src/ZF/Constructible/L_axioms.thy
author wenzelm
Mon Dec 04 22:54:31 2017 +0100 (21 months ago)
changeset 67131 85d10959c2e4
parent 61798 27f3c10b0b50
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     1 (*  Title:      ZF/Constructible/L_axioms.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3 *)
     4 
     5 section \<open>The ZF Axioms (Except Separation) in L\<close>
     6 
     7 theory L_axioms imports Formula Relative Reflection MetaExists begin
     8 
     9 text \<open>The class L satisfies the premises of locale \<open>M_trivial\<close>\<close>
    10 
    11 lemma transL: "[| y\<in>x; L(x) |] ==> L(y)"
    12 apply (insert Transset_Lset)
    13 apply (simp add: Transset_def L_def, blast)
    14 done
    15 
    16 lemma nonempty: "L(0)"
    17 apply (simp add: L_def)
    18 apply (blast intro: zero_in_Lset)
    19 done
    20 
    21 theorem upair_ax: "upair_ax(L)"
    22 apply (simp add: upair_ax_def upair_def, clarify)
    23 apply (rule_tac x="{x,y}" in rexI)
    24 apply (simp_all add: doubleton_in_L)
    25 done
    26 
    27 theorem Union_ax: "Union_ax(L)"
    28 apply (simp add: Union_ax_def big_union_def, clarify)
    29 apply (rule_tac x="\<Union>(x)" in rexI)
    30 apply (simp_all add: Union_in_L, auto)
    31 apply (blast intro: transL)
    32 done
    33 
    34 theorem power_ax: "power_ax(L)"
    35 apply (simp add: power_ax_def powerset_def Relative.subset_def, clarify)
    36 apply (rule_tac x="{y \<in> Pow(x). L(y)}" in rexI)
    37 apply (simp_all add: LPow_in_L, auto)
    38 apply (blast intro: transL)
    39 done
    40 
    41 text\<open>We don't actually need @{term L} to satisfy the foundation axiom.\<close>
    42 theorem foundation_ax: "foundation_ax(L)"
    43 apply (simp add: foundation_ax_def)
    44 apply (rule rallI) 
    45 apply (cut_tac A=x in foundation)
    46 apply (blast intro: transL)
    47 done
    48 
    49 subsection\<open>For L to satisfy Replacement\<close>
    50 
    51 (*Can't move these to Formula unless the definition of univalent is moved
    52 there too!*)
    53 
    54 lemma LReplace_in_Lset:
    55      "[|X \<in> Lset(i); univalent(L,X,Q); Ord(i)|]
    56       ==> \<exists>j. Ord(j) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Lset(j)"
    57 apply (rule_tac x="\<Union>y \<in> Replace(X, %x y. Q(x,y) & L(y)). succ(lrank(y))"
    58        in exI)
    59 apply simp
    60 apply clarify
    61 apply (rule_tac a=x in UN_I)
    62  apply (simp_all add: Replace_iff univalent_def)
    63 apply (blast dest: transL L_I)
    64 done
    65 
    66 lemma LReplace_in_L:
    67      "[|L(X); univalent(L,X,Q)|]
    68       ==> \<exists>Y. L(Y) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Y"
    69 apply (drule L_D, clarify)
    70 apply (drule LReplace_in_Lset, assumption+)
    71 apply (blast intro: L_I Lset_in_Lset_succ)
    72 done
    73 
    74 theorem replacement: "replacement(L,P)"
    75 apply (simp add: replacement_def, clarify)
    76 apply (frule LReplace_in_L, assumption+, clarify)
    77 apply (rule_tac x=Y in rexI)
    78 apply (simp_all add: Replace_iff univalent_def, blast)
    79 done
    80 
    81 subsection\<open>Instantiating the locale \<open>M_trivial\<close>\<close>
    82 text\<open>No instances of Separation yet.\<close>
    83 
    84 lemma Lset_mono_le: "mono_le_subset(Lset)"
    85 by (simp add: mono_le_subset_def le_imp_subset Lset_mono)
    86 
    87 lemma Lset_cont: "cont_Ord(Lset)"
    88 by (simp add: cont_Ord_def Limit_Lset_eq OUnion_def Limit_is_Ord)
    89 
    90 lemmas L_nat = Ord_in_L [OF Ord_nat]
    91 
    92 theorem M_trivial_L: "PROP M_trivial(L)"
    93   apply (rule M_trivial.intro)
    94        apply (erule (1) transL)
    95       apply (rule upair_ax)
    96      apply (rule Union_ax)
    97     apply (rule power_ax)
    98    apply (rule replacement)
    99   apply (rule L_nat)
   100   done
   101 
   102 interpretation L?: M_trivial L by (rule M_trivial_L)
   103 
   104 (* Replaces the following declarations...
   105 lemmas rall_abs = M_trivial.rall_abs [OF M_trivial_L]
   106   and rex_abs = M_trivial.rex_abs [OF M_trivial_L]
   107 ...
   108 declare rall_abs [simp]
   109 declare rex_abs [simp]
   110 ...and dozens of similar ones.
   111 *)
   112 
   113 subsection\<open>Instantiation of the locale \<open>reflection\<close>\<close>
   114 
   115 text\<open>instances of locale constants\<close>
   116 
   117 definition
   118   L_F0 :: "[i=>o,i] => i" where
   119     "L_F0(P,y) == \<mu> b. (\<exists>z. L(z) \<and> P(<y,z>)) \<longrightarrow> (\<exists>z\<in>Lset(b). P(<y,z>))"
   120 
   121 definition
   122   L_FF :: "[i=>o,i] => i" where
   123     "L_FF(P)   == \<lambda>a. \<Union>y\<in>Lset(a). L_F0(P,y)"
   124 
   125 definition
   126   L_ClEx :: "[i=>o,i] => o" where
   127     "L_ClEx(P) == \<lambda>a. Limit(a) \<and> normalize(L_FF(P),a) = a"
   128 
   129 
   130 text\<open>We must use the meta-existential quantifier; otherwise the reflection
   131       terms become enormous!\<close>
   132 definition
   133   L_Reflects :: "[i=>o,[i,i]=>o] => prop"  ("(3REFLECTS/ [_,/ _])") where
   134     "REFLECTS[P,Q] == (\<Or>Cl. Closed_Unbounded(Cl) &
   135                            (\<forall>a. Cl(a) \<longrightarrow> (\<forall>x \<in> Lset(a). P(x) \<longleftrightarrow> Q(a,x))))"
   136 
   137 
   138 theorem Triv_reflection:
   139      "REFLECTS[P, \<lambda>a x. P(x)]"
   140 apply (simp add: L_Reflects_def)
   141 apply (rule meta_exI)
   142 apply (rule Closed_Unbounded_Ord)
   143 done
   144 
   145 theorem Not_reflection:
   146      "REFLECTS[P,Q] ==> REFLECTS[\<lambda>x. ~P(x), \<lambda>a x. ~Q(a,x)]"
   147 apply (unfold L_Reflects_def)
   148 apply (erule meta_exE)
   149 apply (rule_tac x=Cl in meta_exI, simp)
   150 done
   151 
   152 theorem And_reflection:
   153      "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
   154       ==> REFLECTS[\<lambda>x. P(x) \<and> P'(x), \<lambda>a x. Q(a,x) \<and> Q'(a,x)]"
   155 apply (unfold L_Reflects_def)
   156 apply (elim meta_exE)
   157 apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
   158 apply (simp add: Closed_Unbounded_Int, blast)
   159 done
   160 
   161 theorem Or_reflection:
   162      "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
   163       ==> REFLECTS[\<lambda>x. P(x) \<or> P'(x), \<lambda>a x. Q(a,x) \<or> Q'(a,x)]"
   164 apply (unfold L_Reflects_def)
   165 apply (elim meta_exE)
   166 apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
   167 apply (simp add: Closed_Unbounded_Int, blast)
   168 done
   169 
   170 theorem Imp_reflection:
   171      "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
   172       ==> REFLECTS[\<lambda>x. P(x) \<longrightarrow> P'(x), \<lambda>a x. Q(a,x) \<longrightarrow> Q'(a,x)]"
   173 apply (unfold L_Reflects_def)
   174 apply (elim meta_exE)
   175 apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
   176 apply (simp add: Closed_Unbounded_Int, blast)
   177 done
   178 
   179 theorem Iff_reflection:
   180      "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
   181       ==> REFLECTS[\<lambda>x. P(x) \<longleftrightarrow> P'(x), \<lambda>a x. Q(a,x) \<longleftrightarrow> Q'(a,x)]"
   182 apply (unfold L_Reflects_def)
   183 apply (elim meta_exE)
   184 apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
   185 apply (simp add: Closed_Unbounded_Int, blast)
   186 done
   187 
   188 
   189 lemma reflection_Lset: "reflection(Lset)"
   190 by (blast intro: reflection.intro Lset_mono_le Lset_cont 
   191                  Formula.Pair_in_LLimit)+
   192 
   193 
   194 theorem Ex_reflection:
   195      "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
