src/ZF/Ordinal.thy
author paulson
Thu Jan 23 10:30:14 2003 +0100 (2003-01-23)
changeset 13784 b9f6154427a4
parent 13615 449a70d88b38
child 14565 c6dc17aab88a
permissions -rw-r--r--
tidying (by script)
clasohm@1478
     1
(*  Title:      ZF/Ordinal.thy
lcp@435
     2
    ID:         $Id$
clasohm@1478
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
lcp@435
     4
    Copyright   1994  University of Cambridge
lcp@435
     5
lcp@435
     6
*)
lcp@435
     7
paulson@13356
     8
header{*Transitive Sets and Ordinals*}
paulson@13356
     9
paulson@13155
    10
theory Ordinal = WF + Bool + equalities:
paulson@13155
    11
paulson@13155
    12
constdefs
paulson@13155
    13
paulson@13155
    14
  Memrel        :: "i=>i"
paulson@13155
    15
    "Memrel(A)   == {z: A*A . EX x y. z=<x,y> & x:y }"
paulson@13155
    16
paulson@13155
    17
  Transset  :: "i=>o"
paulson@13155
    18
    "Transset(i) == ALL x:i. x<=i"
paulson@13155
    19
paulson@13155
    20
  Ord  :: "i=>o"
paulson@13155
    21
    "Ord(i)      == Transset(i) & (ALL x:i. Transset(x))"
paulson@13155
    22
paulson@13155
    23
  lt        :: "[i,i] => o"  (infixl "<" 50)   (*less-than on ordinals*)
paulson@13155
    24
    "i<j         == i:j & Ord(j)"
paulson@13155
    25
paulson@13155
    26
  Limit         :: "i=>o"
paulson@13155
    27
    "Limit(i)    == Ord(i) & 0<i & (ALL y. y<i --> succ(y)<i)"
wenzelm@2539
    28
wenzelm@2539
    29
syntax
paulson@13155
    30
  "le"          :: "[i,i] => o"  (infixl 50)   (*less-than or equals*)
lcp@435
    31
lcp@435
    32
translations
lcp@435
    33
  "x le y"      == "x < succ(y)"
lcp@435
    34
wenzelm@12114
    35
syntax (xsymbols)
paulson@13155
    36
  "op le"       :: "[i,i] => o"  (infixl "\<le>" 50)  (*less-than or equals*)
paulson@13155
    37
paulson@13155
    38
paulson@13356
    39
subsection{*Rules for Transset*}
paulson@13155
    40
paulson@13356
    41
subsubsection{*Three Neat Characterisations of Transset*}
paulson@13155
    42
paulson@13155
    43
lemma Transset_iff_Pow: "Transset(A) <-> A<=Pow(A)"
paulson@13155
    44
by (unfold Transset_def, blast)
paulson@13155
    45
paulson@13155
    46
lemma Transset_iff_Union_succ: "Transset(A) <-> Union(succ(A)) = A"
paulson@13155
    47
apply (unfold Transset_def)
paulson@13155
    48
apply (blast elim!: equalityE)
paulson@13155
    49
done
paulson@13155
    50
paulson@13155
    51
lemma Transset_iff_Union_subset: "Transset(A) <-> Union(A) <= A"
paulson@13155
    52
by (unfold Transset_def, blast)
paulson@13155
    53
paulson@13356
    54
subsubsection{*Consequences of Downwards Closure*}
paulson@13155
    55
paulson@13155
    56
lemma Transset_doubleton_D: 
paulson@13155
    57
    "[| Transset(C); {a,b}: C |] ==> a:C & b: C"
paulson@13155
    58
by (unfold Transset_def, blast)
paulson@13155
    59
paulson@13155
    60
lemma Transset_Pair_D:
paulson@13155
    61
    "[| Transset(C); <a,b>: C |] ==> a:C & b: C"
paulson@13155
    62
apply (simp add: Pair_def)
paulson@13155
    63
apply (blast dest: Transset_doubleton_D)
paulson@13155
    64
done
paulson@13155
    65
paulson@13155
    66
lemma Transset_includes_domain:
paulson@13155
    67
    "[| Transset(C); A*B <= C; b: B |] ==> A <= C"
paulson@13155
    68
by (blast dest: Transset_Pair_D)
paulson@13155
    69
paulson@13155
    70
lemma Transset_includes_range:
paulson@13155
    71
    "[| Transset(C); A*B <= C; a: A |] ==> B <= C"
paulson@13155
    72
by (blast dest: Transset_Pair_D)
paulson@13155
    73
paulson@13356
    74
subsubsection{*Closure Properties*}
paulson@13155
    75
paulson@13155
    76
lemma Transset_0: "Transset(0)"
paulson@13155
    77
by (unfold Transset_def, blast)
paulson@13155
    78
paulson@13155
    79
lemma Transset_Un: 
paulson@13155
    80
    "[| Transset(i);  Transset(j) |] ==> Transset(i Un j)"
paulson@13155
    81
by (unfold Transset_def, blast)
paulson@13155
    82
paulson@13155
    83
lemma Transset_Int: 
paulson@13155
    84
    "[| Transset(i);  Transset(j) |] ==> Transset(i Int j)"
paulson@13155
    85
by (unfold Transset_def, blast)
paulson@13155
    86
paulson@13155
    87
lemma Transset_succ: "Transset(i) ==> Transset(succ(i))"
paulson@13155
    88
by (unfold Transset_def, blast)
paulson@13155
    89
paulson@13155
    90
lemma Transset_Pow: "Transset(i) ==> Transset(Pow(i))"
paulson@13155
    91
by (unfold Transset_def, blast)
paulson@13155
    92
paulson@13155
    93
lemma Transset_Union: "Transset(A) ==> Transset(Union(A))"
paulson@13155
    94
by (unfold Transset_def, blast)
paulson@13155
    95
paulson@13155
    96
lemma Transset_Union_family: 
paulson@13155
    97
    "[| !!i. i:A ==> Transset(i) |] ==> Transset(Union(A))"
paulson@13155
    98
by (unfold Transset_def, blast)
paulson@13155
    99
paulson@13155
   100
lemma Transset_Inter_family: 
paulson@13203
   101
    "[| !!i. i:A ==> Transset(i) |] ==> Transset(Inter(A))"
paulson@13203
   102
by (unfold Inter_def Transset_def, blast)
paulson@13203
   103
paulson@13203
   104
lemma Transset_UN:
paulson@13615
