src/ZF/Constructible/Internalize.thy
author wenzelm
Mon Dec 04 22:54:31 2017 +0100 (21 months ago)
changeset 67131 85d10959c2e4
parent 61798 27f3c10b0b50
child 69593 3dda49e08b9d
permissions -rw-r--r--
tuned signature;
     1 (*  Title:      ZF/Constructible/Internalize.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3 *)
     4 
     5 theory Internalize imports L_axioms Datatype_absolute begin
     6 
     7 subsection\<open>Internalized Forms of Data Structuring Operators\<close>
     8 
     9 subsubsection\<open>The Formula @{term is_Inl}, Internalized\<close>
    10 
    11 (*  is_Inl(M,a,z) == \<exists>zero[M]. empty(M,zero) & pair(M,zero,a,z) *)
    12 definition
    13   Inl_fm :: "[i,i]=>i" where
    14     "Inl_fm(a,z) == Exists(And(empty_fm(0), pair_fm(0,succ(a),succ(z))))"
    15 
    16 lemma Inl_type [TC]:
    17      "[| x \<in> nat; z \<in> nat |] ==> Inl_fm(x,z) \<in> formula"
    18 by (simp add: Inl_fm_def)
    19 
    20 lemma sats_Inl_fm [simp]:
    21    "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
    22     ==> sats(A, Inl_fm(x,z), env) \<longleftrightarrow> is_Inl(##A, nth(x,env), nth(z,env))"
    23 by (simp add: Inl_fm_def is_Inl_def)
    24 
    25 lemma Inl_iff_sats:
    26       "[| nth(i,env) = x; nth(k,env) = z;
    27           i \<in> nat; k \<in> nat; env \<in> list(A)|]
    28        ==> is_Inl(##A, x, z) \<longleftrightarrow> sats(A, Inl_fm(i,k), env)"
    29 by simp
    30 
    31 theorem Inl_reflection:
    32      "REFLECTS[\<lambda>x. is_Inl(L,f(x),h(x)),
    33                \<lambda>i x. is_Inl(##Lset(i),f(x),h(x))]"
    34 apply (simp only: is_Inl_def)
    35 apply (intro FOL_reflections function_reflections)
    36 done
    37 
    38 
    39 subsubsection\<open>The Formula @{term is_Inr}, Internalized\<close>
    40 
    41 (*  is_Inr(M,a,z) == \<exists>n1[M]. number1(M,n1) & pair(M,n1,a,z) *)
    42 definition
    43   Inr_fm :: "[i,i]=>i" where
    44     "Inr_fm(a,z) == Exists(And(number1_fm(0), pair_fm(0,succ(a),succ(z))))"
    45 
    46 lemma Inr_type [TC]:
    47      "[| x \<in> nat; z \<in> nat |] ==> Inr_fm(x,z) \<in> formula"
    48 by (simp add: Inr_fm_def)
    49 
    50 lemma sats_Inr_fm [simp]:
    51    "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
    52     ==> sats(A, Inr_fm(x,z), env) \<longleftrightarrow> is_Inr(##A, nth(x,env), nth(z,env))"
    53 by (simp add: Inr_fm_def is_Inr_def)
    54 
    55 lemma Inr_iff_sats:
    56       "[| nth(i,env) = x; nth(k,env) = z;
    57           i \<in> nat; k \<in> nat; env \<in> list(A)|]
    58        ==> is_Inr(##A, x, z) \<longleftrightarrow> sats(A, Inr_fm(i,k), env)"
    59 by simp
    60 
    61 theorem Inr_reflection:
    62      "REFLECTS[\<lambda>x. is_Inr(L,f(x),h(x)),
    63                \<lambda>i x. is_Inr(##Lset(i),f(x),h(x))]"
    64 apply (simp only: is_Inr_def)
    65 apply (intro FOL_reflections function_reflections)
    66 done
    67 
    68 
    69 subsubsection\<open>The Formula @{term is_Nil}, Internalized\<close>
    70 
    71 (* is_Nil(M,xs) == \<exists>zero[M]. empty(M,zero) & is_Inl(M,zero,xs) *)
    72 
    73 definition
    74   Nil_fm :: "i=>i" where
    75     "Nil_fm(x) == Exists(And(empty_fm(0), Inl_fm(0,succ(x))))"
    76 
    77 lemma Nil_type [TC]: "x \<in> nat ==> Nil_fm(x) \<in> formula"
    78 by (simp add: Nil_fm_def)
    79 
    80 lemma sats_Nil_fm [simp]:
    81    "[| x \<in> nat; env \<in> list(A)|]
    82     ==> sats(A, Nil_fm(x), env) \<longleftrightarrow> is_Nil(##A, nth(x,env))"
    83 by (simp add: Nil_fm_def is_Nil_def)
    84 
    85 lemma Nil_iff_sats:
    86       "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
    87        ==> is_Nil(##A, x) \<longleftrightarrow> sats(A, Nil_fm(i), env)"
    88 by simp
    89 
    90 theorem Nil_reflection:
    91      "REFLECTS[\<lambda>x. is_Nil(L,f(x)),
    92                \<lambda>i x. is_Nil(##Lset(i),f(x))]"
    93 apply (simp only: is_Nil_def)
    94 apply (intro FOL_reflections function_reflections Inl_reflection)
    95 done
    96 
    97 
    98 subsubsection\<open>The Formula @{term is_Cons}, Internalized\<close>
    99 
   100 
   101 (*  "is_Cons(M,a,l,Z) == \<exists>p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)" *)
   102 definition
   103   Cons_fm :: "[i,i,i]=>i" where
   104     "Cons_fm(a,l,Z) ==
   105        Exists(And(pair_fm(succ(a),succ(l),0), Inr_fm(0,succ(Z))))"
   106 
   107 lemma Cons_type [TC]:
   108      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Cons_fm(x,y,z) \<in> formula"
   109 by (simp add: Cons_fm_def)
   110 
   111 lemma sats_Cons_fm [simp]:
   112    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   113     ==> sats(A, Cons_fm(x,y,z), env) \<longleftrightarrow>
   114        is_Cons(##A, nth(x,env), nth(y,env), nth(z,env))"
   115 by (simp add: Cons_fm_def is_Cons_def)
   116 
   117 lemma Cons_iff_sats:
   118       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   119           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   120        ==>is_Cons(##A, x, y, z) \<longleftrightarrow> sats(A, Cons_fm(i,j,k), env)"
   121 by simp
   122 
   123 theorem Cons_reflection:
   124      "REFLECTS[\<lambda>x. is_Cons(L,f(x),g(x),h(x)),
   125                \<lambda>i x. is_Cons(##Lset(i),f(x),g(x),h(x))]"
   126 apply (simp only: is_Cons_def)
   127 apply (intro FOL_reflections pair_reflection Inr_reflection)
   128 done
   129 
   130 subsubsection\<open>The Formula @{term is_quasilist}, Internalized\<close>
   131 
   132 (* is_quasilist(M,xs) == is_Nil(M,z) | (\<exists>x[M]. \<exists>l[M]. is_Cons(M,x,l,z))" *)
   133 
   134 definition
   135   quasilist_fm :: "i=>i" where
   136     "quasilist_fm(x) ==
   137        Or(Nil_fm(x), Exists(Exists(Cons_fm(1,0,succ(succ(x))))))"
   138 
   139 lemma quasilist_type [TC]: "x \<in> nat ==> quasilist_fm(x) \<in> formula"
   140 by (simp add: quasilist_fm_def)
   141 
   142 lemma sats_quasilist_fm [simp]:
   143    "[| x \<in> nat; env \<in> list(A)|]
   144     ==> sats(A, quasilist_fm(x), env) \<longleftrightarrow> is_quasilist(##A, nth(x,env))"
   145 by (simp add: quasilist_fm_def is_quasilist_def)
   146 
   147 lemma quasilist_iff_sats:
   148       "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
   149        ==> is_quasilist(##A, x) \<longleftrightarrow> sats(A, quasilist_fm(i), env)"
   150 by simp
   151 
   152 theorem quasilist_reflection:
