src/ZF/Constructible/L_axioms.thy
author paulson
Wed Jul 24 17:59:12 2002 +0200 (2002-07-24)
changeset 13418 7c0ba9dba978
parent 13385 31df66ca0780
child 13428 99e52e78eb65
permissions -rw-r--r--
tweaks, aiming towards relativization of "satisfies"
paulson@13339
     1
header {*The ZF Axioms (Except Separation) in L*}
paulson@13223
     2
paulson@13314
     3
theory L_axioms = Formula + Relative + Reflection + MetaExists:
paulson@13223
     4
paulson@13339
     5
text {* The class L satisfies the premises of locale @{text M_triv_axioms} *}
paulson@13223
     6
paulson@13223
     7
lemma transL: "[| y\<in>x; L(x) |] ==> L(y)"
paulson@13223
     8
apply (insert Transset_Lset) 
paulson@13223
     9
apply (simp add: Transset_def L_def, blast) 
paulson@13223
    10
done
paulson@13223
    11
paulson@13223
    12
lemma nonempty: "L(0)"
paulson@13223
    13
apply (simp add: L_def) 
paulson@13223
    14
apply (blast intro: zero_in_Lset) 
paulson@13223
    15
done
paulson@13223
    16
paulson@13223
    17
lemma upair_ax: "upair_ax(L)"
paulson@13223
    18
apply (simp add: upair_ax_def upair_def, clarify)
paulson@13299
    19
apply (rule_tac x="{x,y}" in rexI)  
paulson@13299
    20
apply (simp_all add: doubleton_in_L) 
paulson@13223
    21
done
paulson@13223
    22
paulson@13223
    23
lemma Union_ax: "Union_ax(L)"
paulson@13223
    24
apply (simp add: Union_ax_def big_union_def, clarify)
paulson@13299
    25
apply (rule_tac x="Union(x)" in rexI)  
paulson@13299
    26
apply (simp_all add: Union_in_L, auto) 
paulson@13223
    27
apply (blast intro: transL) 
paulson@13223
    28
done
paulson@13223
    29
paulson@13223
    30
lemma power_ax: "power_ax(L)"
paulson@13223
    31
apply (simp add: power_ax_def powerset_def Relative.subset_def, clarify)
paulson@13299
    32
apply (rule_tac x="{y \<in> Pow(x). L(y)}" in rexI)  
paulson@13299
    33
apply (simp_all add: LPow_in_L, auto)
paulson@13223
    34
apply (blast intro: transL) 
paulson@13223
    35
done
paulson@13223
    36
paulson@13223
    37
subsubsection{*For L to satisfy Replacement *}
paulson@13223
    38
paulson@13223
    39
(*Can't move these to Formula unless the definition of univalent is moved
paulson@13223
    40
there too!*)
paulson@13223
    41
paulson@13223
    42
lemma LReplace_in_Lset:
paulson@13223
    43
     "[|X \<in> Lset(i); univalent(L,X,Q); Ord(i)|] 
paulson@13223
    44
      ==> \<exists>j. Ord(j) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Lset(j)"
paulson@13223
    45
apply (rule_tac x="\<Union>y \<in> Replace(X, %x y. Q(x,y) & L(y)). succ(lrank(y))" 
paulson@13223
    46
       in exI)
paulson@13223
    47
apply simp
paulson@13223
    48
apply clarify 
paulson@13339
    49
apply (rule_tac a=x in UN_I)  
paulson@13223
    50
 apply (simp_all add: Replace_iff univalent_def) 
paulson@13223
    51
apply (blast dest: transL L_I) 
paulson@13223
    52
done
paulson@13223
    53
paulson@13223
    54
lemma LReplace_in_L: 
paulson@13223
    55
     "[|L(X); univalent(L,X,Q)|] 
paulson@13223
    56
      ==> \<exists>Y. L(Y) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Y"
paulson@13223
    57
apply (drule L_D, clarify) 
paulson@13223
    58
apply (drule LReplace_in_Lset, assumption+)
paulson@13223
    59
apply (blast intro: L_I Lset_in_Lset_succ)
paulson@13223
    60
done
paulson@13223
    61
paulson@13223
    62
lemma replacement: "replacement(L,P)"
paulson@13223
    63
apply (simp add: replacement_def, clarify)
paulson@13268
    64
apply (frule LReplace_in_L, assumption+, clarify) 
paulson@13299
    65
apply (rule_tac x=Y in rexI)   
paulson@13299
    66
apply (simp_all add: Replace_iff univalent_def, blast) 
paulson@13223
    67
done
paulson@13223
    68
paulson@13363
    69
subsection{*Instantiating the locale @{text M_triv_axioms}*}
paulson@13363
    70
text{*No instances of Separation yet.*}
paulson@13291
    71
paulson@13291
    72
lemma Lset_mono_le: "mono_le_subset(Lset)"
paulson@13291
    73
by (simp add: mono_le_subset_def le_imp_subset Lset_mono) 
paulson@13291
    74
paulson@13291
    75
lemma Lset_cont: "cont_Ord(Lset)"
paulson@13291
    76
by (simp add: cont_Ord_def Limit_Lset_eq OUnion_def Limit_is_Ord) 
paulson@13291
    77
paulson@13291
    78
lemmas Pair_in_Lset = Formula.Pair_in_LLimit;
paulson@13291
    79
paulson@13291
    80
lemmas L_nat = Ord_in_L [OF Ord_nat];
paulson@13291
    81
paulson@13291
    82
ML
paulson@13291
    83
{*
paulson@13291
    84
val transL = thm "transL";
paulson@13291
    85
val nonempty = thm "nonempty";
paulson@13291
    86
val upair_ax = thm "upair_ax";
paulson@13291
    87
val Union_ax = thm "Union_ax";
paulson@13291
    88
val power_ax = thm "power_ax";
paulson@13291
    89
val replacement = thm "replacement";
paulson@13291
    90
val L_nat = thm "L_nat";
paulson@13291
    91
paulson@13291
    92
fun kill_flex_triv_prems st = Seq.hd ((REPEAT_FIRST assume_tac) st);
paulson@13291
    93
paulson@13363
    94
fun triv_axioms_L th =
paulson@13291
    95
    kill_flex_triv_prems 
paulson@13291
    96
       ([transL, nonempty, upair_ax, Union_ax, power_ax, replacement, L_nat] 
paulson@13291
    97
        MRS (inst "M" "L" th));
paulson@13291
    98
paulson@13363
    99
bind_thm ("rall_abs", triv_axioms_L (thm "M_triv_axioms.rall_abs"));
paulson@13363
   100
bind_thm ("rex_abs", triv_axioms_L (thm "M_triv_axioms.rex_abs"));
paulson@13363
   101
bind_thm ("ball_iff_equiv", triv_axioms_L (thm "M_triv_axioms.ball_iff_equiv"));
paulson@13363
   102
bind_thm ("M_equalityI", triv_axioms_L (thm "M_triv_axioms.M_equalityI"));
paulson@13363
   103
bind_thm ("empty_abs", triv_axioms_L (thm "M_triv_axioms.empty_abs"));
paulson@13363
   104
bind_thm ("subset_abs", triv_axioms_L (thm "M_triv_axioms.subset_abs"));
paulson@13363
   105
bind_thm ("upair_abs", triv_axioms_L (thm "M_triv_axioms.upair_abs"));
paulson@13363
   106
bind_thm ("upair_in_M_iff", triv_axioms_L (thm "M_triv_axioms.upair_in_M_iff"));
paulson@13363
   107
bind_thm ("singleton_in_M_iff", triv_axioms_L (thm "M_triv_axioms.singleton_in_M_iff"));
paulson@13363
   108
bind_thm ("pair_abs", triv_axioms_L (thm "M_triv_axioms.pair_abs"));
paulson@13363
   109
bind_thm ("pair_in_M_iff", triv_axioms_L (thm "M_triv_axioms.pair_in_M_iff"));
paulson@13363
   110
bind_thm ("pair_components_in_M", triv_axioms_L (thm "M_triv_axioms.pair_components_in_M"));
paulson@13363
   111
bind_thm ("cartprod_abs", triv_axioms_L (thm "M_triv_axioms.cartprod_abs"));
paulson@13363
   112
bind_thm ("union_abs", triv_axioms_L (thm "M_triv_axioms.union_abs"));
paulson@13363
   113
bind_thm ("inter_abs", triv_axioms_L (thm "M_triv_axioms.inter_abs"));
paulson@13363
   114
bind_thm ("setdiff_abs", triv_axioms_L (thm "M_triv_axioms.setdiff_abs"));
paulson@13363
   115
bind_thm ("Union_abs", triv_axioms_L (thm "M_triv_axioms.Union_abs"));
paulson@13363
   116
bind_thm ("Union_closed", triv_axioms_L (thm "M_triv_axioms.Union_closed"));
paulson@13363
   117
bind_thm ("Un_closed", triv_axioms_L (thm "M_triv_axioms.Un_closed"));
paulson@13363
   118
bind_thm ("cons_closed", triv_axioms_L (thm "M_triv_axioms.cons_closed"));
paulson@13363
   119
bind_thm ("successor_abs", triv_axioms_L (thm "M_triv_axioms.successor_abs"));
paulson@13363
   120
bind_thm ("succ_in_M_iff", triv_axioms_L (thm "M_triv_axioms.succ_in_M_iff"));
paulson@13363
   121
bind_thm ("separation_closed", triv_axioms_L (thm "M_triv_axioms.separation_closed"));
paulson@13363
   122
bind_thm ("strong_replacementI", triv_axioms_L (thm "M_triv_axioms.strong_replacementI"));
paulson@13363
   123
bind_thm ("strong_replacement_closed", triv_axioms_L (thm "M_triv_axioms.strong_replacement_closed"));
paulson@13363
   124
bind_thm ("RepFun_closed", triv_axioms_L (thm "M_triv_axioms.RepFun_closed"));
paulson@13363
   125
bind_thm ("lam_closed", triv_axioms_L (thm "M_triv_axioms.lam_closed"));
paulson@13363
   126
bind_thm ("image_abs", triv_axioms_L (thm "M_triv_axioms.image_abs"));
paulson@13363
   127
bind_thm ("powerset_Pow", triv_axioms_L (thm "M_triv_axioms.powerset_Pow"));
paulson@13363
   128
bind_thm ("powerset_imp_subset_Pow", triv_axioms_L (thm "M_triv_axioms.powerset_imp_subset_Pow"));
paulson@13363
   129
bind_thm ("nat_into_M", triv_axioms_L (thm "M_triv_axioms.nat_into_M"));
paulson@13363
   130
bind_thm ("nat_case_closed", triv_axioms_L (thm "M_triv_axioms.nat_case_closed"));
paulson@13363
   131
bind_thm ("Inl_in_M_iff", triv_axioms_L (thm "M_triv_axioms.Inl_in_M_iff"));
paulson@13363
   132
bind_thm ("Inr_in_M_iff", triv_axioms_L (thm "M_triv_axioms.Inr_in_M_iff"));
paulson@13363
   133
bind_thm ("lt_closed", triv_axioms_L (thm "M_triv_axioms.lt_closed"));
paulson@13363
   134
bind_thm ("transitive_set_abs", triv_axioms_L (thm "M_triv_axioms.transitive_set_abs"));
paulson@13363
   135
bind_thm ("ordinal_abs", triv_axioms_L (thm "M_triv_axioms.ordinal_abs"));
paulson@13363
   136
bind_thm ("limit_ordinal_abs", triv_axioms_L (thm "M_triv_axioms.limit_ordinal_abs"));
paulson@13363
   137
bind_thm ("successor_ordinal_abs", triv_axioms_L (thm "M_triv_axioms.successor_ordinal_abs"));
paulson@13363
   138
bind_thm ("finite_ordinal_abs", triv_axioms_L (thm "M_triv_axioms.finite_ordinal_abs"));
paulson@13363
   139
bind_thm ("omega_abs", triv_axioms_L (thm "M_triv_axioms.omega_abs"));
paulson@13363
   140
bind_thm ("number1_abs", triv_axioms_L (thm "M_triv_axioms.number1_abs"));
paulson@13363
   141
bind_thm ("number1_abs", triv_axioms_L (thm "M_triv_axioms.number1_abs"));
paulson@13363
   142
bind_thm ("number3_abs", triv_axioms_L (thm "M_triv_axioms.number3_abs"));
paulson@13291
   143
*}
paulson@13291
   144
paulson@13291
   145
declare rall_abs [simp] 
paulson@13291
   146
declare rex_abs [simp] 
paulson@13291
   147
declare empty_abs [simp] 
paulson@13291
   148
declare subset_abs [simp] 
paulson@13291
   149
declare upair_abs [simp] 
paulson@13291
   150
declare upair_in_M_iff [iff]
paulson@13291
   151
declare singleton_in_M_iff [iff]
paulson@13291
   152
declare pair_abs [simp] 
paulson@13291
   153
declare pair_in_M_iff [iff]
paulson@13291
   154
declare cartprod_abs [simp] 
paulson@13291
   155
declare union_abs [simp] 
paulson@13291
   156
declare inter_abs [simp] 
paulson@13291
   157
declare setdiff_abs [simp] 
paulson@13291
   158
declare Union_abs [simp] 
paulson@13291
   159
declare Union_closed [intro,simp]
paulson@13291
   160
declare Un_closed [intro,simp]
paulson@13291
   161
declare cons_closed [intro,simp]
paulson@13291
   162
declare successor_abs [simp] 
paulson@13291
   163
declare succ_in_M_iff [iff]
paulson@13291
   164
declare separation_closed [intro,simp]
paulson@13306
   165
declare strong_replacementI
paulson@13291
   166
declare strong_replacement_closed [intro,simp]
paulson@13291
   167
declare RepFun_closed [intro,simp]
paulson@13291
   168
declare lam_closed [intro,simp]
paulson@13291
   169
declare image_abs [simp] 