   196       ==> REFLECTS[\<lambda>x. \<exists>z. L(z) \<and> P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
   197 apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
   198 apply (elim meta_exE)
   199 apply (rule meta_exI)
   200 apply (erule reflection.Ex_reflection [OF reflection_Lset])
   201 done
   202 
   203 theorem All_reflection:
   204      "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
   205       ==> REFLECTS[\<lambda>x. \<forall>z. L(z) \<longrightarrow> P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]"
   206 apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
   207 apply (elim meta_exE)
   208 apply (rule meta_exI)
   209 apply (erule reflection.All_reflection [OF reflection_Lset])
   210 done
   211 
   212 theorem Rex_reflection:
   213      "REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
   214       ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
   215 apply (unfold rex_def)
   216 apply (intro And_reflection Ex_reflection, assumption)
   217 done
   218 
   219 theorem Rall_reflection:
   220      "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
   221       ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]"
   222 apply (unfold rall_def)
   223 apply (intro Imp_reflection All_reflection, assumption)
   224 done
   225 
   226 text\<open>This version handles an alternative form of the bounded quantifier
   227       in the second argument of \<open>REFLECTS\<close>.\<close>
   228 theorem Rex_reflection':
   229      "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
   230       ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z[##Lset(a)]. Q(a,x,z)]"
   231 apply (unfold setclass_def rex_def)
   232 apply (erule Rex_reflection [unfolded rex_def Bex_def]) 
   233 done
   234 
   235 text\<open>As above.\<close>
   236 theorem Rall_reflection':
   237      "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
   238       ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z[##Lset(a)]. Q(a,x,z)]"
   239 apply (unfold setclass_def rall_def)
   240 apply (erule Rall_reflection [unfolded rall_def Ball_def]) 
   241 done
   242 
   243 lemmas FOL_reflections =
   244         Triv_reflection Not_reflection And_reflection Or_reflection
   245         Imp_reflection Iff_reflection Ex_reflection All_reflection
   246         Rex_reflection Rall_reflection Rex_reflection' Rall_reflection'
   247 
   248 lemma ReflectsD:
   249      "[|REFLECTS[P,Q]; Ord(i)|]
   250       ==> \<exists>j. i<j & (\<forall>x \<in> Lset(j). P(x) \<longleftrightarrow> Q(j,x))"
   251 apply (unfold L_Reflects_def Closed_Unbounded_def)
   252 apply (elim meta_exE, clarify)
   253 apply (blast dest!: UnboundedD)
   254 done
   255 
   256 lemma ReflectsE:
   257      "[| REFLECTS[P,Q]; Ord(i);
   258          !!j. [|i<j;  \<forall>x \<in> Lset(j). P(x) \<longleftrightarrow> Q(j,x)|] ==> R |]
   259       ==> R"
   260 by (drule ReflectsD, assumption, blast)
   261 
   262 lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B"
   263 by blast
   264 
   265 
   266 subsection\<open>Internalized Formulas for some Set-Theoretic Concepts\<close>
   267 
   268 subsubsection\<open>Some numbers to help write de Bruijn indices\<close>
   269 
   270 abbreviation
   271   digit3 :: i   ("3") where "3 == succ(2)"
   272 
   273 abbreviation
   274   digit4 :: i   ("4") where "4 == succ(3)"
   275 
   276 abbreviation
   277   digit5 :: i   ("5") where "5 == succ(4)"
   278 
   279 abbreviation
   280   digit6 :: i   ("6") where "6 == succ(5)"
   281 
   282 abbreviation
   283   digit7 :: i   ("7") where "7 == succ(6)"
   284 
   285 abbreviation
   286   digit8 :: i   ("8") where "8 == succ(7)"
   287 
   288 abbreviation
   289   digit9 :: i   ("9") where "9 == succ(8)"
   290 
   291 
   292 subsubsection\<open>The Empty Set, Internalized\<close>
   293 
   294 definition
   295   empty_fm :: "i=>i" where
   296     "empty_fm(x) == Forall(Neg(Member(0,succ(x))))"
   297 
   298 lemma empty_type [TC]:
   299      "x \<in> nat ==> empty_fm(x) \<in> formula"
   300 by (simp add: empty_fm_def)
   301 
   302 lemma sats_empty_fm [simp]:
   303    "[| x \<in> nat; env \<in> list(A)|]
   304     ==> sats(A, empty_fm(x), env) \<longleftrightarrow> empty(##A, nth(x,env))"
   305 by (simp add: empty_fm_def empty_def)
   306 
   307 lemma empty_iff_sats:
   308       "[| nth(i,env) = x; nth(j,env) = y;
   309           i \<in> nat; env \<in> list(A)|]
   310        ==> empty(##A, x) \<longleftrightarrow> sats(A, empty_fm(i), env)"
   311 by simp
   312 
   313 theorem empty_reflection:
   314      "REFLECTS[\<lambda>x. empty(L,f(x)),
   315                \<lambda>i x. empty(##Lset(i),f(x))]"
   316 apply (simp only: empty_def)
   317 apply (intro FOL_reflections)
   318 done
   319 
   320 text\<open>Not used.  But maybe useful?\<close>
   321 lemma Transset_sats_empty_fm_eq_0:
   322    "[| n \<in> nat; env \<in> list(A); Transset(A)|]
   323     ==> sats(A, empty_fm(n), env) \<longleftrightarrow> nth(n,env) = 0"
   324 apply (simp add: empty_fm_def empty_def Transset_def, auto)
   325 apply (case_tac "n < length(env)")
   326 apply (frule nth_type, assumption+, blast)
   327 apply (simp_all add: not_lt_iff_le nth_eq_0)
   328 done
   329 
   330 
   331 subsubsection\<open>Unordered Pairs, Internalized\<close>
   332 
   333 definition
   334   upair_fm :: "[i,i,i]=>i" where
   335     "upair_fm(x,y,z) ==
   336        And(Member(x,z),
   337            And(Member(y,z),
   338                Forall(Implies(Member(0,succ(z)),
   339                               Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
   340 
   341 lemma upair_type [TC]:
   342      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
   343 by (simp add: upair_fm_def)
   344 
   345 lemma sats_upair_fm [simp]:
   346    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   347     ==> sats(A, upair_fm(x,y,z), env) \<longleftrightarrow>
   348             upair(##A, nth(x,env), nth(y,env), nth(z,env))"
   349 by (simp add: upair_fm_def upair_def)
   350 
   351 lemma upair_iff_sats:
   352       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   353           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   354        ==> upair(##A, x, y, z) \<longleftrightarrow> sats(A, upair_fm(i,j,k), env)"
   355 by (simp add: sats_upair_fm)
   356 
   357 text\<open>Useful? At least it refers to "real" unordered pairs\<close>
   358 lemma sats_upair_fm2 [simp]:
   359    "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
   360     ==> sats(A, upair_fm(x,y,z), env) \<longleftrightarrow>
   361         nth(z,env) = {nth(x,env), nth(y,env)}"
   362 apply (frule lt_length_in_nat, assumption)
   363 apply (simp add: upair_fm_def Transset_def, auto)
   364 apply (blast intro: nth_type)
   365 done
   366 
   367 theorem upair_reflection:
   368      "REFLECTS[\<lambda>x. upair(L,f(x),g(x),h(x)),
   369                \<lambda>i x. upair(##Lset(i),f(x),g(x),h(x))]"
   370 apply (simp add: upair_def)
   371 apply (intro FOL_reflections)
   372 done
   373 
   374 subsubsection\<open>Ordered pairs, Internalized\<close>
   375 
   376 definition
   377   pair_fm :: "[i,i,i]=>i" where
   378     "pair_fm(x,y,z) ==
   379        Exists(And(upair_fm(succ(x),succ(x),0),
   380               Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
   381                          upair_fm(1,0,succ(succ(z)))))))"
   382 
   383 lemma pair_type [TC]:
   384      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
   385 by (simp add: pair_fm_def)
   386 
   387 lemma sats_pair_fm [simp]:
   388    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   389     ==> sats(A, pair_fm(x,y,z), env) \<longleftrightarrow>
