src/ZF/Constructible/L_axioms.thy
 author paulson Thu Jul 04 18:29:50 2002 +0200 (2002-07-04) changeset 13299 3a932abf97e8 parent 13298 b4f370679c65 child 13304 43ef6c6dd906 permissions -rw-r--r--
More use of relativized quantifiers
```     1 header {* The class L satisfies the axioms of ZF*}
```
```     2
```
```     3 theory L_axioms = Formula + Relative + Reflection:
```
```     4
```
```     5
```
```     6 text {* The class L satisfies the premises of locale @{text M_axioms} *}
```
```     7
```
```     8 lemma transL: "[| y\<in>x; L(x) |] ==> L(y)"
```
```     9 apply (insert Transset_Lset)
```
```    10 apply (simp add: Transset_def L_def, blast)
```
```    11 done
```
```    12
```
```    13 lemma nonempty: "L(0)"
```
```    14 apply (simp add: L_def)
```
```    15 apply (blast intro: zero_in_Lset)
```
```    16 done
```
```    17
```
```    18 lemma upair_ax: "upair_ax(L)"
```
```    19 apply (simp add: upair_ax_def upair_def, clarify)
```
```    20 apply (rule_tac x="{x,y}" in rexI)
```
```    21 apply (simp_all add: doubleton_in_L)
```
```    22 done
```
```    23
```
```    24 lemma Union_ax: "Union_ax(L)"
```
```    25 apply (simp add: Union_ax_def big_union_def, clarify)
```
```    26 apply (rule_tac x="Union(x)" in rexI)
```
```    27 apply (simp_all add: Union_in_L, auto)
```
```    28 apply (blast intro: transL)
```
```    29 done
```
```    30
```
```    31 lemma power_ax: "power_ax(L)"
```
```    32 apply (simp add: power_ax_def powerset_def Relative.subset_def, clarify)
```
```    33 apply (rule_tac x="{y \<in> Pow(x). L(y)}" in rexI)
```
```    34 apply (simp_all add: LPow_in_L, auto)
```
```    35 apply (blast intro: transL)
```
```    36 done
```
```    37
```
```    38 subsubsection{*For L to satisfy Replacement *}
```
```    39
```
```    40 (*Can't move these to Formula unless the definition of univalent is moved
```
```    41 there too!*)
```
```    42
```
```    43 lemma LReplace_in_Lset:
```
```    44      "[|X \<in> Lset(i); univalent(L,X,Q); Ord(i)|]
```
```    45       ==> \<exists>j. Ord(j) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Lset(j)"
```
```    46 apply (rule_tac x="\<Union>y \<in> Replace(X, %x y. Q(x,y) & L(y)). succ(lrank(y))"
```
```    47        in exI)
```
```    48 apply simp
```
```    49 apply clarify
```
```    50 apply (rule_tac a="x" in UN_I)
```
```    51  apply (simp_all add: Replace_iff univalent_def)
```
```    52 apply (blast dest: transL L_I)
```
```    53 done
```
```    54
```
```    55 lemma LReplace_in_L:
```
```    56      "[|L(X); univalent(L,X,Q)|]
```
```    57       ==> \<exists>Y. L(Y) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Y"
```
```    58 apply (drule L_D, clarify)
```
```    59 apply (drule LReplace_in_Lset, assumption+)
```
```    60 apply (blast intro: L_I Lset_in_Lset_succ)
```
```    61 done
```
```    62
```
```    63 lemma replacement: "replacement(L,P)"
```
```    64 apply (simp add: replacement_def, clarify)
```
```    65 apply (frule LReplace_in_L, assumption+, clarify)
```
```    66 apply (rule_tac x=Y in rexI)
```
```    67 apply (simp_all add: Replace_iff univalent_def, blast)
```
```    68 done
```
```    69
```
```    70 subsection{*Instantiation of the locale @{text M_triv_axioms}*}
```
```    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 trivaxL 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", trivaxL (thm "M_triv_axioms.ball_abs"));
```
```   100 bind_thm ("rall_abs", trivaxL (thm "M_triv_axioms.rall_abs"));
```
```   101 bind_thm ("bex_abs", trivaxL (thm "M_triv_axioms.bex_abs"));
```
```   102 bind_thm ("rex_abs", trivaxL (thm "M_triv_axioms.rex_abs"));
```
```   103 bind_thm ("ball_iff_equiv", trivaxL (thm "M_triv_axioms.ball_iff_equiv"));
```
```   104 bind_thm ("M_equalityI", trivaxL (thm "M_triv_axioms.M_equalityI"));
```
```   105 bind_thm ("empty_abs", trivaxL (thm "M_triv_axioms.empty_abs"));
```
```   106 bind_thm ("subset_abs", trivaxL (thm "M_triv_axioms.subset_abs"));
```