paulson@13291
   170
declare nat_into_M [intro]
paulson@13291
   171
declare Inl_in_M_iff [iff]
paulson@13291
   172
declare Inr_in_M_iff [iff]
paulson@13291
   173
declare transitive_set_abs [simp] 
paulson@13291
   174
declare ordinal_abs [simp] 
paulson@13291
   175
declare limit_ordinal_abs [simp] 
paulson@13291
   176
declare successor_ordinal_abs [simp] 
paulson@13291
   177
declare finite_ordinal_abs [simp] 
paulson@13291
   178
declare omega_abs [simp] 
paulson@13291
   179
declare number1_abs [simp] 
paulson@13291
   180
declare number1_abs [simp] 
paulson@13291
   181
declare number3_abs [simp]
paulson@13291
   182
paulson@13291
   183
paulson@13291
   184
subsection{*Instantiation of the locale @{text reflection}*}
paulson@13291
   185
paulson@13291
   186
text{*instances of locale constants*}
paulson@13291
   187
constdefs
paulson@13291
   188
  L_F0 :: "[i=>o,i] => i"
paulson@13291
   189
    "L_F0(P,y) == \<mu>b. (\<exists>z. L(z) \<and> P(<y,z>)) --> (\<exists>z\<in>Lset(b). P(<y,z>))"
paulson@13291
   190
paulson@13291
   191
  L_FF :: "[i=>o,i] => i"
paulson@13291
   192
    "L_FF(P)   == \<lambda>a. \<Union>y\<in>Lset(a). L_F0(P,y)"
paulson@13291
   193
paulson@13291
   194
  L_ClEx :: "[i=>o,i] => o"
paulson@13291
   195
    "L_ClEx(P) == \<lambda>a. Limit(a) \<and> normalize(L_FF(P),a) = a"
paulson@13291
   196
paulson@13291
   197
paulson@13314
   198
text{*We must use the meta-existential quantifier; otherwise the reflection
paulson@13314
   199
      terms become enormous!*} 
paulson@13314
   200
constdefs
paulson@13314
   201
  L_Reflects :: "[i=>o,[i,i]=>o] => prop"      ("(3REFLECTS/ [_,/ _])")
paulson@13314
   202
    "REFLECTS[P,Q] == (??Cl. Closed_Unbounded(Cl) &
paulson@13314
   203
                           (\<forall>a. Cl(a) --> (\<forall>x \<in> Lset(a). P(x) <-> Q(a,x))))"
paulson@13291
   204
paulson@13291
   205
paulson@13314
   206
theorem Triv_reflection:
paulson@13314
   207
     "REFLECTS[P, \<lambda>a x. P(x)]"
paulson@13314
   208
apply (simp add: L_Reflects_def) 
paulson@13314
   209
apply (rule meta_exI) 
paulson@13314
   210
apply (rule Closed_Unbounded_Ord) 
paulson@13314
   211
done
paulson@13314
   212
paulson@13314
   213
theorem Not_reflection:
paulson@13314
   214
     "REFLECTS[P,Q] ==> REFLECTS[\<lambda>x. ~P(x), \<lambda>a x. ~Q(a,x)]"
paulson@13314
   215
apply (unfold L_Reflects_def) 
paulson@13314
   216
apply (erule meta_exE) 
paulson@13314
   217
apply (rule_tac x=Cl in meta_exI, simp) 
paulson@13314
   218
done
paulson@13314
   219
paulson@13314
   220
theorem And_reflection:
paulson@13314
   221
     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] 
paulson@13314
   222
      ==> REFLECTS[\<lambda>x. P(x) \<and> P'(x), \<lambda>a x. Q(a,x) \<and> Q'(a,x)]"
paulson@13314
   223
apply (unfold L_Reflects_def) 
paulson@13314
   224
apply (elim meta_exE) 
paulson@13314
   225
apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) 
paulson@13314
   226
apply (simp add: Closed_Unbounded_Int, blast) 
paulson@13314
   227
done
paulson@13314
   228
paulson@13314
   229
theorem Or_reflection:
paulson@13314
   230
     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] 
paulson@13314
   231
      ==> REFLECTS[\<lambda>x. P(x) \<or> P'(x), \<lambda>a x. Q(a,x) \<or> Q'(a,x)]"
paulson@13314
   232
apply (unfold L_Reflects_def) 
paulson@13314
   233
apply (elim meta_exE) 
paulson@13314
   234
apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) 
paulson@13314
   235
apply (simp add: Closed_Unbounded_Int, blast) 
paulson@13314
   236
done
paulson@13314
   237
paulson@13314
   238
theorem Imp_reflection:
paulson@13314
   239
     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] 
paulson@13314
   240
      ==> REFLECTS[\<lambda>x. P(x) --> P'(x), \<lambda>a x. Q(a,x) --> Q'(a,x)]"
paulson@13314
   241
apply (unfold L_Reflects_def) 
paulson@13314
   242
apply (elim meta_exE) 
paulson@13314
   243
apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) 
paulson@13314
   244
apply (simp add: Closed_Unbounded_Int, blast) 
paulson@13314
   245
done
paulson@13314
   246
paulson@13314
   247
theorem Iff_reflection:
paulson@13314
   248
     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] 
paulson@13314
   249
      ==> REFLECTS[\<lambda>x. P(x) <-> P'(x), \<lambda>a x. Q(a,x) <-> Q'(a,x)]"
paulson@13314
   250
apply (unfold L_Reflects_def) 
paulson@13314
   251
apply (elim meta_exE) 
paulson@13314
   252
apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) 
paulson@13314
   253
apply (simp add: Closed_Unbounded_Int, blast) 
paulson@13314
   254
done
paulson@13314
   255
paulson@13314
   256
paulson@13314
   257
theorem Ex_reflection:
paulson@13314
   258
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
paulson@13314
   259
      ==> REFLECTS[\<lambda>x. \<exists>z. L(z) \<and> P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
paulson@13291
   260
apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def) 
paulson@13314
   261
apply (elim meta_exE) 
paulson@13314
   262
apply (rule meta_exI)
paulson@13291
   263
apply (rule reflection.Ex_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset],
paulson@13291
   264
       assumption+)
paulson@13291
   265
done
paulson@13291
   266
paulson@13314
   267
theorem All_reflection:
paulson@13314
   268
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
paulson@13314
   269
      ==> REFLECTS[\<lambda>x. \<forall>z. L(z) --> P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]" 
paulson@13291
   270
apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def) 
paulson@13314
   271
apply (elim meta_exE) 
paulson@13314
   272
apply (rule meta_exI)
paulson@13291
   273
apply (rule reflection.All_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset],
paulson@13291
   274
       assumption+)
paulson@13291
   275
done
paulson@13291
   276
paulson@13314
   277
theorem Rex_reflection:
paulson@13314
   278
     "REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
paulson@13314
   279
      ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
paulson@13314
   280
apply (unfold rex_def) 
paulson@13314
   281
apply (intro And_reflection Ex_reflection, assumption)
paulson@13314
   282
done
paulson@13291
   283
paulson@13314
   284
theorem Rall_reflection:
paulson@13314
   285
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
paulson@13314
   286
      ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]" 
paulson@13314
   287
apply (unfold rall_def) 
paulson@13314
   288
apply (intro Imp_reflection All_reflection, assumption)
paulson@13314
   289
done
paulson@13314
   290
paulson@13323
   291
lemmas FOL_reflections = 
paulson@13314
   292
        Triv_reflection Not_reflection And_reflection Or_reflection
paulson@13314
   293
        Imp_reflection Iff_reflection Ex_reflection All_reflection
paulson@13314
   294
        Rex_reflection Rall_reflection
paulson@13291
   295
paulson@13291
   296
lemma ReflectsD:
paulson@13314
   297
     "[|REFLECTS[P,Q]; Ord(i)|] 
paulson@13291
   298
      ==> \<exists>j. i<j & (\<forall>x \<in> Lset(j). P(x) <-> Q(j,x))"
paulson@13314
   299
apply (unfold L_Reflects_def Closed_Unbounded_def) 
paulson@13314
   300
apply (elim meta_exE, clarify) 
paulson@13291
   301
apply (blast dest!: UnboundedD) 
paulson@13291
   302
done
paulson@13291
   303
paulson@13291
   304
lemma ReflectsE:
paulson@13314
   305
     "[| REFLECTS[P,Q]; Ord(i);
paulson@13291
   306
         !!j. [|i<j;  \<forall>x \<in> Lset(j). P(x) <-> Q(j,x)|] ==> R |]
paulson@13291
   307
      ==> R"
paulson@13316
   308
apply (drule ReflectsD, assumption, blast) 
paulson@13314
   309
done
paulson@13291
   310
paulson@13291
   311
lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B";
paulson@13291
   312
by blast
paulson@13291
   313
paulson@13291
   314
paulson@13339
   315
subsection{*Internalized Formulas for some Set-Theoretic Concepts*}
paulson@13298
   316
paulson@13306
   317
lemmas setclass_simps = rall_setclass_is_ball rex_setclass_is_bex
paulson@13306
   318
paulson@13306
   319
subsubsection{*Some numbers to help write de Bruijn indices*}
paulson@13306
   320
paulson@13306
   321
syntax
paulson@13306
   322
    "3" :: i   ("3")
paulson@13306
   323
    "4" :: i   ("4")
paulson@13306
   324
    "5" :: i   ("5")
paulson@13306
   325
    "6" :: i   ("6")
paulson@13306
   326
    "7" :: i   ("7")
paulson@13306
   327
    "8" :: i   ("8")
paulson@13306
   328
    "9" :: i   ("9")
paulson@13306
   329
paulson@13306
   330
translations
paulson@13306
   331
   "3"  == "succ(2)"
paulson@13306
   332
   "4"  == "succ(3)"
paulson@13306
   333
   "5"  == "succ(4)"
paulson@13306
   334
   "6"  == "succ(5)"
paulson@13306
   335
   "7"  == "succ(6)"
paulson@13306
   336
   "8"  == "succ(7)"
paulson@13306
   337
   "9"  == "succ(8)"
paulson@13306
   338
paulson@13323
   339
paulson@13339
   340
subsubsection{*The Empty Set, Internalized*}
paulson@13323
   341
paulson@13323
   342
constdefs empty_fm :: "i=>i"
paulson@13323
   343
    "empty_fm(x) == Forall(Neg(Member(0,succ(x))))"
paulson@13323
   344
paulson@13323
   345
lemma empty_type [TC]:
paulson@13323
   346
     "x \<in> nat ==> empty_fm(x) \<in> formula"
paulson@13323
   347
by (simp add: empty_fm_def) 
paulson@13323
   348
paulson@13323
   349
lemma arity_empty_fm [simp]:
paulson@13323
   350
     "x \<in> nat ==> arity(empty_fm(x)) = succ(x)"
paulson@13323
   351
by (simp add: empty_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13323
   352
paulson@13323
   353
lemma sats_empty_fm [simp]:
paulson@13323
   354
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13323
   355
    ==> sats(A, empty_fm(x), env) <-> empty(**A, nth(x,env))"
paulson@13323
   356
by (simp add: empty_fm_def empty_def)
paulson@13323
   357
paulson@13323
   358
lemma empty_iff_sats:
paulson@13323
   359
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13323
   360
          i \<in> nat; env \<in> list(A)|]
paulson@13323
   361
       ==> empty(**A, x) <-> sats(A, empty_fm(i), env)"
paulson@13323
   362
by simp
paulson@13323
   363
paulson@13323
   364
theorem empty_reflection:
paulson@13323
   365
     "REFLECTS[\<lambda>x. empty(L,f(x)), 
paulson@13323
   366
               \<lambda>i x. empty(**Lset(i),f(x))]"
paulson@13323
   367
apply (simp only: empty_def setclass_simps)
paulson@13323
   368
apply (intro FOL_reflections)  
paulson@13323
   369
done
paulson@13323
   370
paulson@13385
   371
text{*Not used.  But maybe useful?*}
paulson@13385
   372
lemma Transset_sats_empty_fm_eq_0:
paulson@13385
   373
   "[| n \<in> nat; env \<in> list(A); Transset(A)|]
paulson@13385
   374
    ==> sats(A, empty_fm(n), env) <-> nth(n,env) = 0"
paulson@13385
   375
apply (simp add: empty_fm_def empty_def Transset_def, auto)
paulson@13385
   376
apply (case_tac "n < length(env)") 