   390         pair(##A, nth(x,env), nth(y,env), nth(z,env))"
   391 by (simp add: pair_fm_def pair_def)
   392 
   393 lemma pair_iff_sats:
   394       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   395           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   396        ==> pair(##A, x, y, z) \<longleftrightarrow> sats(A, pair_fm(i,j,k), env)"
   397 by (simp add: sats_pair_fm)
   398 
   399 theorem pair_reflection:
   400      "REFLECTS[\<lambda>x. pair(L,f(x),g(x),h(x)),
   401                \<lambda>i x. pair(##Lset(i),f(x),g(x),h(x))]"
   402 apply (simp only: pair_def)
   403 apply (intro FOL_reflections upair_reflection)
   404 done
   405 
   406 
   407 subsubsection\<open>Binary Unions, Internalized\<close>
   408 
   409 definition
   410   union_fm :: "[i,i,i]=>i" where
   411     "union_fm(x,y,z) ==
   412        Forall(Iff(Member(0,succ(z)),
   413                   Or(Member(0,succ(x)),Member(0,succ(y)))))"
   414 
   415 lemma union_type [TC]:
   416      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
   417 by (simp add: union_fm_def)
   418 
   419 lemma sats_union_fm [simp]:
   420    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   421     ==> sats(A, union_fm(x,y,z), env) \<longleftrightarrow>
   422         union(##A, nth(x,env), nth(y,env), nth(z,env))"
   423 by (simp add: union_fm_def union_def)
   424 
   425 lemma union_iff_sats:
   426       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   427           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   428        ==> union(##A, x, y, z) \<longleftrightarrow> sats(A, union_fm(i,j,k), env)"
   429 by (simp add: sats_union_fm)
   430 
   431 theorem union_reflection:
   432      "REFLECTS[\<lambda>x. union(L,f(x),g(x),h(x)),
   433                \<lambda>i x. union(##Lset(i),f(x),g(x),h(x))]"
   434 apply (simp only: union_def)
   435 apply (intro FOL_reflections)
   436 done
   437 
   438 
   439 subsubsection\<open>Set ``Cons,'' Internalized\<close>
   440 
   441 definition
   442   cons_fm :: "[i,i,i]=>i" where
   443     "cons_fm(x,y,z) ==
   444        Exists(And(upair_fm(succ(x),succ(x),0),
   445                   union_fm(0,succ(y),succ(z))))"
   446 
   447 
   448 lemma cons_type [TC]:
   449      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
   450 by (simp add: cons_fm_def)
   451 
   452 lemma sats_cons_fm [simp]:
   453    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   454     ==> sats(A, cons_fm(x,y,z), env) \<longleftrightarrow>
   455         is_cons(##A, nth(x,env), nth(y,env), nth(z,env))"
   456 by (simp add: cons_fm_def is_cons_def)
   457 
   458 lemma cons_iff_sats:
   459       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   460           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   461        ==> is_cons(##A, x, y, z) \<longleftrightarrow> sats(A, cons_fm(i,j,k), env)"
   462 by simp
   463 
   464 theorem cons_reflection:
   465      "REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)),
   466                \<lambda>i x. is_cons(##Lset(i),f(x),g(x),h(x))]"
   467 apply (simp only: is_cons_def)
   468 apply (intro FOL_reflections upair_reflection union_reflection)
   469 done
   470 
   471 
   472 subsubsection\<open>Successor Function, Internalized\<close>
   473 
   474 definition
   475   succ_fm :: "[i,i]=>i" where
   476     "succ_fm(x,y) == cons_fm(x,x,y)"
   477 
   478 lemma succ_type [TC]:
   479      "[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula"
   480 by (simp add: succ_fm_def)
   481 
   482 lemma sats_succ_fm [simp]:
   483    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   484     ==> sats(A, succ_fm(x,y), env) \<longleftrightarrow>
   485         successor(##A, nth(x,env), nth(y,env))"
   486 by (simp add: succ_fm_def successor_def)
   487 
   488 lemma successor_iff_sats:
   489       "[| nth(i,env) = x; nth(j,env) = y;
   490           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   491        ==> successor(##A, x, y) \<longleftrightarrow> sats(A, succ_fm(i,j), env)"
   492 by simp
   493 
   494 theorem successor_reflection:
   495      "REFLECTS[\<lambda>x. successor(L,f(x),g(x)),
   496                \<lambda>i x. successor(##Lset(i),f(x),g(x))]"
   497 apply (simp only: successor_def)
   498 apply (intro cons_reflection)
   499 done
   500 
   501 
   502 subsubsection\<open>The Number 1, Internalized\<close>
   503 
   504 (* "number1(M,a) == (\<exists>x[M]. empty(M,x) & successor(M,x,a))" *)
   505 definition
   506   number1_fm :: "i=>i" where
   507     "number1_fm(a) == Exists(And(empty_fm(0), succ_fm(0,succ(a))))"
   508 
   509 lemma number1_type [TC]:
   510      "x \<in> nat ==> number1_fm(x) \<in> formula"
   511 by (simp add: number1_fm_def)
   512 
   513 lemma sats_number1_fm [simp]:
   514    "[| x \<in> nat; env \<in> list(A)|]
   515     ==> sats(A, number1_fm(x), env) \<longleftrightarrow> number1(##A, nth(x,env))"
   516 by (simp add: number1_fm_def number1_def)
   517 
   518 lemma number1_iff_sats:
   519       "[| nth(i,env) = x; nth(j,env) = y;
   520           i \<in> nat; env \<in> list(A)|]
   521        ==> number1(##A, x) \<longleftrightarrow> sats(A, number1_fm(i), env)"
   522 by simp
   523 
   524 theorem number1_reflection:
   525      "REFLECTS[\<lambda>x. number1(L,f(x)),
   526                \<lambda>i x. number1(##Lset(i),f(x))]"
   527 apply (simp only: number1_def)
   528 apply (intro FOL_reflections empty_reflection successor_reflection)
   529 done
   530 
   531 
   532 subsubsection\<open>Big Union, Internalized\<close>
   533 
   534 (*  "big_union(M,A,z) == \<forall>x[M]. x \<in> z \<longleftrightarrow> (\<exists>y[M]. y\<in>A & x \<in> y)" *)
   535 definition
   536   big_union_fm :: "[i,i]=>i" where
   537     "big_union_fm(A,z) ==
   538        Forall(Iff(Member(0,succ(z)),
   539                   Exists(And(Member(0,succ(succ(A))), Member(1,0)))))"
   540 
   541 lemma big_union_type [TC]:
   542      "[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula"
   543 by (simp add: big_union_fm_def)
   544 
   545 lemma sats_big_union_fm [simp]:
   546    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   547     ==> sats(A, big_union_fm(x,y), env) \<longleftrightarrow>
   548         big_union(##A, nth(x,env), nth(y,env))"
   549 by (simp add: big_union_fm_def big_union_def)
   550 
   551 lemma big_union_iff_sats:
   552       "[| nth(i,env) = x; nth(j,env) = y;
   553           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   554        ==> big_union(##A, x, y) \<longleftrightarrow> sats(A, big_union_fm(i,j), env)"
   555 by simp
   556 
   557 theorem big_union_reflection:
   558      "REFLECTS[\<lambda>x. big_union(L,f(x),g(x)),
   559                \<lambda>i x. big_union(##Lset(i),f(x),g(x))]"
   560 apply (simp only: big_union_def)
   561 apply (intro FOL_reflections)
   562 done
   563 
   564 
   565 subsubsection\<open>Variants of Satisfaction Definitions for Ordinals, etc.\<close>
   566 
   567 text\<open>The \<open>sats\<close> theorems below are standard versions of the ones proved
   568 in theory \<open>Formula\<close>.  They relate elements of type @{term formula} to
   569 relativized concepts such as @{term subset} or @{term ordinal} rather than to
   570 real concepts such as @{term Ord}.  Now that we have instantiated the locale
   571 \<open>M_trivial\<close>, we no longer require the earlier versions.\<close>
   572 
   573 lemma sats_subset_fm':
   574    "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
   575     ==> sats(A, subset_fm(x,y), env) \<longleftrightarrow> subset(##A, nth(x,env), nth(y,env))"
   576 by (simp add: subset_fm_def Relative.subset_def)