   153      "REFLECTS[\<lambda>x. is_quasilist(L,f(x)),
   154                \<lambda>i x. is_quasilist(##Lset(i),f(x))]"
   155 apply (simp only: is_quasilist_def)
   156 apply (intro FOL_reflections Nil_reflection Cons_reflection)
   157 done
   158 
   159 
   160 subsection\<open>Absoluteness for the Function @{term nth}\<close>
   161 
   162 
   163 subsubsection\<open>The Formula @{term is_hd}, Internalized\<close>
   164 
   165 (*   "is_hd(M,xs,H) == 
   166        (is_Nil(M,xs) \<longrightarrow> empty(M,H)) &
   167        (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | H=x) &
   168        (is_quasilist(M,xs) | empty(M,H))" *)
   169 definition
   170   hd_fm :: "[i,i]=>i" where
   171     "hd_fm(xs,H) == 
   172        And(Implies(Nil_fm(xs), empty_fm(H)),
   173            And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(H#+2,1)))),
   174                Or(quasilist_fm(xs), empty_fm(H))))"
   175 
   176 lemma hd_type [TC]:
   177      "[| x \<in> nat; y \<in> nat |] ==> hd_fm(x,y) \<in> formula"
   178 by (simp add: hd_fm_def) 
   179 
   180 lemma sats_hd_fm [simp]:
   181    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   182     ==> sats(A, hd_fm(x,y), env) \<longleftrightarrow> is_hd(##A, nth(x,env), nth(y,env))"
   183 by (simp add: hd_fm_def is_hd_def)
   184 
   185 lemma hd_iff_sats:
   186       "[| nth(i,env) = x; nth(j,env) = y;
   187           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   188        ==> is_hd(##A, x, y) \<longleftrightarrow> sats(A, hd_fm(i,j), env)"
   189 by simp
   190 
   191 theorem hd_reflection:
   192      "REFLECTS[\<lambda>x. is_hd(L,f(x),g(x)), 
   193                \<lambda>i x. is_hd(##Lset(i),f(x),g(x))]"
   194 apply (simp only: is_hd_def)
   195 apply (intro FOL_reflections Nil_reflection Cons_reflection
   196              quasilist_reflection empty_reflection)  
   197 done
   198 
   199 
   200 subsubsection\<open>The Formula @{term is_tl}, Internalized\<close>
   201 
   202 (*     "is_tl(M,xs,T) ==
   203        (is_Nil(M,xs) \<longrightarrow> T=xs) &
   204        (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | T=l) &
   205        (is_quasilist(M,xs) | empty(M,T))" *)
   206 definition
   207   tl_fm :: "[i,i]=>i" where
   208     "tl_fm(xs,T) ==
   209        And(Implies(Nil_fm(xs), Equal(T,xs)),
   210            And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(T#+2,0)))),
   211                Or(quasilist_fm(xs), empty_fm(T))))"
   212 
   213 lemma tl_type [TC]:
   214      "[| x \<in> nat; y \<in> nat |] ==> tl_fm(x,y) \<in> formula"
   215 by (simp add: tl_fm_def)
   216 
   217 lemma sats_tl_fm [simp]:
   218    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   219     ==> sats(A, tl_fm(x,y), env) \<longleftrightarrow> is_tl(##A, nth(x,env), nth(y,env))"
   220 by (simp add: tl_fm_def is_tl_def)
   221 
   222 lemma tl_iff_sats:
   223       "[| nth(i,env) = x; nth(j,env) = y;
   224           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   225        ==> is_tl(##A, x, y) \<longleftrightarrow> sats(A, tl_fm(i,j), env)"
   226 by simp
   227 
   228 theorem tl_reflection:
   229      "REFLECTS[\<lambda>x. is_tl(L,f(x),g(x)),
   230                \<lambda>i x. is_tl(##Lset(i),f(x),g(x))]"
   231 apply (simp only: is_tl_def)
   232 apply (intro FOL_reflections Nil_reflection Cons_reflection
   233              quasilist_reflection empty_reflection)
   234 done
   235 
   236 
   237 subsubsection\<open>The Operator @{term is_bool_of_o}\<close>
   238 
   239 (*   is_bool_of_o :: "[i=>o, o, i] => o"
   240    "is_bool_of_o(M,P,z) == (P & number1(M,z)) | (~P & empty(M,z))" *)
   241 
   242 text\<open>The formula @{term p} has no free variables.\<close>
   243 definition
   244   bool_of_o_fm :: "[i, i]=>i" where
   245   "bool_of_o_fm(p,z) == 
   246     Or(And(p,number1_fm(z)),
   247        And(Neg(p),empty_fm(z)))"
   248 
   249 lemma is_bool_of_o_type [TC]:
   250      "[| p \<in> formula; z \<in> nat |] ==> bool_of_o_fm(p,z) \<in> formula"
   251 by (simp add: bool_of_o_fm_def)
   252 
   253 lemma sats_bool_of_o_fm:
   254   assumes p_iff_sats: "P \<longleftrightarrow> sats(A, p, env)"
   255   shows 
   256       "[|z \<in> nat; env \<in> list(A)|]
   257        ==> sats(A, bool_of_o_fm(p,z), env) \<longleftrightarrow>
   258            is_bool_of_o(##A, P, nth(z,env))"
   259 by (simp add: bool_of_o_fm_def is_bool_of_o_def p_iff_sats [THEN iff_sym])
   260 
   261 lemma is_bool_of_o_iff_sats:
   262   "[| P \<longleftrightarrow> sats(A, p, env); nth(k,env) = z; k \<in> nat; env \<in> list(A)|]
   263    ==> is_bool_of_o(##A, P, z) \<longleftrightarrow> sats(A, bool_of_o_fm(p,k), env)"
   264 by (simp add: sats_bool_of_o_fm)
   265 
   266 theorem bool_of_o_reflection:
   267      "REFLECTS [P(L), \<lambda>i. P(##Lset(i))] ==>
   268       REFLECTS[\<lambda>x. is_bool_of_o(L, P(L,x), f(x)),  
   269                \<lambda>i x. is_bool_of_o(##Lset(i), P(##Lset(i),x), f(x))]"
   270 apply (simp (no_asm) only: is_bool_of_o_def)
   271 apply (intro FOL_reflections function_reflections, assumption+)
   272 done
   273 
   274 
   275 subsection\<open>More Internalizations\<close>
   276 
   277 subsubsection\<open>The Operator @{term is_lambda}\<close>
   278 
   279 text\<open>The two arguments of @{term p} are always 1, 0. Remember that
   280  @{term p} will be enclosed by three quantifiers.\<close>
   281 
   282 (* is_lambda :: "[i=>o, i, [i,i]=>o, i] => o"
   283     "is_lambda(M, A, is_b, z) == 
   284        \<forall>p[M]. p \<in> z \<longleftrightarrow>
   285         (\<exists>u[M]. \<exists>v[M]. u\<in>A & pair(M,u,v,p) & is_b(u,v))" *)
   286 definition
   287   lambda_fm :: "[i, i, i]=>i" where
   288   "lambda_fm(p,A,z) == 
   289     Forall(Iff(Member(0,succ(z)),
   290             Exists(Exists(And(Member(1,A#+3),
   291                            And(pair_fm(1,0,2), p))))))"
   292 
   293 text\<open>We call @{term p} with arguments x, y by equating them with 
   294   the corresponding quantified variables with de Bruijn indices 1, 0.\<close>
   295 
   296 lemma is_lambda_type [TC]:
   297      "[| p \<in> formula; x \<in> nat; y \<in> nat |] 
   298       ==> lambda_fm(p,x,y) \<in> formula"
   299 by (simp add: lambda_fm_def) 
   300 
   301 lemma sats_lambda_fm:
   302   assumes is_b_iff_sats: 
   303       "!!a0 a1 a2. 
