) = succ(i) \<union> succ(j)"
```
```   531 by (simp add: Ord_Un_if lt_Ord le_Ord2)
```
```   532
```
```   533 lemma lt_Un_iff:
```
```   534      "[| Ord(i); Ord(j) |] ==> k < i \<union> j <-> k < i | k < j";
```
```   535 apply (simp add: Ord_Un_if not_lt_iff_le)
```
```   536 apply (blast intro: leI lt_trans2)+
```
```   537 done
```
```   538
```
```   539 lemma le_Un_iff:
```
```   540      "[| Ord(i); Ord(j) |] ==> k \<le> i \<union> j <-> k \<le> i | k \<le> j";
```
```   541 by (simp add: succ_Un_distrib lt_Un_iff [symmetric])
```
```   542
```
```   543 lemma Un_upper1_lt: "[|k < i; Ord(j)|] ==> k < i Un j"
```
```   544 by (simp add: lt_Un_iff lt_Ord2)
```
```   545
```
```   546 lemma Un_upper2_lt: "[|k < j; Ord(i)|] ==> k < i Un j"
```
```   547 by (simp add: lt_Un_iff lt_Ord2)
```
```   548
```
```   549 (*See also Transset_iff_Union_succ*)
```
```   550 lemma Ord_Union_succ_eq: "Ord(i) ==> \<Union>(succ(i)) = i"
```
```   551 by (blast intro: Ord_trans)
```
```   552
```
```   553
```
```   554 subsection{*Results about Limits*}
```
```   555
```
```   556 lemma Ord_Union [intro,simp,TC]: "[| !!i. i:A ==> Ord(i) |] ==> Ord(Union(A))"
```
```   557 apply (rule Ord_is_Transset [THEN Transset_Union_family, THEN OrdI])
```
```   558 apply (blast intro: Ord_contains_Transset)+
```
```   559 done
```
```   560
```
```   561 lemma Ord_UN [intro,simp,TC]:
```
```   562      "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(\<Union>x\<in>A. B(x))"
```
```   563 by (rule Ord_Union, blast)
```
```   564
```
```   565 lemma Ord_Inter [intro,simp,TC]:
```
```   566     "[| !!i. i:A ==> Ord(i) |] ==> Ord(Inter(A))"
```
```   567 apply (rule Transset_Inter_family [THEN OrdI])
```
```   568 apply (blast intro: Ord_is_Transset)
```
```   569 apply (simp add: Inter_def)
```
```   570 apply (blast intro: Ord_contains_Transset)
```
```   571 done
```
```   572
```
```   573 lemma Ord_INT [intro,simp,TC]:
```
```   574     "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(\<Inter>x\<in>A. B(x))"
```
```   575 by (rule Ord_Inter, blast)
```
```   576
```
```   577
```
```   578 (* No < version; consider (\<Union>i\<in>nat.i)=nat *)
```
```   579 lemma UN_least_le:
```
```   580     "[| Ord(i);  !!x. x:A ==> b(x) le i |] ==> (\<Union>x\<in>A. b(x)) le i"
```
```   581 apply (rule le_imp_subset [THEN UN_least, THEN subset_imp_le])
```
```   582 apply (blast intro: Ord_UN elim: ltE)+
```
```   583 done
```
```   584
```
```   585 lemma UN_succ_least_lt:
```
```   586     "[| j<i;  !!x. x:A ==> b(x)<j |] ==> (\<Union>x\<in>A. succ(b(x))) < i"
```
```   587 apply (rule ltE, assumption)
```
```   588 apply (rule UN_least_le [THEN lt_trans2])
```
```   589 apply (blast intro: succ_leI)+
```
```   590 done
```
```   591
```
```   592 lemma UN_upper_lt:
```
```   593      "[| a\<in>A;  i < b(a);  Ord(\<Union>x\<in>A. b(x)) |] ==> i < (\<Union>x\<in>A. b(x))"
```
```   594 by (unfold lt_def, blast)
```
```   595
```
```   596 lemma UN_upper_le:
```
```   597      "[| a: A;  i le b(a);  Ord(\<Union>x\<in>A. b(x)) |] ==> i le (\<Union>x\<in>A. b(x))"
```
```   598 apply (frule ltD)
```