paulson@13385
   377
apply (frule nth_type, assumption+, blast)  
paulson@13385
   378
apply (simp_all add: not_lt_iff_le nth_eq_0) 
paulson@13385
   379
done
paulson@13385
   380
paulson@13323
   381
paulson@13339
   382
subsubsection{*Unordered Pairs, Internalized*}
paulson@13298
   383
paulson@13298
   384
constdefs upair_fm :: "[i,i,i]=>i"
paulson@13298
   385
    "upair_fm(x,y,z) == 
paulson@13298
   386
       And(Member(x,z), 
paulson@13298
   387
           And(Member(y,z),
paulson@13298
   388
               Forall(Implies(Member(0,succ(z)), 
paulson@13298
   389
                              Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
paulson@13298
   390
paulson@13298
   391
lemma upair_type [TC]:
paulson@13298
   392
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
paulson@13298
   393
by (simp add: upair_fm_def) 
paulson@13298
   394
paulson@13298
   395
lemma arity_upair_fm [simp]:
paulson@13298
   396
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13298
   397
      ==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13298
   398
by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13298
   399
paulson@13298
   400
lemma sats_upair_fm [simp]:
paulson@13298
   401
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13298
   402
    ==> sats(A, upair_fm(x,y,z), env) <-> 
paulson@13298
   403
            upair(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13298
   404
by (simp add: upair_fm_def upair_def)
paulson@13298
   405
paulson@13298
   406
lemma upair_iff_sats:
paulson@13298
   407
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13298
   408
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13298
   409
       ==> upair(**A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
paulson@13298
   410
by (simp add: sats_upair_fm)
paulson@13298
   411
paulson@13298
   412
text{*Useful? At least it refers to "real" unordered pairs*}
paulson@13298
   413
lemma sats_upair_fm2 [simp]:
paulson@13298
   414
   "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
paulson@13298
   415
    ==> sats(A, upair_fm(x,y,z), env) <-> 
paulson@13298
   416
        nth(z,env) = {nth(x,env), nth(y,env)}"
paulson@13298
   417
apply (frule lt_length_in_nat, assumption)  
paulson@13298
   418
apply (simp add: upair_fm_def Transset_def, auto) 
paulson@13298
   419
apply (blast intro: nth_type) 
paulson@13298
   420
done
paulson@13298
   421
paulson@13314
   422
theorem upair_reflection:
paulson@13314
   423
     "REFLECTS[\<lambda>x. upair(L,f(x),g(x),h(x)), 
paulson@13314
   424
               \<lambda>i x. upair(**Lset(i),f(x),g(x),h(x))]" 
paulson@13314
   425
apply (simp add: upair_def)
paulson@13323
   426
apply (intro FOL_reflections)  
paulson@13314
   427
done
paulson@13306
   428
paulson@13339
   429
subsubsection{*Ordered pairs, Internalized*}
paulson@13298
   430
paulson@13298
   431
constdefs pair_fm :: "[i,i,i]=>i"
paulson@13298
   432
    "pair_fm(x,y,z) == 
paulson@13298
   433
       Exists(And(upair_fm(succ(x),succ(x),0),
paulson@13298
   434
              Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
paulson@13298
   435
                         upair_fm(1,0,succ(succ(z)))))))"
paulson@13298
   436
paulson@13298
   437
lemma pair_type [TC]:
paulson@13298
   438
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
paulson@13298
   439
by (simp add: pair_fm_def) 
paulson@13298
   440
paulson@13298
   441
lemma arity_pair_fm [simp]:
paulson@13298
   442
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13298
   443
      ==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13298
   444
by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13298
   445
paulson@13298
   446
lemma sats_pair_fm [simp]:
paulson@13298
   447
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13298
   448
    ==> sats(A, pair_fm(x,y,z), env) <-> 
paulson@13298
   449
        pair(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13298
   450
by (simp add: pair_fm_def pair_def)
paulson@13298
   451
paulson@13298
   452
lemma pair_iff_sats:
paulson@13298
   453
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13298
   454
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13298
   455
       ==> pair(**A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
paulson@13298
   456
by (simp add: sats_pair_fm)
paulson@13298
   457
paulson@13314
   458
theorem pair_reflection:
paulson@13314
   459
     "REFLECTS[\<lambda>x. pair(L,f(x),g(x),h(x)), 
paulson@13314
   460
               \<lambda>i x. pair(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   461
apply (simp only: pair_def setclass_simps)
paulson@13323
   462
apply (intro FOL_reflections upair_reflection)  
paulson@13314
   463
done
paulson@13306
   464
paulson@13306
   465
paulson@13339
   466
subsubsection{*Binary Unions, Internalized*}
paulson@13298
   467
paulson@13306
   468
constdefs union_fm :: "[i,i,i]=>i"
paulson@13306
   469
    "union_fm(x,y,z) == 
paulson@13306
   470
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   471
                  Or(Member(0,succ(x)),Member(0,succ(y)))))"
paulson@13306
   472
paulson@13306
   473
lemma union_type [TC]:
paulson@13306
   474
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
paulson@13306
   475
by (simp add: union_fm_def) 
paulson@13306
   476
paulson@13306
   477
lemma arity_union_fm [simp]:
paulson@13306
   478
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13306
   479
      ==> arity(union_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13306
   480
by (simp add: union_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13298
   481
paulson@13306
   482
lemma sats_union_fm [simp]:
paulson@13306
   483
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13306
   484
    ==> sats(A, union_fm(x,y,z), env) <-> 
paulson@13306
   485
        union(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13306
   486
by (simp add: union_fm_def union_def)
paulson@13306
   487
paulson@13306
   488
lemma union_iff_sats:
paulson@13306
   489
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13306
   490
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   491
       ==> union(**A, x, y, z) <-> sats(A, union_fm(i,j,k), env)"
paulson@13306
   492
by (simp add: sats_union_fm)
paulson@13298
   493
paulson@13314
   494
theorem union_reflection:
paulson@13314
   495
     "REFLECTS[\<lambda>x. union(L,f(x),g(x),h(x)), 
paulson@13314
   496
               \<lambda>i x. union(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   497
apply (simp only: union_def setclass_simps)
paulson@13323
   498
apply (intro FOL_reflections)  
paulson@13314
   499
done
paulson@13306
   500
paulson@13298
   501
paulson@13339
   502
subsubsection{*Set ``Cons,'' Internalized*}
paulson@13306
   503
paulson@13306
   504
constdefs cons_fm :: "[i,i,i]=>i"
paulson@13306
   505
    "cons_fm(x,y,z) == 
paulson@13306
   506
       Exists(And(upair_fm(succ(x),succ(x),0),
paulson@13306
   507
                  union_fm(0,succ(y),succ(z))))"
paulson@13298
   508
paulson@13298
   509
paulson@13306
   510
lemma cons_type [TC]:
paulson@13306
   511
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
paulson@13306
   512
by (simp add: cons_fm_def) 
paulson@13306
   513
paulson@13306
   514
lemma arity_cons_fm [simp]:
paulson@13306
   515
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13306
   516
      ==> arity(cons_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13306
   517
by (simp add: cons_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   518
paulson@13306
   519
lemma sats_cons_fm [simp]:
paulson@13306
   520
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13306
   521
    ==> sats(A, cons_fm(x,y,z), env) <-> 
paulson@13306
   522
        is_cons(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13306
   523
by (simp add: cons_fm_def is_cons_def)
paulson@13306
   524
paulson@13306
   525
lemma cons_iff_sats:
paulson@13306
   526
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13306
   527
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   528
       ==> is_cons(**A, x, y, z) <-> sats(A, cons_fm(i,j,k), env)"
paulson@13306
   529
by simp
paulson@13306
   530
paulson@13314
   531
theorem cons_reflection:
paulson@13314
   532
     "REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)), 
paulson@13314
   533
               \<lambda>i x. is_cons(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   534
apply (simp only: is_cons_def setclass_simps)
paulson@13323
   535
apply (intro FOL_reflections upair_reflection union_reflection)  
paulson@13323
   536
done
paulson@13323
   537
paulson@13323
   538
paulson@13339
   539
subsubsection{*Successor Function, Internalized*}
paulson@13323
   540
paulson@13323
   541
constdefs succ_fm :: "[i,i]=>i"
paulson@13323
   542
    "succ_fm(x,y) == cons_fm(x,x,y)"
paulson@13323
   543
paulson@13323
   544
lemma succ_type [TC]:
paulson@13323
   545
     "[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula"
paulson@13323
   546
by (simp add: succ_fm_def) 
paulson@13323
   547
paulson@13323
   548
lemma arity_succ_fm [simp]:
paulson@13323
   549
     "[| x \<in> nat; y \<in> nat |] 
paulson@13323
   550
      ==> arity(succ_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13323
   551
by (simp add: succ_fm_def)
paulson@13323
   552
paulson@13323
   553
lemma sats_succ_fm [simp]:
paulson@13323
   554
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13323
   555
    ==> sats(A, succ_fm(x,y), env) <-> 
paulson@13323
   556
        successor(**A, nth(x,env), nth(y,env))"
paulson@13323
   557
by (simp add: succ_fm_def successor_def)
paulson@13323
   558
paulson@13323
   559
lemma successor_iff_sats:
paulson@13323
   560
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13323
   561
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13323
   562
       ==> successor(**A, x, y) <-> sats(A, succ_fm(i,j), env)"
paulson@13323
   563
by simp
paulson@13323
   564
paulson@13323
   565
theorem successor_reflection:
paulson@13323
   566
     "REFLECTS[\<lambda>x. successor(L,f(x),g(x)), 
paulson@13323
   567
               \<lambda>i x. successor(**Lset(i),f(x),g(x))]"
paulson@13323
   568
apply (simp only: successor_def setclass_simps)
paulson@13323
   569
apply (intro cons_reflection)  
paulson@13314
   570
done
paulson@13298
   571
paulson@13298
   572
paulson@13363
   573
subsubsection{*The Number 1, Internalized*}
paulson@13363
   574
paulson@13363
   575
(* "number1(M,a) == (\<exists>x[M]. empty(M,x) & successor(M,x,a))" *)
paulson@13363
   576
constdefs number1_fm :: "i=>i"
paulson@13363
   577
    "number1_fm(a) == Exists(And(empty_fm(0), succ_fm(0,succ(a))))"
paulson@13363
   578
paulson@13363
   579
lemma number1_type [TC]:
paulson@13363
   580
     "x \<in> nat ==> number1_fm(x) \<in> formula"