   105
     "(!!x. x \<in> A ==> Transset(B(x))) ==> Transset (\<Union>x\<in>A. B(x))"
paulson@13203
   106
by (rule Transset_Union_family, auto) 
paulson@13203
   107
paulson@13203
   108
lemma Transset_INT:
paulson@13615
   109
     "(!!x. x \<in> A ==> Transset(B(x))) ==> Transset (\<Inter>x\<in>A. B(x))"
paulson@13203
   110
by (rule Transset_Inter_family, auto) 
paulson@13203
   111
paulson@13155
   112
paulson@13356
   113
subsection{*Lemmas for Ordinals*}
paulson@13155
   114
paulson@13155
   115
lemma OrdI:
paulson@13155
   116
    "[| Transset(i);  !!x. x:i ==> Transset(x) |]  ==>  Ord(i)"
paulson@13155
   117
by (simp add: Ord_def) 
paulson@13155
   118
paulson@13155
   119
lemma Ord_is_Transset: "Ord(i) ==> Transset(i)"
paulson@13155
   120
by (simp add: Ord_def) 
paulson@13155
   121
paulson@13155
   122
lemma Ord_contains_Transset: 
paulson@13155
   123
    "[| Ord(i);  j:i |] ==> Transset(j) "
paulson@13155
   124
by (unfold Ord_def, blast)
paulson@13155
   125
paulson@13155
   126
paulson@13155
   127
lemma Ord_in_Ord: "[| Ord(i);  j:i |] ==> Ord(j)"
paulson@13155
   128
by (unfold Ord_def Transset_def, blast)
paulson@13155
   129
paulson@13243
   130
(*suitable for rewriting PROVIDED i has been fixed*)
paulson@13243
   131
lemma Ord_in_Ord': "[| j:i; Ord(i) |] ==> Ord(j)"
paulson@13243
   132
by (blast intro: Ord_in_Ord)
paulson@13243
   133
paulson@13155
   134
(* Ord(succ(j)) ==> Ord(j) *)
paulson@13155
   135
lemmas Ord_succD = Ord_in_Ord [OF _ succI1]
paulson@13155
   136
paulson@13155
   137
lemma Ord_subset_Ord: "[| Ord(i);  Transset(j);  j<=i |] ==> Ord(j)"
paulson@13155
   138
by (simp add: Ord_def Transset_def, blast)
paulson@13155
   139
paulson@13155
   140
lemma OrdmemD: "[| j:i;  Ord(i) |] ==> j<=i"
paulson@13155
   141
by (unfold Ord_def Transset_def, blast)
paulson@13155
   142
paulson@13155
   143
lemma Ord_trans: "[| i:j;  j:k;  Ord(k) |] ==> i:k"
paulson@13155
   144
by (blast dest: OrdmemD)
paulson@13155
   145
paulson@13155
   146
lemma Ord_succ_subsetI: "[| i:j;  Ord(j) |] ==> succ(i) <= j"
paulson@13155
   147
by (blast dest: OrdmemD)
paulson@13155
   148
paulson@13155
   149
paulson@13356
   150
subsection{*The Construction of Ordinals: 0, succ, Union*}
paulson@13155
   151
paulson@13155
   152
lemma Ord_0 [iff,TC]: "Ord(0)"
paulson@13155
   153
by (blast intro: OrdI Transset_0)
paulson@13155
   154
paulson@13155
   155
lemma Ord_succ [TC]: "Ord(i) ==> Ord(succ(i))"
paulson@13155
   156
by (blast intro: OrdI Transset_succ Ord_is_Transset Ord_contains_Transset)
paulson@13155
   157
paulson@13155
   158
lemmas Ord_1 = Ord_0 [THEN Ord_succ]
paulson@13155
   159
paulson@13155
   160
lemma Ord_succ_iff [iff]: "Ord(succ(i)) <-> Ord(i)"
paulson@13155
   161
by (blast intro: Ord_succ dest!: Ord_succD)
paulson@13155
   162
paulson@13172
   163
lemma Ord_Un [intro,simp,TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Un j)"
paulson@13155
   164
apply (unfold Ord_def)
paulson@13155
   165
apply (blast intro!: Transset_Un)
paulson@13155
   166
done
paulson@13155
   167
paulson@13155
   168
lemma Ord_Int [TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Int j)"
paulson@13155
   169
apply (unfold Ord_def)
paulson@13155
   170
apply (blast intro!: Transset_Int)
paulson@13155
   171
done
paulson@13155
   172
paulson@13155
   173
(*There is no set of all ordinals, for then it would contain itself*)
paulson@13155
   174
lemma ON_class: "~ (ALL i. i:X <-> Ord(i))"
paulson@13155
   175
apply (rule notI)
paulson@13784
   176
apply (frule_tac x = X in spec)
paulson@13155
   177
apply (safe elim!: mem_irrefl)
paulson@13155
   178
apply (erule swap, rule OrdI [OF _ Ord_is_Transset])
paulson@13155
   179
apply (simp add: Transset_def)
paulson@13155
   180
apply (blast intro: Ord_in_Ord)+
paulson@13155
   181
done
paulson@13155
   182
paulson@13356
   183
subsection{*< is 'less Than' for Ordinals*}
paulson@13155
   184
paulson@13155
   185
lemma ltI: "[| i:j;  Ord(j) |] ==> i<j"
paulson@13155
   186
by (unfold lt_def, blast)
paulson@13155
   187
paulson@13155
   188
lemma ltE:
paulson@13155
   189
    "[| i<j;  [| i:j;  Ord(i);  Ord(j) |] ==> P |] ==> P"
paulson@13155
   190
apply (unfold lt_def)
paulson@13155
   191
apply (blast intro: Ord_in_Ord)
paulson@13155
   192
done
paulson@13155
   193
paulson@13155
   194
lemma ltD: "i<j ==> i:j"
paulson@13155
   195
by (erule ltE, assumption)
paulson@13155
   196
paulson@13155
   197
lemma not_lt0 [simp]: "~ i<0"
paulson@13155
   198
by (unfold lt_def, blast)
paulson@13155
   199
paulson@13155
   200
lemma lt_Ord: "j<i ==> Ord(j)"
paulson@13155
   201
by (erule ltE, assumption)
paulson@13155
   202
paulson@13155
   203
lemma lt_Ord2: "j<i ==> Ord(i)"
paulson@13155
   204
by (erule ltE, assumption)
paulson@13155
   205
paulson@13155
   206
(* "ja le j ==> Ord(j)" *)
paulson@13155
   207
lemmas le_Ord2 = lt_Ord2 [THEN Ord_succD]
paulson@13155
   208
paulson@13155
   209
(* i<0 ==> R *)
paulson@13155
   210
lemmas lt0E = not_lt0 [THEN notE, elim!]