   577 
   578 theorem subset_reflection:
   579      "REFLECTS[\<lambda>x. subset(L,f(x),g(x)),
   580                \<lambda>i x. subset(##Lset(i),f(x),g(x))]"
   581 apply (simp only: Relative.subset_def)
   582 apply (intro FOL_reflections)
   583 done
   584 
   585 lemma sats_transset_fm':
   586    "[|x \<in> nat; env \<in> list(A)|]
   587     ==> sats(A, transset_fm(x), env) \<longleftrightarrow> transitive_set(##A, nth(x,env))"
   588 by (simp add: sats_subset_fm' transset_fm_def transitive_set_def)
   589 
   590 theorem transitive_set_reflection:
   591      "REFLECTS[\<lambda>x. transitive_set(L,f(x)),
   592                \<lambda>i x. transitive_set(##Lset(i),f(x))]"
   593 apply (simp only: transitive_set_def)
   594 apply (intro FOL_reflections subset_reflection)
   595 done
   596 
   597 lemma sats_ordinal_fm':
   598    "[|x \<in> nat; env \<in> list(A)|]
   599     ==> sats(A, ordinal_fm(x), env) \<longleftrightarrow> ordinal(##A,nth(x,env))"
   600 by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def)
   601 
   602 lemma ordinal_iff_sats:
   603       "[| nth(i,env) = x;  i \<in> nat; env \<in> list(A)|]
   604        ==> ordinal(##A, x) \<longleftrightarrow> sats(A, ordinal_fm(i), env)"
   605 by (simp add: sats_ordinal_fm')
   606 
   607 theorem ordinal_reflection:
   608      "REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(##Lset(i),f(x))]"
   609 apply (simp only: ordinal_def)
   610 apply (intro FOL_reflections transitive_set_reflection)
   611 done
   612 
   613 
   614 subsubsection\<open>Membership Relation, Internalized\<close>
   615 
   616 definition
   617   Memrel_fm :: "[i,i]=>i" where
   618     "Memrel_fm(A,r) ==
   619        Forall(Iff(Member(0,succ(r)),
   620                   Exists(And(Member(0,succ(succ(A))),
   621                              Exists(And(Member(0,succ(succ(succ(A)))),
   622                                         And(Member(1,0),
   623                                             pair_fm(1,0,2))))))))"
   624 
   625 lemma Memrel_type [TC]:
   626      "[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
   627 by (simp add: Memrel_fm_def)
   628 
   629 lemma sats_Memrel_fm [simp]:
   630    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   631     ==> sats(A, Memrel_fm(x,y), env) \<longleftrightarrow>
   632         membership(##A, nth(x,env), nth(y,env))"
   633 by (simp add: Memrel_fm_def membership_def)
   634 
   635 lemma Memrel_iff_sats:
   636       "[| nth(i,env) = x; nth(j,env) = y;
   637           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   638        ==> membership(##A, x, y) \<longleftrightarrow> sats(A, Memrel_fm(i,j), env)"
   639 by simp
   640 
   641 theorem membership_reflection:
   642      "REFLECTS[\<lambda>x. membership(L,f(x),g(x)),
   643                \<lambda>i x. membership(##Lset(i),f(x),g(x))]"
   644 apply (simp only: membership_def)
   645 apply (intro FOL_reflections pair_reflection)
   646 done
   647 
   648 subsubsection\<open>Predecessor Set, Internalized\<close>
   649 
   650 definition
   651   pred_set_fm :: "[i,i,i,i]=>i" where
   652     "pred_set_fm(A,x,r,B) ==
   653        Forall(Iff(Member(0,succ(B)),
   654                   Exists(And(Member(0,succ(succ(r))),
   655                              And(Member(1,succ(succ(A))),
   656                                  pair_fm(1,succ(succ(x)),0))))))"
   657 
   658 
   659 lemma pred_set_type [TC]:
   660      "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |]
   661       ==> pred_set_fm(A,x,r,B) \<in> formula"
   662 by (simp add: pred_set_fm_def)
   663 
   664 lemma sats_pred_set_fm [simp]:
   665    "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|]
   666     ==> sats(A, pred_set_fm(U,x,r,B), env) \<longleftrightarrow>
   667         pred_set(##A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))"
   668 by (simp add: pred_set_fm_def pred_set_def)
   669 
   670 lemma pred_set_iff_sats:
   671       "[| nth(i,env) = U; nth(j,env) = x; nth(k,env) = r; nth(l,env) = B;
   672           i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|]
   673        ==> pred_set(##A,U,x,r,B) \<longleftrightarrow> sats(A, pred_set_fm(i,j,k,l), env)"
   674 by (simp add: sats_pred_set_fm)
   675 
   676 theorem pred_set_reflection:
   677      "REFLECTS[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)),
   678                \<lambda>i x. pred_set(##Lset(i),f(x),g(x),h(x),b(x))]"
   679 apply (simp only: pred_set_def)
   680 apply (intro FOL_reflections pair_reflection)
   681 done
   682 
   683 
   684 
   685 subsubsection\<open>Domain of a Relation, Internalized\<close>
   686 
   687 (* "is_domain(M,r,z) ==
   688         \<forall>x[M]. (x \<in> z \<longleftrightarrow> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w))))" *)
   689 definition
   690   domain_fm :: "[i,i]=>i" where
   691     "domain_fm(r,z) ==
   692        Forall(Iff(Member(0,succ(z)),
   693                   Exists(And(Member(0,succ(succ(r))),
   694                              Exists(pair_fm(2,0,1))))))"
   695 
   696 lemma domain_type [TC]:
   697      "[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
   698 by (simp add: domain_fm_def)
   699 
   700 lemma sats_domain_fm [simp]:
   701    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   702     ==> sats(A, domain_fm(x,y), env) \<longleftrightarrow>
   703         is_domain(##A, nth(x,env), nth(y,env))"
   704 by (simp add: domain_fm_def is_domain_def)
   705 
   706 lemma domain_iff_sats:
   707       "[| nth(i,env) = x; nth(j,env) = y;
   708           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   709        ==> is_domain(##A, x, y) \<longleftrightarrow> sats(A, domain_fm(i,j), env)"
   710 by simp
   711 
   712 theorem domain_reflection:
   713      "REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)),
   714                \<lambda>i x. is_domain(##Lset(i),f(x),g(x))]"
   715 apply (simp only: is_domain_def)
   716 apply (intro FOL_reflections pair_reflection)
   717 done
   718 
   719 
   720 subsubsection\<open>Range of a Relation, Internalized\<close>
   721 
   722 (* "is_range(M,r,z) ==
   723         \<forall>y[M]. (y \<in> z \<longleftrightarrow> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w))))" *)
   724 definition
   725   range_fm :: "[i,i]=>i" where
   726     "range_fm(r,z) ==
   727        Forall(Iff(Member(0,succ(z)),
   728                   Exists(And(Member(0,succ(succ(r))),
   729                              Exists(pair_fm(0,2,1))))))"
   730 
   731 lemma range_type [TC]:
   732      "[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
   733 by (simp add: range_fm_def)
   734 
   735 lemma sats_range_fm [simp]:
   736    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   737     ==> sats(A, range_fm(x,y), env) \<longleftrightarrow>
   738         is_range(##A, nth(x,env), nth(y,env))"
   739 by (simp add: range_fm_def is_range_def)
   740 
   741 lemma range_iff_sats:
   742       "[| nth(i,env) = x; nth(j,env) = y;
   743           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   744        ==> is_range(##A, x, y) \<longleftrightarrow> sats(A, range_fm(i,j), env)"
   745 by simp
   746 
   747 theorem range_reflection:
   748      "REFLECTS[\<lambda>x. is_range(L,f(x),g(x)),
   749                \<lambda>i x. is_range(##Lset(i),f(x),g(x))]"
   750 apply (simp only: is_range_def)
   751 apply (intro FOL_reflections pair_reflection)
   752 done
   753 
   754 
   755 subsubsection\<open>Field of a Relation, Internalized\<close>
   756 
   757 (* "is_field(M,r,z) ==
   758         \<exists>dr[M]. is_domain(M,r,dr) &
   759             (\<exists>rr[M]. is_range(M,r,rr) & union(M,dr,rr,z))" *)
