src/ZF/Constructible/L_axioms.thy
author paulson
Wed Jul 17 15:48:54 2002 +0200 (2002-07-17)
changeset 13385 31df66ca0780
parent 13363 c26eeb000470
child 13418 7c0ba9dba978
permissions -rw-r--r--
Expressing Lset and L without using length and arity; simplifies Separation
proofs
     1 header {*The ZF Axioms (Except Separation) in L*}
     2 
     3 theory L_axioms = Formula + Relative + Reflection + MetaExists:
     4 
     5 text {* The class L satisfies the premises of locale @{text M_triv_axioms} *}
     6 
     7 lemma transL: "[| y\<in>x; L(x) |] ==> L(y)"
     8 apply (insert Transset_Lset) 
     9 apply (simp add: Transset_def L_def, blast) 
    10 done
    11 
    12 lemma nonempty: "L(0)"
    13 apply (simp add: L_def) 
    14 apply (blast intro: zero_in_Lset) 
    15 done
    16 
    17 lemma upair_ax: "upair_ax(L)"
    18 apply (simp add: upair_ax_def upair_def, clarify)
    19 apply (rule_tac x="{x,y}" in rexI)  
    20 apply (simp_all add: doubleton_in_L) 
    21 done
    22 
    23 lemma Union_ax: "Union_ax(L)"
    24 apply (simp add: Union_ax_def big_union_def, clarify)
    25 apply (rule_tac x="Union(x)" in rexI)  
    26 apply (simp_all add: Union_in_L, auto) 
    27 apply (blast intro: transL) 
    28 done
    29 
    30 lemma power_ax: "power_ax(L)"
    31 apply (simp add: power_ax_def powerset_def Relative.subset_def, clarify)
    32 apply (rule_tac x="{y \<in> Pow(x). L(y)}" in rexI)  
    33 apply (simp_all add: LPow_in_L, auto)
    34 apply (blast intro: transL) 
    35 done
    36 
    37 subsubsection{*For L to satisfy Replacement *}
    38 
    39 (*Can't move these to Formula unless the definition of univalent is moved
    40 there too!*)
    41 
    42 lemma LReplace_in_Lset:
    43      "[|X \<in> Lset(i); univalent(L,X,Q); Ord(i)|] 
    44       ==> \<exists>j. Ord(j) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Lset(j)"
    45 apply (rule_tac x="\<Union>y \<in> Replace(X, %x y. Q(x,y) & L(y)). succ(lrank(y))" 
    46        in exI)
    47 apply simp
    48 apply clarify 
    49 apply (rule_tac a=x in UN_I)  
    50  apply (simp_all add: Replace_iff univalent_def) 
    51 apply (blast dest: transL L_I) 
    52 done
    53 
    54 lemma LReplace_in_L: 
    55      "[|L(X); univalent(L,X,Q)|] 
    56       ==> \<exists>Y. L(Y) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Y"
    57 apply (drule L_D, clarify) 
    58 apply (drule LReplace_in_Lset, assumption+)
    59 apply (blast intro: L_I Lset_in_Lset_succ)
    60 done
    61 
    62 lemma replacement: "replacement(L,P)"
    63 apply (simp add: replacement_def, clarify)
    64 apply (frule LReplace_in_L, assumption+, clarify) 
    65 apply (rule_tac x=Y in rexI)   
    66 apply (simp_all add: Replace_iff univalent_def, blast) 
    67 done
    68 
    69 subsection{*Instantiating the locale @{text M_triv_axioms}*}
    70 text{*No instances of Separation yet.*}
    71 
    72 lemma Lset_mono_le: "mono_le_subset(Lset)"
    73 by (simp add: mono_le_subset_def le_imp_subset Lset_mono) 
    74 
    75 lemma Lset_cont: "cont_Ord(Lset)"
    76 by (simp add: cont_Ord_def Limit_Lset_eq OUnion_def Limit_is_Ord) 
    77 
    78 lemmas Pair_in_Lset = Formula.Pair_in_LLimit;
    79 
    80 lemmas L_nat = Ord_in_L [OF Ord_nat];
    81 
    82 ML
    83 {*
    84 val transL = thm "transL";
    85 val nonempty = thm "nonempty";
    86 val upair_ax = thm "upair_ax";
    87 val Union_ax = thm "Union_ax";
    88 val power_ax = thm "power_ax";
    89 val replacement = thm "replacement";
    90 val L_nat = thm "L_nat";
    91 
    92 fun kill_flex_triv_prems st = Seq.hd ((REPEAT_FIRST assume_tac) st);
    93 
    94 fun triv_axioms_L th =
    95     kill_flex_triv_prems 
    96        ([transL, nonempty, upair_ax, Union_ax, power_ax, replacement, L_nat] 
    97         MRS (inst "M" "L" th));
    98 
    99 bind_thm ("ball_abs", triv_axioms_L (thm "M_triv_axioms.ball_abs"));
   100 bind_thm ("rall_abs", triv_axioms_L (thm "M_triv_axioms.rall_abs"));
   101 bind_thm ("bex_abs", triv_axioms_L (thm "M_triv_axioms.bex_abs"));
   102 bind_thm ("rex_abs", triv_axioms_L (thm "M_triv_axioms.rex_abs"));
   103 bind_thm ("ball_iff_equiv", triv_axioms_L (thm "M_triv_axioms.ball_iff_equiv"));
   104 bind_thm ("M_equalityI", triv_axioms_L (thm "M_triv_axioms.M_equalityI"));
   105 bind_thm ("empty_abs", triv_axioms_L (thm "M_triv_axioms.empty_abs"));
   106 bind_thm ("subset_abs", triv_axioms_L (thm "M_triv_axioms.subset_abs"));
   107 bind_thm ("upair_abs", triv_axioms_L (thm "M_triv_axioms.upair_abs"));
   108 bind_thm ("upair_in_M_iff", triv_axioms_L (thm "M_triv_axioms.upair_in_M_iff"));
   109 bind_thm ("singleton_in_M_iff", triv_axioms_L (thm "M_triv_axioms.singleton_in_M_iff"));
   110 bind_thm ("pair_abs", triv_axioms_L (thm "M_triv_axioms.pair_abs"));
   111 bind_thm ("pair_in_M_iff", triv_axioms_L (thm "M_triv_axioms.pair_in_M_iff"));
   112 bind_thm ("pair_components_in_M", triv_axioms_L (thm "M_triv_axioms.pair_components_in_M"));
   113 bind_thm ("cartprod_abs", triv_axioms_L (thm "M_triv_axioms.cartprod_abs"));
   114 bind_thm ("union_abs", triv_axioms_L (thm "M_triv_axioms.union_abs"));
   115 bind_thm ("inter_abs", triv_axioms_L (thm "M_triv_axioms.inter_abs"));
   116 bind_thm ("setdiff_abs", triv_axioms_L (thm "M_triv_axioms.setdiff_abs"));
   117 bind_thm ("Union_abs", triv_axioms_L (thm "M_triv_axioms.Union_abs"));
   118 bind_thm ("Union_closed", triv_axioms_L (thm "M_triv_axioms.Union_closed"));
   119 bind_thm ("Un_closed", triv_axioms_L (thm "M_triv_axioms.Un_closed"));
   120 bind_thm ("cons_closed", triv_axioms_L (thm "M_triv_axioms.cons_closed"));
   121 bind_thm ("successor_abs", triv_axioms_L (thm "M_triv_axioms.successor_abs"));
   122 bind_thm ("succ_in_M_iff", triv_axioms_L (thm "M_triv_axioms.succ_in_M_iff"));
   123 bind_thm ("separation_closed", triv_axioms_L (thm "M_triv_axioms.separation_closed"));
   124 bind_thm ("strong_replacementI", triv_axioms_L (thm "M_triv_axioms.strong_replacementI"));
   125 bind_thm ("strong_replacement_closed", triv_axioms_L (thm "M_triv_axioms.strong_replacement_closed"));
   126 bind_thm ("RepFun_closed", triv_axioms_L (thm "M_triv_axioms.RepFun_closed"));
   127 bind_thm ("lam_closed", triv_axioms_L (thm "M_triv_axioms.lam_closed"));
   128 bind_thm ("image_abs", triv_axioms_L (thm "M_triv_axioms.image_abs"));
   129 bind_thm ("powerset_Pow", triv_axioms_L (thm "M_triv_axioms.powerset_Pow"));
   130 bind_thm ("powerset_imp_subset_Pow", triv_axioms_L (thm "M_triv_axioms.powerset_imp_subset_Pow"));
   131 bind_thm ("nat_into_M", triv_axioms_L (thm "M_triv_axioms.nat_into_M"));
   132 bind_thm ("nat_case_closed", triv_axioms_L (thm "M_triv_axioms.nat_case_closed"));
   133 bind_thm ("Inl_in_M_iff", triv_axioms_L (thm "M_triv_axioms.Inl_in_M_iff"));
   134 bind_thm ("Inr_in_M_iff", triv_axioms_L (thm "M_triv_axioms.Inr_in_M_iff"));
   135 bind_thm ("lt_closed", triv_axioms_L (thm "M_triv_axioms.lt_closed"));
   136 bind_thm ("transitive_set_abs", triv_axioms_L (thm "M_triv_axioms.transitive_set_abs"));
   137 bind_thm ("ordinal_abs", triv_axioms_L (thm "M_triv_axioms.ordinal_abs"));
   138 bind_thm ("limit_ordinal_abs", triv_axioms_L (thm "M_triv_axioms.limit_ordinal_abs"));
   139 bind_thm ("successor_ordinal_abs", triv_axioms_L (thm "M_triv_axioms.successor_ordinal_abs"));
   140 bind_thm ("finite_ordinal_abs", triv_axioms_L (thm "M_triv_axioms.finite_ordinal_abs"));
   141 bind_thm ("omega_abs", triv_axioms_L (thm "M_triv_axioms.omega_abs"));
   142 bind_thm ("number1_abs", triv_axioms_L (thm "M_triv_axioms.number1_abs"));
   143 bind_thm ("number1_abs", triv_axioms_L (thm "M_triv_axioms.number1_abs"));
   144 bind_thm ("number3_abs", triv_axioms_L (thm "M_triv_axioms.number3_abs"));
   145 *}
   146 
   147 declare ball_abs [simp] 
   148 declare rall_abs [simp] 
   149 declare bex_abs [simp] 
   150 declare rex_abs [simp] 
   151 declare empty_abs [simp] 
   152 declare subset_abs [simp] 
   153 declare upair_abs [simp] 
   154 declare upair_in_M_iff [iff]
   155 declare singleton_in_M_iff [iff]
   156 declare pair_abs [simp] 
   157 declare pair_in_M_iff [iff]
   158 declare cartprod_abs [simp] 
   159 declare union_abs [simp] 
   160 declare inter_abs [simp] 
   161 declare setdiff_abs [simp] 
   162 declare Union_abs [simp] 
   163 declare Union_closed [intro,simp]