paulson@13363
   581
by (simp add: number1_fm_def) 
paulson@13363
   582
paulson@13363
   583
lemma arity_number1_fm [simp]:
paulson@13363
   584
     "x \<in> nat ==> arity(number1_fm(x)) = succ(x)"
paulson@13363
   585
by (simp add: number1_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13363
   586
paulson@13363
   587
lemma sats_number1_fm [simp]:
paulson@13363
   588
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13363
   589
    ==> sats(A, number1_fm(x), env) <-> number1(**A, nth(x,env))"
paulson@13363
   590
by (simp add: number1_fm_def number1_def)
paulson@13363
   591
paulson@13363
   592
lemma number1_iff_sats:
paulson@13363
   593
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13363
   594
          i \<in> nat; env \<in> list(A)|]
paulson@13363
   595
       ==> number1(**A, x) <-> sats(A, number1_fm(i), env)"
paulson@13363
   596
by simp
paulson@13363
   597
paulson@13363
   598
theorem number1_reflection:
paulson@13363
   599
     "REFLECTS[\<lambda>x. number1(L,f(x)), 
paulson@13363
   600
               \<lambda>i x. number1(**Lset(i),f(x))]"
paulson@13363
   601
apply (simp only: number1_def setclass_simps)
paulson@13363
   602
apply (intro FOL_reflections empty_reflection successor_reflection)
paulson@13363
   603
done
paulson@13363
   604
paulson@13363
   605
paulson@13352
   606
subsubsection{*Big Union, Internalized*}
paulson@13306
   607
paulson@13352
   608
(*  "big_union(M,A,z) == \<forall>x[M]. x \<in> z <-> (\<exists>y[M]. y\<in>A & x \<in> y)" *)
paulson@13352
   609
constdefs big_union_fm :: "[i,i]=>i"
paulson@13352
   610
    "big_union_fm(A,z) == 
paulson@13352
   611
       Forall(Iff(Member(0,succ(z)),
paulson@13352
   612
                  Exists(And(Member(0,succ(succ(A))), Member(1,0)))))"
paulson@13298
   613
paulson@13352
   614
lemma big_union_type [TC]:
paulson@13352
   615
     "[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula"
paulson@13352
   616
by (simp add: big_union_fm_def) 
paulson@13306
   617
paulson@13352
   618
lemma arity_big_union_fm [simp]:
paulson@13352
   619
     "[| x \<in> nat; y \<in> nat |] 
paulson@13352
   620
      ==> arity(big_union_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13352
   621
by (simp add: big_union_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13298
   622
paulson@13352
   623
lemma sats_big_union_fm [simp]:
paulson@13352
   624
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13352
   625
    ==> sats(A, big_union_fm(x,y), env) <-> 
paulson@13352
   626
        big_union(**A, nth(x,env), nth(y,env))"
paulson@13352
   627
by (simp add: big_union_fm_def big_union_def)
paulson@13306
   628
paulson@13352
   629
lemma big_union_iff_sats:
paulson@13352
   630
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13352
   631
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13352
   632
       ==> big_union(**A, x, y) <-> sats(A, big_union_fm(i,j), env)"
paulson@13306
   633
by simp
paulson@13306
   634
paulson@13352
   635
theorem big_union_reflection:
paulson@13352
   636
     "REFLECTS[\<lambda>x. big_union(L,f(x),g(x)), 
paulson@13352
   637
               \<lambda>i x. big_union(**Lset(i),f(x),g(x))]"
paulson@13352
   638
apply (simp only: big_union_def setclass_simps)
paulson@13352
   639
apply (intro FOL_reflections)  
paulson@13314
   640
done
paulson@13298
   641
paulson@13298
   642
paulson@13306
   643
subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*}
paulson@13306
   644
paulson@13306
   645
text{*Differs from the one in Formula by using "ordinal" rather than "Ord"*}
paulson@13306
   646
paulson@13306
   647
paulson@13306
   648
lemma sats_subset_fm':
paulson@13306
   649
   "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13306
   650
    ==> sats(A, subset_fm(x,y), env) <-> subset(**A, nth(x,env), nth(y,env))" 
paulson@13323
   651
by (simp add: subset_fm_def Relative.subset_def) 
paulson@13298
   652
paulson@13314
   653
theorem subset_reflection:
paulson@13314
   654
     "REFLECTS[\<lambda>x. subset(L,f(x),g(x)), 
paulson@13314
   655
               \<lambda>i x. subset(**Lset(i),f(x),g(x))]" 
paulson@13323
   656
apply (simp only: Relative.subset_def setclass_simps)
paulson@13323
   657
apply (intro FOL_reflections)  
paulson@13314
   658
done
paulson@13306
   659
paulson@13306
   660
lemma sats_transset_fm':
paulson@13306
   661
   "[|x \<in> nat; env \<in> list(A)|]
paulson@13306
   662
    ==> sats(A, transset_fm(x), env) <-> transitive_set(**A, nth(x,env))"
paulson@13306
   663
by (simp add: sats_subset_fm' transset_fm_def transitive_set_def) 
paulson@13298
   664
paulson@13314
   665
theorem transitive_set_reflection:
paulson@13314
   666
     "REFLECTS[\<lambda>x. transitive_set(L,f(x)),
paulson@13314
   667
               \<lambda>i x. transitive_set(**Lset(i),f(x))]"
paulson@13314
   668
apply (simp only: transitive_set_def setclass_simps)
paulson@13323
   669
apply (intro FOL_reflections subset_reflection)  
paulson@13314
   670
done
paulson@13306
   671
paulson@13306
   672
lemma sats_ordinal_fm':
paulson@13306
   673
   "[|x \<in> nat; env \<in> list(A)|]
paulson@13306
   674
    ==> sats(A, ordinal_fm(x), env) <-> ordinal(**A,nth(x,env))"
paulson@13306
   675
by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def)
paulson@13306
   676
paulson@13306
   677
lemma ordinal_iff_sats:
paulson@13306
   678
      "[| nth(i,env) = x;  i \<in> nat; env \<in> list(A)|]
paulson@13306
   679
       ==> ordinal(**A, x) <-> sats(A, ordinal_fm(i), env)"
paulson@13306
   680
by (simp add: sats_ordinal_fm')
paulson@13306
   681
paulson@13314
   682
theorem ordinal_reflection:
paulson@13314
   683
     "REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(**Lset(i),f(x))]"
paulson@13314
   684
apply (simp only: ordinal_def setclass_simps)
paulson@13323
   685
apply (intro FOL_reflections transitive_set_reflection)  
paulson@13314
   686
done
paulson@13298
   687
paulson@13298
   688
paulson@13339
   689
subsubsection{*Membership Relation, Internalized*}
paulson@13298
   690
paulson@13306
   691
constdefs Memrel_fm :: "[i,i]=>i"
paulson@13306
   692
    "Memrel_fm(A,r) == 
paulson@13306
   693
       Forall(Iff(Member(0,succ(r)),
paulson@13306
   694
                  Exists(And(Member(0,succ(succ(A))),
paulson@13306
   695
                             Exists(And(Member(0,succ(succ(succ(A)))),
paulson@13306
   696
                                        And(Member(1,0),
paulson@13306
   697
                                            pair_fm(1,0,2))))))))"
paulson@13306
   698
paulson@13306
   699
lemma Memrel_type [TC]:
paulson@13306
   700
     "[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
paulson@13306
   701
by (simp add: Memrel_fm_def) 
paulson@13298
   702
paulson@13306
   703
lemma arity_Memrel_fm [simp]:
paulson@13306
   704
     "[| x \<in> nat; y \<in> nat |] 
paulson@13306
   705
      ==> arity(Memrel_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13306
   706
by (simp add: Memrel_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   707
paulson@13306
   708
lemma sats_Memrel_fm [simp]:
paulson@13306
   709
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13306
   710
    ==> sats(A, Memrel_fm(x,y), env) <-> 
paulson@13306
   711
        membership(**A, nth(x,env), nth(y,env))"
paulson@13306
   712
by (simp add: Memrel_fm_def membership_def)
paulson@13298
   713
paulson@13306
   714
lemma Memrel_iff_sats:
paulson@13306
   715
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
   716
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13306
   717
       ==> membership(**A, x, y) <-> sats(A, Memrel_fm(i,j), env)"
paulson@13306
   718
by simp
paulson@13304
   719
paulson@13314
   720
theorem membership_reflection:
paulson@13314
   721
     "REFLECTS[\<lambda>x. membership(L,f(x),g(x)), 
paulson@13314
   722
               \<lambda>i x. membership(**Lset(i),f(x),g(x))]"
paulson@13314
   723
apply (simp only: membership_def setclass_simps)
paulson@13323
   724
apply (intro FOL_reflections pair_reflection)  
paulson@13314
   725
done
paulson@13304
   726
paulson@13339
   727
subsubsection{*Predecessor Set, Internalized*}
paulson@13304
   728
paulson@13306
   729
constdefs pred_set_fm :: "[i,i,i,i]=>i"
paulson@13306
   730
    "pred_set_fm(A,x,r,B) == 
paulson@13306
   731
       Forall(Iff(Member(0,succ(B)),
paulson@13306
   732
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   733
                             And(Member(1,succ(succ(A))),
paulson@13306
   734
                                 pair_fm(1,succ(succ(x)),0))))))"
paulson@13306
   735
paulson@13306
   736
paulson@13306
   737
lemma pred_set_type [TC]:
paulson@13306
   738
     "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |] 
paulson@13306
   739
      ==> pred_set_fm(A,x,r,B) \<in> formula"
paulson@13306
   740
by (simp add: pred_set_fm_def) 
paulson@13304
   741
paulson@13306
   742
lemma arity_pred_set_fm [simp]:
paulson@13306
   743
   "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |] 
paulson@13306
   744
    ==> arity(pred_set_fm(A,x,r,B)) = succ(A) \<union> succ(x) \<union> succ(r) \<union> succ(B)"
paulson@13306
   745
by (simp add: pred_set_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   746
paulson@13306
   747
lemma sats_pred_set_fm [simp]:
paulson@13306
   748
   "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|]
paulson@13306
   749
    ==> sats(A, pred_set_fm(U,x,r,B), env) <-> 
paulson@13306
   750
        pred_set(**A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))"
paulson@13306
   751
by (simp add: pred_set_fm_def pred_set_def)
paulson@13306
   752
paulson@13306
   753
lemma pred_set_iff_sats:
paulson@13306
   754
      "[| nth(i,env) = U; nth(j,env) = x; nth(k,env) = r; nth(l,env) = B; 
paulson@13306
   755
          i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|]
paulson@13306
   756
       ==> pred_set(**A,U,x,r,B) <-> sats(A, pred_set_fm(i,j,k,l), env)"
paulson@13306
   757
by (simp add: sats_pred_set_fm)
paulson@13306
   758
paulson@13314
   759
theorem pred_set_reflection:
paulson@13314
   760
     "REFLECTS[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)), 
paulson@13314
   761
               \<lambda>i x. pred_set(**Lset(i),f(x),g(x),h(x),b(x))]" 
paulson@13314
   762
apply (simp only: pred_set_def setclass_simps)
paulson@13323
   763
apply (intro FOL_reflections pair_reflection)  
paulson@13314
   764
done
paulson@13304
   765
paulson@13304
   766
paulson@13298
   767
paulson@13339
   768
subsubsection{*Domain of a Relation, Internalized*}
paulson@13306
   769
paulson@13306
   770
(* "is_domain(M,r,z) == 
paulson@13306
   771
	\<forall>x[M]. (x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w))))" *)
paulson@13306
   772
constdefs domain_fm :: "[i,i]=>i"
paulson@13306
   773
    "domain_fm(r,z) == 