   760 definition
   761   field_fm :: "[i,i]=>i" where
   762     "field_fm(r,z) ==
   763        Exists(And(domain_fm(succ(r),0),
   764               Exists(And(range_fm(succ(succ(r)),0),
   765                          union_fm(1,0,succ(succ(z)))))))"
   766 
   767 lemma field_type [TC]:
   768      "[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula"
   769 by (simp add: field_fm_def)
   770 
   771 lemma sats_field_fm [simp]:
   772    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   773     ==> sats(A, field_fm(x,y), env) \<longleftrightarrow>
   774         is_field(##A, nth(x,env), nth(y,env))"
   775 by (simp add: field_fm_def is_field_def)
   776 
   777 lemma field_iff_sats:
   778       "[| nth(i,env) = x; nth(j,env) = y;
   779           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   780        ==> is_field(##A, x, y) \<longleftrightarrow> sats(A, field_fm(i,j), env)"
   781 by simp
   782 
   783 theorem field_reflection:
   784      "REFLECTS[\<lambda>x. is_field(L,f(x),g(x)),
   785                \<lambda>i x. is_field(##Lset(i),f(x),g(x))]"
   786 apply (simp only: is_field_def)
   787 apply (intro FOL_reflections domain_reflection range_reflection
   788              union_reflection)
   789 done
   790 
   791 
   792 subsubsection\<open>Image under a Relation, Internalized\<close>
   793 
   794 (* "image(M,r,A,z) ==
   795         \<forall>y[M]. (y \<in> z \<longleftrightarrow> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,w))))" *)
   796 definition
   797   image_fm :: "[i,i,i]=>i" where
   798     "image_fm(r,A,z) ==
   799        Forall(Iff(Member(0,succ(z)),
   800                   Exists(And(Member(0,succ(succ(r))),
   801                              Exists(And(Member(0,succ(succ(succ(A)))),
   802                                         pair_fm(0,2,1)))))))"
   803 
   804 lemma image_type [TC]:
   805      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
   806 by (simp add: image_fm_def)
   807 
   808 lemma sats_image_fm [simp]:
   809    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   810     ==> sats(A, image_fm(x,y,z), env) \<longleftrightarrow>
   811         image(##A, nth(x,env), nth(y,env), nth(z,env))"
   812 by (simp add: image_fm_def Relative.image_def)
   813 
   814 lemma image_iff_sats:
   815       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   816           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   817        ==> image(##A, x, y, z) \<longleftrightarrow> sats(A, image_fm(i,j,k), env)"
   818 by (simp add: sats_image_fm)
   819 
   820 theorem image_reflection:
   821      "REFLECTS[\<lambda>x. image(L,f(x),g(x),h(x)),
   822                \<lambda>i x. image(##Lset(i),f(x),g(x),h(x))]"
   823 apply (simp only: Relative.image_def)
   824 apply (intro FOL_reflections pair_reflection)
   825 done
   826 
   827 
   828 subsubsection\<open>Pre-Image under a Relation, Internalized\<close>
   829 
   830 (* "pre_image(M,r,A,z) ==
   831         \<forall>x[M]. x \<in> z \<longleftrightarrow> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" *)
   832 definition
   833   pre_image_fm :: "[i,i,i]=>i" where
   834     "pre_image_fm(r,A,z) ==
   835        Forall(Iff(Member(0,succ(z)),
   836                   Exists(And(Member(0,succ(succ(r))),
   837                              Exists(And(Member(0,succ(succ(succ(A)))),
   838                                         pair_fm(2,0,1)))))))"
   839 
   840 lemma pre_image_type [TC]:
   841      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
   842 by (simp add: pre_image_fm_def)
   843 
   844 lemma sats_pre_image_fm [simp]:
   845    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   846     ==> sats(A, pre_image_fm(x,y,z), env) \<longleftrightarrow>
   847         pre_image(##A, nth(x,env), nth(y,env), nth(z,env))"
   848 by (simp add: pre_image_fm_def Relative.pre_image_def)
   849 
   850 lemma pre_image_iff_sats:
   851       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   852           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   853        ==> pre_image(##A, x, y, z) \<longleftrightarrow> sats(A, pre_image_fm(i,j,k), env)"
   854 by (simp add: sats_pre_image_fm)
   855 
   856 theorem pre_image_reflection:
   857      "REFLECTS[\<lambda>x. pre_image(L,f(x),g(x),h(x)),
   858                \<lambda>i x. pre_image(##Lset(i),f(x),g(x),h(x))]"
   859 apply (simp only: Relative.pre_image_def)
   860 apply (intro FOL_reflections pair_reflection)
   861 done
   862 
   863 
   864 subsubsection\<open>Function Application, Internalized\<close>
   865 
   866 (* "fun_apply(M,f,x,y) ==
   867         (\<exists>xs[M]. \<exists>fxs[M].
   868          upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))" *)
   869 definition
   870   fun_apply_fm :: "[i,i,i]=>i" where
   871     "fun_apply_fm(f,x,y) ==
   872        Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1),
   873                          And(image_fm(succ(succ(f)), 1, 0),
   874                              big_union_fm(0,succ(succ(y)))))))"
   875 
   876 lemma fun_apply_type [TC]:
   877      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
   878 by (simp add: fun_apply_fm_def)
   879 
   880 lemma sats_fun_apply_fm [simp]:
   881    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   882     ==> sats(A, fun_apply_fm(x,y,z), env) \<longleftrightarrow>
   883         fun_apply(##A, nth(x,env), nth(y,env), nth(z,env))"
   884 by (simp add: fun_apply_fm_def fun_apply_def)
   885 
   886 lemma fun_apply_iff_sats:
   887       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   888           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   889        ==> fun_apply(##A, x, y, z) \<longleftrightarrow> sats(A, fun_apply_fm(i,j,k), env)"
   890 by simp
   891 
   892 theorem fun_apply_reflection:
   893      "REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)),
   894                \<lambda>i x. fun_apply(##Lset(i),f(x),g(x),h(x))]"
   895 apply (simp only: fun_apply_def)
   896 apply (intro FOL_reflections upair_reflection image_reflection
   897              big_union_reflection)
   898 done
   899 
   900 
   901 subsubsection\<open>The Concept of Relation, Internalized\<close>
   902 
   903 (* "is_relation(M,r) ==
   904         (\<forall>z[M]. z\<in>r \<longrightarrow> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))" *)
   905 definition
   906   relation_fm :: "i=>i" where
   907     "relation_fm(r) ==
   908        Forall(Implies(Member(0,succ(r)), Exists(Exists(pair_fm(1,0,2)))))"
   909 
   910 lemma relation_type [TC]:
   911      "[| x \<in> nat |] ==> relation_fm(x) \<in> formula"
   912 by (simp add: relation_fm_def)
   913 
   914 lemma sats_relation_fm [simp]:
   915    "[| x \<in> nat; env \<in> list(A)|]
   916     ==> sats(A, relation_fm(x), env) \<longleftrightarrow> is_relation(##A, nth(x,env))"
   917 by (simp add: relation_fm_def is_relation_def)
   918 
   919 lemma relation_iff_sats:
   920       "[| nth(i,env) = x; nth(j,env) = y;
   921           i \<in> nat; env \<in> list(A)|]
   922        ==> is_relation(##A, x) \<longleftrightarrow> sats(A, relation_fm(i), env)"
   923 by simp
   924 
   925 theorem is_relation_reflection:
   926      "REFLECTS[\<lambda>x. is_relation(L,f(x)),
   927                \<lambda>i x. is_relation(##Lset(i),f(x))]"
   928 apply (simp only: is_relation_def)
   929 apply (intro FOL_reflections pair_reflection)
   930 done
   931 
   932 
   933 subsubsection\<open>The Concept of Function, Internalized\<close>
   934 
   935 (* "is_function(M,r) ==
   936         \<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M].