   304         [|a0\<in>A; a1\<in>A; a2\<in>A|] 
   305         ==> is_b(a1, a0) \<longleftrightarrow> sats(A, p, Cons(a0,Cons(a1,Cons(a2,env))))"
   306   shows 
   307       "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
   308        ==> sats(A, lambda_fm(p,x,y), env) \<longleftrightarrow> 
   309            is_lambda(##A, nth(x,env), is_b, nth(y,env))"
   310 by (simp add: lambda_fm_def is_lambda_def is_b_iff_sats [THEN iff_sym]) 
   311 
   312 theorem is_lambda_reflection:
   313   assumes is_b_reflection:
   314     "!!f g h. REFLECTS[\<lambda>x. is_b(L, f(x), g(x), h(x)), 
   315                      \<lambda>i x. is_b(##Lset(i), f(x), g(x), h(x))]"
   316   shows "REFLECTS[\<lambda>x. is_lambda(L, A(x), is_b(L,x), f(x)), 
   317                \<lambda>i x. is_lambda(##Lset(i), A(x), is_b(##Lset(i),x), f(x))]"
   318 apply (simp (no_asm_use) only: is_lambda_def)
   319 apply (intro FOL_reflections is_b_reflection pair_reflection)
   320 done
   321 
   322 subsubsection\<open>The Operator @{term is_Member}, Internalized\<close>
   323 
   324 (*    "is_Member(M,x,y,Z) ==
   325         \<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inl(M,u,Z)" *)
   326 definition
   327   Member_fm :: "[i,i,i]=>i" where
   328     "Member_fm(x,y,Z) ==
   329        Exists(Exists(And(pair_fm(x#+2,y#+2,1), 
   330                       And(Inl_fm(1,0), Inl_fm(0,Z#+2)))))"
   331 
   332 lemma is_Member_type [TC]:
   333      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Member_fm(x,y,z) \<in> formula"
   334 by (simp add: Member_fm_def)
   335 
   336 lemma sats_Member_fm [simp]:
   337    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   338     ==> sats(A, Member_fm(x,y,z), env) \<longleftrightarrow>
   339         is_Member(##A, nth(x,env), nth(y,env), nth(z,env))"
   340 by (simp add: Member_fm_def is_Member_def)
   341 
   342 lemma Member_iff_sats:
   343       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   344           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   345        ==> is_Member(##A, x, y, z) \<longleftrightarrow> sats(A, Member_fm(i,j,k), env)"
   346 by (simp add: sats_Member_fm)
   347 
   348 theorem Member_reflection:
   349      "REFLECTS[\<lambda>x. is_Member(L,f(x),g(x),h(x)),
   350                \<lambda>i x. is_Member(##Lset(i),f(x),g(x),h(x))]"
   351 apply (simp only: is_Member_def)
   352 apply (intro FOL_reflections pair_reflection Inl_reflection)
   353 done
   354 
   355 subsubsection\<open>The Operator @{term is_Equal}, Internalized\<close>
   356 
   357 (*    "is_Equal(M,x,y,Z) ==
   358         \<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inr(M,p,u) & is_Inl(M,u,Z)" *)
   359 definition
   360   Equal_fm :: "[i,i,i]=>i" where
   361     "Equal_fm(x,y,Z) ==
   362        Exists(Exists(And(pair_fm(x#+2,y#+2,1), 
   363                       And(Inr_fm(1,0), Inl_fm(0,Z#+2)))))"
   364 
   365 lemma is_Equal_type [TC]:
   366      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Equal_fm(x,y,z) \<in> formula"
   367 by (simp add: Equal_fm_def)
   368 
   369 lemma sats_Equal_fm [simp]:
   370    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   371     ==> sats(A, Equal_fm(x,y,z), env) \<longleftrightarrow>
   372         is_Equal(##A, nth(x,env), nth(y,env), nth(z,env))"
   373 by (simp add: Equal_fm_def is_Equal_def)
   374 
   375 lemma Equal_iff_sats:
   376       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   377           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   378        ==> is_Equal(##A, x, y, z) \<longleftrightarrow> sats(A, Equal_fm(i,j,k), env)"
   379 by (simp add: sats_Equal_fm)
   380 
   381 theorem Equal_reflection:
   382      "REFLECTS[\<lambda>x. is_Equal(L,f(x),g(x),h(x)),
   383                \<lambda>i x. is_Equal(##Lset(i),f(x),g(x),h(x))]"
   384 apply (simp only: is_Equal_def)
   385 apply (intro FOL_reflections pair_reflection Inl_reflection Inr_reflection)
   386 done
   387 
   388 subsubsection\<open>The Operator @{term is_Nand}, Internalized\<close>
   389 
   390 (*    "is_Nand(M,x,y,Z) ==
   391         \<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inr(M,u,Z)" *)
   392 definition
   393   Nand_fm :: "[i,i,i]=>i" where
   394     "Nand_fm(x,y,Z) ==
   395        Exists(Exists(And(pair_fm(x#+2,y#+2,1), 
   396                       And(Inl_fm(1,0), Inr_fm(0,Z#+2)))))"
   397 
   398 lemma is_Nand_type [TC]:
   399      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Nand_fm(x,y,z) \<in> formula"
   400 by (simp add: Nand_fm_def)
   401 
   402 lemma sats_Nand_fm [simp]:
   403    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   404     ==> sats(A, Nand_fm(x,y,z), env) \<longleftrightarrow>
   405         is_Nand(##A, nth(x,env), nth(y,env), nth(z,env))"
   406 by (simp add: Nand_fm_def is_Nand_def)
   407 
   408 lemma Nand_iff_sats:
   409       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   410           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   411        ==> is_Nand(##A, x, y, z) \<longleftrightarrow> sats(A, Nand_fm(i,j,k), env)"
   412 by (simp add: sats_Nand_fm)
   413 
   414 theorem Nand_reflection:
   415      "REFLECTS[\<lambda>x. is_Nand(L,f(x),g(x),h(x)),
   416                \<lambda>i x. is_Nand(##Lset(i),f(x),g(x),h(x))]"
   417 apply (simp only: is_Nand_def)
   418 apply (intro FOL_reflections pair_reflection Inl_reflection Inr_reflection)
   419 done
   420 
   421 subsubsection\<open>The Operator @{term is_Forall}, Internalized\<close>
   422 
   423 (* "is_Forall(M,p,Z) == \<exists>u[M]. is_Inr(M,p,u) & is_Inr(M,u,Z)" *)
   424 definition
   425   Forall_fm :: "[i,i]=>i" where
   426     "Forall_fm(x,Z) ==
   427        Exists(And(Inr_fm(succ(x),0), Inr_fm(0,succ(Z))))"
   428 
   429 lemma is_Forall_type [TC]:
   430      "[| x \<in> nat; y \<in> nat |] ==> Forall_fm(x,y) \<in> formula"