```   107 bind_thm ("upair_abs", trivaxL (thm "M_triv_axioms.upair_abs"));
```
```   108 bind_thm ("upair_in_M_iff", trivaxL (thm "M_triv_axioms.upair_in_M_iff"));
```
```   109 bind_thm ("singleton_in_M_iff", trivaxL (thm "M_triv_axioms.singleton_in_M_iff"));
```
```   110 bind_thm ("pair_abs", trivaxL (thm "M_triv_axioms.pair_abs"));
```
```   111 bind_thm ("pair_in_M_iff", trivaxL (thm "M_triv_axioms.pair_in_M_iff"));
```
```   112 bind_thm ("pair_components_in_M", trivaxL (thm "M_triv_axioms.pair_components_in_M"));
```
```   113 bind_thm ("cartprod_abs", trivaxL (thm "M_triv_axioms.cartprod_abs"));
```
```   114 bind_thm ("union_abs", trivaxL (thm "M_triv_axioms.union_abs"));
```
```   115 bind_thm ("inter_abs", trivaxL (thm "M_triv_axioms.inter_abs"));
```
```   116 bind_thm ("setdiff_abs", trivaxL (thm "M_triv_axioms.setdiff_abs"));
```
```   117 bind_thm ("Union_abs", trivaxL (thm "M_triv_axioms.Union_abs"));
```
```   118 bind_thm ("Union_closed", trivaxL (thm "M_triv_axioms.Union_closed"));
```
```   119 bind_thm ("Un_closed", trivaxL (thm "M_triv_axioms.Un_closed"));
```
```   120 bind_thm ("cons_closed", trivaxL (thm "M_triv_axioms.cons_closed"));
```
```   121 bind_thm ("successor_abs", trivaxL (thm "M_triv_axioms.successor_abs"));
```
```   122 bind_thm ("succ_in_M_iff", trivaxL (thm "M_triv_axioms.succ_in_M_iff"));
```
```   123 bind_thm ("separation_closed", trivaxL (thm "M_triv_axioms.separation_closed"));
```
```   124 bind_thm ("strong_replacementI", trivaxL (thm "M_triv_axioms.strong_replacementI"));
```
```   125 bind_thm ("strong_replacement_closed", trivaxL (thm "M_triv_axioms.strong_replacement_closed"));
```
```   126 bind_thm ("RepFun_closed", trivaxL (thm "M_triv_axioms.RepFun_closed"));
```
```   127 bind_thm ("lam_closed", trivaxL (thm "M_triv_axioms.lam_closed"));
```
```   128 bind_thm ("image_abs", trivaxL (thm "M_triv_axioms.image_abs"));
```
```   129 bind_thm ("powerset_Pow", trivaxL (thm "M_triv_axioms.powerset_Pow"));
```
```   130 bind_thm ("powerset_imp_subset_Pow", trivaxL (thm "M_triv_axioms.powerset_imp_subset_Pow"));
```
```   131 bind_thm ("nat_into_M", trivaxL (thm "M_triv_axioms.nat_into_M"));
```
```   132 bind_thm ("nat_case_closed", trivaxL (thm "M_triv_axioms.nat_case_closed"));
```
```   133 bind_thm ("Inl_in_M_iff", trivaxL (thm "M_triv_axioms.Inl_in_M_iff"));
```
```   134 bind_thm ("Inr_in_M_iff", trivaxL (thm "M_triv_axioms.Inr_in_M_iff"));
```
```   135 bind_thm ("lt_closed", trivaxL (thm "M_triv_axioms.lt_closed"));
```
```   136 bind_thm ("transitive_set_abs", trivaxL (thm "M_triv_axioms.transitive_set_abs"));
```
```   137 bind_thm ("ordinal_abs", trivaxL (thm "M_triv_axioms.ordinal_abs"));
```
```   138 bind_thm ("limit_ordinal_abs", trivaxL (thm "M_triv_axioms.limit_ordinal_abs"));
```
```   139 bind_thm ("successor_ordinal_abs", trivaxL (thm "M_triv_axioms.successor_ordinal_abs"));
```
```   140 bind_thm ("finite_ordinal_abs", trivaxL (thm "M_triv_axioms.finite_ordinal_abs"));
```
```   141 bind_thm ("omega_abs", trivaxL (thm "M_triv_axioms.omega_abs"));
```
```   142 bind_thm ("number1_abs", trivaxL (thm "M_triv_axioms.number1_abs"));
```
```   143 bind_thm ("number1_abs", trivaxL (thm "M_triv_axioms.number1_abs"));
```
```   144 bind_thm ("number3_abs", trivaxL (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 [rule_format]
```
```   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_Reflects :: "[i=>o,i=>o,[i,i]=>o] => o"
```
```   193     "L_Reflects(Cl,P,Q) == Closed_Unbounded(Cl) &
```
```   194                            (\<forall>a. Cl(a) --> (\<forall>x \<in> Lset(a). P(x) <-> Q(a,x)))"
```
```   195
```
```   196   L_F0 :: "[i=>o,i] => i"
```
```   197     "L_F0(P,y) == \<mu>b. (\<exists>z. L(z) \<and> P(<y,z>)) --> (\<exists>z\<in>Lset(b). P(<y,z>))"
```
```   198
```
```   199   L_FF :: "[i=>o,i] => i"
```
```   200     "L_FF(P)   == \<lambda>a. \<Union>y\<in>Lset(a). L_F0(P,y)"
```
```   201
```
```   202   L_ClEx :: "[i=>o,i] => o"