   937            pair(M,x,y,p) \<longrightarrow> pair(M,x,y',p') \<longrightarrow> p\<in>r \<longrightarrow> p'\<in>r \<longrightarrow> y=y'" *)
   938 definition
   939   function_fm :: "i=>i" where
   940     "function_fm(r) ==
   941        Forall(Forall(Forall(Forall(Forall(
   942          Implies(pair_fm(4,3,1),
   943                  Implies(pair_fm(4,2,0),
   944                          Implies(Member(1,r#+5),
   945                                  Implies(Member(0,r#+5), Equal(3,2))))))))))"
   946 
   947 lemma function_type [TC]:
   948      "[| x \<in> nat |] ==> function_fm(x) \<in> formula"
   949 by (simp add: function_fm_def)
   950 
   951 lemma sats_function_fm [simp]:
   952    "[| x \<in> nat; env \<in> list(A)|]
   953     ==> sats(A, function_fm(x), env) \<longleftrightarrow> is_function(##A, nth(x,env))"
   954 by (simp add: function_fm_def is_function_def)
   955 
   956 lemma is_function_iff_sats:
   957       "[| nth(i,env) = x; nth(j,env) = y;
   958           i \<in> nat; env \<in> list(A)|]
   959        ==> is_function(##A, x) \<longleftrightarrow> sats(A, function_fm(i), env)"
   960 by simp
   961 
   962 theorem is_function_reflection:
   963      "REFLECTS[\<lambda>x. is_function(L,f(x)),
   964                \<lambda>i x. is_function(##Lset(i),f(x))]"
   965 apply (simp only: is_function_def)
   966 apply (intro FOL_reflections pair_reflection)
   967 done
   968 
   969 
   970 subsubsection\<open>Typed Functions, Internalized\<close>
   971 
   972 (* "typed_function(M,A,B,r) ==
   973         is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
   974         (\<forall>u[M]. u\<in>r \<longrightarrow> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) \<longrightarrow> y\<in>B))" *)
   975 
   976 definition
   977   typed_function_fm :: "[i,i,i]=>i" where
   978     "typed_function_fm(A,B,r) ==
   979        And(function_fm(r),
   980          And(relation_fm(r),
   981            And(domain_fm(r,A),
   982              Forall(Implies(Member(0,succ(r)),
   983                   Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))"
   984 
   985 lemma typed_function_type [TC]:
   986      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
   987 by (simp add: typed_function_fm_def)
   988 
   989 lemma sats_typed_function_fm [simp]:
   990    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   991     ==> sats(A, typed_function_fm(x,y,z), env) \<longleftrightarrow>
   992         typed_function(##A, nth(x,env), nth(y,env), nth(z,env))"
   993 by (simp add: typed_function_fm_def typed_function_def)
   994 
   995 lemma typed_function_iff_sats:
   996   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   997       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   998    ==> typed_function(##A, x, y, z) \<longleftrightarrow> sats(A, typed_function_fm(i,j,k), env)"
   999 by simp
  1000 
  1001 lemmas function_reflections =
  1002         empty_reflection number1_reflection
  1003         upair_reflection pair_reflection union_reflection
  1004         big_union_reflection cons_reflection successor_reflection
  1005         fun_apply_reflection subset_reflection
  1006         transitive_set_reflection membership_reflection
  1007         pred_set_reflection domain_reflection range_reflection field_reflection
  1008         image_reflection pre_image_reflection
  1009         is_relation_reflection is_function_reflection
  1010 
  1011 lemmas function_iff_sats =
  1012         empty_iff_sats number1_iff_sats
  1013         upair_iff_sats pair_iff_sats union_iff_sats
  1014         big_union_iff_sats cons_iff_sats successor_iff_sats
  1015         fun_apply_iff_sats  Memrel_iff_sats
  1016         pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats
  1017         image_iff_sats pre_image_iff_sats
  1018         relation_iff_sats is_function_iff_sats
  1019 
  1020 
  1021 theorem typed_function_reflection:
  1022      "REFLECTS[\<lambda>x. typed_function(L,f(x),g(x),h(x)),
  1023                \<lambda>i x. typed_function(##Lset(i),f(x),g(x),h(x))]"
  1024 apply (simp only: typed_function_def)
  1025 apply (intro FOL_reflections function_reflections)
  1026 done
  1027 
  1028 
  1029 subsubsection\<open>Composition of Relations, Internalized\<close>
  1030 
  1031 (* "composition(M,r,s,t) ==
  1032         \<forall>p[M]. p \<in> t \<longleftrightarrow>
  1033                (\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M].
  1034                 pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) &
  1035                 xy \<in> s & yz \<in> r)" *)
  1036 definition
  1037   composition_fm :: "[i,i,i]=>i" where
  1038   "composition_fm(r,s,t) ==
  1039      Forall(Iff(Member(0,succ(t)),
  1040              Exists(Exists(Exists(Exists(Exists(
  1041               And(pair_fm(4,2,5),
  1042                And(pair_fm(4,3,1),
  1043                 And(pair_fm(3,2,0),
  1044                  And(Member(1,s#+6), Member(0,r#+6))))))))))))"
  1045 
  1046 lemma composition_type [TC]:
  1047      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula"
  1048 by (simp add: composition_fm_def)
  1049 
  1050 lemma sats_composition_fm [simp]:
  1051    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
  1052     ==> sats(A, composition_fm(x,y,z), env) \<longleftrightarrow>
  1053         composition(##A, nth(x,env), nth(y,env), nth(z,env))"
  1054 by (simp add: composition_fm_def composition_def)
  1055 
  1056 lemma composition_iff_sats:
  1057       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
  1058           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1059        ==> composition(##A, x, y, z) \<longleftrightarrow> sats(A, composition_fm(i,j,k), env)"
  1060 by simp
  1061 
  1062 theorem composition_reflection:
  1063      "REFLECTS[\<lambda>x. composition(L,f(x),g(x),h(x)),
  1064                \<lambda>i x. composition(##Lset(i),f(x),g(x),h(x))]"
  1065 apply (simp only: composition_def)
  1066 apply (intro FOL_reflections pair_reflection)
  1067 done
  1068 
  1069 
  1070 subsubsection\<open>Injections, Internalized\<close>
  1071 
  1072 (* "injection(M,A,B,f) ==
  1073         typed_function(M,A,B,f) &
  1074         (\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M].