paulson@13306
   774
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   775
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   776
                             Exists(pair_fm(2,0,1))))))"
paulson@13306
   777
paulson@13306
   778
lemma domain_type [TC]:
paulson@13306
   779
     "[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
paulson@13306
   780
by (simp add: domain_fm_def) 
paulson@13306
   781
paulson@13306
   782
lemma arity_domain_fm [simp]:
paulson@13306
   783
     "[| x \<in> nat; y \<in> nat |] 
paulson@13306
   784
      ==> arity(domain_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13306
   785
by (simp add: domain_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   786
paulson@13306
   787
lemma sats_domain_fm [simp]:
paulson@13306
   788
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13306
   789
    ==> sats(A, domain_fm(x,y), env) <-> 
paulson@13306
   790
        is_domain(**A, nth(x,env), nth(y,env))"
paulson@13306
   791
by (simp add: domain_fm_def is_domain_def)
paulson@13306
   792
paulson@13306
   793
lemma domain_iff_sats:
paulson@13306
   794
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
   795
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13306
   796
       ==> is_domain(**A, x, y) <-> sats(A, domain_fm(i,j), env)"
paulson@13306
   797
by simp
paulson@13306
   798
paulson@13314
   799
theorem domain_reflection:
paulson@13314
   800
     "REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)), 
paulson@13314
   801
               \<lambda>i x. is_domain(**Lset(i),f(x),g(x))]"
paulson@13314
   802
apply (simp only: is_domain_def setclass_simps)
paulson@13323
   803
apply (intro FOL_reflections pair_reflection)  
paulson@13314
   804
done
paulson@13306
   805
paulson@13306
   806
paulson@13339
   807
subsubsection{*Range of a Relation, Internalized*}
paulson@13306
   808
paulson@13306
   809
(* "is_range(M,r,z) == 
paulson@13306
   810
	\<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w))))" *)
paulson@13306
   811
constdefs range_fm :: "[i,i]=>i"
paulson@13306
   812
    "range_fm(r,z) == 
paulson@13306
   813
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   814
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   815
                             Exists(pair_fm(0,2,1))))))"
paulson@13306
   816
paulson@13306
   817
lemma range_type [TC]:
paulson@13306
   818
     "[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
paulson@13306
   819
by (simp add: range_fm_def) 
paulson@13306
   820
paulson@13306
   821
lemma arity_range_fm [simp]:
paulson@13306
   822
     "[| x \<in> nat; y \<in> nat |] 
paulson@13306
   823
      ==> arity(range_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13306
   824
by (simp add: range_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   825
paulson@13306
   826
lemma sats_range_fm [simp]:
paulson@13306
   827
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13306
   828
    ==> sats(A, range_fm(x,y), env) <-> 
paulson@13306
   829
        is_range(**A, nth(x,env), nth(y,env))"
paulson@13306
   830
by (simp add: range_fm_def is_range_def)
paulson@13306
   831
paulson@13306
   832
lemma range_iff_sats:
paulson@13306
   833
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
   834
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13306
   835
       ==> is_range(**A, x, y) <-> sats(A, range_fm(i,j), env)"
paulson@13306
   836
by simp
paulson@13306
   837
paulson@13314
   838
theorem range_reflection:
paulson@13314
   839
     "REFLECTS[\<lambda>x. is_range(L,f(x),g(x)), 
paulson@13314
   840
               \<lambda>i x. is_range(**Lset(i),f(x),g(x))]"
paulson@13314
   841
apply (simp only: is_range_def setclass_simps)
paulson@13323
   842
apply (intro FOL_reflections pair_reflection)  
paulson@13314
   843
done
paulson@13306
   844
paulson@13306
   845
 
paulson@13339
   846
subsubsection{*Field of a Relation, Internalized*}
paulson@13323
   847
paulson@13323
   848
(* "is_field(M,r,z) == 
paulson@13323
   849
	\<exists>dr[M]. is_domain(M,r,dr) & 
paulson@13323
   850
            (\<exists>rr[M]. is_range(M,r,rr) & union(M,dr,rr,z))" *)
paulson@13323
   851
constdefs field_fm :: "[i,i]=>i"
paulson@13323
   852
    "field_fm(r,z) == 
paulson@13323
   853
       Exists(And(domain_fm(succ(r),0), 
paulson@13323
   854
              Exists(And(range_fm(succ(succ(r)),0), 
paulson@13323
   855
                         union_fm(1,0,succ(succ(z)))))))"
paulson@13323
   856
paulson@13323
   857
lemma field_type [TC]:
paulson@13323
   858
     "[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula"
paulson@13323
   859
by (simp add: field_fm_def) 
paulson@13323
   860
paulson@13323
   861
lemma arity_field_fm [simp]:
paulson@13323
   862
     "[| x \<in> nat; y \<in> nat |] 
paulson@13323
   863
      ==> arity(field_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13323
   864
by (simp add: field_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13323
   865
paulson@13323
   866
lemma sats_field_fm [simp]:
paulson@13323
   867
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13323
   868
    ==> sats(A, field_fm(x,y), env) <-> 
paulson@13323
   869
        is_field(**A, nth(x,env), nth(y,env))"
paulson@13323
   870
by (simp add: field_fm_def is_field_def)
paulson@13323
   871
paulson@13323
   872
lemma field_iff_sats:
paulson@13323
   873
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13323
   874
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13323
   875
       ==> is_field(**A, x, y) <-> sats(A, field_fm(i,j), env)"
paulson@13323
   876
by simp
paulson@13323
   877
paulson@13323
   878
theorem field_reflection:
paulson@13323
   879
     "REFLECTS[\<lambda>x. is_field(L,f(x),g(x)), 
paulson@13323
   880
               \<lambda>i x. is_field(**Lset(i),f(x),g(x))]"
paulson@13323
   881
apply (simp only: is_field_def setclass_simps)
paulson@13323
   882
apply (intro FOL_reflections domain_reflection range_reflection
paulson@13323
   883
             union_reflection)
paulson@13323
   884
done
paulson@13323
   885
paulson@13323
   886
paulson@13339
   887
subsubsection{*Image under a Relation, Internalized*}
paulson@13306
   888
paulson@13306
   889
(* "image(M,r,A,z) == 
paulson@13306
   890
        \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,w))))" *)
paulson@13306
   891
constdefs image_fm :: "[i,i,i]=>i"
paulson@13306
   892
    "image_fm(r,A,z) == 
paulson@13306
   893
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   894
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   895
                             Exists(And(Member(0,succ(succ(succ(A)))),
paulson@13306
   896
	 			        pair_fm(0,2,1)))))))"
paulson@13306
   897
paulson@13306
   898
lemma image_type [TC]:
paulson@13306
   899
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
paulson@13306
   900
by (simp add: image_fm_def) 
paulson@13306
   901
paulson@13306
   902
lemma arity_image_fm [simp]:
paulson@13306
   903
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13306
   904
      ==> arity(image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13306
   905
by (simp add: image_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
   906
paulson@13306
   907
lemma sats_image_fm [simp]:
paulson@13306
   908
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13306
   909
    ==> sats(A, image_fm(x,y,z), env) <-> 
paulson@13306
   910
        image(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13323
   911
by (simp add: image_fm_def Relative.image_def)
paulson@13306
   912
paulson@13306
   913
lemma image_iff_sats:
paulson@13306
   914
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13306
   915
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   916
       ==> image(**A, x, y, z) <-> sats(A, image_fm(i,j,k), env)"
paulson@13306
   917
by (simp add: sats_image_fm)
paulson@13306
   918
paulson@13314
   919
theorem image_reflection:
paulson@13314
   920
     "REFLECTS[\<lambda>x. image(L,f(x),g(x),h(x)), 
paulson@13314
   921
               \<lambda>i x. image(**Lset(i),f(x),g(x),h(x))]"
paulson@13323
   922
apply (simp only: Relative.image_def setclass_simps)
paulson@13323
   923
apply (intro FOL_reflections pair_reflection)  
paulson@13314
   924
done
paulson@13306
   925
paulson@13306
   926
paulson@13348
   927
subsubsection{*Pre-Image under a Relation, Internalized*}
paulson@13348
   928
paulson@13348
   929
(* "pre_image(M,r,A,z) == 
paulson@13348
   930
	\<forall>x[M]. x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" *)
paulson@13348
   931
constdefs pre_image_fm :: "[i,i,i]=>i"
paulson@13348
   932
    "pre_image_fm(r,A,z) == 
paulson@13348
   933
       Forall(Iff(Member(0,succ(z)),
paulson@13348
   934
                  Exists(And(Member(0,succ(succ(r))),
paulson@13348
   935
                             Exists(And(Member(0,succ(succ(succ(A)))),
paulson@13348
   936
	 			        pair_fm(2,0,1)))))))"
paulson@13348
   937
paulson@13348
   938
lemma pre_image_type [TC]:
paulson@13348
   939
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
paulson@13348
   940
by (simp add: pre_image_fm_def) 
paulson@13348
   941
paulson@13348
   942
lemma arity_pre_image_fm [simp]:
paulson@13348
   943
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13348
   944
      ==> arity(pre_image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13348
   945
by (simp add: pre_image_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13348
   946
paulson@13348
   947
lemma sats_pre_image_fm [simp]:
paulson@13348
   948
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13348
   949
    ==> sats(A, pre_image_fm(x,y,z), env) <-> 
paulson@13348
   950
        pre_image(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13348
   951
by (simp add: pre_image_fm_def Relative.pre_image_def)
paulson@13348
   952
paulson@13348
   953
lemma pre_image_iff_sats:
paulson@13348
   954
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13348
   955
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13348
   956
       ==> pre_image(**A, x, y, z) <-> sats(A, pre_image_fm(i,j,k), env)"
paulson@13348
   957
by (simp add: sats_pre_image_fm)
paulson@13348
   958
paulson@13348
   959
theorem pre_image_reflection:
paulson@13348
   960
     "REFLECTS[\<lambda>x. pre_image(L,f(x),g(x),h(x)), 
paulson@13348
   961
               \<lambda>i x. pre_image(**Lset(i),f(x),g(x),h(x))]"
paulson@13348
   962
apply (simp only: Relative.pre_image_def setclass_simps)
paulson@13348
   963
apply (intro FOL_reflections pair_reflection)  
paulson@13348
   964
done
paulson@13348
   965
paulson@13348
   966
paulson@13352
   967
subsubsection{*Function Application, Internalized*}
paulson@13352
   968
paulson@13352
   969
(* "fun_apply(M,f,x,y) == 
paulson@13352
   970
        (\<exists>xs[M]. \<exists>fxs[M]. 