   431 by (simp add: Forall_fm_def)
   432 
   433 lemma sats_Forall_fm [simp]:
   434    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
   435     ==> sats(A, Forall_fm(x,y), env) \<longleftrightarrow>
   436         is_Forall(##A, nth(x,env), nth(y,env))"
   437 by (simp add: Forall_fm_def is_Forall_def)
   438 
   439 lemma Forall_iff_sats:
   440       "[| nth(i,env) = x; nth(j,env) = y; 
   441           i \<in> nat; j \<in> nat; env \<in> list(A)|]
   442        ==> is_Forall(##A, x, y) \<longleftrightarrow> sats(A, Forall_fm(i,j), env)"
   443 by (simp add: sats_Forall_fm)
   444 
   445 theorem Forall_reflection:
   446      "REFLECTS[\<lambda>x. is_Forall(L,f(x),g(x)),
   447                \<lambda>i x. is_Forall(##Lset(i),f(x),g(x))]"
   448 apply (simp only: is_Forall_def)
   449 apply (intro FOL_reflections pair_reflection Inr_reflection)
   450 done
   451 
   452 
   453 subsubsection\<open>The Operator @{term is_and}, Internalized\<close>
   454 
   455 (* is_and(M,a,b,z) == (number1(M,a)  & z=b) | 
   456                        (~number1(M,a) & empty(M,z)) *)
   457 definition
   458   and_fm :: "[i,i,i]=>i" where
   459     "and_fm(a,b,z) ==
   460        Or(And(number1_fm(a), Equal(z,b)),
   461           And(Neg(number1_fm(a)),empty_fm(z)))"
   462 
   463 lemma is_and_type [TC]:
   464      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> and_fm(x,y,z) \<in> formula"
   465 by (simp add: and_fm_def)
   466 
   467 lemma sats_and_fm [simp]:
   468    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   469     ==> sats(A, and_fm(x,y,z), env) \<longleftrightarrow>
   470         is_and(##A, nth(x,env), nth(y,env), nth(z,env))"
   471 by (simp add: and_fm_def is_and_def)
   472 
   473 lemma is_and_iff_sats:
   474       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   475           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   476        ==> is_and(##A, x, y, z) \<longleftrightarrow> sats(A, and_fm(i,j,k), env)"
   477 by simp
   478 
   479 theorem is_and_reflection:
   480      "REFLECTS[\<lambda>x. is_and(L,f(x),g(x),h(x)),
   481                \<lambda>i x. is_and(##Lset(i),f(x),g(x),h(x))]"
   482 apply (simp only: is_and_def)
   483 apply (intro FOL_reflections function_reflections)
   484 done
   485 
   486 
   487 subsubsection\<open>The Operator @{term is_or}, Internalized\<close>
   488 
   489 (* is_or(M,a,b,z) == (number1(M,a)  & number1(M,z)) | 
   490                      (~number1(M,a) & z=b) *)
   491 
   492 definition
   493   or_fm :: "[i,i,i]=>i" where
   494     "or_fm(a,b,z) ==
   495        Or(And(number1_fm(a), number1_fm(z)),
   496           And(Neg(number1_fm(a)), Equal(z,b)))"
   497 
   498 lemma is_or_type [TC]:
   499      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> or_fm(x,y,z) \<in> formula"
   500 by (simp add: or_fm_def)
   501 
   502 lemma sats_or_fm [simp]:
   503    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   504     ==> sats(A, or_fm(x,y,z), env) \<longleftrightarrow>
   505         is_or(##A, nth(x,env), nth(y,env), nth(z,env))"
   506 by (simp add: or_fm_def is_or_def)
   507 
   508 lemma is_or_iff_sats:
   509       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   510           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   511        ==> is_or(##A, x, y, z) \<longleftrightarrow> sats(A, or_fm(i,j,k), env)"
   512 by simp
   513 
   514 theorem is_or_reflection:
   515      "REFLECTS[\<lambda>x. is_or(L,f(x),g(x),h(x)),
   516                \<lambda>i x. is_or(##Lset(i),f(x),g(x),h(x))]"
   517 apply (simp only: is_or_def)
   518 apply (intro FOL_reflections function_reflections)
   519 done
   520 
   521 
   522 
   523 subsubsection\<open>The Operator @{term is_not}, Internalized\<close>
   524 
   525 (* is_not(M,a,z) == (number1(M,a)  & empty(M,z)) | 
   526                      (~number1(M,a) & number1(M,z)) *)
   527 definition
   528   not_fm :: "[i,i]=>i" where
   529     "not_fm(a,z) ==
   530        Or(And(number1_fm(a), empty_fm(z)),
   531           And(Neg(number1_fm(a)), number1_fm(z)))"
   532 
   533 lemma is_not_type [TC]:
   534      "[| x \<in> nat; z \<in> nat |] ==> not_fm(x,z) \<in> formula"
   535 by (simp add: not_fm_def)
   536 
   537 lemma sats_is_not_fm [simp]:
   538    "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
   539     ==> sats(A, not_fm(x,z), env) \<longleftrightarrow> is_not(##A, nth(x,env), nth(z,env))"
   540 by (simp add: not_fm_def is_not_def)
   541 
   542 lemma is_not_iff_sats:
   543       "[| nth(i,env) = x; nth(k,env) = z;
   544           i \<in> nat; k \<in> nat; env \<in> list(A)|]
   545        ==> is_not(##A, x, z) \<longleftrightarrow> sats(A, not_fm(i,k), env)"
   546 by simp
   547 
   548 theorem is_not_reflection:
   549      "REFLECTS[\<lambda>x. is_not(L,f(x),g(x)),
   550                \<lambda>i x. is_not(##Lset(i),f(x),g(x))]"
   551 apply (simp only: is_not_def)
   552 apply (intro FOL_reflections function_reflections)
   553 done
   554 
   555 
   556 lemmas extra_reflections = 
   557     Inl_reflection Inr_reflection Nil_reflection Cons_reflection
   558     quasilist_reflection hd_reflection tl_reflection bool_of_o_reflection
   559     is_lambda_reflection Member_reflection Equal_reflection Nand_reflection
   560     Forall_reflection is_and_reflection is_or_reflection is_not_reflection
   561 
   562 subsection\<open>Well-Founded Recursion!\<close>
   563 
   564 subsubsection\<open>The Operator @{term M_is_recfun}\<close>
   565 
   566 text\<open>Alternative definition, minimizing nesting of quantifiers around MH\<close>
   567 lemma M_is_recfun_iff:
   568    "M_is_recfun(M,MH,r,a,f) \<longleftrightarrow>
   569     (\<forall>z[M]. z \<in> f \<longleftrightarrow> 
   570      (\<exists>x[M]. \<exists>f_r_sx[M]. \<exists>y[M]. 
