src/ZF/Constructible/L_axioms.thy
 author paulson Wed Jul 31 18:30:25 2002 +0200 (2002-07-31) changeset 13440 cdde97e1db1c parent 13434 78b93a667c01 child 13493 5aa68c051725 permissions -rw-r--r--
some progress towards "satisfies"
2 header {* The ZF Axioms (Except Separation) in L *}
4 theory L_axioms = Formula + Relative + Reflection + MetaExists:
6 text {* The class L satisfies the premises of locale @{text M_triv_axioms} *}
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
13 lemma nonempty: "L(0)"
15 apply (blast intro: zero_in_Lset)
16 done
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)
22 done
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
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
38 subsubsection{*For L to satisfy Replacement *}
40 (*Can't move these to Formula unless the definition of univalent is moved
41 there too!*)
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
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
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
70 subsection{*Instantiating the locale @{text M_triv_axioms}*}
71 text{*No instances of Separation yet.*}
73 lemma Lset_mono_le: "mono_le_subset(Lset)"
74 by (simp add: mono_le_subset_def le_imp_subset Lset_mono)
76 lemma Lset_cont: "cont_Ord(Lset)"
77 by (simp add: cont_Ord_def Limit_Lset_eq OUnion_def Limit_is_Ord)
79 lemmas Pair_in_Lset = Formula.Pair_in_LLimit
81 lemmas L_nat = Ord_in_L [OF Ord_nat]
83 theorem M_triv_axioms_L: "PROP M_triv_axioms(L)"
84   apply (rule M_triv_axioms.intro)
85         apply (erule (1) transL)
86        apply (rule nonempty)
87       apply (rule upair_ax)
88      apply (rule Union_ax)
89     apply (rule power_ax)
90    apply (rule replacement)
91   apply (rule L_nat)
92   done
94 lemmas rall_abs = M_triv_axioms.rall_abs [OF M_triv_axioms_L]
95   and rex_abs = M_triv_axioms.rex_abs [OF M_triv_axioms_L]
96   and ball_iff_equiv = M_triv_axioms.ball_iff_equiv [OF M_triv_axioms_L]
97   and M_equalityI = M_triv_axioms.M_equalityI [OF M_triv_axioms_L]
98   and empty_abs = M_triv_axioms.empty_abs [OF M_triv_axioms_L]
99   and subset_abs = M_triv_axioms.subset_abs [OF M_triv_axioms_L]
100   and upair_abs = M_triv_axioms.upair_abs [OF M_triv_axioms_L]
101   and upair_in_M_iff = M_triv_axioms.upair_in_M_iff [OF M_triv_axioms_L]
102   and singleton_in_M_iff = M_triv_axioms.singleton_in_M_iff [OF M_triv_axioms_L]
103   and pair_abs = M_triv_axioms.pair_abs [OF M_triv_axioms_L]
104   and pair_in_M_iff = M_triv_axioms.pair_in_M_iff [OF M_triv_axioms_L]
105   and pair_components_in_M = M_triv_axioms.pair_components_in_M [OF M_triv_axioms_L]
106   and cartprod_abs = M_triv_axioms.cartprod_abs [OF M_triv_axioms_L]
107   and union_abs = M_triv_axioms.union_abs [OF M_triv_axioms_L]
108   and inter_abs = M_triv_axioms.inter_abs [OF M_triv_axioms_L]
109   and setdiff_abs = M_triv_axioms.setdiff_abs [OF M_triv_axioms_L]
110   and Union_abs = M_triv_axioms.Union_abs [OF M_triv_axioms_L]
111   and Union_closed = M_triv_axioms.Union_closed [OF M_triv_axioms_L]
112   and Un_closed = M_triv_axioms.Un_closed [OF M_triv_axioms_L]
113   and cons_closed = M_triv_axioms.cons_closed [OF M_triv_axioms_L]
114   and successor_abs = M_triv_axioms.successor_abs [OF M_triv_axioms_L]
115   and succ_in_M_iff = M_triv_axioms.succ_in_M_iff [OF M_triv_axioms_L]
116   and separation_closed = M_triv_axioms.separation_closed [OF M_triv_axioms_L]
117   and strong_replacementI = M_triv_axioms.strong_replacementI [OF M_triv_axioms_L]
118   and strong_replacement_closed = M_triv_axioms.strong_replacement_closed [OF M_triv_axioms_L]
119   and RepFun_closed = M_triv_axioms.RepFun_closed [OF M_triv_axioms_L]
120   and lam_closed = M_triv_axioms.lam_closed [OF M_triv_axioms_L]
121   and image_abs = M_triv_axioms.image_abs [OF M_triv_axioms_L]
122   and powerset_Pow = M_triv_axioms.powerset_Pow [OF M_triv_axioms_L]
123   and powerset_imp_subset_Pow = M_triv_axioms.powerset_imp_subset_Pow [OF M_triv_axioms_L]
124   and nat_into_M = M_triv_axioms.nat_into_M [OF M_triv_axioms_L]
125   and nat_case_closed = M_triv_axioms.nat_case_closed [OF M_triv_axioms_L]
126   and Inl_in_M_iff = M_triv_axioms.Inl_in_M_iff [OF M_triv_axioms_L]
127   and Inr_in_M_iff = M_triv_axioms.Inr_in_M_iff [OF M_triv_axioms_L]
128   and lt_closed = M_triv_axioms.lt_closed [OF M_triv_axioms_L]
129   and transitive_set_abs = M_triv_axioms.transitive_set_abs [OF M_triv_axioms_L]
130   and ordinal_abs = M_triv_axioms.ordinal_abs [OF M_triv_axioms_L]
131   and limit_ordinal_abs = M_triv_axioms.limit_ordinal_abs [OF M_triv_axioms_L]
132   and successor_ordinal_abs = M_triv_axioms.successor_ordinal_abs [OF M_triv_axioms_L]
133   and finite_ordinal_abs = M_triv_axioms.finite_ordinal_abs [OF M_triv_axioms_L]
134   and omega_abs = M_triv_axioms.omega_abs [OF M_triv_axioms_L]
135   and number1_abs = M_triv_axioms.number1_abs [OF M_triv_axioms_L]
136   and number2_abs = M_triv_axioms.number2_abs [OF M_triv_axioms_L]
137   and number3_abs = M_triv_axioms.number3_abs [OF M_triv_axioms_L]
139 declare rall_abs [simp]
140 declare rex_abs [simp]
141 declare empty_abs [simp]
142 declare subset_abs [simp]
143 declare upair_abs [simp]
144 declare upair_in_M_iff [iff]
145 declare singleton_in_M_iff [iff]
146 declare pair_abs [simp]
147 declare pair_in_M_iff [iff]
148 declare cartprod_abs [simp]
149 declare union_abs [simp]
150 declare inter_abs [simp]
151 declare setdiff_abs [simp]
152 declare Union_abs [simp]
153 declare Union_closed [intro, simp]
154 declare Un_closed [intro, simp]
155 declare cons_closed [intro, simp]
156 declare successor_abs [simp]
157 declare succ_in_M_iff [iff]
158 declare separation_closed [intro, simp]
159 declare strong_replacementI
160 declare strong_replacement_closed [intro, simp]
161 declare RepFun_closed [intro, simp]
162 declare lam_closed [intro, simp]
163 declare image_abs [simp]
164 declare nat_into_M [intro]
165 declare Inl_in_M_iff [iff]
166 declare Inr_in_M_iff [iff]
167 declare transitive_set_abs [simp]
168 declare ordinal_abs [simp]
169 declare limit_ordinal_abs [simp]
170 declare successor_ordinal_abs [simp]
171 declare finite_ordinal_abs [simp]
172 declare omega_abs [simp]
173 declare number1_abs [simp]
174 declare number2_abs [simp]
175 declare number3_abs [simp]
178 subsection{*Instantiation of the locale @{text reflection}*}
180 text{*instances of locale constants*}
181 constdefs
182   L_F0 :: "[i=>o,i] => i"
183     "L_F0(P,y) == \<mu>b. (\<exists>z. L(z) \<and> P(<y,z>)) --> (\<exists>z\<in>Lset(b). P(<y,z>))"
185   L_FF :: "[i=>o,i] => i"
186     "L_FF(P)   == \<lambda>a. \<Union>y\<in>Lset(a). L_F0(P,y)"
188   L_ClEx :: "[i=>o,i] => o"
189     "L_ClEx(P) == \<lambda>a. Limit(a) \<and> normalize(L_FF(P),a) = a"
192 text{*We must use the meta-existential quantifier; otherwise the reflection
193       terms become enormous!*}
194 constdefs
195   L_Reflects :: "[i=>o,[i,i]=>o] => prop"      ("(3REFLECTS/ [_,/ _])")
196     "REFLECTS[P,Q] == (??Cl. Closed_Unbounded(Cl) &
197                            (\<forall>a. Cl(a) --> (\<forall>x \<in> Lset(a). P(x) <-> Q(a,x))))"