  1075           pair(M,x,y,p) \<longrightarrow> pair(M,x',y,p') \<longrightarrow> p\<in>f \<longrightarrow> p'\<in>f \<longrightarrow> x=x')" *)
  1076 definition
  1077   injection_fm :: "[i,i,i]=>i" where
  1078   "injection_fm(A,B,f) ==
  1079     And(typed_function_fm(A,B,f),
  1080        Forall(Forall(Forall(Forall(Forall(
  1081          Implies(pair_fm(4,2,1),
  1082                  Implies(pair_fm(3,2,0),
  1083                          Implies(Member(1,f#+5),
  1084                                  Implies(Member(0,f#+5), Equal(4,3)))))))))))"
  1085 
  1086 
  1087 lemma injection_type [TC]:
  1088      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
  1089 by (simp add: injection_fm_def)
  1090 
  1091 lemma sats_injection_fm [simp]:
  1092    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
  1093     ==> sats(A, injection_fm(x,y,z), env) \<longleftrightarrow>
  1094         injection(##A, nth(x,env), nth(y,env), nth(z,env))"
  1095 by (simp add: injection_fm_def injection_def)
  1096 
  1097 lemma injection_iff_sats:
  1098   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
  1099       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1100    ==> injection(##A, x, y, z) \<longleftrightarrow> sats(A, injection_fm(i,j,k), env)"
  1101 by simp
  1102 
  1103 theorem injection_reflection:
  1104      "REFLECTS[\<lambda>x. injection(L,f(x),g(x),h(x)),
  1105                \<lambda>i x. injection(##Lset(i),f(x),g(x),h(x))]"
  1106 apply (simp only: injection_def)
  1107 apply (intro FOL_reflections function_reflections typed_function_reflection)
  1108 done
  1109 
  1110 
  1111 subsubsection\<open>Surjections, Internalized\<close>
  1112 
  1113 (*  surjection :: "[i=>o,i,i,i] => o"
  1114     "surjection(M,A,B,f) ==
  1115         typed_function(M,A,B,f) &
  1116         (\<forall>y[M]. y\<in>B \<longrightarrow> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" *)
  1117 definition
  1118   surjection_fm :: "[i,i,i]=>i" where
  1119   "surjection_fm(A,B,f) ==
  1120     And(typed_function_fm(A,B,f),
  1121        Forall(Implies(Member(0,succ(B)),
  1122                       Exists(And(Member(0,succ(succ(A))),
  1123                                  fun_apply_fm(succ(succ(f)),0,1))))))"
  1124 
  1125 lemma surjection_type [TC]:
  1126      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula"
  1127 by (simp add: surjection_fm_def)
  1128 
  1129 lemma sats_surjection_fm [simp]:
  1130    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
  1131     ==> sats(A, surjection_fm(x,y,z), env) \<longleftrightarrow>
  1132         surjection(##A, nth(x,env), nth(y,env), nth(z,env))"
  1133 by (simp add: surjection_fm_def surjection_def)
  1134 
  1135 lemma surjection_iff_sats:
  1136   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
  1137       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1138    ==> surjection(##A, x, y, z) \<longleftrightarrow> sats(A, surjection_fm(i,j,k), env)"
  1139 by simp
  1140 
  1141 theorem surjection_reflection:
  1142      "REFLECTS[\<lambda>x. surjection(L,f(x),g(x),h(x)),
  1143                \<lambda>i x. surjection(##Lset(i),f(x),g(x),h(x))]"
  1144 apply (simp only: surjection_def)
  1145 apply (intro FOL_reflections function_reflections typed_function_reflection)
  1146 done
  1147 
  1148 
  1149 
  1150 subsubsection\<open>Bijections, Internalized\<close>
  1151 
  1152 (*   bijection :: "[i=>o,i,i,i] => o"
  1153     "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *)
  1154 definition
  1155   bijection_fm :: "[i,i,i]=>i" where
  1156   "bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))"
  1157 
  1158 lemma bijection_type [TC]:
  1159      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula"
  1160 by (simp add: bijection_fm_def)
  1161 
  1162 lemma sats_bijection_fm [simp]:
  1163    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
  1164     ==> sats(A, bijection_fm(x,y,z), env) \<longleftrightarrow>
  1165         bijection(##A, nth(x,env), nth(y,env), nth(z,env))"
  1166 by (simp add: bijection_fm_def bijection_def)
  1167 
  1168 lemma bijection_iff_sats:
  1169   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
  1170       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1171    ==> bijection(##A, x, y, z) \<longleftrightarrow> sats(A, bijection_fm(i,j,k), env)"
  1172 by simp
  1173 
  1174 theorem bijection_reflection:
  1175      "REFLECTS[\<lambda>x. bijection(L,f(x),g(x),h(x)),
  1176                \<lambda>i x. bijection(##Lset(i),f(x),g(x),h(x))]"
  1177 apply (simp only: bijection_def)
  1178 apply (intro And_reflection injection_reflection surjection_reflection)
  1179 done
  1180 
  1181 
  1182 subsubsection\<open>Restriction of a Relation, Internalized\<close>
  1183 
  1184 
  1185 (* "restriction(M,r,A,z) ==
  1186         \<forall>x[M]. x \<in> z \<longleftrightarrow> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" *)
  1187 definition
  1188   restriction_fm :: "[i,i,i]=>i" where
  1189     "restriction_fm(r,A,z) ==
  1190        Forall(Iff(Member(0,succ(z)),
  1191                   And(Member(0,succ(r)),
  1192                       Exists(And(Member(0,succ(succ(A))),
  1193                                  Exists(pair_fm(1,0,2)))))))"
  1194 
  1195 lemma restriction_type [TC]:
  1196      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula"
  1197 by (simp add: restriction_fm_def)
  1198 
  1199 lemma sats_restriction_fm [simp]:
  1200    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
  1201     ==> sats(A, restriction_fm(x,y,z), env) \<longleftrightarrow>
  1202         restriction(##A, nth(x,env), nth(y,env), nth(z,env))"
  1203 by (simp add: restriction_fm_def restriction_def)
  1204 
  1205 lemma restriction_iff_sats:
  1206       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
  1207           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1208        ==> restriction(##A, x, y, z) \<longleftrightarrow> sats(A, restriction_fm(i,j,k), env)"
  1209 by simp
  1210 
  1211 theorem restriction_reflection:
  1212      "REFLECTS[\<lambda>x. restriction(L,f(x),g(x),h(x)),
  1213                \<lambda>i x. restriction(##Lset(i),f(x),g(x),h(x))]"
  1214 apply (simp only: restriction_def)
  1215 apply (intro FOL_reflections pair_reflection)
  1216 done
  1217 
  1218 subsubsection\<open>Order-Isomorphisms, Internalized\<close>
  1219 
  1220 (*  order_isomorphism :: "[i=>o,i,i,i,i,i] => o"
  1221    "order_isomorphism(M,A,r,B,s,f) ==
  1222         bijection(M,A,B,f) &
  1223         (\<forall>x[M]. x\<in>A \<longrightarrow> (\<forall>y[M]. y\<in>A \<longrightarrow>
  1224           (\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>q[M].