   571              MH(x, f_r_sx, y) & pair(M,x,y,z) &
   572              (\<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. 
   573                 pair(M,x,a,xa) & upair(M,x,x,sx) &
   574                pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
   575                xa \<in> r)))"
   576 apply (simp add: M_is_recfun_def)
   577 apply (rule rall_cong, blast) 
   578 done
   579 
   580 
   581 (* M_is_recfun :: "[i=>o, [i,i,i]=>o, i, i, i] => o"
   582    "M_is_recfun(M,MH,r,a,f) ==
   583      \<forall>z[M]. z \<in> f \<longleftrightarrow>
   584                2      1           0
   585 new def     (\<exists>x[M]. \<exists>f_r_sx[M]. \<exists>y[M]. 
   586              MH(x, f_r_sx, y) & pair(M,x,y,z) &
   587              (\<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. 
   588                 pair(M,x,a,xa) & upair(M,x,x,sx) &
   589                pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
   590                xa \<in> r)"
   591 *)
   592 
   593 text\<open>The three arguments of @{term p} are always 2, 1, 0 and z\<close>
   594 definition
   595   is_recfun_fm :: "[i, i, i, i]=>i" where
   596   "is_recfun_fm(p,r,a,f) == 
   597    Forall(Iff(Member(0,succ(f)),
   598     Exists(Exists(Exists(
   599      And(p, 
   600       And(pair_fm(2,0,3),
   601        Exists(Exists(Exists(
   602         And(pair_fm(5,a#+7,2),
   603          And(upair_fm(5,5,1),
   604           And(pre_image_fm(r#+7,1,0),
   605            And(restriction_fm(f#+7,0,4), Member(2,r#+7)))))))))))))))"
   606 
   607 lemma is_recfun_type [TC]:
   608      "[| p \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |] 
   609       ==> is_recfun_fm(p,x,y,z) \<in> formula"
   610 by (simp add: is_recfun_fm_def)
   611 
   612 
   613 lemma sats_is_recfun_fm:
   614   assumes MH_iff_sats: 
   615       "!!a0 a1 a2 a3. 
   616         [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A|] 
   617         ==> MH(a2, a1, a0) \<longleftrightarrow> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,env)))))"
   618   shows 
   619       "[|x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   620        ==> sats(A, is_recfun_fm(p,x,y,z), env) \<longleftrightarrow>
   621            M_is_recfun(##A, MH, nth(x,env), nth(y,env), nth(z,env))"
   622 by (simp add: is_recfun_fm_def M_is_recfun_iff MH_iff_sats [THEN iff_sym])
   623 
   624 lemma is_recfun_iff_sats:
   625   assumes MH_iff_sats: 
   626       "!!a0 a1 a2 a3. 
   627         [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A|] 
   628         ==> MH(a2, a1, a0) \<longleftrightarrow> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,env)))))"
   629   shows
   630   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
   631       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   632    ==> M_is_recfun(##A, MH, x, y, z) \<longleftrightarrow> sats(A, is_recfun_fm(p,i,j,k), env)"
   633 by (simp add: sats_is_recfun_fm [OF MH_iff_sats]) 
   634 
   635 text\<open>The additional variable in the premise, namely @{term f'}, is essential.
   636 It lets @{term MH} depend upon @{term x}, which seems often necessary.
   637 The same thing occurs in \<open>is_wfrec_reflection\<close>.\<close>
   638 theorem is_recfun_reflection:
   639   assumes MH_reflection:
   640     "!!f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)), 
   641                      \<lambda>i x. MH(##Lset(i), f'(x), f(x), g(x), h(x))]"
   642   shows "REFLECTS[\<lambda>x. M_is_recfun(L, MH(L,x), f(x), g(x), h(x)), 
   643              \<lambda>i x. M_is_recfun(##Lset(i), MH(##Lset(i),x), f(x), g(x), h(x))]"
   644 apply (simp (no_asm_use) only: M_is_recfun_def)
   645 apply (intro FOL_reflections function_reflections
   646              restriction_reflection MH_reflection)
   647 done
   648 
   649 subsubsection\<open>The Operator @{term is_wfrec}\<close>
   650 
   651 text\<open>The three arguments of @{term p} are always 2, 1, 0;
   652       @{term p} is enclosed by 5 quantifiers.\<close>
   653 
   654 (* is_wfrec :: "[i=>o, i, [i,i,i]=>o, i, i] => o"
   655     "is_wfrec(M,MH,r,a,z) == 
   656       \<exists>f[M]. M_is_recfun(M,MH,r,a,f) & MH(a,f,z)" *)
   657 definition
   658   is_wfrec_fm :: "[i, i, i, i]=>i" where
   659   "is_wfrec_fm(p,r,a,z) == 
   660     Exists(And(is_recfun_fm(p, succ(r), succ(a), 0),
   661            Exists(Exists(Exists(Exists(
   662              And(Equal(2,a#+5), And(Equal(1,4), And(Equal(0,z#+5), p)))))))))"
   663 
   664 text\<open>We call @{term p} with arguments a, f, z by equating them with 
   665   the corresponding quantified variables with de Bruijn indices 2, 1, 0.\<close>
   666 
   667 text\<open>There's an additional existential quantifier to ensure that the
   668       environments in both calls to MH have the same length.\<close>
   669 
   670 lemma is_wfrec_type [TC]:
   671      "[| p \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |] 
   672       ==> is_wfrec_fm(p,x,y,z) \<in> formula"
   673 by (simp add: is_wfrec_fm_def) 
   674 
   675 lemma sats_is_wfrec_fm:
   676   assumes MH_iff_sats: 
   677       "!!a0 a1 a2 a3 a4. 
   678         [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A|] 
   679         ==> MH(a2, a1, a0) \<longleftrightarrow> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,env))))))"
   680   shows 
   681       "[|x \<in> nat; y < length(env); z < length(env); env \<in> list(A)|]
   682        ==> sats(A, is_wfrec_fm(p,x,y,z), env) \<longleftrightarrow> 
   683            is_wfrec(##A, MH, nth(x,env), nth(y,env), nth(z,env))"
   684 apply (frule_tac x=z in lt_length_in_nat, assumption)  
   685 apply (frule lt_length_in_nat, assumption)  
   686 apply (simp add: is_wfrec_fm_def sats_is_recfun_fm is_wfrec_def MH_iff_sats [THEN iff_sym], blast) 
   687 done
   688 
   689 
   690 lemma is_wfrec_iff_sats:
   691   assumes MH_iff_sats: 
   692       "!!a0 a1 a2 a3 a4. 