200 theorem Triv_reflection:
201      "REFLECTS[P, \<lambda>a x. P(x)]"
203 apply (rule meta_exI)
204 apply (rule Closed_Unbounded_Ord)
205 done
207 theorem Not_reflection:
208      "REFLECTS[P,Q] ==> REFLECTS[\<lambda>x. ~P(x), \<lambda>a x. ~Q(a,x)]"
209 apply (unfold L_Reflects_def)
210 apply (erule meta_exE)
211 apply (rule_tac x=Cl in meta_exI, simp)
212 done
214 theorem And_reflection:
215      "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
216       ==> REFLECTS[\<lambda>x. P(x) \<and> P'(x), \<lambda>a x. Q(a,x) \<and> Q'(a,x)]"
217 apply (unfold L_Reflects_def)
218 apply (elim meta_exE)
219 apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
220 apply (simp add: Closed_Unbounded_Int, blast)
221 done
223 theorem Or_reflection:
224      "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
225       ==> REFLECTS[\<lambda>x. P(x) \<or> P'(x), \<lambda>a x. Q(a,x) \<or> Q'(a,x)]"
226 apply (unfold L_Reflects_def)
227 apply (elim meta_exE)
228 apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
229 apply (simp add: Closed_Unbounded_Int, blast)
230 done
232 theorem Imp_reflection:
233      "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
234       ==> REFLECTS[\<lambda>x. P(x) --> P'(x), \<lambda>a x. Q(a,x) --> Q'(a,x)]"
235 apply (unfold L_Reflects_def)
236 apply (elim meta_exE)
237 apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
238 apply (simp add: Closed_Unbounded_Int, blast)
239 done
241 theorem Iff_reflection:
242      "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
243       ==> REFLECTS[\<lambda>x. P(x) <-> P'(x), \<lambda>a x. Q(a,x) <-> Q'(a,x)]"
244 apply (unfold L_Reflects_def)
245 apply (elim meta_exE)
246 apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
247 apply (simp add: Closed_Unbounded_Int, blast)
248 done
251 lemma reflection_Lset: "reflection(Lset)"
252 apply (blast intro: reflection.intro Lset_mono_le Lset_cont Pair_in_Lset) +
253 done
255 theorem Ex_reflection:
256      "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
257       ==> REFLECTS[\<lambda>x. \<exists>z. L(z) \<and> P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
258 apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
259 apply (elim meta_exE)
260 apply (rule meta_exI)
261 apply (erule reflection.Ex_reflection [OF reflection_Lset])
262 done
264 theorem All_reflection:
265      "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
266       ==> REFLECTS[\<lambda>x. \<forall>z. L(z) --> P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]"
267 apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
268 apply (elim meta_exE)
269 apply (rule meta_exI)
270 apply (erule reflection.All_reflection [OF reflection_Lset])
271 done
273 theorem Rex_reflection:
274      "REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
275       ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
276 apply (unfold rex_def)
277 apply (intro And_reflection Ex_reflection, assumption)
278 done
280 theorem Rall_reflection:
281      "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
282       ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]"
283 apply (unfold rall_def)
284 apply (intro Imp_reflection All_reflection, assumption)
285 done
287 text{*This version handles an alternative form of the bounded quantifier
288       in the second argument of @{text REFLECTS}.*}
289 theorem Rex_reflection':
290      "REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
291       ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z[**Lset(a)]. Q(a,x,z)]"
292 apply (unfold setclass_def rex_def)
293 apply (erule Rex_reflection [unfolded rex_def Bex_def])
294 done
296 text{*As above.*}
297 theorem Rall_reflection':
298      "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
299       ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z[**Lset(a)]. Q(a,x,z)]"
300 apply (unfold setclass_def rall_def)
301 apply (erule Rall_reflection [unfolded rall_def Ball_def])
302 done
304 lemmas FOL_reflections =
305         Triv_reflection Not_reflection And_reflection Or_reflection
306         Imp_reflection Iff_reflection Ex_reflection All_reflection
307         Rex_reflection Rall_reflection Rex_reflection' Rall_reflection'
309 lemma ReflectsD:
310      "[|REFLECTS[P,Q]; Ord(i)|]
311       ==> \<exists>j. i<j & (\<forall>x \<in> Lset(j). P(x) <-> Q(j,x))"
312 apply (unfold L_Reflects_def Closed_Unbounded_def)
313 apply (elim meta_exE, clarify)
314 apply (blast dest!: UnboundedD)
315 done
317 lemma ReflectsE:
318      "[| REFLECTS[P,Q]; Ord(i);
319          !!j. [|i<j;  \<forall>x \<in> Lset(j). P(x) <-> Q(j,x)|] ==> R |]
320       ==> R"
321 apply (drule ReflectsD, assumption, blast)
322 done
324 lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B"
325 by blast
328 subsection{*Internalized Formulas for some Set-Theoretic Concepts*}
330 lemmas setclass_simps = rall_setclass_is_ball rex_setclass_is_bex
332 subsubsection{*Some numbers to help write de Bruijn indices*}
334 syntax
335     "3" :: i   ("3")
336     "4" :: i   ("4")
337     "5" :: i   ("5")
338     "6" :: i   ("6")
339     "7" :: i   ("7")
340     "8" :: i   ("8")
341     "9" :: i   ("9")
343 translations
344    "3"  == "succ(2)"
345    "4"  == "succ(3)"
346    "5"  == "succ(4)"
347    "6"  == "succ(5)"
348    "7"  == "succ(6)"
349    "8"  == "succ(7)"
350    "9"  == "succ(8)"
353 subsubsection{*The Empty Set, Internalized*}
355 constdefs empty_fm :: "i=>i"
356     "empty_fm(x) == Forall(Neg(Member(0,succ(x))))"
358 lemma empty_type [TC]:
359      "x \<in> nat ==> empty_fm(x) \<in> formula"
362 lemma arity_empty_fm [simp]:
363      "x \<in> nat ==> arity(empty_fm(x)) = succ(x)"
364 by (simp add: empty_fm_def succ_Un_distrib [symmetric] Un_ac)
366 lemma sats_empty_fm [simp]:
367    "[| x \<in> nat; env \<in> list(A)|]
368     ==> sats(A, empty_fm(x), env) <-> empty(**A, nth(x,env))"
369 by (simp add: empty_fm_def empty_def)
371 lemma empty_iff_sats:
372       "[| nth(i,env) = x; nth(j,env) = y;
373           i \<in> nat; env \<in> list(A)|]
374        ==> empty(**A, x) <-> sats(A, empty_fm(i), env)"
375 by simp
377 theorem empty_reflection:
378      "REFLECTS[\<lambda>x. empty(L,f(x)),
379                \<lambda>i x. empty(**Lset(i),f(x))]"
380 apply (simp only: empty_def setclass_simps)
381 apply (intro FOL_reflections)
382 done
384 text{*Not used.  But maybe useful?*}
385 lemma Transset_sats_empty_fm_eq_0:
386    "[| n \<in> nat; env \<in> list(A); Transset(A)|]
387     ==> sats(A, empty_fm(n), env) <-> nth(n,env) = 0"
388 apply (simp add: empty_fm_def empty_def Transset_def, auto)
389 apply (case_tac "n < length(env)")
390 apply (frule nth_type, assumption+, blast)
391 apply (simp_all add: not_lt_iff_le nth_eq_0)
392 done
395 subsubsection{*Unordered Pairs, Internalized*}
397 constdefs upair_fm :: "[i,i,i]=>i"
398     "upair_fm(x,y,z) ==
399        And(Member(x,z),
400            And(Member(y,z),
401                Forall(Implies(Member(0,succ(z)),
402                               Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
404 lemma upair_type [TC]:
405      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
408 lemma arity_upair_fm [simp]:
409      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
410       ==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
411 by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac)
413 lemma sats_upair_fm [simp]:
414    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
415     ==> sats(A, upair_fm(x,y,z), env) <->
416             upair(**A, nth(x,env), nth(y,env), nth(z,env))"
417 by (simp add: upair_fm_def upair_def)
419 lemma upair_iff_sats:
420       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
421           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
422        ==> upair(**A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
425 text{*Useful? At least it refers to "real" unordered pairs*}
426 lemma sats_upair_fm2 [simp]:
427    "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
428     ==> sats(A, upair_fm(x,y,z), env) <->
429         nth(z,env) = {nth(x,env), nth(y,env)}"
430 apply (frule lt_length_in_nat, assumption)
431 apply (simp add: upair_fm_def Transset_def, auto)
432 apply (blast intro: nth_type)
433 done
435 theorem upair_reflection:
436      "REFLECTS[\<lambda>x. upair(L,f(x),g(x),h(x)),
437                \<lambda>i x. upair(**Lset(i),f(x),g(x),h(x))]"
439 apply (intro FOL_reflections)
440 done
442 subsubsection{*Ordered pairs, Internalized*}
444 constdefs pair_fm :: "[i,i,i]=>i"
445     "pair_fm(x,y,z) ==
446        Exists(And(upair_fm(succ(x),succ(x),0),
447               Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
448                          upair_fm(1,0,succ(succ(z)))))))"
450 lemma pair_type [TC]:
451      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
454 lemma arity_pair_fm [simp]:
455      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
456       ==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
457 by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac)
459 lemma sats_pair_fm [simp]:
460    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
461     ==> sats(A, pair_fm(x,y,z), env) <->
462         pair(**A, nth(x,env), nth(y,env), nth(z,env))"
463 by (simp add: pair_fm_def pair_def)
465 lemma pair_iff_sats:
466       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
467           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
468        ==> pair(**A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
471 theorem pair_reflection:
472      "REFLECTS[\<lambda>x. pair(L,f(x),g(x),h(x)),
473                \<lambda>i x. pair(**Lset(i),f(x),g(x),h(x))]"
474 apply (simp only: pair_def setclass_simps)
475 apply (intro FOL_reflections upair_reflection)
476 done
479 subsubsection{*Binary Unions, Internalized*}
481 constdefs union_fm :: "[i,i,i]=>i"
482     "union_fm(x,y,z) ==
483        Forall(Iff(Member(0,succ(z)),
484                   Or(Member(0,succ(x)),Member(0,succ(y)))))"
486 lemma union_type [TC]:
487      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
490 lemma arity_union_fm [simp]:
491      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
492       ==> arity(union_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
493 by (simp add: union_fm_def succ_Un_distrib [symmetric] Un_ac)
495 lemma sats_union_fm [simp]:
496    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
497     ==> sats(A, union_fm(x,y,z), env) <->
498         union(**A, nth(x,env), nth(y,env), nth(z,env))"
499 by (simp add: union_fm_def union_def)
501 lemma union_iff_sats:
502       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
503           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
504        ==> union(**A, x, y, z) <-> sats(A, union_fm(i,j,k), env)"
507 theorem union_reflection:
508      "REFLECTS[\<lambda>x. union(L,f(x),g(x),h(x)),
509                \<lambda>i x. union(**Lset(i),f(x),g(x),h(x))]"
510 apply (simp only: union_def setclass_simps)
511 apply (intro FOL_reflections)
512 done
515 subsubsection{*Set ``Cons,'' Internalized*}
517 constdefs cons_fm :: "[i,i,i]=>i"
518     "cons_fm(x,y,z) ==
519        Exists(And(upair_fm(succ(x),succ(x),0),
520                   union_fm(0,succ(y),succ(z))))"
523 lemma cons_type [TC]:
524      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
527 lemma arity_cons_fm [simp]:
528      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
529       ==> arity(cons_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
530 by (simp add: cons_fm_def succ_Un_distrib [symmetric] Un_ac)
532 lemma sats_cons_fm [simp]:
533    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
534     ==> sats(A, cons_fm(x,y,z), env) <->
535         is_cons(**A, nth(x,env), nth(y,env), nth(z,env))"
536 by (simp add: cons_fm_def is_cons_def)
538 lemma cons_iff_sats:
539       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
540           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
541        ==> is_cons(**A, x, y, z) <-> sats(A, cons_fm(i,j,k), env)"
542 by simp
544 theorem cons_reflection:
545      "REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)),
546                \<lambda>i x. is_cons(**Lset(i),f(x),g(x),h(x))]"
547 apply (simp only: is_cons_def setclass_simps)
548 apply (intro FOL_reflections upair_reflection union_reflection)
549 done
552 subsubsection{*Successor Function, Internalized*}
554 constdefs succ_fm :: "[i,i]=>i"
555     "succ_fm(x,y) == cons_fm(x,x,y)"
557 lemma succ_type [TC]:
558      "[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula"
561 lemma arity_succ_fm [simp]:
562      "[| x \<in> nat; y \<in> nat |]
563       ==> arity(succ_fm(x,y)) = succ(x) \<union> succ(y)"
566 lemma sats_succ_fm [simp]:
567    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
568     ==> sats(A, succ_fm(x,y), env) <->
569         successor(**A, nth(x,env), nth(y,env))"
570 by (simp add: succ_fm_def successor_def)
572 lemma successor_iff_sats:
573       "[| nth(i,env) = x; nth(j,env) = y;
574           i \<in> nat; j \<in> nat; env \<in> list(A)|]
575        ==> successor(**A, x, y) <-> sats(A, succ_fm(i,j), env)"
576 by simp
578 theorem successor_reflection:
579      "REFLECTS[\<lambda>x. successor(L,f(x),g(x)),
580                \<lambda>i x. successor(**Lset(i),f(x),g(x))]"
581 apply (simp only: successor_def setclass_simps)
582 apply (intro cons_reflection)
583 done
586 subsubsection{*The Number 1, Internalized*}
588 (* "number1(M,a) == (\<exists>x[M]. empty(M,x) & successor(M,x,a))" *)
589 constdefs number1_fm :: "i=>i"
590     "number1_fm(a) == Exists(And(empty_fm(0), succ_fm(0,succ(a))))"
592 lemma number1_type [TC]:
593      "x \<in> nat ==> number1_fm(x) \<in> formula"
596 lemma arity_number1_fm [simp]:
597      "x \<in> nat ==> arity(number1_fm(x)) = succ(x)"
598 by (simp add: number1_fm_def succ_Un_distrib [symmetric] Un_ac)
600 lemma sats_number1_fm [simp]:
601    "[| x \<in> nat; env \<in> list(A)|]
602     ==> sats(A, number1_fm(x), env) <-> number1(**A, nth(x,env))"
603 by (simp add: number1_fm_def number1_def)
605 lemma number1_iff_sats:
606       "[| nth(i,env) = x; nth(j,env) = y;
607           i \<in> nat; env \<in> list(A)|]
608        ==> number1(**A, x) <-> sats(A, number1_fm(i), env)"
609 by simp
611 theorem number1_reflection:
612      "REFLECTS[\<lambda>x. number1(L,f(x)),
613                \<lambda>i x. number1(**Lset(i),f(x))]"
614 apply (simp only: number1_def setclass_simps)
615 apply (intro FOL_reflections empty_reflection successor_reflection)
616 done
619 subsubsection{*Big Union, Internalized*}
621 (*  "big_union(M,A,z) == \<forall>x[M]. x \<in> z <-> (\<exists>y[M]. y\<in>A & x \<in> y)" *)
622 constdefs big_union_fm :: "[i,i]=>i"
623     "big_union_fm(A,z) ==
624        Forall(Iff(Member(0,succ(z)),
625                   Exists(And(Member(0,succ(succ(A))), Member(1,0)))))"
627 lemma big_union_type [TC]:
628      "[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula"
631 lemma arity_big_union_fm [simp]:
632      "[| x \<in> nat; y \<in> nat |]
633       ==> arity(big_union_fm(x,y)) = succ(x) \<union> succ(y)"
634 by (simp add: big_union_fm_def succ_Un_distrib [symmetric] Un_ac)
636 lemma sats_big_union_fm [simp]:
637    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
638     ==> sats(A, big_union_fm(x,y), env) <->
639         big_union(**A, nth(x,env), nth(y,env))"
640 by (simp add: big_union_fm_def big_union_def)
642 lemma big_union_iff_sats:
643       "[| nth(i,env) = x; nth(j,env) = y;
644           i \<in> nat; j \<in> nat; env \<in> list(A)|]
645        ==> big_union(**A, x, y) <-> sats(A, big_union_fm(i,j), env)"
646 by simp
648 theorem big_union_reflection:
649      "REFLECTS[\<lambda>x. big_union(L,f(x),g(x)),
650                \<lambda>i x. big_union(**Lset(i),f(x),g(x))]"
651 apply (simp only: big_union_def setclass_simps)
652 apply (intro FOL_reflections)
653 done
656 subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*}
658 text{*Differs from the one in Formula by using "ordinal" rather than "Ord"*}
661 lemma sats_subset_fm':
662    "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
663     ==> sats(A, subset_fm(x,y), env) <-> subset(**A, nth(x,env), nth(y,env))"
664 by (simp add: subset_fm_def Relative.subset_def)
666 theorem subset_reflection:
667      "REFLECTS[\<lambda>x. subset(L,f(x),g(x)),
668                \<lambda>i x. subset(**Lset(i),f(x),g(x))]"
669 apply (simp only: Relative.subset_def setclass_simps)
670 apply (intro FOL_reflections)
671 done
673 lemma sats_transset_fm':
674    "[|x \<in> nat; env \<in> list(A)|]
675     ==> sats(A, transset_fm(x), env) <-> transitive_set(**A, nth(x,env))"
676 by (simp add: sats_subset_fm' transset_fm_def transitive_set_def)
678 theorem transitive_set_reflection:
679      "REFLECTS[\<lambda>x. transitive_set(L,f(x)),
680                \<lambda>i x. transitive_set(**Lset(i),f(x))]"
681 apply (simp only: transitive_set_def setclass_simps)
682 apply (intro FOL_reflections subset_reflection)
683 done
685 lemma sats_ordinal_fm':
686    "[|x \<in> nat; env \<in> list(A)|]
687     ==> sats(A, ordinal_fm(x), env) <-> ordinal(**A,nth(x,env))"
688 by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def)
690 lemma ordinal_iff_sats:
691       "[| nth(i,env) = x;  i \<in> nat; env \<in> list(A)|]
692        ==> ordinal(**A, x) <-> sats(A, ordinal_fm(i), env)"
695 theorem ordinal_reflection:
696      "REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(**Lset(i),f(x))]"
697 apply (simp only: ordinal_def setclass_simps)
698 apply (intro FOL_reflections transitive_set_reflection)
699 done
702 subsubsection{*Membership Relation, Internalized*}
704 constdefs Memrel_fm :: "[i,i]=>i"
705     "Memrel_fm(A,r) ==
706        Forall(Iff(Member(0,succ(r)),
707                   Exists(And(Member(0,succ(succ(A))),
708                              Exists(And(Member(0,succ(succ(succ(A)))),
709                                         And(Member(1,0),
710                                             pair_fm(1,0,2))))))))"
712 lemma Memrel_type [TC]:
713      "[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
716 lemma arity_Memrel_fm [simp]:
717      "[| x \<in> nat; y \<in> nat |]
718       ==> arity(Memrel_fm(x,y)) = succ(x) \<union> succ(y)"
719 by (simp add: Memrel_fm_def succ_Un_distrib [symmetric] Un_ac)
721 lemma sats_Memrel_fm [simp]:
722    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
723     ==> sats(A, Memrel_fm(x,y), env) <->
724         membership(**A, nth(x,env), nth(y,env))"
725 by (simp add: Memrel_fm_def membership_def)
727 lemma Memrel_iff_sats:
728       "[| nth(i,env) = x; nth(j,env) = y;
729           i \<in> nat; j \<in> nat; env \<in> list(A)|]
730        ==> membership(**A, x, y) <-> sats(A, Memrel_fm(i,j), env)"
731 by simp
733 theorem membership_reflection:
734      "REFLECTS[\<lambda>x. membership(L,f(x),g(x)),
735                \<lambda>i x. membership(**Lset(i),f(x),g(x))]"
736 apply (simp only: membership_def setclass_simps)
737 apply (intro FOL_reflections pair_reflection)
738 done
740 subsubsection{*Predecessor Set, Internalized*}
742 constdefs pred_set_fm :: "[i,i,i,i]=>i"
743     "pred_set_fm(A,x,r,B) ==
744        Forall(Iff(Member(0,succ(B)),
745                   Exists(And(Member(0,succ(succ(r))),
746                              And(Member(1,succ(succ(A))),
747                                  pair_fm(1,succ(succ(x)),0))))))"
750 lemma pred_set_type [TC]:
751      "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |]
752       ==> pred_set_fm(A,x,r,B) \<in> formula"
755 lemma arity_pred_set_fm [simp]:
756    "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |]
757     ==> arity(pred_set_fm(A,x,r,B)) = succ(A) \<union> succ(x) \<union> succ(r) \<union> succ(B)"
758 by (simp add: pred_set_fm_def succ_Un_distrib [symmetric] Un_ac)
760 lemma sats_pred_set_fm [simp]:
761    "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|]
762     ==> sats(A, pred_set_fm(U,x,r,B), env) <->
763         pred_set(**A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))"
764 by (simp add: pred_set_fm_def pred_set_def)
766 lemma pred_set_iff_sats:
767       "[| nth(i,env) = U; nth(j,env) = x; nth(k,env) = r; nth(l,env) = B;
768           i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|]
769        ==> pred_set(**A,U,x,r,B) <-> sats(A, pred_set_fm(i,j,k,l), env)"
772 theorem pred_set_reflection:
773      "REFLECTS[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)),
774                \<lambda>i x. pred_set(**Lset(i),f(x),g(x),h(x),b(x))]"
775 apply (simp only: pred_set_def setclass_simps)
776 apply (intro FOL_reflections pair_reflection)
777 done
781 subsubsection{*Domain of a Relation, Internalized*}
783 (* "is_domain(M,r,z) ==
784         \<forall>x[M]. (x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w))))" *)
785 constdefs domain_fm :: "[i,i]=>i"
786     "domain_fm(r,z) ==
787        Forall(Iff(Member(0,succ(z)),
788                   Exists(And(Member(0,succ(succ(r))),
789                              Exists(pair_fm(2,0,1))))))"
791 lemma domain_type [TC]:
792      "[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
795 lemma arity_domain_fm [simp]:
796      "[| x \<in> nat; y \<in> nat |]
797       ==> arity(domain_fm(x,y)) = succ(x) \<union> succ(y)"
798 by (simp add: domain_fm_def succ_Un_distrib [symmetric] Un_ac)
800 lemma sats_domain_fm [simp]:
801    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
802     ==> sats(A, domain_fm(x,y), env) <->
803         is_domain(**A, nth(x,env), nth(y,env))"
804 by (simp add: domain_fm_def is_domain_def)
806 lemma domain_iff_sats:
807       "[| nth(i,env) = x; nth(j,env) = y;
808           i \<in> nat; j \<in> nat; env \<in> list(A)|]
809        ==> is_domain(**A, x, y) <-> sats(A, domain_fm(i,j), env)"
810 by simp
812 theorem domain_reflection:
813      "REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)),
814                \<lambda>i x. is_domain(**Lset(i),f(x),g(x))]"
815 apply (simp only: is_domain_def setclass_simps)
816 apply (intro FOL_reflections pair_reflection)
817 done
820 subsubsection{*Range of a Relation, Internalized*}
822 (* "is_range(M,r,z) ==
823         \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w))))" *)
824 constdefs range_fm :: "[i,i]=>i"
825     "range_fm(r,z) ==
826        Forall(Iff(Member(0,succ(z)),
827                   Exists(And(Member(0,succ(succ(r))),
828                              Exists(pair_fm(0,2,1))))))"
830 lemma range_type [TC]:
831      "[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
834 lemma arity_range_fm [simp]:
835      "[| x \<in> nat; y \<in> nat |]
836       ==> arity(range_fm(x,y)) = succ(x) \<union> succ(y)"
837 by (simp add: range_fm_def succ_Un_distrib [symmetric] Un_ac)
839 lemma sats_range_fm [simp]:
840    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
841     ==> sats(A, range_fm(x,y), env) <->
842         is_range(**A, nth(x,env), nth(y,env))"
843 by (simp add: range_fm_def is_range_def)
845 lemma range_iff_sats:
846       "[| nth(i,env) = x; nth(j,env) = y;
847           i \<in> nat; j \<in> nat; env \<in> list(A)|]
848        ==> is_range(**A, x, y) <-> sats(A, range_fm(i,j), env)"
849 by simp
851 theorem range_reflection:
852      "REFLECTS[\<lambda>x. is_range(L,f(x),g(x)),
853                \<lambda>i x. is_range(**Lset(i),f(x),g(x))]"
854 apply (simp only: is_range_def setclass_simps)
855 apply (intro FOL_reflections pair_reflection)
856 done
859 subsubsection{*Field of a Relation, Internalized*}
861 (* "is_field(M,r,z) ==
862         \<exists>dr[M]. is_domain(M,r,dr) &
863             (\<exists>rr[M]. is_range(M,r,rr) & union(M,dr,rr,z))" *)
864 constdefs field_fm :: "[i,i]=>i"
865     "field_fm(r,z) ==
866        Exists(And(domain_fm(succ(r),0),
867               Exists(And(range_fm(succ(succ(r)),0),
868                          union_fm(1,0,succ(succ(z)))))))"
870 lemma field_type [TC]:
871      "[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula"
874 lemma arity_field_fm [simp]:
875      "[| x \<in> nat; y \<in> nat |]
876       ==> arity(field_fm(x,y)) = succ(x) \<union> succ(y)"
877 by (simp add: field_fm_def succ_Un_distrib [symmetric] Un_ac)
879 lemma sats_field_fm [simp]:
880    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
881     ==> sats(A, field_fm(x,y), env) <->
882         is_field(**A, nth(x,env), nth(y,env))"
883 by (simp add: field_fm_def is_field_def)
885 lemma field_iff_sats:
886       "[| nth(i,env) = x; nth(j,env) = y;
887           i \<in> nat; j \<in> nat; env \<in> list(A)|]
888        ==> is_field(**A, x, y) <-> sats(A, field_fm(i,j), env)"
889 by simp
891 theorem field_reflection:
892      "REFLECTS[\<lambda>x. is_field(L,f(x),g(x)),
893                \<lambda>i x. is_field(**Lset(i),f(x),g(x))]"
894 apply (simp only: is_field_def setclass_simps)
895 apply (intro FOL_reflections domain_reflection range_reflection
896              union_reflection)
897 done
900 subsubsection{*Image under a Relation, Internalized*}
902 (* "image(M,r,A,z) ==
903         \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,w))))" *)
904 constdefs image_fm :: "[i,i,i]=>i"
905     "image_fm(r,A,z) ==
906        Forall(Iff(Member(0,succ(z)),
907                   Exists(And(Member(0,succ(succ(r))),
908                              Exists(And(Member(0,succ(succ(succ(A)))),
909                                         pair_fm(0,2,1)))))))"
911 lemma image_type [TC]:
912      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
915 lemma arity_image_fm [simp]:
916      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
917       ==> arity(image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
918 by (simp add: image_fm_def succ_Un_distrib [symmetric] Un_ac)
920 lemma sats_image_fm [simp]:
921    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
922     ==> sats(A, image_fm(x,y,z), env) <->
923         image(**A, nth(x,env), nth(y,env), nth(z,env))"
924 by (simp add: image_fm_def Relative.image_def)
926 lemma image_iff_sats:
927       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
928           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
929        ==> image(**A, x, y, z) <-> sats(A, image_fm(i,j,k), env)"
932 theorem image_reflection:
933      "REFLECTS[\<lambda>x. image(L,f(x),g(x),h(x)),
934                \<lambda>i x. image(**Lset(i),f(x),g(x),h(x))]"
935 apply (simp only: Relative.image_def setclass_simps)
936 apply (intro FOL_reflections pair_reflection)
937 done
940 subsubsection{*Pre-Image under a Relation, Internalized*}
942 (* "pre_image(M,r,A,z) ==
943         \<forall>x[M]. x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" *)
944 constdefs pre_image_fm :: "[i,i,i]=>i"
945     "pre_image_fm(r,A,z) ==
946        Forall(Iff(Member(0,succ(z)),
947                   Exists(And(Member(0,succ(succ(r))),
948                              Exists(And(Member(0,succ(succ(succ(A)))),
949                                         pair_fm(2,0,1)))))))"
951 lemma pre_image_type [TC]:
952      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
955 lemma arity_pre_image_fm [simp]:
956      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
957       ==> arity(pre_image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
958 by (simp add: pre_image_fm_def succ_Un_distrib [symmetric] Un_ac)
960 lemma sats_pre_image_fm [simp]:
961    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
962     ==> sats(A, pre_image_fm(x,y,z), env) <->
963         pre_image(**A, nth(x,env), nth(y,env), nth(z,env))"
964 by (simp add: pre_image_fm_def Relative.pre_image_def)
966 lemma pre_image_iff_sats:
967       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
968           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
969        ==> pre_image(**A, x, y, z) <-> sats(A, pre_image_fm(i,j,k), env)"
972 theorem pre_image_reflection:
973      "REFLECTS[\<lambda>x. pre_image(L,f(x),g(x),h(x)),
974                \<lambda>i x. pre_image(**Lset(i),f(x),g(x),h(x))]"
975 apply (simp only: Relative.pre_image_def setclass_simps)
976 apply (intro FOL_reflections pair_reflection)
977 done
980 subsubsection{*Function Application, Internalized*}
982 (* "fun_apply(M,f,x,y) ==
983         (\<exists>xs[M]. \<exists>fxs[M].