paulson@13155
   211
paulson@13155
   212
lemma lt_trans: "[| i<j;  j<k |] ==> i<k"
paulson@13155
   213
by (blast intro!: ltI elim!: ltE intro: Ord_trans)
paulson@13155
   214
paulson@13155
   215
lemma lt_not_sym: "i<j ==> ~ (j<i)"
paulson@13155
   216
apply (unfold lt_def)
paulson@13155
   217
apply (blast elim: mem_asym)
paulson@13155
   218
done
paulson@13155
   219
paulson@13155
   220
(* [| i<j;  ~P ==> j<i |] ==> P *)
paulson@13155
   221
lemmas lt_asym = lt_not_sym [THEN swap]
paulson@13155
   222
paulson@13155
   223
lemma lt_irrefl [elim!]: "i<i ==> P"
paulson@13155
   224
by (blast intro: lt_asym)
paulson@13155
   225
paulson@13155
   226
lemma lt_not_refl: "~ i<i"
paulson@13155
   227
apply (rule notI)
paulson@13155
   228
apply (erule lt_irrefl)
paulson@13155
   229
done
paulson@13155
   230
paulson@13155
   231
paulson@13155
   232
(** le is less than or equals;  recall  i le j  abbrevs  i<succ(j) !! **)
paulson@13155
   233
paulson@13155
   234
lemma le_iff: "i le j <-> i<j | (i=j & Ord(j))"
paulson@13155
   235
by (unfold lt_def, blast)
paulson@13155
   236
paulson@13155
   237
(*Equivalently, i<j ==> i < succ(j)*)
paulson@13155
   238
lemma leI: "i<j ==> i le j"
paulson@13155
   239
by (simp (no_asm_simp) add: le_iff)
paulson@13155
   240
paulson@13155
   241
lemma le_eqI: "[| i=j;  Ord(j) |] ==> i le j"
paulson@13155
   242
by (simp (no_asm_simp) add: le_iff)
paulson@13155
   243
paulson@13155
   244
lemmas le_refl = refl [THEN le_eqI]
paulson@13155
   245
paulson@13155
   246
lemma le_refl_iff [iff]: "i le i <-> Ord(i)"
paulson@13155
   247
by (simp (no_asm_simp) add: lt_not_refl le_iff)
paulson@13155
   248
paulson@13155
   249
lemma leCI: "(~ (i=j & Ord(j)) ==> i<j) ==> i le j"
paulson@13155
   250
by (simp add: le_iff, blast)
paulson@13155
   251
paulson@13155
   252
lemma leE:
paulson@13155
   253
    "[| i le j;  i<j ==> P;  [| i=j;  Ord(j) |] ==> P |] ==> P"
paulson@13155
   254
by (simp add: le_iff, blast)
paulson@13155
   255
paulson@13155
   256
lemma le_anti_sym: "[| i le j;  j le i |] ==> i=j"
paulson@13155
   257
apply (simp add: le_iff)
paulson@13155
   258
apply (blast elim: lt_asym)
paulson@13155
   259
done
paulson@13155
   260
paulson@13155
   261
lemma le0_iff [simp]: "i le 0 <-> i=0"
paulson@13155
   262
by (blast elim!: leE)
paulson@13155
   263
paulson@13155
   264
lemmas le0D = le0_iff [THEN iffD1, dest!]
paulson@13155
   265
paulson@13356
   266
subsection{*Natural Deduction Rules for Memrel*}
paulson@13155
   267
paulson@13155
   268
(*The lemmas MemrelI/E give better speed than [iff] here*)
paulson@13155
   269
lemma Memrel_iff [simp]: "<a,b> : Memrel(A) <-> a:b & a:A & b:A"
paulson@13155
   270
by (unfold Memrel_def, blast)
paulson@13155
   271
paulson@13155
   272
lemma MemrelI [intro!]: "[| a: b;  a: A;  b: A |] ==> <a,b> : Memrel(A)"
paulson@13155
   273
by auto
paulson@13155
   274
paulson@13155
   275
lemma MemrelE [elim!]:
paulson@13155
   276
    "[| <a,b> : Memrel(A);   
paulson@13155
   277
        [| a: A;  b: A;  a:b |]  ==> P |]  
paulson@13155
   278
     ==> P"
paulson@13155
   279
by auto
paulson@13155
   280
paulson@13155
   281
lemma Memrel_type: "Memrel(A) <= A*A"
paulson@13155
   282
by (unfold Memrel_def, blast)
paulson@13155
   283
paulson@13155
   284
lemma Memrel_mono: "A<=B ==> Memrel(A) <= Memrel(B)"
paulson@13155
   285
by (unfold Memrel_def, blast)
paulson@13155
   286
paulson@13155
   287
lemma Memrel_0 [simp]: "Memrel(0) = 0"
paulson@13155
   288
by (unfold Memrel_def, blast)
paulson@13155
   289
paulson@13155
   290
lemma Memrel_1 [simp]: "Memrel(1) = 0"
paulson@13155
   291
by (unfold Memrel_def, blast)
paulson@13155
   292
paulson@13269
   293
lemma relation_Memrel: "relation(Memrel(A))"
paulson@13269
   294
by (simp add: relation_def Memrel_def, blast)
paulson@13269
   295
paulson@13155
   296
(*The membership relation (as a set) is well-founded.
paulson@13155
   297
  Proof idea: show A<=B by applying the foundation axiom to A-B *)
paulson@13155
   298
lemma wf_Memrel: "wf(Memrel(A))"
paulson@13155
   299
apply (unfold wf_def)
paulson@13155
   300
apply (rule foundation [THEN disjE, THEN allI], erule disjI1, blast) 
paulson@13155
   301
done
paulson@13155
   302
paulson@13396
   303
text{*The premise @{term "Ord(i)"} does not suffice.*}
paulson@13155
   304
lemma trans_Memrel: 
paulson@13155
   305
    "Ord(i) ==> trans(Memrel(i))"
paulson@13155
   306
by (unfold Ord_def Transset_def trans_def, blast)
paulson@13155
   307
paulson@13396
   308
text{*However, the following premise is strong enough.*}
paulson@13396
   309
lemma Transset_trans_Memrel: 
paulson@13396
   310
    "\<forall>j\<in>i. Transset(j) ==> trans(Memrel(i))"
paulson@13396
   311
by (unfold Transset_def trans_def, blast)
paulson@13396
   312
paulson@13155
   313
(*If Transset(A) then Memrel(A) internalizes the membership relation below A*)
paulson@13155
   314
lemma Transset_Memrel_iff: 
paulson@13155
   315
    "Transset(A) ==> <a,b> : Memrel(A) <-> a:b & b:A"
paulson@13155
   316
by (unfold Transset_def, blast)
paulson@13155
   317
paulson@13155
   318
paulson@13356
   319
subsection{*Transfinite Induction*}
paulson@13155
   320
paulson@13155
   321
(*Epsilon induction over a transitive set*)
paulson@13155
   322
lemma Transset_induct: 
paulson@13155
   323
    "[| i: k;  Transset(k);                           
paulson@13155
   324
        !!x.[| x: k;  ALL y:x. P(y) |] ==> P(x) |]
paulson@13155
   325
     ==>  P(i)"
paulson@13155
   326
apply (simp add: Transset_def) 
paulson@13269
   327
apply (erule wf_Memrel [THEN wf_induct2], blast+)
paulson@13155
   328
done
paulson@13155
   329
paulson@13155
   330
(*Induction over an ordinal*)