```   599 apply (rule le_imp_subset [THEN subset_trans, THEN subset_imp_le])
```
```   600 apply (blast intro: lt_Ord UN_upper)+
```
```   601 done
```
```   602
```
```   603 lemma lt_Union_iff: "\<forall>i\<in>A. Ord(i) ==> (j < \<Union>(A)) <-> (\<exists>i\<in>A. j<i)"
```
```   604 by (auto simp: lt_def Ord_Union)
```
```   605
```
```   606 lemma Union_upper_le:
```
```   607      "[| j: J;  i\<le>j;  Ord(\<Union>(J)) |] ==> i \<le> \<Union>J"
```
```   608 apply (subst Union_eq_UN)
```
```   609 apply (rule UN_upper_le, auto)
```
```   610 done
```
```   611
```
```   612 lemma le_implies_UN_le_UN:
```
```   613     "[| !!x. x:A ==> c(x) le d(x) |] ==> (\<Union>x\<in>A. c(x)) le (\<Union>x\<in>A. d(x))"
```
```   614 apply (rule UN_least_le)
```
```   615 apply (rule_tac [2] UN_upper_le)
```
```   616 apply (blast intro: Ord_UN le_Ord2)+
```
```   617 done
```
```   618
```
```   619 lemma Ord_equality: "Ord(i) ==> (\<Union>y\<in>i. succ(y)) = i"
```
```   620 by (blast intro: Ord_trans)
```
```   621
```
```   622 (*Holds for all transitive sets, not just ordinals*)
```
```   623 lemma Ord_Union_subset: "Ord(i) ==> Union(i) <= i"
```
```   624 by (blast intro: Ord_trans)
```
```   625
```
```   626
```
```   627 subsection{*Limit Ordinals -- General Properties*}
```
```   628
```
```   629 lemma Limit_Union_eq: "Limit(i) ==> Union(i) = i"
```
```   630 apply (unfold Limit_def)
```
```   631 apply (fast intro!: ltI elim!: ltE elim: Ord_trans)
```
```   632 done
```
```   633
```
```   634 lemma Limit_is_Ord: "Limit(i) ==> Ord(i)"
```
```   635 apply (unfold Limit_def)
```
```   636 apply (erule conjunct1)
```
```   637 done
```
```   638
```
```   639 lemma Limit_has_0: "Limit(i) ==> 0 < i"
```
```   640 apply (unfold Limit_def)
```
```   641 apply (erule conjunct2 [THEN conjunct1])
```
```   642 done
```
```   643
```
```   644 lemma Limit_nonzero: "Limit(i) ==> i ~= 0"
```
```   645 by (drule Limit_has_0, blast)
```
```   646
```
```   647 lemma Limit_has_succ: "[| Limit(i);  j<i |] ==> succ(j) < i"
```
```   648 by (unfold Limit_def, blast)
```
```   649
```
```   650 lemma Limit_succ_lt_iff [simp]: "Limit(i) ==> succ(j) < i <-> (j<i)"
```
```   651 apply (safe intro!: Limit_has_succ)
```
```   652 apply (frule lt_Ord)
```
```   653 apply (blast intro: lt_trans)
```
```   654 done
```
```   655
```
```   656 lemma zero_not_Limit [iff]: "~ Limit(0)"
```
```   657 by (simp add: Limit_def)
```
```   658
```
```   659 lemma Limit_has_1: "Limit(i) ==> 1 < i"
```
```   660 by (blast intro: Limit_has_0 Limit_has_succ)
```
```   661
```
```   662 lemma increasing_LimitI: "[| 0<l; \<forall>x\<in>l. \<exists>y\<in>l. x<y |] ==> Limit(l)"
```
```   663 apply (unfold Limit_def, simp add: lt_Ord2, clarify)
```
```   664 apply (drule_tac i=y in ltD)
```
```   665 apply (blast intro: lt_trans1 [OF _ ltI] lt_Ord2)
```
```   666 done
```
```   667
```
```   668 lemma non_succ_LimitI:
```
```   669     "[| 0<i;  ALL y. succ(y) ~= i |] ==> Limit(i)"
```
```   670 apply (unfold Limit_def)
```
```   671 apply (safe del: subsetI)
```
```   672 apply (rule_tac [2] not_le_iff_lt [THEN iffD1])
```