   164 declare Un_closed [intro,simp]
   165 declare cons_closed [intro,simp]
   166 declare successor_abs [simp] 
   167 declare succ_in_M_iff [iff]
   168 declare separation_closed [intro,simp]
   169 declare strong_replacementI
   170 declare strong_replacement_closed [intro,simp]
   171 declare RepFun_closed [intro,simp]
   172 declare lam_closed [intro,simp]
   173 declare image_abs [simp] 
   174 declare nat_into_M [intro]
   175 declare Inl_in_M_iff [iff]
   176 declare Inr_in_M_iff [iff]
   177 declare transitive_set_abs [simp] 
   178 declare ordinal_abs [simp] 
   179 declare limit_ordinal_abs [simp] 
   180 declare successor_ordinal_abs [simp] 
   181 declare finite_ordinal_abs [simp] 
   182 declare omega_abs [simp] 
   183 declare number1_abs [simp] 
   184 declare number1_abs [simp] 
   185 declare number3_abs [simp]
   186 
   187 
   188 subsection{*Instantiation of the locale @{text reflection}*}
   189 
   190 text{*instances of locale constants*}
   191 constdefs
   192   L_F0 :: "[i=>o,i] => i"
   193     "L_F0(P,y) == \<mu>b. (\<exists>z. L(z) \<and> P(<y,z>)) --> (\<exists>z\<in>Lset(b). P(<y,z>))"
   194 
   195   L_FF :: "[i=>o,i] => i"
   196     "L_FF(P)   == \<lambda>a. \<Union>y\<in>Lset(a). L_F0(P,y)"
   197 
   198   L_ClEx :: "[i=>o,i] => o"
   199     "L_ClEx(P) == \<lambda>a. Limit(a) \<and> normalize(L_FF(P),a) = a"
   200 
   201 
   202 text{*We must use the meta-existential quantifier; otherwise the reflection
   203       terms become enormous!*} 
   204 constdefs
   205   L_Reflects :: "[i=>o,[i,i]=>o] => prop"      ("(3REFLECTS/ [_,/ _])")
   206     "REFLECTS[P,Q] == (??Cl. Closed_Unbounded(Cl) &
   207                            (\<forall>a. Cl(a) --> (\<forall>x \<in> Lset(a). P(x) <-> Q(a,x))))"
   208 
   209 
   210 theorem Triv_reflection:
   211      "REFLECTS[P, \<lambda>a x. P(x)]"
   212 apply (simp add: L_Reflects_def) 
   213 apply (rule meta_exI) 
   214 apply (rule Closed_Unbounded_Ord) 
   215 done
   216 
   217 theorem Not_reflection:
   218      "REFLECTS[P,Q] ==> REFLECTS[\<lambda>x. ~P(x), \<lambda>a x. ~Q(a,x)]"
   219 apply (unfold L_Reflects_def) 
   220 apply (erule meta_exE) 
   221 apply (rule_tac x=Cl in meta_exI, simp) 
   222 done
   223 
   224 theorem And_reflection:
   225      "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] 
   226       ==> REFLECTS[\<lambda>x. P(x) \<and> P'(x), \<lambda>a x. Q(a,x) \<and> Q'(a,x)]"
   227 apply (unfold L_Reflects_def) 
   228 apply (elim meta_exE) 
   229 apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) 
   230 apply (simp add: Closed_Unbounded_Int, blast) 
   231 done
   232 
   233 theorem Or_reflection:
   234      "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] 
   235       ==> REFLECTS[\<lambda>x. P(x) \<or> P'(x), \<lambda>a x. Q(a,x) \<or> Q'(a,x)]"
   236 apply (unfold L_Reflects_def) 
   237 apply (elim meta_exE) 
   238 apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) 
   239 apply (simp add: Closed_Unbounded_Int, blast) 
   240 done
   241 
   242 theorem Imp_reflection:
   243      "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] 
   244       ==> REFLECTS[\<lambda>x. P(x) --> P'(x), \<lambda>a x. Q(a,x) --> Q'(a,x)]"
   245 apply (unfold L_Reflects_def) 
   246 apply (elim meta_exE) 
   247 apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) 
   248 apply (simp add: Closed_Unbounded_Int, blast) 
   249 done
   250 
   251 theorem Iff_reflection:
   252      "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |] 
   253       ==> REFLECTS[\<lambda>x. P(x) <-> P'(x), \<lambda>a x. Q(a,x) <-> Q'(a,x)]"
   254 apply (unfold L_Reflects_def) 
   255 apply (elim meta_exE) 
   256 apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI) 
   257 apply (simp add: Closed_Unbounded_Int, blast) 
   258 done
   259 
   260 
   261 theorem Ex_reflection:
   262      "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
   263       ==> REFLECTS[\<lambda>x. \<exists>z. L(z) \<and> P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
   264 apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def) 
   265 apply (elim meta_exE) 
   266 apply (rule meta_exI)
   267 apply (rule reflection.Ex_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset],
   268        assumption+)
   269 done
   270 
   271 theorem All_reflection:
   272      "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
   273       ==> REFLECTS[\<lambda>x. \<forall>z. L(z) --> P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]" 
   274 apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def) 
   275 apply (elim meta_exE) 
   276 apply (rule meta_exI)
   277 apply (rule reflection.All_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset],
   278        assumption+)
   279 done
   280 
   281 theorem Rex_reflection:
   282      "REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
   283       ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
   284 apply (unfold rex_def) 
   285 apply (intro And_reflection Ex_reflection, assumption)
   286 done
   287 
   288 theorem Rall_reflection:
   289      "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
   290       ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]" 
   291 apply (unfold rall_def) 
   292 apply (intro Imp_reflection All_reflection, assumption)
   293 done
   294 
   295 lemmas FOL_reflections = 
   296         Triv_reflection Not_reflection And_reflection Or_reflection
   297         Imp_reflection Iff_reflection Ex_reflection All_reflection
   298         Rex_reflection Rall_reflection
   299 
   300 lemma ReflectsD:
   301      "[|REFLECTS[P,Q]; Ord(i)|] 
   302       ==> \<exists>j. i<j & (\<forall>x \<in> Lset(j). P(x) <-> Q(j,x))"
   303 apply (unfold L_Reflects_def Closed_Unbounded_def) 
   304 apply (elim meta_exE, clarify) 
   305 apply (blast dest!: UnboundedD) 
   306 done
   307 
   308 lemma ReflectsE:
   309      "[| REFLECTS[P,Q]; Ord(i);
   310          !!j. [|i<j;  \<forall>x \<in> Lset(j). P(x) <-> Q(j,x)|] ==> R |]
   311       ==> R"
   312 apply (drule ReflectsD, assumption, blast) 
   313 done
   314 
   315 lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B";
   316 by blast
   317 
   318 
   319 subsection{*Internalized Formulas for some Set-Theoretic Concepts*}
   320 
   321 lemmas setclass_simps = rall_setclass_is_ball rex_setclass_is_bex
   322 
   323 subsubsection{*Some numbers to help write de Bruijn indices*}
   324 
   325 syntax
   326     "3" :: i   ("3")
   327     "4" :: i   ("4")
   328     "5" :: i   ("5")
   329     "6" :: i   ("6")
   330     "7" :: i   ("7")
   331     "8" :: i   ("8")
   332     "9" :: i   ("9")
   333 
   334 translations
   335    "3"  == "succ(2)"
   336    "4"  == "succ(3)"
   337    "5"  == "succ(4)"
   338    "6"  == "succ(5)"
   339    "7"  == "succ(6)"
   340    "8"  == "succ(7)"
   341    "9"  == "succ(8)"
   342 
   343 
   344 subsubsection{*The Empty Set, Internalized*}
   345 
   346 constdefs empty_fm :: "i=>i"
   347     "empty_fm(x) == Forall(Neg(Member(0,succ(x))))"
   348 
   349 lemma empty_type [TC]:
   350      "x \<in> nat ==> empty_fm(x) \<in> formula"
   351 by (simp add: empty_fm_def) 
   352 
   353 lemma arity_empty_fm [simp]:
   354      "x \<in> nat ==> arity(empty_fm(x)) = succ(x)"
   355 by (simp add: empty_fm_def succ_Un_distrib [symmetric] Un_ac) 
   356 
   357 lemma sats_empty_fm [simp]:
   358    "[| x \<in> nat; env \<in> list(A)|]
   359     ==> sats(A, empty_fm(x), env) <-> empty(**A, nth(x,env))"
   360 by (simp add: empty_fm_def empty_def)
   361 
   362 lemma empty_iff_sats:
   363       "[| nth(i,env) = x; nth(j,env) = y; 
   364           i \<in> nat; env \<in> list(A)|]
   365        ==> empty(**A, x) <-> sats(A, empty_fm(i), env)"
   366 by simp
   367 
   368 theorem empty_reflection:
   369      "REFLECTS[\<lambda>x. empty(L,f(x)), 
   370                \<lambda>i x. empty(**Lset(i),f(x))]"
   371 apply (simp only: empty_def setclass_simps)
   372 apply (intro FOL_reflections)  
   373 done
   374 
   375 text{*Not used.  But maybe useful?*}
   376 lemma Transset_sats_empty_fm_eq_0:
   377    "[| n \<in> nat; env \<in> list(A); Transset(A)|]
   378     ==> sats(A, empty_fm(n), env) <-> nth(n,env) = 0"
   379 apply (simp add: empty_fm_def empty_def Transset_def, auto)