```
```   203     "L_ClEx(P) == \<lambda>a. Limit(a) \<and> normalize(L_FF(P),a) = a"
```
```   204
```
```   205 theorem Triv_reflection [intro]:
```
```   206      "L_Reflects(Ord, P, \<lambda>a x. P(x))"
```
```   207 by (simp add: L_Reflects_def)
```
```   208
```
```   209 theorem Not_reflection [intro]:
```
```   210      "L_Reflects(Cl,P,Q) ==> L_Reflects(Cl, \<lambda>x. ~P(x), \<lambda>a x. ~Q(a,x))"
```
```   211 by (simp add: L_Reflects_def)
```
```   212
```
```   213 theorem And_reflection [intro]:
```
```   214      "[| L_Reflects(Cl,P,Q); L_Reflects(C',P',Q') |]
```
```   215       ==> L_Reflects(\<lambda>a. Cl(a) \<and> C'(a), \<lambda>x. P(x) \<and> P'(x),
```
```   216                                       \<lambda>a x. Q(a,x) \<and> Q'(a,x))"
```
```   217 by (simp add: L_Reflects_def Closed_Unbounded_Int, blast)
```
```   218
```
```   219 theorem Or_reflection [intro]:
```
```   220      "[| L_Reflects(Cl,P,Q); L_Reflects(C',P',Q') |]
```
```   221       ==> L_Reflects(\<lambda>a. Cl(a) \<and> C'(a), \<lambda>x. P(x) \<or> P'(x),
```
```   222                                       \<lambda>a x. Q(a,x) \<or> Q'(a,x))"
```
```   223 by (simp add: L_Reflects_def Closed_Unbounded_Int, blast)
```
```   224
```
```   225 theorem Imp_reflection [intro]:
```
```   226      "[| L_Reflects(Cl,P,Q); L_Reflects(C',P',Q') |]
```
```   227       ==> L_Reflects(\<lambda>a. Cl(a) \<and> C'(a),
```
```   228                    \<lambda>x. P(x) --> P'(x),
```
```   229                    \<lambda>a x. Q(a,x) --> Q'(a,x))"
```
```   230 by (simp add: L_Reflects_def Closed_Unbounded_Int, blast)
```
```   231
```
```   232 theorem Iff_reflection [intro]:
```
```   233      "[| L_Reflects(Cl,P,Q); L_Reflects(C',P',Q') |]
```
```   234       ==> L_Reflects(\<lambda>a. Cl(a) \<and> C'(a),
```
```   235                    \<lambda>x. P(x) <-> P'(x),
```
```   236                    \<lambda>a x. Q(a,x) <-> Q'(a,x))"
```
```   237 by (simp add: L_Reflects_def Closed_Unbounded_Int, blast)
```
```   238
```
```   239
```
```   240 theorem Ex_reflection [intro]:
```
```   241      "L_Reflects(Cl, \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x)))
```
```   242       ==> L_Reflects(\<lambda>a. Cl(a) \<and> L_ClEx(\<lambda>x. P(fst(x),snd(x)), a),
```
```   243                    \<lambda>x. \<exists>z. L(z) \<and> P(x,z),
```
```   244                    \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z))"
```
```   245 apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
```
```   246 apply (rule reflection.Ex_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset],
```
```   247        assumption+)
```
```   248 done
```
```   249
```
```   250 theorem All_reflection [intro]:
```
```   251      "L_Reflects(Cl,  \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x)))
```
```   252       ==> L_Reflects(\<lambda>a. Cl(a) \<and> L_ClEx(\<lambda>x. ~P(fst(x),snd(x)), a),
```
```   253                    \<lambda>x. \<forall>z. L(z) --> P(x,z),
```
```   254                    \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z))"
```
```   255 apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
```
```   256 apply (rule reflection.All_reflection [OF Lset_mono_le Lset_cont Pair_in_Lset],
```
```   257        assumption+)
```
```   258 done
```
```   259
```
```   260 theorem Rex_reflection [intro]:
```
```   261      "L_Reflects(Cl, \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x)))
```
```   262       ==> L_Reflects(\<lambda>a. Cl(a) \<and> L_ClEx(\<lambda>x. P(fst(x),snd(x)), a),
```
```   263                    \<lambda>x. \<exists>z[L]. P(x,z),
```
```   264                    \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z))"
```
```   265 by (unfold rex_def, blast)
```
```   266
```
```   267 theorem Rall_reflection [intro]:
```
```   268      "L_Reflects(Cl,  \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x)))
```