```   673 apply (simp_all add: lt_Ord lt_Ord2)
```
```   674 apply (blast elim: leE lt_asym)
```
```   675 done
```
```   676
```
```   677 lemma succ_LimitE [elim!]: "Limit(succ(i)) ==> P"
```
```   678 apply (rule lt_irrefl)
```
```   679 apply (rule Limit_has_succ, assumption)
```
```   680 apply (erule Limit_is_Ord [THEN Ord_succD, THEN le_refl])
```
```   681 done
```
```   682
```
```   683 lemma not_succ_Limit [simp]: "~ Limit(succ(i))"
```
```   684 by blast
```
```   685
```
```   686 lemma Limit_le_succD: "[| Limit(i);  i le succ(j) |] ==> i le j"
```
```   687 by (blast elim!: leE)
```
```   688
```
```   689
```
```   690 subsubsection{*Traditional 3-Way Case Analysis on Ordinals*}
```
```   691
```
```   692 lemma Ord_cases_disj: "Ord(i) ==> i=0 | (EX j. Ord(j) & i=succ(j)) | Limit(i)"
```
```   693 by (blast intro!: non_succ_LimitI Ord_0_lt)
```
```   694
```
```   695 lemma Ord_cases:
```
```   696     "[| Ord(i);
```
```   697         i=0                          ==> P;
```
```   698         !!j. [| Ord(j); i=succ(j) |] ==> P;
```
```   699         Limit(i)                     ==> P
```
```   700      |] ==> P"
```
```   701 by (drule Ord_cases_disj, blast)
```
```   702
```
```   703 lemma trans_induct3 [case_names 0 succ limit, consumes 1]:
```
```   704      "[| Ord(i);
```
```   705          P(0);
```
```   706          !!x. [| Ord(x);  P(x) |] ==> P(succ(x));
```
```   707          !!x. [| Limit(x);  ALL y:x. P(y) |] ==> P(x)
```
```   708       |] ==> P(i)"
```
```   709 apply (erule trans_induct)
```
```   710 apply (erule Ord_cases, blast+)
```
```   711 done
```
```   712
```
```   713 lemmas trans_induct3_rule = trans_induct3 [rule_format, case_names 0 succ limit, consumes 1]
```
```   714
```
```   715 text{*A set of ordinals is either empty, contains its own union, or its
```
```   716 union is a limit ordinal.*}
```
```   717 lemma Ord_set_cases:
```
```   718    "\<forall>i\<in>I. Ord(i) ==> I=0 \<or> \<Union>(I) \<in> I \<or> (\<Union>(I) \<notin> I \<and> Limit(\<Union>(I)))"
```
```   719 apply (clarify elim!: not_emptyE)
```
```   720 apply (cases "\<Union>(I)" rule: Ord_cases)
```
```   721    apply (blast intro: Ord_Union)
```
```   722   apply (blast intro: subst_elem)
```
```   723  apply auto
```
```   724 apply (clarify elim!: equalityE succ_subsetE)
```
```   725 apply (simp add: Union_subset_iff)
```
```   726 apply (subgoal_tac "B = succ(j)", blast)
```
```   727 apply (rule le_anti_sym)
```
```   728  apply (simp add: le_subset_iff)
```
```   729 apply (simp add: ltI)
```
```   730 done
```
```   731
```
```   732 text{*If the union of a set of ordinals is a successor, then it is
```
```   733 an element of that set.*}
```
```   734 lemma Ord_Union_eq_succD: "[|\<forall>x\<in>X. Ord(x);  \<Union>X = succ(j)|] ==> succ(j) \<in> X"
```
```   735 by (drule Ord_set_cases, auto)
```
```   736
```
```   737 lemma Limit_Union [rule_format]: "[| I \<noteq> 0;  \<forall>i\<in>I. Limit(i) |] ==> Limit(\<Union>I)"
```
```   738 apply (simp add: Limit_def lt_def)
```
```   739 apply (blast intro!: equalityI)
```
```   740 done
```
```   741
```
```   742 end
```