wenzelm@13534
   331
lemmas Ord_induct [consumes 2] = Transset_induct [OF _ Ord_is_Transset]
wenzelm@13534
   332
lemmas Ord_induct_rule = Ord_induct [rule_format, consumes 2]
paulson@13155
   333
paulson@13155
   334
(*Induction over the class of ordinals -- a useful corollary of Ord_induct*)
paulson@13155
   335
wenzelm@13534
   336
lemma trans_induct [consumes 1]:
paulson@13155
   337
    "[| Ord(i);  
paulson@13155
   338
        !!x.[| Ord(x);  ALL y:x. P(y) |] ==> P(x) |]
paulson@13155
   339
     ==>  P(i)"
paulson@13155
   340
apply (rule Ord_succ [THEN succI1 [THEN Ord_induct]], assumption)
paulson@13155
   341
apply (blast intro: Ord_succ [THEN Ord_in_Ord]) 
paulson@13155
   342
done
paulson@13155
   343
wenzelm@13534
   344
lemmas trans_induct_rule = trans_induct [rule_format, consumes 1]
wenzelm@13534
   345
paulson@13155
   346
paulson@13155
   347
(*** Fundamental properties of the epsilon ordering (< on ordinals) ***)
paulson@13155
   348
paulson@13155
   349
paulson@13356
   350
subsubsection{*Proving That < is a Linear Ordering on the Ordinals*}
paulson@13155
   351
paulson@13155
   352
lemma Ord_linear [rule_format]:
paulson@13155
   353
     "Ord(i) ==> (ALL j. Ord(j) --> i:j | i=j | j:i)"
paulson@13155
   354
apply (erule trans_induct)
paulson@13155
   355
apply (rule impI [THEN allI])
paulson@13155
   356
apply (erule_tac i=j in trans_induct) 
paulson@13155
   357
apply (blast dest: Ord_trans) 
paulson@13155
   358
done
paulson@13155
   359
paulson@13155
   360
(*The trichotomy law for ordinals!*)
paulson@13155
   361
lemma Ord_linear_lt:
paulson@13155
   362
    "[| Ord(i);  Ord(j);  i<j ==> P;  i=j ==> P;  j<i ==> P |] ==> P"
paulson@13155
   363
apply (simp add: lt_def) 
paulson@13155
   364
apply (rule_tac i1=i and j1=j in Ord_linear [THEN disjE], blast+)
paulson@13155
   365
done
paulson@13155
   366
paulson@13155
   367
lemma Ord_linear2:
paulson@13155
   368
    "[| Ord(i);  Ord(j);  i<j ==> P;  j le i ==> P |]  ==> P"
paulson@13784
   369
apply (rule_tac i = i and j = j in Ord_linear_lt)
paulson@13155
   370
apply (blast intro: leI le_eqI sym ) +
paulson@13155
   371
done
paulson@13155
   372
paulson@13155
   373
lemma Ord_linear_le:
paulson@13155
   374
    "[| Ord(i);  Ord(j);  i le j ==> P;  j le i ==> P |]  ==> P"
paulson@13784
   375
apply (rule_tac i = i and j = j in Ord_linear_lt)
paulson@13155
   376
apply (blast intro: leI le_eqI ) +
paulson@13155
   377
done
paulson@13155
   378
paulson@13155
   379
lemma le_imp_not_lt: "j le i ==> ~ i<j"
paulson@13155
   380
by (blast elim!: leE elim: lt_asym)
paulson@13155
   381
paulson@13155
   382
lemma not_lt_imp_le: "[| ~ i<j;  Ord(i);  Ord(j) |] ==> j le i"
paulson@13784
   383
by (rule_tac i = i and j = j in Ord_linear2, auto)
paulson@13155
   384
paulson@13356
   385
subsubsection{*Some Rewrite Rules for <, le*}
paulson@13155
   386
paulson@13155
   387
lemma Ord_mem_iff_lt: "Ord(j) ==> i:j <-> i<j"
paulson@13155
   388
by (unfold lt_def, blast)
paulson@13155
   389
paulson@13155
   390
lemma not_lt_iff_le: "[| Ord(i);  Ord(j) |] ==> ~ i<j <-> j le i"
paulson@13155
   391
by (blast dest: le_imp_not_lt not_lt_imp_le)
wenzelm@2540
   392
paulson@13155
   393
lemma not_le_iff_lt: "[| Ord(i);  Ord(j) |] ==> ~ i le j <-> j<i"
paulson@13155
   394
by (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym])
paulson@13155
   395
paulson@13155
   396
(*This is identical to 0<succ(i) *)
paulson@13155
   397
lemma Ord_0_le: "Ord(i) ==> 0 le i"
paulson@13155
   398
by (erule not_lt_iff_le [THEN iffD1], auto)
paulson@13155
   399
paulson@13155
   400
lemma Ord_0_lt: "[| Ord(i);  i~=0 |] ==> 0<i"
paulson@13155
   401
apply (erule not_le_iff_lt [THEN iffD1])
paulson@13155
   402
apply (rule Ord_0, blast)
paulson@13155
   403
done
paulson@13155
   404
paulson@13155
   405
lemma Ord_0_lt_iff: "Ord(i) ==> i~=0 <-> 0<i"
paulson@13155
   406
by (blast intro: Ord_0_lt)
paulson@13155
   407
paulson@13155
   408
paulson@13356
   409
subsection{*Results about Less-Than or Equals*}
paulson@13155
   410
paulson@13155
   411
(** For ordinals, j<=i (subset) implies j le i (less-than or equals) **)
paulson@13155
   412
paulson@13155
   413
lemma zero_le_succ_iff [iff]: "0 le succ(x) <-> Ord(x)"
paulson@13155
   414
by (blast intro: Ord_0_le elim: ltE)
paulson@13155
   415
paulson@13155
   416
lemma subset_imp_le: "[| j<=i;  Ord(i);  Ord(j) |] ==> j le i"
paulson@13269
   417
apply (rule not_lt_iff_le [THEN iffD1], assumption+)
paulson@13155
   418
apply (blast elim: ltE mem_irrefl)
paulson@13155
   419
done
paulson@13155
   420
paulson@13155
   421
lemma le_imp_subset: "i le j ==> i<=j"
paulson@13155
   422
by (blast dest: OrdmemD elim: ltE leE)
paulson@13155
   423
paulson@13155
   424
lemma le_subset_iff: "j le i <-> j<=i & Ord(i) & Ord(j)"
paulson@13155
   425
by (blast dest: subset_imp_le le_imp_subset elim: ltE)
paulson@13155
   426
paulson@13155
   427
lemma le_succ_iff: "i le succ(j) <-> i le j | i=succ(j) & Ord(i)"
paulson@13155
   428
apply (simp (no_asm) add: le_iff)
paulson@13155
   429
apply blast
paulson@13155
   430
done
paulson@13155
   431
paulson@13155
   432
(*Just a variant of subset_imp_le*)
paulson@13155
   433
lemma all_lt_imp_le: "[| Ord(i);  Ord(j);  !!x. x<j ==> x<i |] ==> j le i"
paulson@13155
   434
by (blast intro: not_lt_imp_le dest: lt_irrefl)
paulson@13155
   435
paulson@13356
   436
subsubsection{*Transitivity Laws*}
paulson@13155
   437
paulson@13155
   438
lemma lt_trans1: "[| i le j;  j<k |] ==> i<k"
paulson@13155
   439
by (blast elim!: leE intro: lt_trans)
paulson@13155
   440
paulson@13155
   441
lemma lt_trans2: "[| i<j;  j le k |] ==> i<k"
paulson@13155
   442
by (blast elim!: leE intro: lt_trans)