   380 apply (case_tac "n < length(env)") 
   381 apply (frule nth_type, assumption+, blast)  
   382 apply (simp_all add: not_lt_iff_le nth_eq_0) 
   383 done
   384 
   385 
   386 subsubsection{*Unordered Pairs, Internalized*}
   387 
   388 constdefs upair_fm :: "[i,i,i]=>i"
   389     "upair_fm(x,y,z) == 
   390        And(Member(x,z), 
   391            And(Member(y,z),
   392                Forall(Implies(Member(0,succ(z)), 
   393                               Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
   394 
   395 lemma upair_type [TC]:
   396      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
   397 by (simp add: upair_fm_def) 
   398 
   399 lemma arity_upair_fm [simp]:
   400      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
   401       ==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
   402 by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac) 
   403 
   404 lemma sats_upair_fm [simp]:
   405    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   406     ==> sats(A, upair_fm(x,y,z), env) <-> 
   407             upair(**A, nth(x,env), nth(y,env), nth(z,env))"
   408 by (simp add: upair_fm_def upair_def)
   409 
   410 lemma upair_iff_sats:
   411       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
   412           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   413        ==> upair(**A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
   414 by (simp add: sats_upair_fm)
   415 
   416 text{*Useful? At least it refers to "real" unordered pairs*}
   417 lemma sats_upair_fm2 [simp]:
   418    "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
   419     ==> sats(A, upair_fm(x,y,z), env) <-> 
   420         nth(z,env) = {nth(x,env), nth(y,env)}"
   421 apply (frule lt_length_in_nat, assumption)  
   422 apply (simp add: upair_fm_def Transset_def, auto) 
   423 apply (blast intro: nth_type) 
   424 done
   425 
   426 theorem upair_reflection:
   427      "REFLECTS[\<lambda>x. upair(L,f(x),g(x),h(x)), 
   428                \<lambda>i x. upair(**Lset(i),f(x),g(x),h(x))]" 
   429 apply (simp add: upair_def)
   430 apply (intro FOL_reflections)  
   431 done
   432 
   433 subsubsection{*Ordered pairs, Internalized*}
   434 
   435 constdefs pair_fm :: "[i,i,i]=>i"
   436     "pair_fm(x,y,z) == 
   437        Exists(And(upair_fm(succ(x),succ(x),0),
   438               Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
   439                          upair_fm(1,0,succ(succ(z)))))))"
   440 
   441 lemma pair_type [TC]:
   442      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
   443 by (simp add: pair_fm_def) 
   444 
   445 lemma arity_pair_fm [simp]:
   446      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
   447       ==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
   448 by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac) 
   449 
   450 lemma sats_pair_fm [simp]:
   451    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   452     ==> sats(A, pair_fm(x,y,z), env) <-> 
   453         pair(**A, nth(x,env), nth(y,env), nth(z,env))"
   454 by (simp add: pair_fm_def pair_def)
   455 
   456 lemma pair_iff_sats:
   457       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
   458           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   459        ==> pair(**A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
   460 by (simp add: sats_pair_fm)
   461 
   462 theorem pair_reflection:
   463      "REFLECTS[\<lambda>x. pair(L,f(x),g(x),h(x)), 
   464                \<lambda>i x. pair(**Lset(i),f(x),g(x),h(x))]"
   465 apply (simp only: pair_def setclass_simps)
   466 apply (intro FOL_reflections upair_reflection)  
   467 done
   468 
   469 
   470 subsubsection{*Binary Unions, Internalized*}
   471 
   472 constdefs union_fm :: "[i,i,i]=>i"
   473     "union_fm(x,y,z) == 
   474        Forall(Iff(Member(0,succ(z)),
   475                   Or(Member(0,succ(x)),Member(0,succ(y)))))"
   476 
   477 lemma union_type [TC]:
   478      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
   479 by (simp add: union_fm_def) 
   480 
   481 lemma arity_union_fm [simp]:
   482      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
   483       ==> arity(union_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
   484 by (simp add: union_fm_def succ_Un_distrib [symmetric] Un_ac) 
   485 
   486 lemma sats_union_fm [simp]:
   487    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   488     ==> sats(A, union_fm(x,y,z), env) <-> 
   489         union(**A, nth(x,env), nth(y,env), nth(z,env))"
   490 by (simp add: union_fm_def union_def)
   491 
   492 lemma union_iff_sats:
   493       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
   494           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   495        ==> union(**A, x, y, z) <-> sats(A, union_fm(i,j,k), env)"
   496 by (simp add: sats_union_fm)
   497 
   498 theorem union_reflection:
   499      "REFLECTS[\<lambda>x. union(L,f(x),g(x),h(x)), 
   500                \<lambda>i x. union(**Lset(i),f(x),g(x),h(x))]"
   501 apply (simp only: union_def setclass_simps)
   502 apply (intro FOL_reflections)  
   503 done
   504 
   505 
   506 subsubsection{*Set ``Cons,'' Internalized*}
   507 
   508 constdefs cons_fm :: "[i,i,i]=>i"
   509     "cons_fm(x,y,z) == 
   510        Exists(And(upair_fm(succ(x),succ(x),0),
   511                   union_fm(0,succ(y),succ(z))))"
   512 
   513 
   514 lemma cons_type [TC]:
   515      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
   516 by (simp add: cons_fm_def) 
   517 
   518 lemma arity_cons_fm [simp]:
   519      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
   520       ==> arity(cons_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
   521 by (simp add: cons_fm_def succ_Un_distrib [symmetric] Un_ac) 
   522 
   523 lemma sats_cons_fm [simp]:
   524    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   525     ==> sats(A, cons_fm(x,y,z), env) <-> 
   526         is_cons(**A, nth(x,env), nth(y,env), nth(z,env))"
   527 by (simp add: cons_fm_def is_cons_def)
   528 
   529 lemma cons_iff_sats:
   530       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
   531           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   532        ==> is_cons(**A, x, y, z) <-> sats(A, cons_fm(i,j,k), env)"
   533 by simp
   534 
   535 theorem cons_reflection:
   536      "REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)), 
   537                \<lambda>i x. is_cons(**Lset(i),f(x),g(x),h(x))]"
   538 apply (simp only: is_cons_def setclass_simps)
   539 apply (intro FOL_reflections upair_reflection union_reflection)  
   540 done
   541 
   542 
   543 subsubsection{*Successor Function, Internalized*}
   544 
   545 constdefs succ_fm :: "[i,i]=>i"
   546     "succ_fm(x,y) == cons_fm(x,x,y)"
   547 
   548 lemma succ_type [TC]:
   549      "[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula"
   550 by (simp add: succ_fm_def) 
   551 
   552 lemma arity_succ_fm [simp]:
   553      "[| x \<in> nat; y \<in> nat |] 
   554       ==> arity(succ_fm(x,y)) = succ(x) \<union> succ(y)"
   555 by (simp add: succ_fm_def)
   556 
   557 lemma sats_succ_fm [simp]:
   558    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   559     ==> sats(A, succ_fm(x,y), env) <-> 
   560         successor(**A, nth(x,env), nth(y,env))"
   561 by (simp add: succ_fm_def successor_def)
   562 
   563 lemma successor_iff_sats:
   564       "[| nth(i,env) = x; nth(j,env) = y; 
   565           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   566        ==> successor(**A, x, y) <-> sats(A, succ_fm(i,j), env)"
   567 by simp
   568 
   569 theorem successor_reflection:
   570      "REFLECTS[\<lambda>x. successor(L,f(x),g(x)), 
   571                \<lambda>i x. successor(**Lset(i),f(x),g(x))]"
   572 apply (simp only: successor_def setclass_simps)
   573 apply (intro cons_reflection)  
   574 done
   575 
   576 
   577 subsubsection{*The Number 1, Internalized*}
   578 
   579 (* "number1(M,a) == (\<exists>x[M]. empty(M,x) & successor(M,x,a))" *)
   580 constdefs number1_fm :: "i=>i"
   581     "number1_fm(a) == Exists(And(empty_fm(0), succ_fm(0,succ(a))))"
   582 
   583 lemma number1_type [TC]:
   584      "x \<in> nat ==> number1_fm(x) \<in> formula"
   585 by (simp add: number1_fm_def) 
   586 
   587 lemma arity_number1_fm [simp]:
   588      "x \<in> nat ==> arity(number1_fm(x)) = succ(x)"
   589 by (simp add: number1_fm_def succ_Un_distrib [symmetric] Un_ac) 
   590 
   591 lemma sats_number1_fm [simp]:
   592    "[| x \<in> nat; env \<in> list(A)|]
   593     ==> sats(A, number1_fm(x), env) <-> number1(**A, nth(x,env))"
   594 by (simp add: number1_fm_def number1_def)
   595 
   596 lemma number1_iff_sats:
   597       "[| nth(i,env) = x; nth(j,env) = y; 
   598           i \<in> nat; env \<in> list(A)|]
   599        ==> number1(**A, x) <-> sats(A, number1_fm(i), env)"
   600 by simp
   601 
   602 theorem number1_reflection:
   603      "REFLECTS[\<lambda>x. number1(L,f(x)), 
   604                \<lambda>i x. number1(**Lset(i),f(x))]"
   605 apply (simp only: number1_def setclass_simps)
   606 apply (intro FOL_reflections empty_reflection successor_reflection)
   607 done
   608 
   609 
   610 subsubsection{*Big Union, Internalized*}
   611 
   612 (*  "big_union(M,A,z) == \<forall>x[M]. x \<in> z <-> (\<exists>y[M]. y\<in>A & x \<in> y)" *)
   613 constdefs big_union_fm :: "[i,i]=>i"
   614     "big_union_fm(A,z) == 
   615        Forall(Iff(Member(0,succ(z)),
   616                   Exists(And(Member(0,succ(succ(A))), Member(1,0)))))"
   617 
   618 lemma big_union_type [TC]:
   619      "[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula"
   620 by (simp add: big_union_fm_def) 
   621 
   622 lemma arity_big_union_fm [simp]:
   623      "[| x \<in> nat; y \<in> nat |] 
   624       ==> arity(big_union_fm(x,y)) = succ(x) \<union> succ(y)"
   625 by (simp add: big_union_fm_def succ_Un_distrib [symmetric] Un_ac)
   626 
   627 lemma sats_big_union_fm [simp]:
   628    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   629     ==> sats(A, big_union_fm(x,y), env) <-> 
   630         big_union(**A, nth(x,env), nth(y,env))"
   631 by (simp add: big_union_fm_def big_union_def)
   632 
   633 lemma big_union_iff_sats:
   634       "[| nth(i,env) = x; nth(j,env) = y; 
   635           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   636        ==> big_union(**A, x, y) <-> sats(A, big_union_fm(i,j), env)"
   637 by simp
   638 
   639 theorem big_union_reflection:
   640      "REFLECTS[\<lambda>x. big_union(L,f(x),g(x)), 
   641                \<lambda>i x. big_union(**Lset(i),f(x),g(x))]"
   642 apply (simp only: big_union_def setclass_simps)
   643 apply (intro FOL_reflections)  
   644 done
   645 
   646 
   647 subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*}
   648 
   649 text{*Differs from the one in Formula by using "ordinal" rather than "Ord"*}
   650 
   651 
   652 lemma sats_subset_fm':
   653    "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
   654     ==> sats(A, subset_fm(x,y), env) <-> subset(**A, nth(x,env), nth(y,env))" 
   655 by (simp add: subset_fm_def Relative.subset_def) 
   656 
   657 theorem subset_reflection:
   658      "REFLECTS[\<lambda>x. subset(L,f(x),g(x)), 
   659                \<lambda>i x. subset(**Lset(i),f(x),g(x))]" 
   660 apply (simp only: Relative.subset_def setclass_simps)
   661 apply (intro FOL_reflections)  
   662 done
   663 
   664 lemma sats_transset_fm':
   665    "[|x \<in> nat; env \<in> list(A)|]
   666     ==> sats(A, transset_fm(x), env) <-> transitive_set(**A, nth(x,env))"
   667 by (simp add: sats_subset_fm' transset_fm_def transitive_set_def) 
   668 
   669 theorem transitive_set_reflection:
   670      "REFLECTS[\<lambda>x. transitive_set(L,f(x)),
   671                \<lambda>i x. transitive_set(**Lset(i),f(x))]"
   672 apply (simp only: transitive_set_def setclass_simps)
   673 apply (intro FOL_reflections subset_reflection)  
   674 done
   675 
   676 lemma sats_ordinal_fm':
   677    "[|x \<in> nat; env \<in> list(A)|]
   678     ==> sats(A, ordinal_fm(x), env) <-> ordinal(**A,nth(x,env))"
   679 by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def)
   680 
   681 lemma ordinal_iff_sats:
   682       "[| nth(i,env) = x;  i \<in> nat; env \<in> list(A)|]
   683        ==> ordinal(**A, x) <-> sats(A, ordinal_fm(i), env)"
   684 by (simp add: sats_ordinal_fm')
   685 
   686 theorem ordinal_reflection:
   687      "REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(**Lset(i),f(x))]"
   688 apply (simp only: ordinal_def setclass_simps)
   689 apply (intro FOL_reflections transitive_set_reflection)  
   690 done
   691 
   692 
   693 subsubsection{*Membership Relation, Internalized*}
   694 
   695 constdefs Memrel_fm :: "[i,i]=>i"
   696     "Memrel_fm(A,r) == 
   697        Forall(Iff(Member(0,succ(r)),
   698                   Exists(And(Member(0,succ(succ(A))),
   699                              Exists(And(Member(0,succ(succ(succ(A)))),
   700                                         And(Member(1,0),
   701                                             pair_fm(1,0,2))))))))"
   702 
   703 lemma Memrel_type [TC]:
   704      "[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
   705 by (simp add: Memrel_fm_def) 
   706 
   707 lemma arity_Memrel_fm [simp]:
   708      "[| x \<in> nat; y \<in> nat |] 
   709       ==> arity(Memrel_fm(x,y)) = succ(x) \<union> succ(y)"
   710 by (simp add: Memrel_fm_def succ_Un_distrib [symmetric] Un_ac) 
   711 
   712 lemma sats_Memrel_fm [simp]:
   713    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   714     ==> sats(A, Memrel_fm(x,y), env) <-> 
   715         membership(**A, nth(x,env), nth(y,env))"
   716 by (simp add: Memrel_fm_def membership_def)
   717 
   718 lemma Memrel_iff_sats:
   719       "[| nth(i,env) = x; nth(j,env) = y; 
   720           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   721        ==> membership(**A, x, y) <-> sats(A, Memrel_fm(i,j), env)"
   722 by simp
   723 
   724 theorem membership_reflection:
   725      "REFLECTS[\<lambda>x. membership(L,f(x),g(x)), 
   726                \<lambda>i x. membership(**Lset(i),f(x),g(x))]"
   727 apply (simp only: membership_def setclass_simps)
   728 apply (intro FOL_reflections pair_reflection)  
   729 done
   730 
   731 subsubsection{*Predecessor Set, Internalized*}
   732 
   733 constdefs pred_set_fm :: "[i,i,i,i]=>i"
   734     "pred_set_fm(A,x,r,B) == 
   735        Forall(Iff(Member(0,succ(B)),
   736                   Exists(And(Member(0,succ(succ(r))),
   737                              And(Member(1,succ(succ(A))),
   738                                  pair_fm(1,succ(succ(x)),0))))))"
   739 
   740 
   741 lemma pred_set_type [TC]:
   742      "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |] 
   743       ==> pred_set_fm(A,x,r,B) \<in> formula"
   744 by (simp add: pred_set_fm_def) 
   745 
   746 lemma arity_pred_set_fm [simp]:
   747    "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |] 
   748     ==> arity(pred_set_fm(A,x,r,B)) = succ(A) \<union> succ(x) \<union> succ(r) \<union> succ(B)"
   749 by (simp add: pred_set_fm_def succ_Un_distrib [symmetric] Un_ac) 
   750 
   751 lemma sats_pred_set_fm [simp]:
   752    "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|]
   753     ==> sats(A, pred_set_fm(U,x,r,B), env) <-> 
   754         pred_set(**A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))"
   755 by (simp add: pred_set_fm_def pred_set_def)
   756 
   757 lemma pred_set_iff_sats:
   758       "[| nth(i,env) = U; nth(j,env) = x; nth(k,env) = r; nth(l,env) = B; 
   759           i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|]
   760        ==> pred_set(**A,U,x,r,B) <-> sats(A, pred_set_fm(i,j,k,l), env)"
   761 by (simp add: sats_pred_set_fm)
   762 
   763 theorem pred_set_reflection:
   764      "REFLECTS[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)), 
   765                \<lambda>i x. pred_set(**Lset(i),f(x),g(x),h(x),b(x))]" 
   766 apply (simp only: pred_set_def setclass_simps)
   767 apply (intro FOL_reflections pair_reflection)  
   768 done
   769 
   770 
   771 
   772 subsubsection{*Domain of a Relation, Internalized*}
   773 
   774 (* "is_domain(M,r,z) == 
   775 	\<forall>x[M]. (x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w))))" *)
   776 constdefs domain_fm :: "[i,i]=>i"
   777     "domain_fm(r,z) == 
   778        Forall(Iff(Member(0,succ(z)),