984          upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))" *)
985 constdefs fun_apply_fm :: "[i,i,i]=>i"
986     "fun_apply_fm(f,x,y) ==
987        Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1),
988                          And(image_fm(succ(succ(f)), 1, 0),
989                              big_union_fm(0,succ(succ(y)))))))"
991 lemma fun_apply_type [TC]:
992      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
995 lemma arity_fun_apply_fm [simp]:
996      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
997       ==> arity(fun_apply_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
998 by (simp add: fun_apply_fm_def succ_Un_distrib [symmetric] Un_ac)
1000 lemma sats_fun_apply_fm [simp]:
1001    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
1002     ==> sats(A, fun_apply_fm(x,y,z), env) <->
1003         fun_apply(**A, nth(x,env), nth(y,env), nth(z,env))"
1004 by (simp add: fun_apply_fm_def fun_apply_def)
1006 lemma fun_apply_iff_sats:
1007       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
1008           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
1009        ==> fun_apply(**A, x, y, z) <-> sats(A, fun_apply_fm(i,j,k), env)"
1010 by simp
1012 theorem fun_apply_reflection:
1013      "REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)),
1014                \<lambda>i x. fun_apply(**Lset(i),f(x),g(x),h(x))]"
1015 apply (simp only: fun_apply_def setclass_simps)
1016 apply (intro FOL_reflections upair_reflection image_reflection
1017              big_union_reflection)
1018 done
1021 subsubsection{*The Concept of Relation, Internalized*}
1023 (* "is_relation(M,r) ==
1024         (\<forall>z[M]. z\<in>r --> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))" *)
1025 constdefs relation_fm :: "i=>i"
1026     "relation_fm(r) ==
1027        Forall(Implies(Member(0,succ(r)), Exists(Exists(pair_fm(1,0,2)))))"
1029 lemma relation_type [TC]:
1030      "[| x \<in> nat |] ==> relation_fm(x) \<in> formula"
1033 lemma arity_relation_fm [simp]:
1034      "x \<in> nat ==> arity(relation_fm(x)) = succ(x)"
1035 by (simp add: relation_fm_def succ_Un_distrib [symmetric] Un_ac)
1037 lemma sats_relation_fm [simp]:
1038    "[| x \<in> nat; env \<in> list(A)|]
1039     ==> sats(A, relation_fm(x), env) <-> is_relation(**A, nth(x,env))"
1040 by (simp add: relation_fm_def is_relation_def)
1042 lemma relation_iff_sats:
1043       "[| nth(i,env) = x; nth(j,env) = y;
1044           i \<in> nat; env \<in> list(A)|]
1045        ==> is_relation(**A, x) <-> sats(A, relation_fm(i), env)"
1046 by simp
1048 theorem is_relation_reflection:
1049      "REFLECTS[\<lambda>x. is_relation(L,f(x)),
1050                \<lambda>i x. is_relation(**Lset(i),f(x))]"
1051 apply (simp only: is_relation_def setclass_simps)
1052 apply (intro FOL_reflections pair_reflection)
1053 done
1056 subsubsection{*The Concept of Function, Internalized*}
1058 (* "is_function(M,r) ==
1059         \<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M].
1060            pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>r --> y=y'" *)
1061 constdefs function_fm :: "i=>i"
1062     "function_fm(r) ==
1063        Forall(Forall(Forall(Forall(Forall(
1064          Implies(pair_fm(4,3,1),
1065                  Implies(pair_fm(4,2,0),
1066                          Implies(Member(1,r#+5),
1067                                  Implies(Member(0,r#+5), Equal(3,2))))))))))"
1069 lemma function_type [TC]:
1070      "[| x \<in> nat |] ==> function_fm(x) \<in> formula"
1073 lemma arity_function_fm [simp]:
1074      "x \<in> nat ==> arity(function_fm(x)) = succ(x)"
1075 by (simp add: function_fm_def succ_Un_distrib [symmetric] Un_ac)
1077 lemma sats_function_fm [simp]:
1078    "[| x \<in> nat; env \<in> list(A)|]
1079     ==> sats(A, function_fm(x), env) <-> is_function(**A, nth(x,env))"
1080 by (simp add: function_fm_def is_function_def)
1082 lemma function_iff_sats:
1083       "[| nth(i,env) = x; nth(j,env) = y;
1084           i \<in> nat; env \<in> list(A)|]
1085        ==> is_function(**A, x) <-> sats(A, function_fm(i), env)"
1086 by simp
1088 theorem is_function_reflection:
1089      "REFLECTS[\<lambda>x. is_function(L,f(x)),
1090                \<lambda>i x. is_function(**Lset(i),f(x))]"
1091 apply (simp only: is_function_def setclass_simps)
1092 apply (intro FOL_reflections pair_reflection)
1093 done
1096 subsubsection{*Typed Functions, Internalized*}
1098 (* "typed_function(M,A,B,r) ==
1099         is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
1100         (\<forall>u[M]. u\<in>r --> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) --> y\<in>B))" *)
1102 constdefs typed_function_fm :: "[i,i,i]=>i"
1103     "typed_function_fm(A,B,r) ==
1104        And(function_fm(r),
1105          And(relation_fm(r),
1106            And(domain_fm(r,A),
1107              Forall(Implies(Member(0,succ(r)),
1108                   Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))"
1110 lemma typed_function_type [TC]:
1111      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
1114 lemma arity_typed_function_fm [simp]:
1115      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
1116       ==> arity(typed_function_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
1117 by (simp add: typed_function_fm_def succ_Un_distrib [symmetric] Un_ac)
1119 lemma sats_typed_function_fm [simp]:
1120    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
1121     ==> sats(A, typed_function_fm(x,y,z), env) <->
1122         typed_function(**A, nth(x,env), nth(y,env), nth(z,env))"
1123 by (simp add: typed_function_fm_def typed_function_def)
1125 lemma typed_function_iff_sats:
1126   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
1127       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
1128    ==> typed_function(**A, x, y, z) <-> sats(A, typed_function_fm(i,j,k), env)"
1129 by simp
1131 lemmas function_reflections =
1132         empty_reflection number1_reflection
1133         upair_reflection pair_reflection union_reflection
1134         big_union_reflection cons_reflection successor_reflection
1135         fun_apply_reflection subset_reflection
1136         transitive_set_reflection membership_reflection
1137         pred_set_reflection domain_reflection range_reflection field_reflection
1138         image_reflection pre_image_reflection
1139         is_relation_reflection is_function_reflection
1141 lemmas function_iff_sats =
1142         empty_iff_sats number1_iff_sats
1143         upair_iff_sats pair_iff_sats union_iff_sats
1144         cons_iff_sats successor_iff_sats
1145         fun_apply_iff_sats  Memrel_iff_sats
1146         pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats
1147         image_iff_sats pre_image_iff_sats
1148         relation_iff_sats function_iff_sats
1151 theorem typed_function_reflection:
1152      "REFLECTS[\<lambda>x. typed_function(L,f(x),g(x),h(x)),
1153                \<lambda>i x. typed_function(**Lset(i),f(x),g(x),h(x))]"
1154 apply (simp only: typed_function_def setclass_simps)
1155 apply (intro FOL_reflections function_reflections)
1156 done
1159 subsubsection{*Composition of Relations, Internalized*}
1161 (* "composition(M,r,s,t) ==
1162         \<forall>p[M]. p \<in> t <->
1163                (\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M].