```   269       ==> L_Reflects(\<lambda>a. Cl(a) \<and> L_ClEx(\<lambda>x. ~P(fst(x),snd(x)), a),
```
```   270                    \<lambda>x. \<forall>z[L]. P(x,z),
```
```   271                    \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z))"
```
```   272 by (unfold rall_def, blast)
```
```   273
```
```   274 lemma ReflectsD:
```
```   275      "[|L_Reflects(Cl,P,Q); Ord(i)|]
```
```   276       ==> \<exists>j. i<j & (\<forall>x \<in> Lset(j). P(x) <-> Q(j,x))"
```
```   277 apply (simp add: L_Reflects_def Closed_Unbounded_def, clarify)
```
```   278 apply (blast dest!: UnboundedD)
```
```   279 done
```
```   280
```
```   281 lemma ReflectsE:
```
```   282      "[| L_Reflects(Cl,P,Q); Ord(i);
```
```   283          !!j. [|i<j;  \<forall>x \<in> Lset(j). P(x) <-> Q(j,x)|] ==> R |]
```
```   284       ==> R"
```
```   285 by (blast dest!: ReflectsD)
```
```   286
```
```   287 lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B";
```
```   288 by blast
```
```   289
```
```   290
```
```   291 subsection{*Internalized formulas for some relativized ones*}
```
```   292
```
```   293 subsubsection{*Unordered pairs*}
```
```   294
```
```   295 constdefs upair_fm :: "[i,i,i]=>i"
```
```   296     "upair_fm(x,y,z) ==
```
```   297        And(Member(x,z),
```
```   298            And(Member(y,z),
```
```   299                Forall(Implies(Member(0,succ(z)),
```
```   300                               Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
```
```   301
```
```   302 lemma upair_type [TC]:
```
```   303      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
```
```   304 by (simp add: upair_fm_def)
```
```   305
```
```   306 lemma arity_upair_fm [simp]:
```
```   307      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   308       ==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   309 by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   310
```
```   311 lemma sats_upair_fm [simp]:
```
```   312    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   313     ==> sats(A, upair_fm(x,y,z), env) <->
```
```   314             upair(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   315 by (simp add: upair_fm_def upair_def)
```
```   316
```
```   317 lemma upair_iff_sats:
```
```   318       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   319           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   320        ==> upair(**A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
```
```   321 by (simp add: sats_upair_fm)
```
```   322
```
```   323 text{*Useful? At least it refers to "real" unordered pairs*}
```
```   324 lemma sats_upair_fm2 [simp]:
```
```   325    "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
```
```   326     ==> sats(A, upair_fm(x,y,z), env) <->
```
```   327         nth(z,env) = {nth(x,env), nth(y,env)}"
```
```   328 apply (frule lt_length_in_nat, assumption)
```
```   329 apply (simp add: upair_fm_def Transset_def, auto)
```
```   330 apply (blast intro: nth_type)
```
```   331 done
```
```   332
```
```   333 subsubsection{*Ordered pairs*}
```
```   334
```
```   335 constdefs pair_fm :: "[i,i,i]=>i"
```
```   336     "pair_fm(x,y,z) ==
```
```   337        Exists(And(upair_fm(succ(x),succ(x),0),
```
```   338               Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
```
```   339                          upair_fm(1,0,succ(succ(z)))))))"
```
```   340
```
```   341 lemma pair_type [TC]:
```
```   342      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
```
```   343 by (simp add: pair_fm_def)
```
```   344
```
```   345 lemma arity_pair_fm [simp]:
```
```   346      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   347       ==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   348 by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   349
```
```   350 lemma sats_pair_fm [simp]:
```
```   351    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   352     ==> sats(A, pair_fm(x,y,z), env) <->
```