   779                   Exists(And(Member(0,succ(succ(r))),
   780                              Exists(pair_fm(2,0,1))))))"
   781 
   782 lemma domain_type [TC]:
   783      "[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
   784 by (simp add: domain_fm_def) 
   785 
   786 lemma arity_domain_fm [simp]:
   787      "[| x \<in> nat; y \<in> nat |] 
   788       ==> arity(domain_fm(x,y)) = succ(x) \<union> succ(y)"
   789 by (simp add: domain_fm_def succ_Un_distrib [symmetric] Un_ac) 
   790 
   791 lemma sats_domain_fm [simp]:
   792    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   793     ==> sats(A, domain_fm(x,y), env) <-> 
   794         is_domain(**A, nth(x,env), nth(y,env))"
   795 by (simp add: domain_fm_def is_domain_def)
   796 
   797 lemma domain_iff_sats:
   798       "[| nth(i,env) = x; nth(j,env) = y; 
   799           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   800        ==> is_domain(**A, x, y) <-> sats(A, domain_fm(i,j), env)"
   801 by simp
   802 
   803 theorem domain_reflection:
   804      "REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)), 
   805                \<lambda>i x. is_domain(**Lset(i),f(x),g(x))]"
   806 apply (simp only: is_domain_def setclass_simps)
   807 apply (intro FOL_reflections pair_reflection)  
   808 done
   809 
   810 
   811 subsubsection{*Range of a Relation, Internalized*}
   812 
   813 (* "is_range(M,r,z) == 
   814 	\<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w))))" *)
   815 constdefs range_fm :: "[i,i]=>i"
   816     "range_fm(r,z) == 
   817        Forall(Iff(Member(0,succ(z)),
   818                   Exists(And(Member(0,succ(succ(r))),
   819                              Exists(pair_fm(0,2,1))))))"
   820 
   821 lemma range_type [TC]:
   822      "[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
   823 by (simp add: range_fm_def) 
   824 
   825 lemma arity_range_fm [simp]:
   826      "[| x \<in> nat; y \<in> nat |] 
   827       ==> arity(range_fm(x,y)) = succ(x) \<union> succ(y)"
   828 by (simp add: range_fm_def succ_Un_distrib [symmetric] Un_ac) 
   829 
   830 lemma sats_range_fm [simp]:
   831    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   832     ==> sats(A, range_fm(x,y), env) <-> 
   833         is_range(**A, nth(x,env), nth(y,env))"
   834 by (simp add: range_fm_def is_range_def)
   835 
   836 lemma range_iff_sats:
   837       "[| nth(i,env) = x; nth(j,env) = y; 
   838           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   839        ==> is_range(**A, x, y) <-> sats(A, range_fm(i,j), env)"
   840 by simp
   841 
   842 theorem range_reflection:
   843      "REFLECTS[\<lambda>x. is_range(L,f(x),g(x)), 
   844                \<lambda>i x. is_range(**Lset(i),f(x),g(x))]"
   845 apply (simp only: is_range_def setclass_simps)
   846 apply (intro FOL_reflections pair_reflection)  
   847 done
   848 
   849  
   850 subsubsection{*Field of a Relation, Internalized*}
   851 
   852 (* "is_field(M,r,z) == 
   853 	\<exists>dr[M]. is_domain(M,r,dr) & 
   854             (\<exists>rr[M]. is_range(M,r,rr) & union(M,dr,rr,z))" *)
   855 constdefs field_fm :: "[i,i]=>i"
   856     "field_fm(r,z) == 
   857        Exists(And(domain_fm(succ(r),0), 
   858               Exists(And(range_fm(succ(succ(r)),0), 
   859                          union_fm(1,0,succ(succ(z)))))))"
   860 
   861 lemma field_type [TC]:
   862      "[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula"
   863 by (simp add: field_fm_def) 
   864 
   865 lemma arity_field_fm [simp]:
   866      "[| x \<in> nat; y \<in> nat |] 
   867       ==> arity(field_fm(x,y)) = succ(x) \<union> succ(y)"
   868 by (simp add: field_fm_def succ_Un_distrib [symmetric] Un_ac) 
   869 
   870 lemma sats_field_fm [simp]:
   871    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   872     ==> sats(A, field_fm(x,y), env) <-> 
   873         is_field(**A, nth(x,env), nth(y,env))"
   874 by (simp add: field_fm_def is_field_def)
   875 
   876 lemma field_iff_sats:
   877       "[| nth(i,env) = x; nth(j,env) = y; 
   878           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   879        ==> is_field(**A, x, y) <-> sats(A, field_fm(i,j), env)"
   880 by simp
   881 
   882 theorem field_reflection:
   883      "REFLECTS[\<lambda>x. is_field(L,f(x),g(x)), 
   884                \<lambda>i x. is_field(**Lset(i),f(x),g(x))]"
   885 apply (simp only: is_field_def setclass_simps)
   886 apply (intro FOL_reflections domain_reflection range_reflection
   887              union_reflection)
   888 done
   889 
   890 
   891 subsubsection{*Image under a Relation, Internalized*}
   892 
   893 (* "image(M,r,A,z) == 
   894         \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,w))))" *)
   895 constdefs image_fm :: "[i,i,i]=>i"
   896     "image_fm(r,A,z) == 
   897        Forall(Iff(Member(0,succ(z)),
   898                   Exists(And(Member(0,succ(succ(r))),
   899                              Exists(And(Member(0,succ(succ(succ(A)))),
   900 	 			        pair_fm(0,2,1)))))))"
   901 
   902 lemma image_type [TC]:
   903      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
   904 by (simp add: image_fm_def) 
   905 
   906 lemma arity_image_fm [simp]:
   907      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
   908       ==> arity(image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
   909 by (simp add: image_fm_def succ_Un_distrib [symmetric] Un_ac) 
   910 
   911 lemma sats_image_fm [simp]:
   912    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   913     ==> sats(A, image_fm(x,y,z), env) <-> 
   914         image(**A, nth(x,env), nth(y,env), nth(z,env))"
   915 by (simp add: image_fm_def Relative.image_def)
   916 
   917 lemma image_iff_sats:
   918       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
   919           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   920        ==> image(**A, x, y, z) <-> sats(A, image_fm(i,j,k), env)"
   921 by (simp add: sats_image_fm)
   922 
   923 theorem image_reflection:
   924      "REFLECTS[\<lambda>x. image(L,f(x),g(x),h(x)), 
   925                \<lambda>i x. image(**Lset(i),f(x),g(x),h(x))]"
   926 apply (simp only: Relative.image_def setclass_simps)
   927 apply (intro FOL_reflections pair_reflection)  
   928 done
   929 
   930 
   931 subsubsection{*Pre-Image under a Relation, Internalized*}
   932 
   933 (* "pre_image(M,r,A,z) == 
   934 	\<forall>x[M]. x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" *)
   935 constdefs pre_image_fm :: "[i,i,i]=>i"
   936     "pre_image_fm(r,A,z) == 
   937        Forall(Iff(Member(0,succ(z)),
   938                   Exists(And(Member(0,succ(succ(r))),
   939                              Exists(And(Member(0,succ(succ(succ(A)))),
   940 	 			        pair_fm(2,0,1)))))))"
   941 
   942 lemma pre_image_type [TC]:
   943      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
   944 by (simp add: pre_image_fm_def) 
   945 
   946 lemma arity_pre_image_fm [simp]:
   947      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
   948       ==> arity(pre_image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
   949 by (simp add: pre_image_fm_def succ_Un_distrib [symmetric] Un_ac) 
   950 
   951 lemma sats_pre_image_fm [simp]:
   952    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   953     ==> sats(A, pre_image_fm(x,y,z), env) <-> 
   954         pre_image(**A, nth(x,env), nth(y,env), nth(z,env))"
   955 by (simp add: pre_image_fm_def Relative.pre_image_def)
   956 
   957 lemma pre_image_iff_sats:
   958       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
   959           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   960        ==> pre_image(**A, x, y, z) <-> sats(A, pre_image_fm(i,j,k), env)"
   961 by (simp add: sats_pre_image_fm)
   962 
   963 theorem pre_image_reflection:
   964      "REFLECTS[\<lambda>x. pre_image(L,f(x),g(x),h(x)), 
   965                \<lambda>i x. pre_image(**Lset(i),f(x),g(x),h(x))]"
   966 apply (simp only: Relative.pre_image_def setclass_simps)
   967 apply (intro FOL_reflections pair_reflection)  
   968 done
   969 
   970 
   971 subsubsection{*Function Application, Internalized*}
   972 
   973 (* "fun_apply(M,f,x,y) == 
   974         (\<exists>xs[M]. \<exists>fxs[M]. 