   693         [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A|] 
   694         ==> MH(a2, a1, a0) \<longleftrightarrow> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,env))))))"
   695   shows
   696   "[|nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
   697       i \<in> nat; j < length(env); k < length(env); env \<in> list(A)|]
   698    ==> is_wfrec(##A, MH, x, y, z) \<longleftrightarrow> sats(A, is_wfrec_fm(p,i,j,k), env)" 
   699 by (simp add: sats_is_wfrec_fm [OF MH_iff_sats])
   700 
   701 theorem is_wfrec_reflection:
   702   assumes MH_reflection:
   703     "!!f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)), 
   704                      \<lambda>i x. MH(##Lset(i), f'(x), f(x), g(x), h(x))]"
   705   shows "REFLECTS[\<lambda>x. is_wfrec(L, MH(L,x), f(x), g(x), h(x)), 
   706                \<lambda>i x. is_wfrec(##Lset(i), MH(##Lset(i),x), f(x), g(x), h(x))]"
   707 apply (simp (no_asm_use) only: is_wfrec_def)
   708 apply (intro FOL_reflections MH_reflection is_recfun_reflection)
   709 done
   710 
   711 
   712 subsection\<open>For Datatypes\<close>
   713 
   714 subsubsection\<open>Binary Products, Internalized\<close>
   715 
   716 definition
   717   cartprod_fm :: "[i,i,i]=>i" where
   718 (* "cartprod(M,A,B,z) ==
   719         \<forall>u[M]. u \<in> z \<longleftrightarrow> (\<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>B & pair(M,x,y,u)))" *)
   720     "cartprod_fm(A,B,z) ==
   721        Forall(Iff(Member(0,succ(z)),
   722                   Exists(And(Member(0,succ(succ(A))),
   723                          Exists(And(Member(0,succ(succ(succ(B)))),
   724                                     pair_fm(1,0,2)))))))"
   725 
   726 lemma cartprod_type [TC]:
   727      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cartprod_fm(x,y,z) \<in> formula"
   728 by (simp add: cartprod_fm_def)
   729 
   730 lemma sats_cartprod_fm [simp]:
   731    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   732     ==> sats(A, cartprod_fm(x,y,z), env) \<longleftrightarrow>
   733         cartprod(##A, nth(x,env), nth(y,env), nth(z,env))"
   734 by (simp add: cartprod_fm_def cartprod_def)
   735 
   736 lemma cartprod_iff_sats:
   737       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   738           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   739        ==> cartprod(##A, x, y, z) \<longleftrightarrow> sats(A, cartprod_fm(i,j,k), env)"
   740 by (simp add: sats_cartprod_fm)
   741 
   742 theorem cartprod_reflection:
   743      "REFLECTS[\<lambda>x. cartprod(L,f(x),g(x),h(x)),
   744                \<lambda>i x. cartprod(##Lset(i),f(x),g(x),h(x))]"
   745 apply (simp only: cartprod_def)
   746 apply (intro FOL_reflections pair_reflection)
   747 done
   748 
   749 
   750 subsubsection\<open>Binary Sums, Internalized\<close>
   751 
   752 (* "is_sum(M,A,B,Z) ==
   753        \<exists>A0[M]. \<exists>n1[M]. \<exists>s1[M]. \<exists>B1[M].
   754          3      2       1        0
   755        number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) &
   756        cartprod(M,s1,B,B1) & union(M,A0,B1,Z)"  *)
   757 definition
   758   sum_fm :: "[i,i,i]=>i" where
   759     "sum_fm(A,B,Z) ==
   760        Exists(Exists(Exists(Exists(
   761         And(number1_fm(2),
   762             And(cartprod_fm(2,A#+4,3),
   763                 And(upair_fm(2,2,1),
   764                     And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))"
   765 
   766 lemma sum_type [TC]:
   767      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> sum_fm(x,y,z) \<in> formula"
   768 by (simp add: sum_fm_def)
   769 
   770 lemma sats_sum_fm [simp]:
   771    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
   772     ==> sats(A, sum_fm(x,y,z), env) \<longleftrightarrow>
   773         is_sum(##A, nth(x,env), nth(y,env), nth(z,env))"
   774 by (simp add: sum_fm_def is_sum_def)
   775 
   776 lemma sum_iff_sats:
   777       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
   778           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
   779        ==> is_sum(##A, x, y, z) \<longleftrightarrow> sats(A, sum_fm(i,j,k), env)"
   780 by simp
   781 
   782 theorem sum_reflection:
   783      "REFLECTS[\<lambda>x. is_sum(L,f(x),g(x),h(x)),
   784                \<lambda>i x. is_sum(##Lset(i),f(x),g(x),h(x))]"
   785 apply (simp only: is_sum_def)
   786 apply (intro FOL_reflections function_reflections cartprod_reflection)
   787 done
   788 
   789 
   790 subsubsection\<open>The Operator @{term quasinat}\<close>
   791 
   792 (* "is_quasinat(M,z) == empty(M,z) | (\<exists>m[M]. successor(M,m,z))" *)
   793 definition
   794   quasinat_fm :: "i=>i" where
   795     "quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))"
   796 
   797 lemma quasinat_type [TC]:
   798      "x \<in> nat ==> quasinat_fm(x) \<in> formula"
   799 by (simp add: quasinat_fm_def)
   800 
   801 lemma sats_quasinat_fm [simp]:
   802    "[| x \<in> nat; env \<in> list(A)|]
   803     ==> sats(A, quasinat_fm(x), env) \<longleftrightarrow> is_quasinat(##A, nth(x,env))"
   804 by (simp add: quasinat_fm_def is_quasinat_def)
   805 
   806 lemma quasinat_iff_sats:
   807       "[| nth(i,env) = x; nth(j,env) = y;
   808           i \<in> nat; env \<in> list(A)|]
   809        ==> is_quasinat(##A, x) \<longleftrightarrow> sats(A, quasinat_fm(i), env)"
   810 by simp
   811 
   812 theorem quasinat_reflection:
   813      "REFLECTS[\<lambda>x. is_quasinat(L,f(x)),
   814                \<lambda>i x. is_quasinat(##Lset(i),f(x))]"
   815 apply (simp only: is_quasinat_def)
   816 apply (intro FOL_reflections function_reflections)
   817 done
   818 
   819 
   820 subsubsection\<open>The Operator @{term is_nat_case}\<close>
   821 text\<open>I could not get it to work with the more natural assumption that 
   822  @{term is_b} takes two arguments.  Instead it must be a formula where 1 and 0
   823  stand for @{term m} and @{term b}, respectively.\<close>
   824 
   825 (* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o"
   826     "is_nat_case(M, a, is_b, k, z) ==
   827        (empty(M,k) \<longrightarrow> z=a) &
   828        (\<forall>m[M]. successor(M,m,k) \<longrightarrow> is_b(m,z)) &
   829        (is_quasinat(M,k) | empty(M,z))" *)
   830 text\<open>The formula @{term is_b} has free variables 1 and 0.\<close>
   831 definition
   832   is_nat_case_fm :: "[i, i, i, i]=>i" where
   833  "is_nat_case_fm(a,is_b,k,z) == 
   834     And(Implies(empty_fm(k), Equal(z,a)),
   835         And(Forall(Implies(succ_fm(0,succ(k)), 
   836                    Forall(Implies(Equal(0,succ(succ(z))), is_b)))),