paulson@13155
   443
paulson@13155
   444
lemma le_trans: "[| i le j;  j le k |] ==> i le k"
paulson@13155
   445
by (blast intro: lt_trans1)
paulson@13155
   446
paulson@13155
   447
lemma succ_leI: "i<j ==> succ(i) le j"
paulson@13155
   448
apply (rule not_lt_iff_le [THEN iffD1]) 
paulson@13155
   449
apply (blast elim: ltE leE lt_asym)+
paulson@13155
   450
done
paulson@13155
   451
paulson@13155
   452
(*Identical to  succ(i) < succ(j) ==> i<j  *)
paulson@13155
   453
lemma succ_leE: "succ(i) le j ==> i<j"
paulson@13155
   454
apply (rule not_le_iff_lt [THEN iffD1])
paulson@13155
   455
apply (blast elim: ltE leE lt_asym)+
paulson@13155
   456
done
paulson@13155
   457
paulson@13155
   458
lemma succ_le_iff [iff]: "succ(i) le j <-> i<j"
paulson@13155
   459
by (blast intro: succ_leI succ_leE)
paulson@13155
   460
paulson@13155
   461
lemma succ_le_imp_le: "succ(i) le succ(j) ==> i le j"
paulson@13155
   462
by (blast dest!: succ_leE)
paulson@13155
   463
paulson@13155
   464
lemma lt_subset_trans: "[| i <= j;  j<k;  Ord(i) |] ==> i<k"
paulson@13155
   465
apply (rule subset_imp_le [THEN lt_trans1]) 
paulson@13155
   466
apply (blast intro: elim: ltE) +
paulson@13155
   467
done
paulson@13155
   468
paulson@13172
   469
lemma lt_imp_0_lt: "j<i ==> 0<i"
paulson@13172
   470
by (blast intro: lt_trans1 Ord_0_le [OF lt_Ord]) 
paulson@13172
   471
paulson@13243
   472
lemma succ_lt_iff: "succ(i) < j <-> i<j & succ(i) \<noteq> j"
paulson@13162
   473
apply auto 
paulson@13162
   474
apply (blast intro: lt_trans le_refl dest: lt_Ord) 
paulson@13162
   475
apply (frule lt_Ord) 
paulson@13162
   476
apply (rule not_le_iff_lt [THEN iffD1]) 
paulson@13162
   477
  apply (blast intro: lt_Ord2)
paulson@13162
   478
 apply blast  
paulson@13162
   479
apply (simp add: lt_Ord lt_Ord2 le_iff) 
paulson@13162
   480
apply (blast dest: lt_asym) 
paulson@13162
   481
done
paulson@13162
   482
paulson@13243
   483
lemma Ord_succ_mem_iff: "Ord(j) ==> succ(i) \<in> succ(j) <-> i\<in>j"
paulson@13243
   484
apply (insert succ_le_iff [of i j]) 
paulson@13243
   485
apply (simp add: lt_def) 
paulson@13243
   486
done
paulson@13243
   487
paulson@13356
   488
subsubsection{*Union and Intersection*}
paulson@13155
   489
paulson@13155
   490
lemma Un_upper1_le: "[| Ord(i); Ord(j) |] ==> i le i Un j"
paulson@13155
   491
by (rule Un_upper1 [THEN subset_imp_le], auto)
paulson@13155
   492
paulson@13155
   493
lemma Un_upper2_le: "[| Ord(i); Ord(j) |] ==> j le i Un j"
paulson@13155
   494
by (rule Un_upper2 [THEN subset_imp_le], auto)
paulson@13155
   495
paulson@13155
   496
(*Replacing k by succ(k') yields the similar rule for le!*)
paulson@13155
   497
lemma Un_least_lt: "[| i<k;  j<k |] ==> i Un j < k"
paulson@13784
   498
apply (rule_tac i = i and j = j in Ord_linear_le)
paulson@13155
   499
apply (auto simp add: Un_commute le_subset_iff subset_Un_iff lt_Ord) 
paulson@13155
   500
done
paulson@13155
   501
paulson@13155
   502
lemma Un_least_lt_iff: "[| Ord(i); Ord(j) |] ==> i Un j < k  <->  i<k & j<k"
paulson@13155
   503
apply (safe intro!: Un_least_lt)
paulson@13155
   504
apply (rule_tac [2] Un_upper2_le [THEN lt_trans1])
paulson@13155
   505
apply (rule Un_upper1_le [THEN lt_trans1], auto) 
paulson@13155
   506
done
paulson@13155
   507
paulson@13155
   508
lemma Un_least_mem_iff:
paulson@13155
   509
    "[| Ord(i); Ord(j); Ord(k) |] ==> i Un j : k  <->  i:k & j:k"
paulson@13155
   510
apply (insert Un_least_lt_iff [of i j k]) 
paulson@13155
   511
apply (simp add: lt_def)
paulson@13155
   512
done
paulson@13155
   513
paulson@13155
   514
(*Replacing k by succ(k') yields the similar rule for le!*)
paulson@13155
   515
lemma Int_greatest_lt: "[| i<k;  j<k |] ==> i Int j < k"
paulson@13784
   516
apply (rule_tac i = i and j = j in Ord_linear_le)
paulson@13155
   517
apply (auto simp add: Int_commute le_subset_iff subset_Int_iff lt_Ord) 
paulson@13155
   518
done
paulson@13155
   519
paulson@13162
   520
lemma Ord_Un_if:
paulson@13162
   521
     "[| Ord(i); Ord(j) |] ==> i \<union> j = (if j<i then i else j)"
paulson@13162
   522
by (simp add: not_lt_iff_le le_imp_subset leI
paulson@13162
   523
              subset_Un_iff [symmetric]  subset_Un_iff2 [symmetric]) 
paulson@13162
   524
paulson@13162
   525
lemma succ_Un_distrib:
paulson@13162
   526
     "[| Ord(i); Ord(j) |] ==> succ(i \<union> j) = succ(i) \<union> succ(j)"
paulson@13162
   527
by (simp add: Ord_Un_if lt_Ord le_Ord2) 
paulson@13162
   528
paulson@13162
   529
lemma lt_Un_iff:
paulson@13162
   530
     "[| Ord(i); Ord(j) |] ==> k < i \<union> j <-> k < i | k < j";
paulson@13162
   531
apply (simp add: Ord_Un_if not_lt_iff_le) 
paulson@13162
   532
apply (blast intro: leI lt_trans2)+ 
paulson@13162
   533
done
paulson@13162
   534
paulson@13162
   535
lemma le_Un_iff:
paulson@13162
   536
     "[| Ord(i); Ord(j) |] ==> k \<le> i \<union> j <-> k \<le> i | k \<le> j";
paulson@13162
   537
by (simp add: succ_Un_distrib lt_Un_iff [symmetric]) 
paulson@13162
   538
paulson@13172
   539
lemma Un_upper1_lt: "[|k < i; Ord(j)|] ==> k < i Un j"
paulson@13172
   540
by (simp add: lt_Un_iff lt_Ord2) 
paulson@13172
   541
paulson@13172
   542
lemma Un_upper2_lt: "[|k < j; Ord(i)|] ==> k < i Un j"
paulson@13172
   543
by (simp add: lt_Un_iff lt_Ord2) 
paulson@13172
   544
paulson@13172
   545
(*See also Transset_iff_Union_succ*)
paulson@13172
   546
lemma Ord_Union_succ_eq: "Ord(i) ==> \<Union>(succ(i)) = i"
paulson@13172
   547
by (blast intro: Ord_trans)
paulson@13172
   548
paulson@13162
   549
paulson@13356
   550
subsection{*Results about Limits*}
paulson@13155
   551
paulson@13172
   552
lemma Ord_Union [intro,simp,TC]: "[| !!i. i:A ==> Ord(i) |] ==> Ord(Union(A))"