  1225             pair(M,x,y,p) \<longrightarrow> fun_apply(M,f,x,fx) \<longrightarrow> fun_apply(M,f,y,fy) \<longrightarrow>
  1226             pair(M,fx,fy,q) \<longrightarrow> (p\<in>r \<longleftrightarrow> q\<in>s))))"
  1227   *)
  1228 
  1229 definition
  1230   order_isomorphism_fm :: "[i,i,i,i,i]=>i" where
  1231  "order_isomorphism_fm(A,r,B,s,f) ==
  1232    And(bijection_fm(A,B,f),
  1233      Forall(Implies(Member(0,succ(A)),
  1234        Forall(Implies(Member(0,succ(succ(A))),
  1235          Forall(Forall(Forall(Forall(
  1236            Implies(pair_fm(5,4,3),
  1237              Implies(fun_apply_fm(f#+6,5,2),
  1238                Implies(fun_apply_fm(f#+6,4,1),
  1239                  Implies(pair_fm(2,1,0),
  1240                    Iff(Member(3,r#+6), Member(0,s#+6)))))))))))))))"
  1241 
  1242 lemma order_isomorphism_type [TC]:
  1243      "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]
  1244       ==> order_isomorphism_fm(A,r,B,s,f) \<in> formula"
  1245 by (simp add: order_isomorphism_fm_def)
  1246 
  1247 lemma sats_order_isomorphism_fm [simp]:
  1248    "[| U \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat; env \<in> list(A)|]
  1249     ==> sats(A, order_isomorphism_fm(U,r,B,s,f), env) \<longleftrightarrow>
  1250         order_isomorphism(##A, nth(U,env), nth(r,env), nth(B,env),
  1251                                nth(s,env), nth(f,env))"
  1252 by (simp add: order_isomorphism_fm_def order_isomorphism_def)
  1253 
  1254 lemma order_isomorphism_iff_sats:
  1255   "[| nth(i,env) = U; nth(j,env) = r; nth(k,env) = B; nth(j',env) = s;
  1256       nth(k',env) = f;
  1257       i \<in> nat; j \<in> nat; k \<in> nat; j' \<in> nat; k' \<in> nat; env \<in> list(A)|]
  1258    ==> order_isomorphism(##A,U,r,B,s,f) \<longleftrightarrow>
  1259        sats(A, order_isomorphism_fm(i,j,k,j',k'), env)"
  1260 by simp
  1261 
  1262 theorem order_isomorphism_reflection:
  1263      "REFLECTS[\<lambda>x. order_isomorphism(L,f(x),g(x),h(x),g'(x),h'(x)),
  1264                \<lambda>i x. order_isomorphism(##Lset(i),f(x),g(x),h(x),g'(x),h'(x))]"
  1265 apply (simp only: order_isomorphism_def)
  1266 apply (intro FOL_reflections function_reflections bijection_reflection)
  1267 done
  1268 
  1269 subsubsection\<open>Limit Ordinals, Internalized\<close>
  1270 
  1271 text\<open>A limit ordinal is a non-empty, successor-closed ordinal\<close>
  1272 
  1273 (* "limit_ordinal(M,a) ==
  1274         ordinal(M,a) & ~ empty(M,a) &
  1275         (\<forall>x[M]. x\<in>a \<longrightarrow> (\<exists>y[M]. y\<in>a & successor(M,x,y)))" *)
  1276 
  1277 definition
  1278   limit_ordinal_fm :: "i=>i" where
  1279     "limit_ordinal_fm(x) ==
  1280         And(ordinal_fm(x),
  1281             And(Neg(empty_fm(x)),
  1282                 Forall(Implies(Member(0,succ(x)),
  1283                                Exists(And(Member(0,succ(succ(x))),
  1284                                           succ_fm(1,0)))))))"
  1285 
  1286 lemma limit_ordinal_type [TC]:
  1287      "x \<in> nat ==> limit_ordinal_fm(x) \<in> formula"
  1288 by (simp add: limit_ordinal_fm_def)
  1289 
  1290 lemma sats_limit_ordinal_fm [simp]:
  1291    "[| x \<in> nat; env \<in> list(A)|]
  1292     ==> sats(A, limit_ordinal_fm(x), env) \<longleftrightarrow> limit_ordinal(##A, nth(x,env))"
  1293 by (simp add: limit_ordinal_fm_def limit_ordinal_def sats_ordinal_fm')
  1294 
  1295 lemma limit_ordinal_iff_sats:
  1296       "[| nth(i,env) = x; nth(j,env) = y;
  1297           i \<in> nat; env \<in> list(A)|]
  1298        ==> limit_ordinal(##A, x) \<longleftrightarrow> sats(A, limit_ordinal_fm(i), env)"
  1299 by simp
  1300 
  1301 theorem limit_ordinal_reflection:
  1302      "REFLECTS[\<lambda>x. limit_ordinal(L,f(x)),
  1303                \<lambda>i x. limit_ordinal(##Lset(i),f(x))]"
  1304 apply (simp only: limit_ordinal_def)
  1305 apply (intro FOL_reflections ordinal_reflection
  1306              empty_reflection successor_reflection)
  1307 done
  1308 
  1309 subsubsection\<open>Finite Ordinals: The Predicate ``Is A Natural Number''\<close>
  1310 
  1311 (*     "finite_ordinal(M,a) == 
  1312         ordinal(M,a) & ~ limit_ordinal(M,a) & 
  1313         (\<forall>x[M]. x\<in>a \<longrightarrow> ~ limit_ordinal(M,x))" *)
  1314 definition
  1315   finite_ordinal_fm :: "i=>i" where
  1316     "finite_ordinal_fm(x) ==
  1317        And(ordinal_fm(x),
  1318           And(Neg(limit_ordinal_fm(x)),
  1319            Forall(Implies(Member(0,succ(x)),
  1320                           Neg(limit_ordinal_fm(0))))))"
  1321 
  1322 lemma finite_ordinal_type [TC]:
  1323      "x \<in> nat ==> finite_ordinal_fm(x) \<in> formula"
  1324 by (simp add: finite_ordinal_fm_def)
  1325 
  1326 lemma sats_finite_ordinal_fm [simp]:
  1327    "[| x \<in> nat; env \<in> list(A)|]
  1328     ==> sats(A, finite_ordinal_fm(x), env) \<longleftrightarrow> finite_ordinal(##A, nth(x,env))"
  1329 by (simp add: finite_ordinal_fm_def sats_ordinal_fm' finite_ordinal_def)
  1330 
  1331 lemma finite_ordinal_iff_sats:
  1332       "[| nth(i,env) = x; nth(j,env) = y;
  1333           i \<in> nat; env \<in> list(A)|]
  1334        ==> finite_ordinal(##A, x) \<longleftrightarrow> sats(A, finite_ordinal_fm(i), env)"
  1335 by simp
  1336 
  1337 theorem finite_ordinal_reflection:
  1338      "REFLECTS[\<lambda>x. finite_ordinal(L,f(x)),
  1339                \<lambda>i x. finite_ordinal(##Lset(i),f(x))]"
  1340 apply (simp only: finite_ordinal_def)
  1341 apply (intro FOL_reflections ordinal_reflection limit_ordinal_reflection)
  1342 done
  1343 
  1344 
  1345 subsubsection\<open>Omega: The Set of Natural Numbers\<close>
  1346 
  1347 (* omega(M,a) == limit_ordinal(M,a) & (\<forall>x[M]. x\<in>a \<longrightarrow> ~ limit_ordinal(M,x)) *)
  1348 definition
  1349   omega_fm :: "i=>i" where
  1350     "omega_fm(x) ==
  1351        And(limit_ordinal_fm(x),
  1352            Forall(Implies(Member(0,succ(x)),
  1353                           Neg(limit_ordinal_fm(0)))))"
  1354 
  1355 lemma omega_type [TC]:
  1356      "x \<in> nat ==> omega_fm(x) \<in> formula"
  1357 by (simp add: omega_fm_def)
  1358 
  1359 lemma sats_omega_fm [simp]:
  1360    "[| x \<in> nat; env \<in> list(A)|]
  1361     ==> sats(A, omega_fm(x), env) \<longleftrightarrow> omega(##A, nth(x,env))"
  1362 by (simp add: omega_fm_def omega_def)
  1363 
  1364 lemma omega_iff_sats:
  1365       "[| nth(i,env) = x; nth(j,env) = y;
  1366           i \<in> nat; env \<in> list(A)|]
  1367        ==> omega(##A, x) \<longleftrightarrow> sats(A, omega_fm(i), env)"
  1368 by simp
  1369 
  1370 theorem omega_reflection:
  1371      "REFLECTS[\<lambda>x. omega(L,f(x)),
  1372                \<lambda>i x. omega(##Lset(i),f(x))]"
  1373 apply (simp only: omega_def)
  1374 apply (intro FOL_reflections limit_ordinal_reflection)
  1375 done
  1376 
  1377 
  1378 lemmas fun_plus_reflections =
  1379         typed_function_reflection composition_reflection
  1380         injection_reflection surjection_reflection
  1381         bijection_reflection restriction_reflection
  1382         order_isomorphism_reflection finite_ordinal_reflection 
  1383         ordinal_reflection limit_ordinal_reflection omega_reflection
  1384 
  1385 lemmas fun_plus_iff_sats =
  1386         typed_function_iff_sats composition_iff_sats
  1387         injection_iff_sats surjection_iff_sats
  1388         bijection_iff_sats restriction_iff_sats
  1389         order_isomorphism_iff_sats finite_ordinal_iff_sats
  1390         ordinal_iff_sats limit_ordinal_iff_sats omega_iff_sats
  1391 
  1392 end