```   353         pair(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   354 by (simp add: pair_fm_def pair_def)
```
```   355
```
```   356 lemma pair_iff_sats:
```
```   357       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   358           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   359        ==> pair(**A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
```
```   360 by (simp add: sats_pair_fm)
```
```   361
```
```   362
```
```   363
```
```   364 subsection{*Proving instances of Separation using Reflection!*}
```
```   365
```
```   366 text{*Helps us solve for de Bruijn indices!*}
```
```   367 lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x"
```
```   368 by simp
```
```   369
```
```   370
```
```   371 lemma Collect_conj_in_DPow:
```
```   372      "[| {x\<in>A. P(x)} \<in> DPow(A);  {x\<in>A. Q(x)} \<in> DPow(A) |]
```
```   373       ==> {x\<in>A. P(x) & Q(x)} \<in> DPow(A)"
```
```   374 by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric])
```
```   375
```
```   376 lemma Collect_conj_in_DPow_Lset:
```
```   377      "[|z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))|]
```
```   378       ==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))"
```
```   379 apply (frule mem_Lset_imp_subset_Lset)
```
```   380 apply (simp add: Collect_conj_in_DPow Collect_mem_eq
```
```   381                  subset_Int_iff2 elem_subset_in_DPow)
```
```   382 done
```
```   383
```
```   384 lemma separation_CollectI:
```
```   385      "(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))"
```
```   386 apply (unfold separation_def, clarify)
```
```   387 apply (rule_tac x="{x\<in>z. P(x)}" in rexI)
```
```   388 apply simp_all
```
```   389 done
```
```   390
```
```   391 text{*Reduces the original comprehension to the reflected one*}
```
```   392 lemma reflection_imp_L_separation:
```
```   393       "[| \<forall>x\<in>Lset(j). P(x) <-> Q(x);
```
```   394           {x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j));
```
```   395           Ord(j);  z \<in> Lset(j)|] ==> L({x \<in> z . P(x)})"
```
```   396 apply (rule_tac i = "succ(j)" in L_I)
```
```   397  prefer 2 apply simp
```
```   398 apply (subgoal_tac "{x \<in> z. P(x)} = {x \<in> Lset(j). x \<in> z & (Q(x))}")
```
```   399  prefer 2
```
```   400  apply (blast dest: mem_Lset_imp_subset_Lset)
```
```   401 apply (simp add: Lset_succ Collect_conj_in_DPow_Lset)
```
```   402 done
```
```   403
```
```   404
```
```   405 subsubsection{*Separation for Intersection*}
```
```   406
```
```   407 lemma Inter_Reflects:
```
```   408      "L_Reflects(?Cl, \<lambda>x. \<forall>y[L]. y\<in>A --> x \<in> y,
```
```   409                \<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A --> x \<in> y)"
```
```   410 by fast
```
```   411
```
```   412 lemma Inter_separation:
```
```   413      "L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A --> x\<in>y)"
```
```   414 apply (rule separation_CollectI)
```
```   415 apply (rule_tac A="{A,z}" in subset_LsetE, blast )
```
```   416 apply (rule ReflectsE [OF Inter_Reflects], assumption)
```
```   417 apply (drule subset_Lset_ltD, assumption)
```
```   418 apply (erule reflection_imp_L_separation)
```
```   419   apply (simp_all add: lt_Ord2, clarify)
```
```   420 apply (rule DPowI2)
```
```   421 apply (rule ball_iff_sats)
```
```   422 apply (rule imp_iff_sats)
```
```   423 apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats)
```
```   424 apply (rule_tac i=0 and j=2 in mem_iff_sats)
```
```   425 apply (simp_all add: succ_Un_distrib [symmetric])
```
```   426 done
```
```   427
```
```   428 subsubsection{*Separation for Cartesian Product*}
```
```   429
```
```   430 text{*The @{text simplified} attribute tidies up the reflecting class.*}
```
```   431 theorem upair_reflection [simplified,intro]:
```
```   432      "L_Reflects(?Cl, \<lambda>x. upair(L,f(x),g(x),h(x)),
```
```   433                     \<lambda>i x. upair(**Lset(i),f(x),g(x),h(x)))"