   975          upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))" *)
   976 constdefs fun_apply_fm :: "[i,i,i]=>i"
   977     "fun_apply_fm(f,x,y) == 
   978        Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1),
   979                          And(image_fm(succ(succ(f)), 1, 0), 
   980                              big_union_fm(0,succ(succ(y)))))))"
   981 
   982 lemma fun_apply_type [TC]:
   983      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
   984 by (simp add: fun_apply_fm_def) 
   985 
   986 lemma arity_fun_apply_fm [simp]:
   987      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
   988       ==> arity(fun_apply_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
   989 by (simp add: fun_apply_fm_def succ_Un_distrib [symmetric] Un_ac) 
   990 
   991 lemma sats_fun_apply_fm [simp]:
   992    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   993     ==> sats(A, fun_apply_fm(x,y,z), env) <-> 
   994         fun_apply(**A, nth(x,env), nth(y,env), nth(z,env))"
   995 by (simp add: fun_apply_fm_def fun_apply_def)
   996 
   997 lemma fun_apply_iff_sats:
   998       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
   999           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1000        ==> fun_apply(**A, x, y, z) <-> sats(A, fun_apply_fm(i,j,k), env)"
  1001 by simp
  1002 
  1003 theorem fun_apply_reflection:
  1004      "REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)), 
  1005                \<lambda>i x. fun_apply(**Lset(i),f(x),g(x),h(x))]" 
  1006 apply (simp only: fun_apply_def setclass_simps)
  1007 apply (intro FOL_reflections upair_reflection image_reflection
  1008              big_union_reflection)  
  1009 done
  1010 
  1011 
  1012 subsubsection{*The Concept of Relation, Internalized*}
  1013 
  1014 (* "is_relation(M,r) == 
  1015         (\<forall>z[M]. z\<in>r --> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))" *)
  1016 constdefs relation_fm :: "i=>i"
  1017     "relation_fm(r) == 
  1018        Forall(Implies(Member(0,succ(r)), Exists(Exists(pair_fm(1,0,2)))))"
  1019 
  1020 lemma relation_type [TC]:
  1021      "[| x \<in> nat |] ==> relation_fm(x) \<in> formula"
  1022 by (simp add: relation_fm_def) 
  1023 
  1024 lemma arity_relation_fm [simp]:
  1025      "x \<in> nat ==> arity(relation_fm(x)) = succ(x)"
  1026 by (simp add: relation_fm_def succ_Un_distrib [symmetric] Un_ac) 
  1027 
  1028 lemma sats_relation_fm [simp]:
  1029    "[| x \<in> nat; env \<in> list(A)|]
  1030     ==> sats(A, relation_fm(x), env) <-> is_relation(**A, nth(x,env))"
  1031 by (simp add: relation_fm_def is_relation_def)
  1032 
  1033 lemma relation_iff_sats:
  1034       "[| nth(i,env) = x; nth(j,env) = y; 
  1035           i \<in> nat; env \<in> list(A)|]
  1036        ==> is_relation(**A, x) <-> sats(A, relation_fm(i), env)"
  1037 by simp
  1038 
  1039 theorem is_relation_reflection:
  1040      "REFLECTS[\<lambda>x. is_relation(L,f(x)), 
  1041                \<lambda>i x. is_relation(**Lset(i),f(x))]"
  1042 apply (simp only: is_relation_def setclass_simps)
  1043 apply (intro FOL_reflections pair_reflection)  
  1044 done
  1045 
  1046 
  1047 subsubsection{*The Concept of Function, Internalized*}
  1048 
  1049 (* "is_function(M,r) == 
  1050 	\<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M]. 
  1051            pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>r --> y=y'" *)
  1052 constdefs function_fm :: "i=>i"
  1053     "function_fm(r) == 
  1054        Forall(Forall(Forall(Forall(Forall(
  1055          Implies(pair_fm(4,3,1),
  1056                  Implies(pair_fm(4,2,0),
  1057                          Implies(Member(1,r#+5),
  1058                                  Implies(Member(0,r#+5), Equal(3,2))))))))))"
  1059 
  1060 lemma function_type [TC]:
  1061      "[| x \<in> nat |] ==> function_fm(x) \<in> formula"
  1062 by (simp add: function_fm_def) 
  1063 
  1064 lemma arity_function_fm [simp]:
  1065      "x \<in> nat ==> arity(function_fm(x)) = succ(x)"
  1066 by (simp add: function_fm_def succ_Un_distrib [symmetric] Un_ac) 
  1067 
  1068 lemma sats_function_fm [simp]:
  1069    "[| x \<in> nat; env \<in> list(A)|]
  1070     ==> sats(A, function_fm(x), env) <-> is_function(**A, nth(x,env))"
  1071 by (simp add: function_fm_def is_function_def)
  1072 
  1073 lemma function_iff_sats:
  1074       "[| nth(i,env) = x; nth(j,env) = y; 
  1075           i \<in> nat; env \<in> list(A)|]
  1076        ==> is_function(**A, x) <-> sats(A, function_fm(i), env)"
  1077 by simp
  1078 
  1079 theorem is_function_reflection:
  1080      "REFLECTS[\<lambda>x. is_function(L,f(x)), 
  1081                \<lambda>i x. is_function(**Lset(i),f(x))]"
  1082 apply (simp only: is_function_def setclass_simps)
  1083 apply (intro FOL_reflections pair_reflection)  
  1084 done
  1085 
  1086 
  1087 subsubsection{*Typed Functions, Internalized*}
  1088 
  1089 (* "typed_function(M,A,B,r) == 
  1090         is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
  1091         (\<forall>u[M]. u\<in>r --> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) --> y\<in>B))" *)
  1092 
  1093 constdefs typed_function_fm :: "[i,i,i]=>i"
  1094     "typed_function_fm(A,B,r) == 
  1095        And(function_fm(r),
  1096          And(relation_fm(r),
  1097            And(domain_fm(r,A),
  1098              Forall(Implies(Member(0,succ(r)),
  1099                   Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))"
  1100 
  1101 lemma typed_function_type [TC]:
  1102      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
  1103 by (simp add: typed_function_fm_def) 
  1104 
  1105 lemma arity_typed_function_fm [simp]:
  1106      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
  1107       ==> arity(typed_function_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
  1108 by (simp add: typed_function_fm_def succ_Un_distrib [symmetric] Un_ac) 
  1109 
  1110 lemma sats_typed_function_fm [simp]:
  1111    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
  1112     ==> sats(A, typed_function_fm(x,y,z), env) <-> 
  1113         typed_function(**A, nth(x,env), nth(y,env), nth(z,env))"
  1114 by (simp add: typed_function_fm_def typed_function_def)
  1115 
  1116 lemma typed_function_iff_sats:
  1117   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
  1118       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1119    ==> typed_function(**A, x, y, z) <-> sats(A, typed_function_fm(i,j,k), env)"
  1120 by simp
  1121 
  1122 lemmas function_reflections = 
  1123         empty_reflection number1_reflection
  1124 	upair_reflection pair_reflection union_reflection
  1125 	big_union_reflection cons_reflection successor_reflection 
  1126         fun_apply_reflection subset_reflection
  1127 	transitive_set_reflection membership_reflection
  1128 	pred_set_reflection domain_reflection range_reflection field_reflection
  1129         image_reflection pre_image_reflection
  1130 	is_relation_reflection is_function_reflection
  1131 
  1132 lemmas function_iff_sats = 
  1133         empty_iff_sats number1_iff_sats 
  1134 	upair_iff_sats pair_iff_sats union_iff_sats
  1135 	cons_iff_sats successor_iff_sats
  1136         fun_apply_iff_sats  Memrel_iff_sats
  1137 	pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats
  1138         image_iff_sats pre_image_iff_sats 
  1139 	relation_iff_sats function_iff_sats
  1140 
  1141 
  1142 theorem typed_function_reflection:
  1143      "REFLECTS[\<lambda>x. typed_function(L,f(x),g(x),h(x)), 
  1144                \<lambda>i x. typed_function(**Lset(i),f(x),g(x),h(x))]"
  1145 apply (simp only: typed_function_def setclass_simps)
  1146 apply (intro FOL_reflections function_reflections)  
  1147 done
  1148 
  1149 
  1150 subsubsection{*Composition of Relations, Internalized*}
  1151 
  1152 (* "composition(M,r,s,t) == 
  1153         \<forall>p[M]. p \<in> t <-> 
  1154                (\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M]. 