   837             Or(quasinat_fm(k), empty_fm(z))))"
   838 
   839 lemma is_nat_case_type [TC]:
   840      "[| is_b \<in> formula;  
   841          x \<in> nat; y \<in> nat; z \<in> nat |] 
   842       ==> is_nat_case_fm(x,is_b,y,z) \<in> formula"
   843 by (simp add: is_nat_case_fm_def)
   844 
   845 lemma sats_is_nat_case_fm:
   846   assumes is_b_iff_sats: 
   847       "!!a. a \<in> A ==> is_b(a,nth(z, env)) \<longleftrightarrow> 
   848                       sats(A, p, Cons(nth(z,env), Cons(a, env)))"
   849   shows 
   850       "[|x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
   851        ==> sats(A, is_nat_case_fm(x,p,y,z), env) \<longleftrightarrow>
   852            is_nat_case(##A, nth(x,env), is_b, nth(y,env), nth(z,env))"
   853 apply (frule lt_length_in_nat, assumption)
   854 apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym])
   855 done
   856 
   857 lemma is_nat_case_iff_sats:
   858   "[| (!!a. a \<in> A ==> is_b(a,z) \<longleftrightarrow>
   859                       sats(A, p, Cons(z, Cons(a,env))));
   860       nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
   861       i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
   862    ==> is_nat_case(##A, x, is_b, y, z) \<longleftrightarrow> sats(A, is_nat_case_fm(i,p,j,k), env)"
   863 by (simp add: sats_is_nat_case_fm [of A is_b])
   864 
   865 
   866 text\<open>The second argument of @{term is_b} gives it direct access to @{term x},
   867   which is essential for handling free variable references.  Without this
   868   argument, we cannot prove reflection for @{term iterates_MH}.\<close>
   869 theorem is_nat_case_reflection:
   870   assumes is_b_reflection:
   871     "!!h f g. REFLECTS[\<lambda>x. is_b(L, h(x), f(x), g(x)),
   872                      \<lambda>i x. is_b(##Lset(i), h(x), f(x), g(x))]"
   873   shows "REFLECTS[\<lambda>x. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)),
   874                \<lambda>i x. is_nat_case(##Lset(i), f(x), is_b(##Lset(i), x), g(x), h(x))]"
   875 apply (simp (no_asm_use) only: is_nat_case_def)
   876 apply (intro FOL_reflections function_reflections
   877              restriction_reflection is_b_reflection quasinat_reflection)
   878 done
   879 
   880 
   881 subsection\<open>The Operator @{term iterates_MH}, Needed for Iteration\<close>
   882 
   883 (*  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
   884    "iterates_MH(M,isF,v,n,g,z) ==
   885         is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
   886                     n, z)" *)
   887 definition
   888   iterates_MH_fm :: "[i, i, i, i, i]=>i" where
   889  "iterates_MH_fm(isF,v,n,g,z) == 
   890     is_nat_case_fm(v, 
   891       Exists(And(fun_apply_fm(succ(succ(succ(g))),2,0), 
   892                      Forall(Implies(Equal(0,2), isF)))), 
   893       n, z)"
   894 
   895 lemma iterates_MH_type [TC]:
   896      "[| p \<in> formula;  
   897          v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |] 
   898       ==> iterates_MH_fm(p,v,x,y,z) \<in> formula"
   899 by (simp add: iterates_MH_fm_def)
   900 
   901 lemma sats_iterates_MH_fm:
   902   assumes is_F_iff_sats:
   903       "!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
   904               ==> is_F(a,b) \<longleftrightarrow>
   905                   sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env)))))"
   906   shows 
   907       "[|v \<in> nat; x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
   908        ==> sats(A, iterates_MH_fm(p,v,x,y,z), env) \<longleftrightarrow>
   909            iterates_MH(##A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))"
   910 apply (frule lt_length_in_nat, assumption)  
   911 apply (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm 
   912               is_F_iff_sats [symmetric])
   913 apply (rule is_nat_case_cong) 
   914 apply (simp_all add: setclass_def)
   915 done
   916 
   917 lemma iterates_MH_iff_sats:
   918   assumes is_F_iff_sats:
   919       "!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
   920               ==> is_F(a,b) \<longleftrightarrow>
   921                   sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env)))))"
   922   shows 
   923   "[| nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
   924       i' \<in> nat; i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
   925    ==> iterates_MH(##A, is_F, v, x, y, z) \<longleftrightarrow>
   926        sats(A, iterates_MH_fm(p,i',i,j,k), env)"
   927 by (simp add: sats_iterates_MH_fm [OF is_F_iff_sats]) 
   928 
   929 text\<open>The second argument of @{term p} gives it direct access to @{term x},
   930   which is essential for handling free variable references.  Without this
   931   argument, we cannot prove reflection for @{term list_N}.\<close>
   932 theorem iterates_MH_reflection:
   933   assumes p_reflection:
   934     "!!f g h. REFLECTS[\<lambda>x. p(L, h(x), f(x), g(x)),
   935                      \<lambda>i x. p(##Lset(i), h(x), f(x), g(x))]"
   936  shows "REFLECTS[\<lambda>x. iterates_MH(L, p(L,x), e(x), f(x), g(x), h(x)),
   937                \<lambda>i x. iterates_MH(##Lset(i), p(##Lset(i),x), e(x), f(x), g(x), h(x))]"
   938 apply (simp (no_asm_use) only: iterates_MH_def)
   939 apply (intro FOL_reflections function_reflections is_nat_case_reflection
   940              restriction_reflection p_reflection)
   941 done
   942 
   943 
   944 subsubsection\<open>The Operator @{term is_iterates}\<close>
   945 
   946 text\<open>The three arguments of @{term p} are always 2, 1, 0;
   947       @{term p} is enclosed by 9 (??) quantifiers.\<close>
   948 
   949 (*    "is_iterates(M,isF,v,n,Z) == 
   950       \<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
   951        1       0       is_wfrec(M, iterates_MH(M,isF,v), msn, n, Z)"*)
   952 
   953 definition
   954   is_iterates_fm :: "[i, i, i, i]=>i" where
   955   "is_iterates_fm(p,v,n,Z) == 
   956      Exists(Exists(
   957       And(succ_fm(n#+2,1),
   958        And(Memrel_fm(1,0),
   959               is_wfrec_fm(iterates_MH_fm(p, v#+7, 2, 1, 0), 
   960                           0, n#+2, Z#+2)))))"
   961 
   962 text\<open>We call @{term p} with arguments a, f, z by equating them with 
   963   the corresponding quantified variables with de Bruijn indices 2, 1, 0.\<close>
   964 
   965 
   966 lemma is_iterates_type [TC]:
   967      "[| p \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |] 
   968       ==> is_iterates_fm(p,x,y,z) \<in> formula"
   969 by (simp add: is_iterates_fm_def) 
   970 
   971 lemma sats_is_iterates_fm:
   972   assumes is_F_iff_sats:
   973       "!!a b c d e f g h i j k. 