```
```   434 by (simp add: upair_def, fast)
```
```   435
```
```   436 theorem pair_reflection [simplified,intro]:
```
```   437      "L_Reflects(?Cl, \<lambda>x. pair(L,f(x),g(x),h(x)),
```
```   438                     \<lambda>i x. pair(**Lset(i),f(x),g(x),h(x)))"
```
```   439 by (simp only: pair_def rex_setclass_is_bex, fast)
```
```   440
```
```   441 lemma cartprod_Reflects [simplified]:
```
```   442      "L_Reflects(?Cl, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)),
```
```   443                 \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B &
```
```   444                                pair(**Lset(i),x,y,z)))"
```
```   445 by fast
```
```   446
```
```   447 lemma cartprod_separation:
```
```   448      "[| L(A); L(B) |]
```
```   449       ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)))"
```
```   450 apply (rule separation_CollectI)
```
```   451 apply (rule_tac A="{A,B,z}" in subset_LsetE, blast )
```
```   452 apply (rule ReflectsE [OF cartprod_Reflects], assumption)
```
```   453 apply (drule subset_Lset_ltD, assumption)
```
```   454 apply (erule reflection_imp_L_separation)
```
```   455   apply (simp_all add: lt_Ord2, clarify)
```
```   456 apply (rule DPowI2)
```
```   457 apply (rename_tac u)
```
```   458 apply (rule bex_iff_sats)
```
```   459 apply (rule conj_iff_sats)
```
```   460 apply (rule_tac i=0 and j=2 and env="[x,u,A,B]" in mem_iff_sats, simp_all)
```
```   461 apply (rule bex_iff_sats)
```
```   462 apply (rule conj_iff_sats)
```
```   463 apply (rule mem_iff_sats)
```
```   464 apply (blast intro: nth_0 nth_ConsI)
```
```   465 apply (blast intro: nth_0 nth_ConsI, simp_all)
```
```   466 apply (rule_tac i=1 and j=0 and k=2 in pair_iff_sats)
```
```   467 apply (simp_all add: succ_Un_distrib [symmetric])
```
```   468 done
```
```   469
```
```   470 subsubsection{*Separation for Image*}
```
```   471
```
```   472 text{*No @{text simplified} here: it simplifies the occurrence of
```
```   473       the predicate @{term pair}!*}
```
```   474 lemma image_Reflects:
```
```   475      "L_Reflects(?Cl, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)),
```
```   476            \<lambda>i y. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). x\<in>A & pair(**Lset(i),x,y,p)))"
```
```   477 by fast
```
```   478
```
```   479
```
```   480 lemma image_separation:
```
```   481      "[| L(A); L(r) |]
```
```   482       ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)))"
```
```   483 apply (rule separation_CollectI)
```
```   484 apply (rule_tac A="{A,r,z}" in subset_LsetE, blast )
```
```   485 apply (rule ReflectsE [OF image_Reflects], assumption)
```
```   486 apply (drule subset_Lset_ltD, assumption)
```
```   487 apply (erule reflection_imp_L_separation)
```
```   488   apply (simp_all add: lt_Ord2, clarify)
```
```   489 apply (rule DPowI2)
```
```   490 apply (rule bex_iff_sats)
```
```   491 apply (rule conj_iff_sats)
```
```   492 apply (rule_tac env="[p,y,A,r]" in mem_iff_sats)
```
```   493 apply (blast intro: nth_0 nth_ConsI)
```
```   494 apply (blast intro: nth_0 nth_ConsI, simp_all)
```
```   495 apply (rule bex_iff_sats)
```
```   496 apply (rule conj_iff_sats)
```
```   497 apply (rule mem_iff_sats)
```
```   498 apply (blast intro: nth_0 nth_ConsI)
```
```   499 apply (blast intro: nth_0 nth_ConsI, simp_all)
```
```   500 apply (rule pair_iff_sats)
```
```   501 apply (blast intro: nth_0 nth_ConsI)
```
```   502 apply (blast intro: nth_0 nth_ConsI)
```
```   503 apply (blast intro: nth_0 nth_ConsI)
```
```   504 apply (simp_all add: succ_Un_distrib [symmetric])
```
```   505 done
```
```   506
```
```   507
```
```   508 subsubsection{*Separation for Converse*}
```
```   509
```
```   510 lemma converse_Reflects:
```
```   511      "L_Reflects(?Cl,
```
```   512         \<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)),
```
```   513      \<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i).