  1155                 pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) & 
  1156                 xy \<in> s & yz \<in> r)" *)
  1157 constdefs composition_fm :: "[i,i,i]=>i"
  1158   "composition_fm(r,s,t) == 
  1159      Forall(Iff(Member(0,succ(t)),
  1160              Exists(Exists(Exists(Exists(Exists( 
  1161               And(pair_fm(4,2,5),
  1162                And(pair_fm(4,3,1),
  1163                 And(pair_fm(3,2,0),
  1164                  And(Member(1,s#+6), Member(0,r#+6))))))))))))"
  1165 
  1166 lemma composition_type [TC]:
  1167      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula"
  1168 by (simp add: composition_fm_def) 
  1169 
  1170 lemma arity_composition_fm [simp]:
  1171      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
  1172       ==> arity(composition_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
  1173 by (simp add: composition_fm_def succ_Un_distrib [symmetric] Un_ac) 
  1174 
  1175 lemma sats_composition_fm [simp]:
  1176    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
  1177     ==> sats(A, composition_fm(x,y,z), env) <-> 
  1178         composition(**A, nth(x,env), nth(y,env), nth(z,env))"
  1179 by (simp add: composition_fm_def composition_def)
  1180 
  1181 lemma composition_iff_sats:
  1182       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
  1183           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1184        ==> composition(**A, x, y, z) <-> sats(A, composition_fm(i,j,k), env)"
  1185 by simp
  1186 
  1187 theorem composition_reflection:
  1188      "REFLECTS[\<lambda>x. composition(L,f(x),g(x),h(x)), 
  1189                \<lambda>i x. composition(**Lset(i),f(x),g(x),h(x))]"
  1190 apply (simp only: composition_def setclass_simps)
  1191 apply (intro FOL_reflections pair_reflection)  
  1192 done
  1193 
  1194 
  1195 subsubsection{*Injections, Internalized*}
  1196 
  1197 (* "injection(M,A,B,f) == 
  1198 	typed_function(M,A,B,f) &
  1199         (\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M]. 
  1200           pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>f --> x=x')" *)
  1201 constdefs injection_fm :: "[i,i,i]=>i"
  1202  "injection_fm(A,B,f) == 
  1203     And(typed_function_fm(A,B,f),
  1204        Forall(Forall(Forall(Forall(Forall(
  1205          Implies(pair_fm(4,2,1),
  1206                  Implies(pair_fm(3,2,0),
  1207                          Implies(Member(1,f#+5),
  1208                                  Implies(Member(0,f#+5), Equal(4,3)))))))))))"
  1209 
  1210 
  1211 lemma injection_type [TC]:
  1212      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
  1213 by (simp add: injection_fm_def) 
  1214 
  1215 lemma arity_injection_fm [simp]:
  1216      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
  1217       ==> arity(injection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
  1218 by (simp add: injection_fm_def succ_Un_distrib [symmetric] Un_ac) 
  1219 
  1220 lemma sats_injection_fm [simp]:
  1221    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
  1222     ==> sats(A, injection_fm(x,y,z), env) <-> 
  1223         injection(**A, nth(x,env), nth(y,env), nth(z,env))"
  1224 by (simp add: injection_fm_def injection_def)
  1225 
  1226 lemma injection_iff_sats:
  1227   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
  1228       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1229    ==> injection(**A, x, y, z) <-> sats(A, injection_fm(i,j,k), env)"
  1230 by simp
  1231 
  1232 theorem injection_reflection:
  1233      "REFLECTS[\<lambda>x. injection(L,f(x),g(x),h(x)), 
  1234                \<lambda>i x. injection(**Lset(i),f(x),g(x),h(x))]"
  1235 apply (simp only: injection_def setclass_simps)
  1236 apply (intro FOL_reflections function_reflections typed_function_reflection)  
  1237 done
  1238 
  1239 
  1240 subsubsection{*Surjections, Internalized*}
  1241 
  1242 (*  surjection :: "[i=>o,i,i,i] => o"
  1243     "surjection(M,A,B,f) == 
  1244         typed_function(M,A,B,f) &
  1245         (\<forall>y[M]. y\<in>B --> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" *)
  1246 constdefs surjection_fm :: "[i,i,i]=>i"
  1247  "surjection_fm(A,B,f) == 
  1248     And(typed_function_fm(A,B,f),
  1249        Forall(Implies(Member(0,succ(B)),
  1250                       Exists(And(Member(0,succ(succ(A))),
  1251                                  fun_apply_fm(succ(succ(f)),0,1))))))"
  1252 
  1253 lemma surjection_type [TC]:
  1254      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula"
  1255 by (simp add: surjection_fm_def) 
  1256 
  1257 lemma arity_surjection_fm [simp]:
  1258      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
  1259       ==> arity(surjection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
  1260 by (simp add: surjection_fm_def succ_Un_distrib [symmetric] Un_ac) 
  1261 
  1262 lemma sats_surjection_fm [simp]:
  1263    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
  1264     ==> sats(A, surjection_fm(x,y,z), env) <-> 
  1265         surjection(**A, nth(x,env), nth(y,env), nth(z,env))"
  1266 by (simp add: surjection_fm_def surjection_def)
  1267 
  1268 lemma surjection_iff_sats:
  1269   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
  1270       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1271    ==> surjection(**A, x, y, z) <-> sats(A, surjection_fm(i,j,k), env)"
  1272 by simp
  1273 
  1274 theorem surjection_reflection:
  1275      "REFLECTS[\<lambda>x. surjection(L,f(x),g(x),h(x)), 
  1276                \<lambda>i x. surjection(**Lset(i),f(x),g(x),h(x))]"
  1277 apply (simp only: surjection_def setclass_simps)
  1278 apply (intro FOL_reflections function_reflections typed_function_reflection)  
  1279 done
  1280 
  1281 
  1282 
  1283 subsubsection{*Bijections, Internalized*}
  1284 
  1285 (*   bijection :: "[i=>o,i,i,i] => o"
  1286     "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *)
  1287 constdefs bijection_fm :: "[i,i,i]=>i"
  1288  "bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))"
  1289 
  1290 lemma bijection_type [TC]:
  1291      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula"
  1292 by (simp add: bijection_fm_def) 
  1293 
  1294 lemma arity_bijection_fm [simp]:
  1295      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
  1296       ==> arity(bijection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
  1297 by (simp add: bijection_fm_def succ_Un_distrib [symmetric] Un_ac) 
  1298 
  1299 lemma sats_bijection_fm [simp]:
  1300    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
  1301     ==> sats(A, bijection_fm(x,y,z), env) <-> 
  1302         bijection(**A, nth(x,env), nth(y,env), nth(z,env))"
  1303 by (simp add: bijection_fm_def bijection_def)
  1304 
  1305 lemma bijection_iff_sats:
  1306   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
  1307       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1308    ==> bijection(**A, x, y, z) <-> sats(A, bijection_fm(i,j,k), env)"
  1309 by simp
  1310 
  1311 theorem bijection_reflection:
  1312      "REFLECTS[\<lambda>x. bijection(L,f(x),g(x),h(x)), 
  1313                \<lambda>i x. bijection(**Lset(i),f(x),g(x),h(x))]"
  1314 apply (simp only: bijection_def setclass_simps)
  1315 apply (intro And_reflection injection_reflection surjection_reflection)  
  1316 done
  1317 
  1318 
  1319 subsubsection{*Restriction of a Relation, Internalized*}
  1320 
  1321 
  1322 (* "restriction(M,r,A,z) == 
  1323 	\<forall>x[M]. x \<in> z <-> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" *)
  1324 constdefs restriction_fm :: "[i,i,i]=>i"
  1325     "restriction_fm(r,A,z) == 
  1326        Forall(Iff(Member(0,succ(z)),
  1327                   And(Member(0,succ(r)),
  1328                       Exists(And(Member(0,succ(succ(A))),
  1329                                  Exists(pair_fm(1,0,2)))))))"
  1330 
  1331 lemma restriction_type [TC]:
  1332      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula"
  1333 by (simp add: restriction_fm_def) 
  1334 
  1335 lemma arity_restriction_fm [simp]:
  1336      "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
  1337       ==> arity(restriction_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
  1338 by (simp add: restriction_fm_def succ_Un_distrib [symmetric] Un_ac) 
  1339 
  1340 lemma sats_restriction_fm [simp]:
  1341    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
  1342     ==> sats(A, restriction_fm(x,y,z), env) <-> 
  1343         restriction(**A, nth(x,env), nth(y,env), nth(z,env))"
  1344 by (simp add: restriction_fm_def restriction_def)
  1345 
  1346 lemma restriction_iff_sats:
  1347       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
  1348           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1349        ==> restriction(**A, x, y, z) <-> sats(A, restriction_fm(i,j,k), env)"
  1350 by simp
  1351 
  1352 theorem restriction_reflection:
  1353      "REFLECTS[\<lambda>x. restriction(L,f(x),g(x),h(x)), 
  1354                \<lambda>i x. restriction(**Lset(i),f(x),g(x),h(x))]"
  1355 apply (simp only: restriction_def setclass_simps)
  1356 apply (intro FOL_reflections pair_reflection)  
  1357 done
  1358 
  1359 subsubsection{*Order-Isomorphisms, Internalized*}
  1360 
  1361 (*  order_isomorphism :: "[i=>o,i,i,i,i,i] => o"
  1362    "order_isomorphism(M,A,r,B,s,f) == 
  1363         bijection(M,A,B,f) & 
  1364         (\<forall>x[M]. x\<in>A --> (\<forall>y[M]. y\<in>A -->
  1365           (\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>q[M].