paulson@13352
   971
         upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))" *)
paulson@13352
   972
constdefs fun_apply_fm :: "[i,i,i]=>i"
paulson@13352
   973
    "fun_apply_fm(f,x,y) == 
paulson@13352
   974
       Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1),
paulson@13352
   975
                         And(image_fm(succ(succ(f)), 1, 0), 
paulson@13352
   976
                             big_union_fm(0,succ(succ(y)))))))"
paulson@13352
   977
paulson@13352
   978
lemma fun_apply_type [TC]:
paulson@13352
   979
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
paulson@13352
   980
by (simp add: fun_apply_fm_def) 
paulson@13352
   981
paulson@13352
   982
lemma arity_fun_apply_fm [simp]:
paulson@13352
   983
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13352
   984
      ==> arity(fun_apply_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13352
   985
by (simp add: fun_apply_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13352
   986
paulson@13352
   987
lemma sats_fun_apply_fm [simp]:
paulson@13352
   988
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13352
   989
    ==> sats(A, fun_apply_fm(x,y,z), env) <-> 
paulson@13352
   990
        fun_apply(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13352
   991
by (simp add: fun_apply_fm_def fun_apply_def)
paulson@13352
   992
paulson@13352
   993
lemma fun_apply_iff_sats:
paulson@13352
   994
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13352
   995
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13352
   996
       ==> fun_apply(**A, x, y, z) <-> sats(A, fun_apply_fm(i,j,k), env)"
paulson@13352
   997
by simp
paulson@13352
   998
paulson@13352
   999
theorem fun_apply_reflection:
paulson@13352
  1000
     "REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)), 
paulson@13352
  1001
               \<lambda>i x. fun_apply(**Lset(i),f(x),g(x),h(x))]" 
paulson@13352
  1002
apply (simp only: fun_apply_def setclass_simps)
paulson@13352
  1003
apply (intro FOL_reflections upair_reflection image_reflection
paulson@13352
  1004
             big_union_reflection)  
paulson@13352
  1005
done
paulson@13352
  1006
paulson@13352
  1007
paulson@13339
  1008
subsubsection{*The Concept of Relation, Internalized*}
paulson@13306
  1009
paulson@13306
  1010
(* "is_relation(M,r) == 
paulson@13306
  1011
        (\<forall>z[M]. z\<in>r --> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))" *)
paulson@13306
  1012
constdefs relation_fm :: "i=>i"
paulson@13306
  1013
    "relation_fm(r) == 
paulson@13306
  1014
       Forall(Implies(Member(0,succ(r)), Exists(Exists(pair_fm(1,0,2)))))"
paulson@13306
  1015
paulson@13306
  1016
lemma relation_type [TC]:
paulson@13306
  1017
     "[| x \<in> nat |] ==> relation_fm(x) \<in> formula"
paulson@13306
  1018
by (simp add: relation_fm_def) 
paulson@13306
  1019
paulson@13306
  1020
lemma arity_relation_fm [simp]:
paulson@13306
  1021
     "x \<in> nat ==> arity(relation_fm(x)) = succ(x)"
paulson@13306
  1022
by (simp add: relation_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
  1023
paulson@13306
  1024
lemma sats_relation_fm [simp]:
paulson@13306
  1025
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13306
  1026
    ==> sats(A, relation_fm(x), env) <-> is_relation(**A, nth(x,env))"
paulson@13306
  1027
by (simp add: relation_fm_def is_relation_def)
paulson@13306
  1028
paulson@13306
  1029
lemma relation_iff_sats:
paulson@13306
  1030
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
  1031
          i \<in> nat; env \<in> list(A)|]
paulson@13306
  1032
       ==> is_relation(**A, x) <-> sats(A, relation_fm(i), env)"
paulson@13306
  1033
by simp
paulson@13306
  1034
paulson@13314
  1035
theorem is_relation_reflection:
paulson@13314
  1036
     "REFLECTS[\<lambda>x. is_relation(L,f(x)), 
paulson@13314
  1037
               \<lambda>i x. is_relation(**Lset(i),f(x))]"
paulson@13314
  1038
apply (simp only: is_relation_def setclass_simps)
paulson@13323
  1039
apply (intro FOL_reflections pair_reflection)  
paulson@13314
  1040
done
paulson@13306
  1041
paulson@13306
  1042
paulson@13339
  1043
subsubsection{*The Concept of Function, Internalized*}
paulson@13306
  1044
paulson@13306
  1045
(* "is_function(M,r) == 
paulson@13306
  1046
	\<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M]. 
paulson@13306
  1047
           pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>r --> y=y'" *)
paulson@13306
  1048
constdefs function_fm :: "i=>i"
paulson@13306
  1049
    "function_fm(r) == 
paulson@13306
  1050
       Forall(Forall(Forall(Forall(Forall(
paulson@13306
  1051
         Implies(pair_fm(4,3,1),
paulson@13306
  1052
                 Implies(pair_fm(4,2,0),
paulson@13306
  1053
                         Implies(Member(1,r#+5),
paulson@13306
  1054
                                 Implies(Member(0,r#+5), Equal(3,2))))))))))"
paulson@13306
  1055
paulson@13306
  1056
lemma function_type [TC]:
paulson@13306
  1057
     "[| x \<in> nat |] ==> function_fm(x) \<in> formula"
paulson@13306
  1058
by (simp add: function_fm_def) 
paulson@13306
  1059
paulson@13306
  1060
lemma arity_function_fm [simp]:
paulson@13306
  1061
     "x \<in> nat ==> arity(function_fm(x)) = succ(x)"
paulson@13306
  1062
by (simp add: function_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13306
  1063
paulson@13306
  1064
lemma sats_function_fm [simp]:
paulson@13306
  1065
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13306
  1066
    ==> sats(A, function_fm(x), env) <-> is_function(**A, nth(x,env))"
paulson@13306
  1067
by (simp add: function_fm_def is_function_def)
paulson@13306
  1068
paulson@13306
  1069
lemma function_iff_sats:
paulson@13306
  1070
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13306
  1071
          i \<in> nat; env \<in> list(A)|]
paulson@13306
  1072
       ==> is_function(**A, x) <-> sats(A, function_fm(i), env)"
paulson@13306
  1073
by simp
paulson@13306
  1074
paulson@13314
  1075
theorem is_function_reflection:
paulson@13314
  1076
     "REFLECTS[\<lambda>x. is_function(L,f(x)), 
paulson@13314
  1077
               \<lambda>i x. is_function(**Lset(i),f(x))]"
paulson@13314
  1078
apply (simp only: is_function_def setclass_simps)
paulson@13323
  1079
apply (intro FOL_reflections pair_reflection)  
paulson@13314
  1080
done
paulson@13298
  1081
paulson@13298
  1082
paulson@13339
  1083
subsubsection{*Typed Functions, Internalized*}
paulson@13309
  1084
paulson@13309
  1085
(* "typed_function(M,A,B,r) == 
paulson@13309
  1086
        is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
paulson@13309
  1087
        (\<forall>u[M]. u\<in>r --> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) --> y\<in>B))" *)
paulson@13309
  1088
paulson@13309
  1089
constdefs typed_function_fm :: "[i,i,i]=>i"
paulson@13309
  1090
    "typed_function_fm(A,B,r) == 
paulson@13309
  1091
       And(function_fm(r),
paulson@13309
  1092
         And(relation_fm(r),
paulson@13309
  1093
           And(domain_fm(r,A),
paulson@13309
  1094
             Forall(Implies(Member(0,succ(r)),
paulson@13309
  1095
                  Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))"
paulson@13309
  1096
paulson@13309
  1097
lemma typed_function_type [TC]:
paulson@13309
  1098
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
paulson@13309
  1099
by (simp add: typed_function_fm_def) 
paulson@13309
  1100
paulson@13309
  1101
lemma arity_typed_function_fm [simp]:
paulson@13309
  1102
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13309
  1103
      ==> arity(typed_function_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13309
  1104
by (simp add: typed_function_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
  1105
paulson@13309
  1106
lemma sats_typed_function_fm [simp]:
paulson@13309
  1107
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13309
  1108
    ==> sats(A, typed_function_fm(x,y,z), env) <-> 
paulson@13309
  1109
        typed_function(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1110
by (simp add: typed_function_fm_def typed_function_def)
paulson@13309
  1111
paulson@13309
  1112
lemma typed_function_iff_sats:
paulson@13309
  1113
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13309
  1114
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1115
   ==> typed_function(**A, x, y, z) <-> sats(A, typed_function_fm(i,j,k), env)"
paulson@13309
  1116
by simp
paulson@13309
  1117
paulson@13323
  1118
lemmas function_reflections = 
paulson@13363
  1119
        empty_reflection number1_reflection
paulson@13363
  1120
	upair_reflection pair_reflection union_reflection
paulson@13352
  1121
	big_union_reflection cons_reflection successor_reflection 
paulson@13323
  1122
        fun_apply_reflection subset_reflection
paulson@13323
  1123
	transitive_set_reflection membership_reflection
paulson@13323
  1124
	pred_set_reflection domain_reflection range_reflection field_reflection
paulson@13348
  1125
        image_reflection pre_image_reflection
paulson@13314
  1126
	is_relation_reflection is_function_reflection
paulson@13309
  1127
paulson@13323
  1128
lemmas function_iff_sats = 
paulson@13363
  1129
        empty_iff_sats number1_iff_sats 
paulson@13363
  1130
	upair_iff_sats pair_iff_sats union_iff_sats
paulson@13323
  1131
	cons_iff_sats successor_iff_sats
paulson@13323
  1132
        fun_apply_iff_sats  Memrel_iff_sats
paulson@13323
  1133
	pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats
paulson@13348
  1134
        image_iff_sats pre_image_iff_sats 
paulson@13323
  1135
	relation_iff_sats function_iff_sats
paulson@13323
  1136
paulson@13309
  1137
paulson@13314
  1138
theorem typed_function_reflection:
paulson@13314
  1139
     "REFLECTS[\<lambda>x. typed_function(L,f(x),g(x),h(x)), 
paulson@13314
  1140
               \<lambda>i x. typed_function(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1141
apply (simp only: typed_function_def setclass_simps)
paulson@13323
  1142
apply (intro FOL_reflections function_reflections)  
paulson@13323
  1143
done
paulson@13323
  1144
paulson@13323
  1145
paulson@13339
  1146
subsubsection{*Composition of Relations, Internalized*}
paulson@13323
  1147
paulson@13323
  1148
(* "composition(M,r,s,t) == 
paulson@13323
  1149
        \<forall>p[M]. p \<in> t <-> 
paulson@13323
  1150
               (\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M]. 