```
```   514                      pair(**Lset(i),x,y,p) & pair(**Lset(i),y,x,z)))"
```
```   515 by fast
```
```   516
```
```   517 lemma converse_separation:
```
```   518      "L(r) ==> separation(L,
```
```   519          \<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)))"
```
```   520 apply (rule separation_CollectI)
```
```   521 apply (rule_tac A="{r,z}" in subset_LsetE, blast )
```
```   522 apply (rule ReflectsE [OF converse_Reflects], assumption)
```
```   523 apply (drule subset_Lset_ltD, assumption)
```
```   524 apply (erule reflection_imp_L_separation)
```
```   525   apply (simp_all add: lt_Ord2, clarify)
```
```   526 apply (rule DPowI2)
```
```   527 apply (rename_tac u)
```
```   528 apply (rule bex_iff_sats)
```
```   529 apply (rule conj_iff_sats)
```
```   530 apply (rule_tac i=0 and j="2" and env="[p,u,r]" in mem_iff_sats, simp_all)
```
```   531 apply (rule bex_iff_sats)
```
```   532 apply (rule bex_iff_sats)
```
```   533 apply (rule conj_iff_sats)
```
```   534 apply (rule_tac i=1 and j=0 and k=2 in pair_iff_sats, simp_all)
```
```   535 apply (rule pair_iff_sats)
```
```   536 apply (blast intro: nth_0 nth_ConsI)
```
```   537 apply (blast intro: nth_0 nth_ConsI)
```
```   538 apply (blast intro: nth_0 nth_ConsI)
```
```   539 apply (simp_all add: succ_Un_distrib [symmetric])
```
```   540 done
```
```   541
```
```   542
```
```   543 subsubsection{*Separation for Restriction*}
```
```   544
```
```   545 lemma restrict_Reflects:
```
```   546      "L_Reflects(?Cl, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)),
```
```   547         \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(**Lset(i),x,y,z)))"
```
```   548 by fast
```
```   549
```
```   550 lemma restrict_separation:
```
```   551    "L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))"
```
```   552 apply (rule separation_CollectI)
```
```   553 apply (rule_tac A="{A,z}" in subset_LsetE, blast )
```
```   554 apply (rule ReflectsE [OF restrict_Reflects], assumption)
```
```   555 apply (drule subset_Lset_ltD, assumption)
```
```   556 apply (erule reflection_imp_L_separation)
```
```   557   apply (simp_all add: lt_Ord2, clarify)
```
```   558 apply (rule DPowI2)
```
```   559 apply (rename_tac u)
```
```   560 apply (rule bex_iff_sats)
```
```   561 apply (rule conj_iff_sats)
```
```   562 apply (rule_tac i=0 and j="2" and env="[x,u,A]" in mem_iff_sats, simp_all)
```
```   563 apply (rule bex_iff_sats)
```
```   564 apply (rule_tac i=1 and j=0 and k=2 in pair_iff_sats)
```
```   565 apply (simp_all add: succ_Un_distrib [symmetric])
```
```   566 done
```
```   567
```
```   568
```
```   569 subsubsection{*Separation for Composition*}
```
```   570
```
```   571 lemma comp_Reflects:
```
```   572      "L_Reflects(?Cl, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
```
```   573 		  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
```
```   574                   xy\<in>s & yz\<in>r,
```
```   575         \<lambda>i xz. \<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). \<exists>z\<in>Lset(i). \<exists>xy\<in>Lset(i). \<exists>yz\<in>Lset(i).