1164                 pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) &
1165                 xy \<in> s & yz \<in> r)" *)
1166 constdefs composition_fm :: "[i,i,i]=>i"
1167   "composition_fm(r,s,t) ==
1168      Forall(Iff(Member(0,succ(t)),
1169              Exists(Exists(Exists(Exists(Exists(
1170               And(pair_fm(4,2,5),
1171                And(pair_fm(4,3,1),
1172                 And(pair_fm(3,2,0),
1173                  And(Member(1,s#+6), Member(0,r#+6))))))))))))"
1175 lemma composition_type [TC]:
1176      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula"
1179 lemma arity_composition_fm [simp]:
1180      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
1181       ==> arity(composition_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
1182 by (simp add: composition_fm_def succ_Un_distrib [symmetric] Un_ac)
1184 lemma sats_composition_fm [simp]:
1185    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
1186     ==> sats(A, composition_fm(x,y,z), env) <->
1187         composition(**A, nth(x,env), nth(y,env), nth(z,env))"
1188 by (simp add: composition_fm_def composition_def)
1190 lemma composition_iff_sats:
1191       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
1192           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
1193        ==> composition(**A, x, y, z) <-> sats(A, composition_fm(i,j,k), env)"
1194 by simp
1196 theorem composition_reflection:
1197      "REFLECTS[\<lambda>x. composition(L,f(x),g(x),h(x)),
1198                \<lambda>i x. composition(**Lset(i),f(x),g(x),h(x))]"
1199 apply (simp only: composition_def setclass_simps)
1200 apply (intro FOL_reflections pair_reflection)
1201 done
1204 subsubsection{*Injections, Internalized*}
1206 (* "injection(M,A,B,f) ==
1207         typed_function(M,A,B,f) &
1208         (\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M].
1209           pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>f --> x=x')" *)
1210 constdefs injection_fm :: "[i,i,i]=>i"
1211  "injection_fm(A,B,f) ==
1212     And(typed_function_fm(A,B,f),
1213        Forall(Forall(Forall(Forall(Forall(
1214          Implies(pair_fm(4,2,1),
1215                  Implies(pair_fm(3,2,0),
1216                          Implies(Member(1,f#+5),
1217                                  Implies(Member(0,f#+5), Equal(4,3)))))))))))"
1220 lemma injection_type [TC]:
1221      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
1224 lemma arity_injection_fm [simp]:
1225      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
1226       ==> arity(injection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
1227 by (simp add: injection_fm_def succ_Un_distrib [symmetric] Un_ac)
1229 lemma sats_injection_fm [simp]:
1230    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
1231     ==> sats(A, injection_fm(x,y,z), env) <->
1232         injection(**A, nth(x,env), nth(y,env), nth(z,env))"
1233 by (simp add: injection_fm_def injection_def)
1235 lemma injection_iff_sats:
1236   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
1237       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
1238    ==> injection(**A, x, y, z) <-> sats(A, injection_fm(i,j,k), env)"
1239 by simp
1241 theorem injection_reflection:
1242      "REFLECTS[\<lambda>x. injection(L,f(x),g(x),h(x)),
1243                \<lambda>i x. injection(**Lset(i),f(x),g(x),h(x))]"
1244 apply (simp only: injection_def setclass_simps)
1245 apply (intro FOL_reflections function_reflections typed_function_reflection)
1246 done
1249 subsubsection{*Surjections, Internalized*}
1251 (*  surjection :: "[i=>o,i,i,i] => o"
1252     "surjection(M,A,B,f) ==
1253         typed_function(M,A,B,f) &
1254         (\<forall>y[M]. y\<in>B --> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" *)
1255 constdefs surjection_fm :: "[i,i,i]=>i"
1256  "surjection_fm(A,B,f) ==
1257     And(typed_function_fm(A,B,f),
1258        Forall(Implies(Member(0,succ(B)),
1259                       Exists(And(Member(0,succ(succ(A))),
1260                                  fun_apply_fm(succ(succ(f)),0,1))))))"
1262 lemma surjection_type [TC]:
1263      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula"
1266 lemma arity_surjection_fm [simp]:
1267      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
1268       ==> arity(surjection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
1269 by (simp add: surjection_fm_def succ_Un_distrib [symmetric] Un_ac)
1271 lemma sats_surjection_fm [simp]:
1272    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
1273     ==> sats(A, surjection_fm(x,y,z), env) <->
1274         surjection(**A, nth(x,env), nth(y,env), nth(z,env))"
1275 by (simp add: surjection_fm_def surjection_def)
1277 lemma surjection_iff_sats:
1278   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
1279       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
1280    ==> surjection(**A, x, y, z) <-> sats(A, surjection_fm(i,j,k), env)"
1281 by simp
1283 theorem surjection_reflection:
1284      "REFLECTS[\<lambda>x. surjection(L,f(x),g(x),h(x)),
1285                \<lambda>i x. surjection(**Lset(i),f(x),g(x),h(x))]"
1286 apply (simp only: surjection_def setclass_simps)
1287 apply (intro FOL_reflections function_reflections typed_function_reflection)
1288 done
1292 subsubsection{*Bijections, Internalized*}
1294 (*   bijection :: "[i=>o,i,i,i] => o"
1295     "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *)
1296 constdefs bijection_fm :: "[i,i,i]=>i"
1297  "bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))"
1299 lemma bijection_type [TC]:
1300      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula"
1303 lemma arity_bijection_fm [simp]:
1304      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
1305       ==> arity(bijection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
1306 by (simp add: bijection_fm_def succ_Un_distrib [symmetric] Un_ac)
1308 lemma sats_bijection_fm [simp]:
1309    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
1310     ==> sats(A, bijection_fm(x,y,z), env) <->
1311         bijection(**A, nth(x,env), nth(y,env), nth(z,env))"
1312 by (simp add: bijection_fm_def bijection_def)
1314 lemma bijection_iff_sats:
1315   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
1316       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
1317    ==> bijection(**A, x, y, z) <-> sats(A, bijection_fm(i,j,k), env)"
1318 by simp
1320 theorem bijection_reflection:
1321      "REFLECTS[\<lambda>x. bijection(L,f(x),g(x),h(x)),
1322                \<lambda>i x. bijection(**Lset(i),f(x),g(x),h(x))]"
1323 apply (simp only: bijection_def setclass_simps)
1324 apply (intro And_reflection injection_reflection surjection_reflection)
1325 done
1328 subsubsection{*Restriction of a Relation, Internalized*}
1331 (* "restriction(M,r,A,z) ==
1332         \<forall>x[M]. x \<in> z <-> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" *)
1333 constdefs restriction_fm :: "[i,i,i]=>i"
1334     "restriction_fm(r,A,z) ==
1335        Forall(Iff(Member(0,succ(z)),
1336                   And(Member(0,succ(r)),
1337                       Exists(And(Member(0,succ(succ(A))),
1338                                  Exists(pair_fm(1,0,2)))))))"
1340 lemma restriction_type [TC]:
1341      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula"
1344 lemma arity_restriction_fm [simp]:
1345      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
1346       ==> arity(restriction_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
1347 by (simp add: restriction_fm_def succ_Un_distrib [symmetric] Un_ac)
1349 lemma sats_restriction_fm [simp]:
1350    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
1351     ==> sats(A, restriction_fm(x,y,z), env) <->
1352         restriction(**A, nth(x,env), nth(y,env), nth(z,env))"
1353 by (simp add: restriction_fm_def restriction_def)
1355 lemma restriction_iff_sats:
1356       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
1357           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
1358        ==> restriction(**A, x, y, z) <-> sats(A, restriction_fm(i,j,k), env)"
1359 by simp
1361 theorem restriction_reflection:
1362      "REFLECTS[\<lambda>x. restriction(L,f(x),g(x),h(x)),
1363                \<lambda>i x. restriction(**Lset(i),f(x),g(x),h(x))]"
1364 apply (simp only: restriction_def setclass_simps)
1365 apply (intro FOL_reflections pair_reflection)
1366 done
1368 subsubsection{*Order-Isomorphisms, Internalized*}
1370 (*  order_isomorphism :: "[i=>o,i,i,i,i,i] => o"
1371    "order_isomorphism(M,A,r,B,s,f) ==
1372         bijection(M,A,B,f) &
1373         (\<forall>x[M]. x\<in>A --> (\<forall>y[M]. y\<in>A -->
1374           (\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>q[M].