paulson@13323
  1151
                pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) & 
paulson@13323
  1152
                xy \<in> s & yz \<in> r)" *)
paulson@13323
  1153
constdefs composition_fm :: "[i,i,i]=>i"
paulson@13323
  1154
  "composition_fm(r,s,t) == 
paulson@13323
  1155
     Forall(Iff(Member(0,succ(t)),
paulson@13323
  1156
             Exists(Exists(Exists(Exists(Exists( 
paulson@13323
  1157
              And(pair_fm(4,2,5),
paulson@13323
  1158
               And(pair_fm(4,3,1),
paulson@13323
  1159
                And(pair_fm(3,2,0),
paulson@13323
  1160
                 And(Member(1,s#+6), Member(0,r#+6))))))))))))"
paulson@13323
  1161
paulson@13323
  1162
lemma composition_type [TC]:
paulson@13323
  1163
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula"
paulson@13323
  1164
by (simp add: composition_fm_def) 
paulson@13323
  1165
paulson@13323
  1166
lemma arity_composition_fm [simp]:
paulson@13323
  1167
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13323
  1168
      ==> arity(composition_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13323
  1169
by (simp add: composition_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13323
  1170
paulson@13323
  1171
lemma sats_composition_fm [simp]:
paulson@13323
  1172
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13323
  1173
    ==> sats(A, composition_fm(x,y,z), env) <-> 
paulson@13323
  1174
        composition(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13323
  1175
by (simp add: composition_fm_def composition_def)
paulson@13323
  1176
paulson@13323
  1177
lemma composition_iff_sats:
paulson@13323
  1178
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13323
  1179
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13323
  1180
       ==> composition(**A, x, y, z) <-> sats(A, composition_fm(i,j,k), env)"
paulson@13323
  1181
by simp
paulson@13323
  1182
paulson@13323
  1183
theorem composition_reflection:
paulson@13323
  1184
     "REFLECTS[\<lambda>x. composition(L,f(x),g(x),h(x)), 
paulson@13323
  1185
               \<lambda>i x. composition(**Lset(i),f(x),g(x),h(x))]"
paulson@13323
  1186
apply (simp only: composition_def setclass_simps)
paulson@13323
  1187
apply (intro FOL_reflections pair_reflection)  
paulson@13314
  1188
done
paulson@13314
  1189
paulson@13309
  1190
paulson@13339
  1191
subsubsection{*Injections, Internalized*}
paulson@13309
  1192
paulson@13309
  1193
(* "injection(M,A,B,f) == 
paulson@13309
  1194
	typed_function(M,A,B,f) &
paulson@13309
  1195
        (\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M]. 
paulson@13309
  1196
          pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>f --> x=x')" *)
paulson@13309
  1197
constdefs injection_fm :: "[i,i,i]=>i"
paulson@13309
  1198
 "injection_fm(A,B,f) == 
paulson@13309
  1199
    And(typed_function_fm(A,B,f),
paulson@13309
  1200
       Forall(Forall(Forall(Forall(Forall(
paulson@13309
  1201
         Implies(pair_fm(4,2,1),
paulson@13309
  1202
                 Implies(pair_fm(3,2,0),
paulson@13309
  1203
                         Implies(Member(1,f#+5),
paulson@13309
  1204
                                 Implies(Member(0,f#+5), Equal(4,3)))))))))))"
paulson@13309
  1205
paulson@13309
  1206
paulson@13309
  1207
lemma injection_type [TC]:
paulson@13309
  1208
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
paulson@13309
  1209
by (simp add: injection_fm_def) 
paulson@13309
  1210
paulson@13309
  1211
lemma arity_injection_fm [simp]:
paulson@13309
  1212
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13309
  1213
      ==> arity(injection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13309
  1214
by (simp add: injection_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
  1215
paulson@13309
  1216
lemma sats_injection_fm [simp]:
paulson@13309
  1217
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13309
  1218
    ==> sats(A, injection_fm(x,y,z), env) <-> 
paulson@13309
  1219
        injection(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1220
by (simp add: injection_fm_def injection_def)
paulson@13309
  1221
paulson@13309
  1222
lemma injection_iff_sats:
paulson@13309
  1223
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13309
  1224
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1225
   ==> injection(**A, x, y, z) <-> sats(A, injection_fm(i,j,k), env)"
paulson@13309
  1226
by simp
paulson@13309
  1227
paulson@13314
  1228
theorem injection_reflection:
paulson@13314
  1229
     "REFLECTS[\<lambda>x. injection(L,f(x),g(x),h(x)), 
paulson@13314
  1230
               \<lambda>i x. injection(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1231
apply (simp only: injection_def setclass_simps)
paulson@13323
  1232
apply (intro FOL_reflections function_reflections typed_function_reflection)  
paulson@13314
  1233
done
paulson@13309
  1234
paulson@13309
  1235
paulson@13339
  1236
subsubsection{*Surjections, Internalized*}
paulson@13309
  1237
paulson@13309
  1238
(*  surjection :: "[i=>o,i,i,i] => o"
paulson@13309
  1239
    "surjection(M,A,B,f) == 
paulson@13309
  1240
        typed_function(M,A,B,f) &
paulson@13309
  1241
        (\<forall>y[M]. y\<in>B --> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" *)
paulson@13309
  1242
constdefs surjection_fm :: "[i,i,i]=>i"
paulson@13309
  1243
 "surjection_fm(A,B,f) == 
paulson@13309
  1244
    And(typed_function_fm(A,B,f),
paulson@13309
  1245
       Forall(Implies(Member(0,succ(B)),
paulson@13309
  1246
                      Exists(And(Member(0,succ(succ(A))),
paulson@13309
  1247
                                 fun_apply_fm(succ(succ(f)),0,1))))))"
paulson@13309
  1248
paulson@13309
  1249
lemma surjection_type [TC]:
paulson@13309
  1250
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula"
paulson@13309
  1251
by (simp add: surjection_fm_def) 
paulson@13309
  1252
paulson@13309
  1253
lemma arity_surjection_fm [simp]:
paulson@13309
  1254
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13309
  1255
      ==> arity(surjection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13309
  1256
by (simp add: surjection_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
  1257
paulson@13309
  1258
lemma sats_surjection_fm [simp]:
paulson@13309
  1259
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13309
  1260
    ==> sats(A, surjection_fm(x,y,z), env) <-> 
paulson@13309
  1261
        surjection(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1262
by (simp add: surjection_fm_def surjection_def)
paulson@13309
  1263
paulson@13309
  1264
lemma surjection_iff_sats:
paulson@13309
  1265
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13309
  1266
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1267
   ==> surjection(**A, x, y, z) <-> sats(A, surjection_fm(i,j,k), env)"
paulson@13309
  1268
by simp
paulson@13309
  1269
paulson@13314
  1270
theorem surjection_reflection:
paulson@13314
  1271
     "REFLECTS[\<lambda>x. surjection(L,f(x),g(x),h(x)), 
paulson@13314
  1272
               \<lambda>i x. surjection(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1273
apply (simp only: surjection_def setclass_simps)
paulson@13323
  1274
apply (intro FOL_reflections function_reflections typed_function_reflection)  
paulson@13314
  1275
done
paulson@13309
  1276
paulson@13309
  1277
paulson@13309
  1278
paulson@13339
  1279
subsubsection{*Bijections, Internalized*}
paulson@13309
  1280
paulson@13309
  1281
(*   bijection :: "[i=>o,i,i,i] => o"
paulson@13309
  1282
    "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *)
paulson@13309
  1283
constdefs bijection_fm :: "[i,i,i]=>i"
paulson@13309
  1284
 "bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))"
paulson@13309
  1285
paulson@13309
  1286
lemma bijection_type [TC]:
paulson@13309
  1287
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula"
paulson@13309
  1288
by (simp add: bijection_fm_def) 
paulson@13309
  1289
paulson@13309
  1290
lemma arity_bijection_fm [simp]:
paulson@13309
  1291
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13309
  1292
      ==> arity(bijection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13309
  1293
by (simp add: bijection_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
  1294
paulson@13309
  1295
lemma sats_bijection_fm [simp]:
paulson@13309
  1296
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13309
  1297
    ==> sats(A, bijection_fm(x,y,z), env) <-> 
paulson@13309
  1298
        bijection(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1299
by (simp add: bijection_fm_def bijection_def)
paulson@13309
  1300
paulson@13309
  1301
lemma bijection_iff_sats:
paulson@13309
  1302
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13309
  1303
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1304
   ==> bijection(**A, x, y, z) <-> sats(A, bijection_fm(i,j,k), env)"
paulson@13309
  1305
by simp
paulson@13309
  1306
paulson@13314
  1307
theorem bijection_reflection:
paulson@13314
  1308
     "REFLECTS[\<lambda>x. bijection(L,f(x),g(x),h(x)), 
paulson@13314
  1309
               \<lambda>i x. bijection(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1310
apply (simp only: bijection_def setclass_simps)
paulson@13314
  1311
apply (intro And_reflection injection_reflection surjection_reflection)  
paulson@13314
  1312
done
paulson@13309
  1313
paulson@13309
  1314
paulson@13348
  1315
subsubsection{*Restriction of a Relation, Internalized*}
paulson@13348
  1316
paulson@13348
  1317
paulson@13348
  1318
(* "restriction(M,r,A,z) == 
paulson@13348
  1319
	\<forall>x[M]. x \<in> z <-> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" *)
paulson@13348
  1320
constdefs restriction_fm :: "[i,i,i]=>i"
paulson@13348
  1321
    "restriction_fm(r,A,z) == 
paulson@13348
  1322
       Forall(Iff(Member(0,succ(z)),
paulson@13348
  1323
                  And(Member(0,succ(r)),
paulson@13348
  1324
                      Exists(And(Member(0,succ(succ(A))),
paulson@13348
  1325
                                 Exists(pair_fm(1,0,2)))))))"
paulson@13348
  1326
paulson@13348
  1327
lemma restriction_type [TC]:
paulson@13348
  1328
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula"
paulson@13348
  1329
by (simp add: restriction_fm_def) 
paulson@13348
  1330
paulson@13348
  1331
lemma arity_restriction_fm [simp]:
paulson@13348
  1332
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13348
  1333
      ==> arity(restriction_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13348
  1334
by (simp add: restriction_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13348
  1335
paulson@13348
  1336
lemma sats_restriction_fm [simp]:
paulson@13348
  1337
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13348
  1338
    ==> sats(A, restriction_fm(x,y,z), env) <-> 
paulson@13348
  1339
        restriction(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13348
  1340
by (simp add: restriction_fm_def restriction_def)
paulson@13348
  1341
paulson@13348
  1342
lemma restriction_iff_sats:
paulson@13348
  1343
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13348
  1344
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13348
  1345
       ==> restriction(**A, x, y, z) <-> sats(A, restriction_fm(i,j,k), env)"
paulson@13348
  1346
by simp
paulson@13348
  1347
paulson@13348
  1348
theorem restriction_reflection:
paulson@13348
  1349
     "REFLECTS[\<lambda>x. restriction(L,f(x),g(x),h(x)), 
paulson@13348
  1350
               \<lambda>i x. restriction(**Lset(i),f(x),g(x),h(x))]"
paulson@13348
  1351
apply (simp only: restriction_def setclass_simps)
paulson@13348
  1352
apply (intro FOL_reflections pair_reflection)  
paulson@13348
  1353
done
paulson@13348
  1354
paulson@13339
  1355
subsubsection{*Order-Isomorphisms, Internalized*}
paulson@13309
  1356
paulson@13309
  1357
(*  order_isomorphism :: "[i=>o,i,i,i,i,i] => o"
paulson@13309
  1358
   "order_isomorphism(M,A,r,B,s,f) == 
paulson@13309
  1359
        bijection(M,A,B,f) & 
paulson@13309
  1360
        (\<forall>x[M]. x\<in>A --> (\<forall>y[M]. y\<in>A -->
paulson@13309
  1361
          (\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>q[M].