```
```   576 		  pair(**Lset(i),x,z,xz) & pair(**Lset(i),x,y,xy) &
```
```   577                   pair(**Lset(i),y,z,yz) & xy\<in>s & yz\<in>r)"
```
```   578 by fast
```
```   579
```
```   580 lemma comp_separation:
```
```   581      "[| L(r); L(s) |]
```
```   582       ==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
```
```   583 		  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
```
```   584                   xy\<in>s & yz\<in>r)"
```
```   585 apply (rule separation_CollectI)
```
```   586 apply (rule_tac A="{r,s,z}" in subset_LsetE, blast )
```
```   587 apply (rule ReflectsE [OF comp_Reflects], assumption)
```
```   588 apply (drule subset_Lset_ltD, assumption)
```
```   589 apply (erule reflection_imp_L_separation)
```
```   590   apply (simp_all add: lt_Ord2, clarify)
```
```   591 apply (rule DPowI2)
```
```   592 apply (rename_tac u)
```
```   593 apply (rule bex_iff_sats)+
```
```   594 apply (rename_tac x y z)
```
```   595 apply (rule conj_iff_sats)
```
```   596 apply (rule_tac env="[z,y,x,u,r,s]" in pair_iff_sats)
```
```   597 apply (blast intro: nth_0 nth_ConsI)
```
```   598 apply (blast intro: nth_0 nth_ConsI)
```
```   599 apply (blast intro: nth_0 nth_ConsI, simp_all)
```
```   600 apply (rule bex_iff_sats)
```
```   601 apply (rule conj_iff_sats)
```
```   602 apply (rule pair_iff_sats)
```
```   603 apply (blast intro: nth_0 nth_ConsI)
```
```   604 apply (blast intro: nth_0 nth_ConsI)
```
```   605 apply (blast intro: nth_0 nth_ConsI, simp_all)
```
```   606 apply (rule bex_iff_sats)
```
```   607 apply (rule conj_iff_sats)
```
```   608 apply (rule pair_iff_sats)
```
```   609 apply (blast intro: nth_0 nth_ConsI)
```
```   610 apply (blast intro: nth_0 nth_ConsI)
```
```   611 apply (blast intro: nth_0 nth_ConsI, simp_all)
```
```   612 apply (rule conj_iff_sats)
```
```   613 apply (rule mem_iff_sats)
```
```   614 apply (blast intro: nth_0 nth_ConsI)
```
```   615 apply (blast intro: nth_0 nth_ConsI, simp)
```
```   616 apply (rule mem_iff_sats)
```
```   617 apply (blast intro: nth_0 nth_ConsI)
```
```   618 apply (blast intro: nth_0 nth_ConsI)
```
```   619 apply (simp_all add: succ_Un_distrib [symmetric])
```
```   620 done
```
```   621
```
```   622
```
```   623
```
```   624
```
```   625 end
```
```   626
```
```   627 (*
```
```   628
```
```   629   and pred_separation:
```
```   630      "[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p\<in>r. L(p) & pair(L,y,x,p))"
```
```   631   and Memrel_separation:
```
```   632      "separation(L, \<lambda>z. \<exists>x y. L(x) & L(y) & pair(L,x,y,z) \<and> x \<in> y)"
```
```   633   and obase_separation:
```
```   634      --{*part of the order type formalization*}
```
```   635      "[| L(A); L(r) |]
```
```   636       ==> separation(L, \<lambda>a. \<exists>x g mx par. L(x) & L(g) & L(mx) & L(par) &
```
```   637 	     ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
```
```   638 	     order_isomorphism(L,par,r,x,mx,g))"
```
```   639   and well_ord_iso_separation:
```
```   640      "[| L(A); L(f); L(r) |]
```
```   641       ==> separation (L, \<lambda>x. x\<in>A --> (\<exists>y. L(y) \<and> (\<exists>p. L(p) \<and>
```
```   642 		     fun_apply(L,f,x,y) \<and> pair(L,y,x,p) \<and> p \<in> r)))"
```
```   643   and obase_equals_separation:
```
```   644      "[| L(A); L(r) |]
```
```   645       ==> separation
```
```   646       (L, \<lambda>x. x\<in>A --> ~(\<exists>y. L(y) & (\<exists>g. L(g) &
```
```   647 	      ordinal(L,y) & (\<exists>my pxr. L(my) & L(pxr) &
```
```   648 	      membership(L,y,my) & pred_set(L,A,x,r,pxr) &
```
```   649 	      order_isomorphism(L,pxr,r,y,my,g)))))"
```
```   650   and is_recfun_separation:
```
```   651      --{*for well-founded recursion.  NEEDS RELATIVIZATION*}
```
```   652      "[| L(A); L(f); L(g); L(a); L(b) |]
```
```   653      ==> separation(L, \<lambda>x. x \<in> A --> \<langle>x,a\<rangle> \<in> r \<and> \<langle>x,b\<rangle> \<in> r \<and> f`x \<noteq> g`x)"
```
```   654   and omap_replacement:
```
```   655      "[| L(A); L(r) |]
```
```   656       ==> strong_replacement(L,
```
```   657              \<lambda>a z. \<exists>x g mx par. L(x) & L(g) & L(mx) & L(par) &
```
```   658 	     ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
```
```   659 	     pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))"
```
```   660
```
`   661 *)`