paulson@13155
   553
apply (rule Ord_is_Transset [THEN Transset_Union_family, THEN OrdI])
paulson@13155
   554
apply (blast intro: Ord_contains_Transset)+
paulson@13155
   555
done
paulson@13155
   556
paulson@13172
   557
lemma Ord_UN [intro,simp,TC]:
paulson@13615
   558
     "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(\<Union>x\<in>A. B(x))"
paulson@13155
   559
by (rule Ord_Union, blast)
paulson@13155
   560
paulson@13203
   561
lemma Ord_Inter [intro,simp,TC]:
paulson@13203
   562
    "[| !!i. i:A ==> Ord(i) |] ==> Ord(Inter(A))" 
paulson@13203
   563
apply (rule Transset_Inter_family [THEN OrdI])
paulson@13203
   564
apply (blast intro: Ord_is_Transset) 
paulson@13203
   565
apply (simp add: Inter_def) 
paulson@13203
   566
apply (blast intro: Ord_contains_Transset) 
paulson@13203
   567
done
paulson@13203
   568
paulson@13203
   569
lemma Ord_INT [intro,simp,TC]:
paulson@13615
   570
    "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(\<Inter>x\<in>A. B(x))"
paulson@13203
   571
by (rule Ord_Inter, blast) 
paulson@13203
   572
paulson@13203
   573
paulson@13615
   574
(* No < version; consider (\<Union>i\<in>nat.i)=nat *)
paulson@13155
   575
lemma UN_least_le:
paulson@13615
   576
    "[| Ord(i);  !!x. x:A ==> b(x) le i |] ==> (\<Union>x\<in>A. b(x)) le i"
paulson@13155
   577
apply (rule le_imp_subset [THEN UN_least, THEN subset_imp_le])
paulson@13155
   578
apply (blast intro: Ord_UN elim: ltE)+
paulson@13155
   579
done
paulson@13155
   580
paulson@13155
   581
lemma UN_succ_least_lt:
paulson@13615
   582
    "[| j<i;  !!x. x:A ==> b(x)<j |] ==> (\<Union>x\<in>A. succ(b(x))) < i"
paulson@13155
   583
apply (rule ltE, assumption)
paulson@13155
   584
apply (rule UN_least_le [THEN lt_trans2])
paulson@13155
   585
apply (blast intro: succ_leI)+
paulson@13155
   586
done
paulson@13155
   587
paulson@13172
   588
lemma UN_upper_lt:
paulson@13172
   589
     "[| a\<in>A;  i < b(a);  Ord(\<Union>x\<in>A. b(x)) |] ==> i < (\<Union>x\<in>A. b(x))"
paulson@13172
   590
by (unfold lt_def, blast) 
paulson@13172
   591
paulson@13155
   592
lemma UN_upper_le:
paulson@13615
   593
     "[| a: A;  i le b(a);  Ord(\<Union>x\<in>A. b(x)) |] ==> i le (\<Union>x\<in>A. b(x))"
paulson@13155
   594
apply (frule ltD)
paulson@13155
   595
apply (rule le_imp_subset [THEN subset_trans, THEN subset_imp_le])
paulson@13155
   596
apply (blast intro: lt_Ord UN_upper)+
paulson@13155
   597
done
paulson@13155
   598
paulson@13172
   599
lemma lt_Union_iff: "\<forall>i\<in>A. Ord(i) ==> (j < \<Union>(A)) <-> (\<exists>i\<in>A. j<i)"
paulson@13172
   600
by (auto simp: lt_def Ord_Union)
paulson@13172
   601
paulson@13172
   602
lemma Union_upper_le:
paulson@13172
   603
     "[| j: J;  i\<le>j;  Ord(\<Union>(J)) |] ==> i \<le> \<Union>J"
paulson@13172
   604
apply (subst Union_eq_UN)  
paulson@13172
   605
apply (rule UN_upper_le, auto)
paulson@13172
   606
done
paulson@13172
   607
paulson@13155
   608
lemma le_implies_UN_le_UN:
paulson@13615
   609
    "[| !!x. x:A ==> c(x) le d(x) |] ==> (\<Union>x\<in>A. c(x)) le (\<Union>x\<in>A. d(x))"
paulson@13155
   610
apply (rule UN_least_le)
paulson@13155
   611
apply (rule_tac [2] UN_upper_le)
paulson@13155
   612
apply (blast intro: Ord_UN le_Ord2)+ 
paulson@13155
   613
done
paulson@13155
   614
paulson@13615
   615
lemma Ord_equality: "Ord(i) ==> (\<Union>y\<in>i. succ(y)) = i"
paulson@13155
   616
by (blast intro: Ord_trans)
paulson@13155
   617
paulson@13155
   618
(*Holds for all transitive sets, not just ordinals*)
paulson@13155
   619
lemma Ord_Union_subset: "Ord(i) ==> Union(i) <= i"
paulson@13155
   620
by (blast intro: Ord_trans)
paulson@13155
   621
paulson@13155
   622
paulson@13356
   623
subsection{*Limit Ordinals -- General Properties*}
paulson@13155
   624
paulson@13155
   625
lemma Limit_Union_eq: "Limit(i) ==> Union(i) = i"
paulson@13155
   626
apply (unfold Limit_def)
paulson@13155
   627
apply (fast intro!: ltI elim!: ltE elim: Ord_trans)
paulson@13155
   628
done
paulson@13155
   629
paulson@13155
   630
lemma Limit_is_Ord: "Limit(i) ==> Ord(i)"
paulson@13155
   631
apply (unfold Limit_def)
paulson@13155
   632
apply (erule conjunct1)
paulson@13155
   633
done
paulson@13155
   634
paulson@13155
   635
lemma Limit_has_0: "Limit(i) ==> 0 < i"
paulson@13155
   636
apply (unfold Limit_def)
paulson@13155
   637
apply (erule conjunct2 [THEN conjunct1])
paulson@13155
   638
done
paulson@13155
   639
paulson@13544
   640
lemma Limit_nonzero: "Limit(i) ==> i ~= 0"
paulson@13544
   641
by (drule Limit_has_0, blast)
paulson@13544
   642
paulson@13155
   643
lemma Limit_has_succ: "[| Limit(i);  j<i |] ==> succ(j) < i"
paulson@13155
   644
by (unfold Limit_def, blast)
paulson@13155
   645
paulson@13544
   646
lemma Limit_succ_lt_iff [simp]: "Limit(i) ==> succ(j) < i <-> (j<i)"
paulson@13544
   647
apply (safe intro!: Limit_has_succ)
paulson@13544
   648
apply (frule lt_Ord)
paulson@13544
   649
apply (blast intro: lt_trans)   
paulson@13544
   650
done
paulson@13544
   651
paulson@13172
   652
lemma zero_not_Limit [iff]: "~ Limit(0)"
paulson@13172
   653
by (simp add: Limit_def)
paulson@13172
   654
paulson@13172
   655
lemma Limit_has_1: "Limit(i) ==> 1 < i"
paulson@13172
   656
by (blast intro: Limit_has_0 Limit_has_succ)
paulson@13172
   657
paulson@13172
   658
lemma increasing_LimitI: "[| 0<l; \<forall>x\<in>l. \<exists>y\<in>l. x<y |] ==> Limit(l)"
paulson@13544
   659
apply (unfold Limit_def, simp add: lt_Ord2, clarify)
paulson@13172
   660
apply (drule_tac i=y in ltD) 
paulson@13172
   661
apply (blast intro: lt_trans1 [OF _ ltI] lt_Ord2)
paulson@13172
   662
done
paulson@13172
   663
paulson@13155