paulson@13309
  1362
            pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) --> 
paulson@13309
  1363
            pair(M,fx,fy,q) --> (p\<in>r <-> q\<in>s))))"
paulson@13309
  1364
  *)
paulson@13309
  1365
paulson@13309
  1366
constdefs order_isomorphism_fm :: "[i,i,i,i,i]=>i"
paulson@13309
  1367
 "order_isomorphism_fm(A,r,B,s,f) == 
paulson@13309
  1368
   And(bijection_fm(A,B,f), 
paulson@13309
  1369
     Forall(Implies(Member(0,succ(A)),
paulson@13309
  1370
       Forall(Implies(Member(0,succ(succ(A))),
paulson@13309
  1371
         Forall(Forall(Forall(Forall(
paulson@13309
  1372
           Implies(pair_fm(5,4,3),
paulson@13309
  1373
             Implies(fun_apply_fm(f#+6,5,2),
paulson@13309
  1374
               Implies(fun_apply_fm(f#+6,4,1),
paulson@13309
  1375
                 Implies(pair_fm(2,1,0), 
paulson@13309
  1376
                   Iff(Member(3,r#+6), Member(0,s#+6)))))))))))))))"
paulson@13309
  1377
paulson@13309
  1378
lemma order_isomorphism_type [TC]:
paulson@13309
  1379
     "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]  
paulson@13309
  1380
      ==> order_isomorphism_fm(A,r,B,s,f) \<in> formula"
paulson@13309
  1381
by (simp add: order_isomorphism_fm_def) 
paulson@13309
  1382
paulson@13309
  1383
lemma arity_order_isomorphism_fm [simp]:
paulson@13309
  1384
     "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |] 
paulson@13309
  1385
      ==> arity(order_isomorphism_fm(A,r,B,s,f)) = 
paulson@13309
  1386
          succ(A) \<union> succ(r) \<union> succ(B) \<union> succ(s) \<union> succ(f)" 
paulson@13309
  1387
by (simp add: order_isomorphism_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13309
  1388
paulson@13309
  1389
lemma sats_order_isomorphism_fm [simp]:
paulson@13309
  1390
   "[| U \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat; env \<in> list(A)|]
paulson@13309
  1391
    ==> sats(A, order_isomorphism_fm(U,r,B,s,f), env) <-> 
paulson@13309
  1392
        order_isomorphism(**A, nth(U,env), nth(r,env), nth(B,env), 
paulson@13309
  1393
                               nth(s,env), nth(f,env))"
paulson@13309
  1394
by (simp add: order_isomorphism_fm_def order_isomorphism_def)
paulson@13309
  1395
paulson@13309
  1396
lemma order_isomorphism_iff_sats:
paulson@13309
  1397
  "[| nth(i,env) = U; nth(j,env) = r; nth(k,env) = B; nth(j',env) = s; 
paulson@13309
  1398
      nth(k',env) = f; 
paulson@13309
  1399
      i \<in> nat; j \<in> nat; k \<in> nat; j' \<in> nat; k' \<in> nat; env \<in> list(A)|]
paulson@13309
  1400
   ==> order_isomorphism(**A,U,r,B,s,f) <-> 
paulson@13309
  1401
       sats(A, order_isomorphism_fm(i,j,k,j',k'), env)" 
paulson@13309
  1402
by simp
paulson@13309
  1403
paulson@13314
  1404
theorem order_isomorphism_reflection:
paulson@13314
  1405
     "REFLECTS[\<lambda>x. order_isomorphism(L,f(x),g(x),h(x),g'(x),h'(x)), 
paulson@13314
  1406
               \<lambda>i x. order_isomorphism(**Lset(i),f(x),g(x),h(x),g'(x),h'(x))]"
paulson@13314
  1407
apply (simp only: order_isomorphism_def setclass_simps)
paulson@13323
  1408
apply (intro FOL_reflections function_reflections bijection_reflection)  
paulson@13323
  1409
done
paulson@13323
  1410
paulson@13339
  1411
subsubsection{*Limit Ordinals, Internalized*}
paulson@13323
  1412
paulson@13323
  1413
text{*A limit ordinal is a non-empty, successor-closed ordinal*}
paulson@13323
  1414
paulson@13323
  1415
(* "limit_ordinal(M,a) == 
paulson@13323
  1416
	ordinal(M,a) & ~ empty(M,a) & 
paulson@13323
  1417
        (\<forall>x[M]. x\<in>a --> (\<exists>y[M]. y\<in>a & successor(M,x,y)))" *)
paulson@13323
  1418
paulson@13323
  1419
constdefs limit_ordinal_fm :: "i=>i"
paulson@13323
  1420
    "limit_ordinal_fm(x) == 
paulson@13323
  1421
        And(ordinal_fm(x),
paulson@13323
  1422
            And(Neg(empty_fm(x)),
paulson@13323
  1423
	        Forall(Implies(Member(0,succ(x)),
paulson@13323
  1424
                               Exists(And(Member(0,succ(succ(x))),
paulson@13323
  1425
                                          succ_fm(1,0)))))))"
paulson@13323
  1426
paulson@13323
  1427
lemma limit_ordinal_type [TC]:
paulson@13323
  1428
     "x \<in> nat ==> limit_ordinal_fm(x) \<in> formula"
paulson@13323
  1429
by (simp add: limit_ordinal_fm_def) 
paulson@13323
  1430
paulson@13323
  1431
lemma arity_limit_ordinal_fm [simp]:
paulson@13323
  1432
     "x \<in> nat ==> arity(limit_ordinal_fm(x)) = succ(x)"
paulson@13323
  1433
by (simp add: limit_ordinal_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13323
  1434
paulson@13323
  1435
lemma sats_limit_ordinal_fm [simp]:
paulson@13323
  1436
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13323
  1437
    ==> sats(A, limit_ordinal_fm(x), env) <-> limit_ordinal(**A, nth(x,env))"
paulson@13323
  1438
by (simp add: limit_ordinal_fm_def limit_ordinal_def sats_ordinal_fm')
paulson@13323
  1439
paulson@13323
  1440
lemma limit_ordinal_iff_sats:
paulson@13323
  1441
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13323
  1442
          i \<in> nat; env \<in> list(A)|]
paulson@13323
  1443
       ==> limit_ordinal(**A, x) <-> sats(A, limit_ordinal_fm(i), env)"
paulson@13323
  1444
by simp
paulson@13323
  1445
paulson@13323
  1446
theorem limit_ordinal_reflection:
paulson@13323
  1447
     "REFLECTS[\<lambda>x. limit_ordinal(L,f(x)), 
paulson@13323
  1448
               \<lambda>i x. limit_ordinal(**Lset(i),f(x))]"
paulson@13323
  1449
apply (simp only: limit_ordinal_def setclass_simps)
paulson@13323
  1450
apply (intro FOL_reflections ordinal_reflection 
paulson@13323
  1451
             empty_reflection successor_reflection)  
paulson@13314
  1452
done
paulson@13309
  1453
paulson@13323
  1454
subsubsection{*Omega: The Set of Natural Numbers*}
paulson@13323
  1455
paulson@13323
  1456
(* omega(M,a) == limit_ordinal(M,a) & (\<forall>x[M]. x\<in>a --> ~ limit_ordinal(M,x)) *)
paulson@13323
  1457
constdefs omega_fm :: "i=>i"
paulson@13323
  1458
    "omega_fm(x) == 
paulson@13323
  1459
       And(limit_ordinal_fm(x),
paulson@13323
  1460
           Forall(Implies(Member(0,succ(x)),
paulson@13323
  1461
                          Neg(limit_ordinal_fm(0)))))"
paulson@13323
  1462
paulson@13323
  1463
lemma omega_type [TC]:
paulson@13323
  1464
     "x \<in> nat ==> omega_fm(x) \<in> formula"
paulson@13323
  1465
by (simp add: omega_fm_def) 
paulson@13323
  1466
paulson@13323
  1467
lemma arity_omega_fm [simp]:
paulson@13323
  1468
     "x \<in> nat ==> arity(omega_fm(x)) = succ(x)"
paulson@13323
  1469
by (simp add: omega_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13323
  1470
paulson@13323
  1471
lemma sats_omega_fm [simp]:
paulson@13323
  1472
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13323
  1473
    ==> sats(A, omega_fm(x), env) <-> omega(**A, nth(x,env))"
paulson@13323
  1474
by (simp add: omega_fm_def omega_def)
paulson@13316
  1475
paulson@13323
  1476
lemma omega_iff_sats:
paulson@13323
  1477
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13323
  1478
          i \<in> nat; env \<in> list(A)|]
paulson@13323
  1479
       ==> omega(**A, x) <-> sats(A, omega_fm(i), env)"
paulson@13323
  1480
by simp
paulson@13323
  1481
paulson@13323
  1482
theorem omega_reflection:
paulson@13323
  1483
     "REFLECTS[\<lambda>x. omega(L,f(x)), 
paulson@13323
  1484
               \<lambda>i x. omega(**Lset(i),f(x))]"
paulson@13323
  1485
apply (simp only: omega_def setclass_simps)
paulson@13323
  1486
apply (intro FOL_reflections limit_ordinal_reflection)  
paulson@13323
  1487
done
paulson@13323
  1488
paulson@13323
  1489
paulson@13323
  1490
lemmas fun_plus_reflections =
paulson@13323
  1491
        typed_function_reflection composition_reflection
paulson@13323
  1492
        injection_reflection surjection_reflection
paulson@13348
  1493
        bijection_reflection restriction_reflection
paulson@13348
  1494
        order_isomorphism_reflection
paulson@13323
  1495
        ordinal_reflection limit_ordinal_reflection omega_reflection
paulson@13323
  1496
paulson@13323
  1497
lemmas fun_plus_iff_sats = 
paulson@13323
  1498
	typed_function_iff_sats composition_iff_sats
paulson@13348
  1499
        injection_iff_sats surjection_iff_sats 
paulson@13348
  1500
        bijection_iff_sats restriction_iff_sats 
paulson@13316
  1501
        order_isomorphism_iff_sats
paulson@13323
  1502
        ordinal_iff_sats limit_ordinal_iff_sats omega_iff_sats
paulson@13316
  1503
paulson@13223
  1504
end