  1366             pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) --> 
  1367             pair(M,fx,fy,q) --> (p\<in>r <-> q\<in>s))))"
  1368   *)
  1369 
  1370 constdefs order_isomorphism_fm :: "[i,i,i,i,i]=>i"
  1371  "order_isomorphism_fm(A,r,B,s,f) == 
  1372    And(bijection_fm(A,B,f), 
  1373      Forall(Implies(Member(0,succ(A)),
  1374        Forall(Implies(Member(0,succ(succ(A))),
  1375          Forall(Forall(Forall(Forall(
  1376            Implies(pair_fm(5,4,3),
  1377              Implies(fun_apply_fm(f#+6,5,2),
  1378                Implies(fun_apply_fm(f#+6,4,1),
  1379                  Implies(pair_fm(2,1,0), 
  1380                    Iff(Member(3,r#+6), Member(0,s#+6)))))))))))))))"
  1381 
  1382 lemma order_isomorphism_type [TC]:
  1383      "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]  
  1384       ==> order_isomorphism_fm(A,r,B,s,f) \<in> formula"
  1385 by (simp add: order_isomorphism_fm_def) 
  1386 
  1387 lemma arity_order_isomorphism_fm [simp]:
  1388      "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |] 
  1389       ==> arity(order_isomorphism_fm(A,r,B,s,f)) = 
  1390           succ(A) \<union> succ(r) \<union> succ(B) \<union> succ(s) \<union> succ(f)" 
  1391 by (simp add: order_isomorphism_fm_def succ_Un_distrib [symmetric] Un_ac) 
  1392 
  1393 lemma sats_order_isomorphism_fm [simp]:
  1394    "[| U \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat; env \<in> list(A)|]
  1395     ==> sats(A, order_isomorphism_fm(U,r,B,s,f), env) <-> 
  1396         order_isomorphism(**A, nth(U,env), nth(r,env), nth(B,env), 
  1397                                nth(s,env), nth(f,env))"
  1398 by (simp add: order_isomorphism_fm_def order_isomorphism_def)
  1399 
  1400 lemma order_isomorphism_iff_sats:
  1401   "[| nth(i,env) = U; nth(j,env) = r; nth(k,env) = B; nth(j',env) = s; 
  1402       nth(k',env) = f; 
  1403       i \<in> nat; j \<in> nat; k \<in> nat; j' \<in> nat; k' \<in> nat; env \<in> list(A)|]
  1404    ==> order_isomorphism(**A,U,r,B,s,f) <-> 
  1405        sats(A, order_isomorphism_fm(i,j,k,j',k'), env)" 
  1406 by simp
  1407 
  1408 theorem order_isomorphism_reflection:
  1409      "REFLECTS[\<lambda>x. order_isomorphism(L,f(x),g(x),h(x),g'(x),h'(x)), 
  1410                \<lambda>i x. order_isomorphism(**Lset(i),f(x),g(x),h(x),g'(x),h'(x))]"
  1411 apply (simp only: order_isomorphism_def setclass_simps)
  1412 apply (intro FOL_reflections function_reflections bijection_reflection)  
  1413 done
  1414 
  1415 subsubsection{*Limit Ordinals, Internalized*}
  1416 
  1417 text{*A limit ordinal is a non-empty, successor-closed ordinal*}
  1418 
  1419 (* "limit_ordinal(M,a) == 
  1420 	ordinal(M,a) & ~ empty(M,a) & 
  1421         (\<forall>x[M]. x\<in>a --> (\<exists>y[M]. y\<in>a & successor(M,x,y)))" *)
  1422 
  1423 constdefs limit_ordinal_fm :: "i=>i"
  1424     "limit_ordinal_fm(x) == 
  1425         And(ordinal_fm(x),
  1426             And(Neg(empty_fm(x)),
  1427 	        Forall(Implies(Member(0,succ(x)),
  1428                                Exists(And(Member(0,succ(succ(x))),
  1429                                           succ_fm(1,0)))))))"
  1430 
  1431 lemma limit_ordinal_type [TC]:
  1432      "x \<in> nat ==> limit_ordinal_fm(x) \<in> formula"
  1433 by (simp add: limit_ordinal_fm_def) 
  1434 
  1435 lemma arity_limit_ordinal_fm [simp]:
  1436      "x \<in> nat ==> arity(limit_ordinal_fm(x)) = succ(x)"
  1437 by (simp add: limit_ordinal_fm_def succ_Un_distrib [symmetric] Un_ac) 
  1438 
  1439 lemma sats_limit_ordinal_fm [simp]:
  1440    "[| x \<in> nat; env \<in> list(A)|]
  1441     ==> sats(A, limit_ordinal_fm(x), env) <-> limit_ordinal(**A, nth(x,env))"
  1442 by (simp add: limit_ordinal_fm_def limit_ordinal_def sats_ordinal_fm')
  1443 
  1444 lemma limit_ordinal_iff_sats:
  1445       "[| nth(i,env) = x; nth(j,env) = y; 
  1446           i \<in> nat; env \<in> list(A)|]
  1447        ==> limit_ordinal(**A, x) <-> sats(A, limit_ordinal_fm(i), env)"
  1448 by simp
  1449 
  1450 theorem limit_ordinal_reflection:
  1451      "REFLECTS[\<lambda>x. limit_ordinal(L,f(x)), 
  1452                \<lambda>i x. limit_ordinal(**Lset(i),f(x))]"
  1453 apply (simp only: limit_ordinal_def setclass_simps)
  1454 apply (intro FOL_reflections ordinal_reflection 
  1455              empty_reflection successor_reflection)  
  1456 done
  1457 
  1458 subsubsection{*Omega: The Set of Natural Numbers*}
  1459 
  1460 (* omega(M,a) == limit_ordinal(M,a) & (\<forall>x[M]. x\<in>a --> ~ limit_ordinal(M,x)) *)
  1461 constdefs omega_fm :: "i=>i"
  1462     "omega_fm(x) == 
  1463        And(limit_ordinal_fm(x),
  1464            Forall(Implies(Member(0,succ(x)),
  1465                           Neg(limit_ordinal_fm(0)))))"
  1466 
  1467 lemma omega_type [TC]:
  1468      "x \<in> nat ==> omega_fm(x) \<in> formula"
  1469 by (simp add: omega_fm_def) 
  1470 
  1471 lemma arity_omega_fm [simp]:
  1472      "x \<in> nat ==> arity(omega_fm(x)) = succ(x)"
  1473 by (simp add: omega_fm_def succ_Un_distrib [symmetric] Un_ac) 
  1474 
  1475 lemma sats_omega_fm [simp]:
  1476    "[| x \<in> nat; env \<in> list(A)|]
  1477     ==> sats(A, omega_fm(x), env) <-> omega(**A, nth(x,env))"
  1478 by (simp add: omega_fm_def omega_def)
  1479 
  1480 lemma omega_iff_sats:
  1481       "[| nth(i,env) = x; nth(j,env) = y; 
  1482           i \<in> nat; env \<in> list(A)|]
  1483        ==> omega(**A, x) <-> sats(A, omega_fm(i), env)"
  1484 by simp
  1485 
  1486 theorem omega_reflection:
  1487      "REFLECTS[\<lambda>x. omega(L,f(x)), 
  1488                \<lambda>i x. omega(**Lset(i),f(x))]"
  1489 apply (simp only: omega_def setclass_simps)
  1490 apply (intro FOL_reflections limit_ordinal_reflection)  
  1491 done
  1492 
  1493 
  1494 lemmas fun_plus_reflections =
  1495         typed_function_reflection composition_reflection
  1496         injection_reflection surjection_reflection
  1497         bijection_reflection restriction_reflection
  1498         order_isomorphism_reflection
  1499         ordinal_reflection limit_ordinal_reflection omega_reflection
  1500 
  1501 lemmas fun_plus_iff_sats = 
  1502 	typed_function_iff_sats composition_iff_sats
  1503         injection_iff_sats surjection_iff_sats 
  1504         bijection_iff_sats restriction_iff_sats 
  1505         order_isomorphism_iff_sats
  1506         ordinal_iff_sats limit_ordinal_iff_sats omega_iff_sats
  1507 
  1508 end