   664
lemma non_succ_LimitI: 
paulson@13155
   665
    "[| 0<i;  ALL y. succ(y) ~= i |] ==> Limit(i)"
paulson@13155
   666
apply (unfold Limit_def)
paulson@13155
   667
apply (safe del: subsetI)
paulson@13155
   668
apply (rule_tac [2] not_le_iff_lt [THEN iffD1])
paulson@13155
   669
apply (simp_all add: lt_Ord lt_Ord2) 
paulson@13155
   670
apply (blast elim: leE lt_asym)
paulson@13155
   671
done
paulson@13155
   672
paulson@13155
   673
lemma succ_LimitE [elim!]: "Limit(succ(i)) ==> P"
paulson@13155
   674
apply (rule lt_irrefl)
paulson@13155
   675
apply (rule Limit_has_succ, assumption)
paulson@13155
   676
apply (erule Limit_is_Ord [THEN Ord_succD, THEN le_refl])
paulson@13155
   677
done
paulson@13155
   678
paulson@13155
   679
lemma not_succ_Limit [simp]: "~ Limit(succ(i))"
paulson@13155
   680
by blast
paulson@13155
   681
paulson@13155
   682
lemma Limit_le_succD: "[| Limit(i);  i le succ(j) |] ==> i le j"
paulson@13155
   683
by (blast elim!: leE)
paulson@13155
   684
paulson@13172
   685
paulson@13356
   686
subsubsection{*Traditional 3-Way Case Analysis on Ordinals*}
paulson@13155
   687
paulson@13155
   688
lemma Ord_cases_disj: "Ord(i) ==> i=0 | (EX j. Ord(j) & i=succ(j)) | Limit(i)"
paulson@13155
   689
by (blast intro!: non_succ_LimitI Ord_0_lt)
paulson@13155
   690
paulson@13155
   691
lemma Ord_cases:
paulson@13155
   692
    "[| Ord(i);                  
paulson@13155
   693
        i=0                          ==> P;      
paulson@13155
   694
        !!j. [| Ord(j); i=succ(j) |] ==> P;      
paulson@13155
   695
        Limit(i)                     ==> P       
paulson@13155
   696
     |] ==> P"
paulson@13155
   697
by (drule Ord_cases_disj, blast)  
paulson@13155
   698
wenzelm@13534
   699
lemma trans_induct3 [case_names 0 succ limit, consumes 1]:
paulson@13155
   700
     "[| Ord(i);                 
paulson@13155
   701
         P(0);                   
paulson@13155
   702
         !!x. [| Ord(x);  P(x) |] ==> P(succ(x));        
paulson@13155
   703
         !!x. [| Limit(x);  ALL y:x. P(y) |] ==> P(x)    
paulson@13155
   704
      |] ==> P(i)"
paulson@13155
   705
apply (erule trans_induct)
paulson@13155
   706
apply (erule Ord_cases, blast+)
paulson@13155
   707
done
paulson@13155
   708
wenzelm@13534
   709
lemmas trans_induct3_rule = trans_induct3 [rule_format, case_names 0 succ limit, consumes 1]
wenzelm@13534
   710
paulson@13172
   711
text{*A set of ordinals is either empty, contains its own union, or its
paulson@13172
   712
union is a limit ordinal.*}
paulson@13172
   713
lemma Ord_set_cases:
paulson@13172
   714
   "\<forall>i\<in>I. Ord(i) ==> I=0 \<or> \<Union>(I) \<in> I \<or> (\<Union>(I) \<notin> I \<and> Limit(\<Union>(I)))"
paulson@13172
   715
apply (clarify elim!: not_emptyE) 
paulson@13172
   716
apply (cases "\<Union>(I)" rule: Ord_cases) 
paulson@13172
   717
   apply (blast intro: Ord_Union)
paulson@13172
   718
  apply (blast intro: subst_elem)
paulson@13172
   719
 apply auto 
paulson@13172
   720
apply (clarify elim!: equalityE succ_subsetE)
paulson@13172
   721
apply (simp add: Union_subset_iff)
paulson@13172
   722
apply (subgoal_tac "B = succ(j)", blast)
paulson@13172
   723
apply (rule le_anti_sym) 
paulson@13172
   724
 apply (simp add: le_subset_iff) 
paulson@13172
   725
apply (simp add: ltI)
paulson@13172
   726
done
paulson@13172
   727
paulson@13172
   728
text{*If the union of a set of ordinals is a successor, then it is
paulson@13172
   729
an element of that set.*}
paulson@13172
   730
lemma Ord_Union_eq_succD: "[|\<forall>x\<in>X. Ord(x);  \<Union>X = succ(j)|] ==> succ(j) \<in> X"
paulson@13172
   731
by (drule Ord_set_cases, auto)
paulson@13172
   732
paulson@13172
   733
lemma Limit_Union [rule_format]: "[| I \<noteq> 0;  \<forall>i\<in>I. Limit(i) |] ==> Limit(\<Union>I)"
paulson@13172
   734
apply (simp add: Limit_def lt_def)
paulson@13172
   735
apply (blast intro!: equalityI)
paulson@13172
   736
done
paulson@13172
   737
paulson@13155
   738
ML 
paulson@13155
   739
{*
paulson@13155
   740
val Memrel_def = thm "Memrel_def";
paulson@13155
   741
val Transset_def = thm "Transset_def";
paulson@13155
   742
val Ord_def = thm "Ord_def";
paulson@13155
   743
val lt_def = thm "lt_def";
paulson@13155
   744
val Limit_def = thm "Limit_def";
paulson@13155
   745
paulson@13155
   746
val Transset_iff_Pow = thm "Transset_iff_Pow";
paulson@13155
   747
val Transset_iff_Union_succ = thm "Transset_iff_Union_succ";
paulson@13155
   748
val Transset_iff_Union_subset = thm "Transset_iff_Union_subset";
paulson@13155
   749
val Transset_doubleton_D = thm "Transset_doubleton_D";
paulson@13155
   750
val Transset_Pair_D = thm "Transset_Pair_D";
paulson@13155
   751
val Transset_includes_domain = thm "Transset_includes_domain";
paulson@13155
   752
val Transset_includes_range = thm "Transset_includes_range";
paulson@13155
   753
val Transset_0 = thm "Transset_0";
paulson@13155
   754
val Transset_Un = thm "Transset_Un";
paulson@13155
   755
val Transset_Int = thm "Transset_Int";
paulson@13155
   756
val Transset_succ = thm "Transset_succ";
paulson@13155
   757
val Transset_Pow = thm "Transset_Pow";
paulson@13155
   758
val Transset_Union = thm "Transset_Union";
paulson@13155
   759
val Transset_Union_family = thm "Transset_Union_family";
paulson@13155
   760
val Transset_Inter_family = thm "Transset_Inter_family";
paulson@13155
   761
val OrdI = thm "OrdI";
paulson@13155
   762
val Ord_is_Transset = thm "Ord_is_Transset";
paulson@13155
   763
val Ord_contains_Transset = thm "Ord_contains_Transset";
paulson@13155
   764
val Ord_in_Ord = thm "Ord_in_Ord";
paulson@13155
   765
val Ord_succD = thm "Ord_succD";
paulson@13155