1375             pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) -->
1376             pair(M,fx,fy,q) --> (p\<in>r <-> q\<in>s))))"
1377   *)
1379 constdefs order_isomorphism_fm :: "[i,i,i,i,i]=>i"
1380  "order_isomorphism_fm(A,r,B,s,f) ==
1381    And(bijection_fm(A,B,f),
1382      Forall(Implies(Member(0,succ(A)),
1383        Forall(Implies(Member(0,succ(succ(A))),
1384          Forall(Forall(Forall(Forall(
1385            Implies(pair_fm(5,4,3),
1386              Implies(fun_apply_fm(f#+6,5,2),
1387                Implies(fun_apply_fm(f#+6,4,1),
1388                  Implies(pair_fm(2,1,0),
1389                    Iff(Member(3,r#+6), Member(0,s#+6)))))))))))))))"
1391 lemma order_isomorphism_type [TC]:
1392      "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]
1393       ==> order_isomorphism_fm(A,r,B,s,f) \<in> formula"
1396 lemma arity_order_isomorphism_fm [simp]:
1397      "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]
1398       ==> arity(order_isomorphism_fm(A,r,B,s,f)) =
1399           succ(A) \<union> succ(r) \<union> succ(B) \<union> succ(s) \<union> succ(f)"
1400 by (simp add: order_isomorphism_fm_def succ_Un_distrib [symmetric] Un_ac)
1402 lemma sats_order_isomorphism_fm [simp]:
1403    "[| U \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat; env \<in> list(A)|]
1404     ==> sats(A, order_isomorphism_fm(U,r,B,s,f), env) <->
1405         order_isomorphism(**A, nth(U,env), nth(r,env), nth(B,env),
1406                                nth(s,env), nth(f,env))"
1407 by (simp add: order_isomorphism_fm_def order_isomorphism_def)
1409 lemma order_isomorphism_iff_sats:
1410   "[| nth(i,env) = U; nth(j,env) = r; nth(k,env) = B; nth(j',env) = s;
1411       nth(k',env) = f;
1412       i \<in> nat; j \<in> nat; k \<in> nat; j' \<in> nat; k' \<in> nat; env \<in> list(A)|]
1413    ==> order_isomorphism(**A,U,r,B,s,f) <->
1414        sats(A, order_isomorphism_fm(i,j,k,j',k'), env)"
1415 by simp
1417 theorem order_isomorphism_reflection:
1418      "REFLECTS[\<lambda>x. order_isomorphism(L,f(x),g(x),h(x),g'(x),h'(x)),
1419                \<lambda>i x. order_isomorphism(**Lset(i),f(x),g(x),h(x),g'(x),h'(x))]"
1420 apply (simp only: order_isomorphism_def setclass_simps)
1421 apply (intro FOL_reflections function_reflections bijection_reflection)
1422 done
1424 subsubsection{*Limit Ordinals, Internalized*}
1426 text{*A limit ordinal is a non-empty, successor-closed ordinal*}
1428 (* "limit_ordinal(M,a) ==
1429         ordinal(M,a) & ~ empty(M,a) &
1430         (\<forall>x[M]. x\<in>a --> (\<exists>y[M]. y\<in>a & successor(M,x,y)))" *)
1432 constdefs limit_ordinal_fm :: "i=>i"
1433     "limit_ordinal_fm(x) ==
1434         And(ordinal_fm(x),
1435             And(Neg(empty_fm(x)),
1436                 Forall(Implies(Member(0,succ(x)),
1437                                Exists(And(Member(0,succ(succ(x))),
1438                                           succ_fm(1,0)))))))"
1440 lemma limit_ordinal_type [TC]:
1441      "x \<in> nat ==> limit_ordinal_fm(x) \<in> formula"
1444 lemma arity_limit_ordinal_fm [simp]:
1445      "x \<in> nat ==> arity(limit_ordinal_fm(x)) = succ(x)"
1446 by (simp add: limit_ordinal_fm_def succ_Un_distrib [symmetric] Un_ac)
1448 lemma sats_limit_ordinal_fm [simp]:
1449    "[| x \<in> nat; env \<in> list(A)|]
1450     ==> sats(A, limit_ordinal_fm(x), env) <-> limit_ordinal(**A, nth(x,env))"
1451 by (simp add: limit_ordinal_fm_def limit_ordinal_def sats_ordinal_fm')
1453 lemma limit_ordinal_iff_sats:
1454       "[| nth(i,env) = x; nth(j,env) = y;
1455           i \<in> nat; env \<in> list(A)|]
1456        ==> limit_ordinal(**A, x) <-> sats(A, limit_ordinal_fm(i), env)"
1457 by simp
1459 theorem limit_ordinal_reflection:
1460      "REFLECTS[\<lambda>x. limit_ordinal(L,f(x)),
1461                \<lambda>i x. limit_ordinal(**Lset(i),f(x))]"
1462 apply (simp only: limit_ordinal_def setclass_simps)
1463 apply (intro FOL_reflections ordinal_reflection
1464              empty_reflection successor_reflection)
1465 done
1467 subsubsection{*Omega: The Set of Natural Numbers*}
1469 (* omega(M,a) == limit_ordinal(M,a) & (\<forall>x[M]. x\<in>a --> ~ limit_ordinal(M,x)) *)
1470 constdefs omega_fm :: "i=>i"
1471     "omega_fm(x) ==
1472        And(limit_ordinal_fm(x),
1473            Forall(Implies(Member(0,succ(x)),
1474                           Neg(limit_ordinal_fm(0)))))"
1476 lemma omega_type [TC]:
1477      "x \<in> nat ==> omega_fm(x) \<in> formula"
1480 lemma arity_omega_fm [simp]:
1481      "x \<in> nat ==> arity(omega_fm(x)) = succ(x)"
1482 by (simp add: omega_fm_def succ_Un_distrib [symmetric] Un_ac)
1484 lemma sats_omega_fm [simp]:
1485    "[| x \<in> nat; env \<in> list(A)|]
1486     ==> sats(A, omega_fm(x), env) <-> omega(**A, nth(x,env))"
1487 by (simp add: omega_fm_def omega_def)
1489 lemma omega_iff_sats:
1490       "[| nth(i,env) = x; nth(j,env) = y;
1491           i \<in> nat; env \<in> list(A)|]
1492        ==> omega(**A, x) <-> sats(A, omega_fm(i), env)"
1493 by simp
1495 theorem omega_reflection:
1496      "REFLECTS[\<lambda>x. omega(L,f(x)),
1497                \<lambda>i x. omega(**Lset(i),f(x))]"
1498 apply (simp only: omega_def setclass_simps)
1499 apply (intro FOL_reflections limit_ordinal_reflection)
1500 done
1503 lemmas fun_plus_reflections =
1504         typed_function_reflection composition_reflection
1505         injection_reflection surjection_reflection
1506         bijection_reflection restriction_reflection
1507         order_isomorphism_reflection
1508         ordinal_reflection limit_ordinal_reflection omega_reflection
1510 lemmas fun_plus_iff_sats =
1511         typed_function_iff_sats composition_iff_sats
1512         injection_iff_sats surjection_iff_sats
1513         bijection_iff_sats restriction_iff_sats
1514         order_isomorphism_iff_sats
1515         ordinal_iff_sats limit_ordinal_iff_sats omega_iff_sats
1517 end