   766
val Ord_subset_Ord = thm "Ord_subset_Ord";
paulson@13155
   767
val OrdmemD = thm "OrdmemD";
paulson@13155
   768
val Ord_trans = thm "Ord_trans";
paulson@13155
   769
val Ord_succ_subsetI = thm "Ord_succ_subsetI";
paulson@13155
   770
val Ord_0 = thm "Ord_0";
paulson@13155
   771
val Ord_succ = thm "Ord_succ";
paulson@13155
   772
val Ord_1 = thm "Ord_1";
paulson@13155
   773
val Ord_succ_iff = thm "Ord_succ_iff";
paulson@13155
   774
val Ord_Un = thm "Ord_Un";
paulson@13155
   775
val Ord_Int = thm "Ord_Int";
paulson@13155
   776
val Ord_Inter = thm "Ord_Inter";
paulson@13155
   777
val Ord_INT = thm "Ord_INT";
paulson@13155
   778
val ON_class = thm "ON_class";
paulson@13155
   779
val ltI = thm "ltI";
paulson@13155
   780
val ltE = thm "ltE";
paulson@13155
   781
val ltD = thm "ltD";
paulson@13155
   782
val not_lt0 = thm "not_lt0";
paulson@13155
   783
val lt_Ord = thm "lt_Ord";
paulson@13155
   784
val lt_Ord2 = thm "lt_Ord2";
paulson@13155
   785
val le_Ord2 = thm "le_Ord2";
paulson@13155
   786
val lt0E = thm "lt0E";
paulson@13155
   787
val lt_trans = thm "lt_trans";
paulson@13155
   788
val lt_not_sym = thm "lt_not_sym";
paulson@13155
   789
val lt_asym = thm "lt_asym";
paulson@13155
   790
val lt_irrefl = thm "lt_irrefl";
paulson@13155
   791
val lt_not_refl = thm "lt_not_refl";
paulson@13155
   792
val le_iff = thm "le_iff";
paulson@13155
   793
val leI = thm "leI";
paulson@13155
   794
val le_eqI = thm "le_eqI";
paulson@13155
   795
val le_refl = thm "le_refl";
paulson@13155
   796
val le_refl_iff = thm "le_refl_iff";
paulson@13155
   797
val leCI = thm "leCI";
paulson@13155
   798
val leE = thm "leE";
paulson@13155
   799
val le_anti_sym = thm "le_anti_sym";
paulson@13155
   800
val le0_iff = thm "le0_iff";
paulson@13155
   801
val le0D = thm "le0D";
paulson@13155
   802
val Memrel_iff = thm "Memrel_iff";
paulson@13155
   803
val MemrelI = thm "MemrelI";
paulson@13155
   804
val MemrelE = thm "MemrelE";
paulson@13155
   805
val Memrel_type = thm "Memrel_type";
paulson@13155
   806
val Memrel_mono = thm "Memrel_mono";
paulson@13155
   807
val Memrel_0 = thm "Memrel_0";
paulson@13155
   808
val Memrel_1 = thm "Memrel_1";
paulson@13155
   809
val wf_Memrel = thm "wf_Memrel";
paulson@13155
   810
val trans_Memrel = thm "trans_Memrel";
paulson@13155
   811
val Transset_Memrel_iff = thm "Transset_Memrel_iff";
paulson@13155
   812
val Transset_induct = thm "Transset_induct";
paulson@13155
   813
val Ord_induct = thm "Ord_induct";
paulson@13155
   814
val trans_induct = thm "trans_induct";
paulson@13155
   815
val Ord_linear = thm "Ord_linear";
paulson@13155
   816
val Ord_linear_lt = thm "Ord_linear_lt";
paulson@13155
   817
val Ord_linear2 = thm "Ord_linear2";
paulson@13155
   818
val Ord_linear_le = thm "Ord_linear_le";
paulson@13155
   819
val le_imp_not_lt = thm "le_imp_not_lt";
paulson@13155
   820
val not_lt_imp_le = thm "not_lt_imp_le";
paulson@13155
   821
val Ord_mem_iff_lt = thm "Ord_mem_iff_lt";
paulson@13155
   822
val not_lt_iff_le = thm "not_lt_iff_le";
paulson@13155
   823
val not_le_iff_lt = thm "not_le_iff_lt";
paulson@13155
   824
val Ord_0_le = thm "Ord_0_le";
paulson@13155
   825
val Ord_0_lt = thm "Ord_0_lt";
paulson@13155
   826
val Ord_0_lt_iff = thm "Ord_0_lt_iff";
paulson@13155
   827
val zero_le_succ_iff = thm "zero_le_succ_iff";
paulson@13155
   828
val subset_imp_le = thm "subset_imp_le";
paulson@13155
   829
val le_imp_subset = thm "le_imp_subset";
paulson@13155
   830
val le_subset_iff = thm "le_subset_iff";
paulson@13155
   831
val le_succ_iff = thm "le_succ_iff";
paulson@13155
   832
val all_lt_imp_le = thm "all_lt_imp_le";
paulson@13155
   833
val lt_trans1 = thm "lt_trans1";
paulson@13155
   834
val lt_trans2 = thm "lt_trans2";
paulson@13155
   835
val le_trans = thm "le_trans";
paulson@13155
   836
val succ_leI = thm "succ_leI";
paulson@13155
   837
val succ_leE = thm "succ_leE";
paulson@13155
   838
val succ_le_iff = thm "succ_le_iff";
paulson@13155
   839
val succ_le_imp_le = thm "succ_le_imp_le";
paulson@13155
   840
val lt_subset_trans = thm "lt_subset_trans";
paulson@13155
   841
val Un_upper1_le = thm "Un_upper1_le";
paulson@13155
   842
val Un_upper2_le = thm "Un_upper2_le";
paulson@13155
   843
val Un_least_lt = thm "Un_least_lt";
paulson@13155
   844
val Un_least_lt_iff = thm "Un_least_lt_iff";
paulson@13155
   845
val Un_least_mem_iff = thm "Un_least_mem_iff";
paulson@13155
   846
val Int_greatest_lt = thm "Int_greatest_lt";
paulson@13155
   847
val Ord_Union = thm "Ord_Union";
paulson@13155
   848
val Ord_UN = thm "Ord_UN";
paulson@13155
   849
val UN_least_le = thm "UN_least_le";
paulson@13155
   850
val UN_succ_least_lt = thm "UN_succ_least_lt";
paulson@13155
   851
val UN_upper_le = thm "UN_upper_le";
paulson@13155
   852
val le_implies_UN_le_UN = thm "le_implies_UN_le_UN";
paulson@13155
   853
val Ord_equality = thm "Ord_equality";
paulson@13155
   854
val Ord_Union_subset = thm "Ord_Union_subset";
paulson@13155
   855
val Limit_Union_eq = thm "Limit_Union_eq";
paulson@13155
   856
val Limit_is_Ord = thm "Limit_is_Ord";
paulson@13155
   857
val Limit_has_0 = thm "Limit_has_0";
paulson@13155
   858
val Limit_has_succ = thm "Limit_has_succ";
paulson@13155
   859
val non_succ_LimitI = thm "non_succ_LimitI";
paulson@13155
   860
val succ_LimitE = thm "succ_LimitE";
paulson@13155
   861
val not_succ_Limit = thm "not_succ_Limit";
paulson@13155
   862
val Limit_le_succD = thm "Limit_le_succD";
paulson@13155
   863
val Ord_cases_disj = thm "Ord_cases_disj";
paulson@13155
   864
val Ord_cases = thm "Ord_cases";
paulson@13155
   865
val trans_induct3 = thm "trans_induct3";
paulson@13155
   866
*}
lcp@435
   867
lcp@435
   868
end