   974               [| a \<in> A; b \<in> A; c \<in> A; d \<in> A; e \<in> A; f \<in> A; 
   975                  g \<in> A; h \<in> A; i \<in> A; j \<in> A; k \<in> A|]
   976               ==> is_F(a,b) \<longleftrightarrow>
   977                   sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d, Cons(e, Cons(f, 
   978                       Cons(g, Cons(h, Cons(i, Cons(j, Cons(k, env))))))))))))"
   979   shows 
   980       "[|x \<in> nat; y < length(env); z < length(env); env \<in> list(A)|]
   981        ==> sats(A, is_iterates_fm(p,x,y,z), env) \<longleftrightarrow>
   982            is_iterates(##A, is_F, nth(x,env), nth(y,env), nth(z,env))"
   983 apply (frule_tac x=z in lt_length_in_nat, assumption)  
   984 apply (frule lt_length_in_nat, assumption)  
   985 apply (simp add: is_iterates_fm_def is_iterates_def sats_is_nat_case_fm 
   986               is_F_iff_sats [symmetric] sats_is_wfrec_fm sats_iterates_MH_fm)
   987 done
   988 
   989 
   990 lemma is_iterates_iff_sats:
   991   assumes is_F_iff_sats:
   992       "!!a b c d e f g h i j k. 
   993               [| a \<in> A; b \<in> A; c \<in> A; d \<in> A; e \<in> A; f \<in> A; 
   994                  g \<in> A; h \<in> A; i \<in> A; j \<in> A; k \<in> A|]
   995               ==> is_F(a,b) \<longleftrightarrow>
   996                   sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d, Cons(e, Cons(f, 
   997                       Cons(g, Cons(h, Cons(i, Cons(j, Cons(k, env))))))))))))"
   998   shows 
   999   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
  1000       i \<in> nat; j < length(env); k < length(env); env \<in> list(A)|]
  1001    ==> is_iterates(##A, is_F, x, y, z) \<longleftrightarrow>
  1002        sats(A, is_iterates_fm(p,i,j,k), env)"
  1003 by (simp add: sats_is_iterates_fm [OF is_F_iff_sats]) 
  1004 
  1005 text\<open>The second argument of @{term p} gives it direct access to @{term x},
  1006   which is essential for handling free variable references.  Without this
  1007   argument, we cannot prove reflection for @{term list_N}.\<close>
  1008 theorem is_iterates_reflection:
  1009   assumes p_reflection:
  1010     "!!f g h. REFLECTS[\<lambda>x. p(L, h(x), f(x), g(x)),
  1011                      \<lambda>i x. p(##Lset(i), h(x), f(x), g(x))]"
  1012  shows "REFLECTS[\<lambda>x. is_iterates(L, p(L,x), f(x), g(x), h(x)),
  1013                \<lambda>i x. is_iterates(##Lset(i), p(##Lset(i),x), f(x), g(x), h(x))]"
  1014 apply (simp (no_asm_use) only: is_iterates_def)
  1015 apply (intro FOL_reflections function_reflections p_reflection
  1016              is_wfrec_reflection iterates_MH_reflection)
  1017 done
  1018 
  1019 
  1020 subsubsection\<open>The Formula @{term is_eclose_n}, Internalized\<close>
  1021 
  1022 (* is_eclose_n(M,A,n,Z) == is_iterates(M, big_union(M), A, n, Z) *)
  1023 
  1024 definition
  1025   eclose_n_fm :: "[i,i,i]=>i" where
  1026   "eclose_n_fm(A,n,Z) == is_iterates_fm(big_union_fm(1,0), A, n, Z)"
  1027 
  1028 lemma eclose_n_fm_type [TC]:
  1029  "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> eclose_n_fm(x,y,z) \<in> formula"
  1030 by (simp add: eclose_n_fm_def)
  1031 
  1032 lemma sats_eclose_n_fm [simp]:
  1033    "[| x \<in> nat; y < length(env); z < length(env); env \<in> list(A)|]
  1034     ==> sats(A, eclose_n_fm(x,y,z), env) \<longleftrightarrow>
  1035         is_eclose_n(##A, nth(x,env), nth(y,env), nth(z,env))"
  1036 apply (frule_tac x=z in lt_length_in_nat, assumption)  
  1037 apply (frule_tac x=y in lt_length_in_nat, assumption)  
  1038 apply (simp add: eclose_n_fm_def is_eclose_n_def 
  1039                  sats_is_iterates_fm) 
  1040 done
  1041 
  1042 lemma eclose_n_iff_sats:
  1043       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
  1044           i \<in> nat; j < length(env); k < length(env); env \<in> list(A)|]
  1045        ==> is_eclose_n(##A, x, y, z) \<longleftrightarrow> sats(A, eclose_n_fm(i,j,k), env)"
  1046 by (simp add: sats_eclose_n_fm)
  1047 
  1048 theorem eclose_n_reflection:
  1049      "REFLECTS[\<lambda>x. is_eclose_n(L, f(x), g(x), h(x)),  
  1050                \<lambda>i x. is_eclose_n(##Lset(i), f(x), g(x), h(x))]"
  1051 apply (simp only: is_eclose_n_def)
  1052 apply (intro FOL_reflections function_reflections is_iterates_reflection) 
  1053 done
  1054 
  1055 
  1056 subsubsection\<open>Membership in @{term "eclose(A)"}\<close>
  1057 
  1058 (* mem_eclose(M,A,l) == 
  1059       \<exists>n[M]. \<exists>eclosen[M]. 
  1060        finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l \<in> eclosen *)
  1061 definition
  1062   mem_eclose_fm :: "[i,i]=>i" where
  1063     "mem_eclose_fm(x,y) ==
  1064        Exists(Exists(
  1065          And(finite_ordinal_fm(1),
  1066            And(eclose_n_fm(x#+2,1,0), Member(y#+2,0)))))"
  1067 
  1068 lemma mem_eclose_type [TC]:
  1069      "[| x \<in> nat; y \<in> nat |] ==> mem_eclose_fm(x,y) \<in> formula"
  1070 by (simp add: mem_eclose_fm_def)
  1071 
  1072 lemma sats_mem_eclose_fm [simp]:
  1073    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
  1074     ==> sats(A, mem_eclose_fm(x,y), env) \<longleftrightarrow> mem_eclose(##A, nth(x,env), nth(y,env))"
  1075 by (simp add: mem_eclose_fm_def mem_eclose_def)
  1076 
  1077 lemma mem_eclose_iff_sats:
  1078       "[| nth(i,env) = x; nth(j,env) = y;
  1079           i \<in> nat; j \<in> nat; env \<in> list(A)|]
  1080        ==> mem_eclose(##A, x, y) \<longleftrightarrow> sats(A, mem_eclose_fm(i,j), env)"
  1081 by simp
  1082 
  1083 theorem mem_eclose_reflection:
  1084      "REFLECTS[\<lambda>x. mem_eclose(L,f(x),g(x)),
  1085                \<lambda>i x. mem_eclose(##Lset(i),f(x),g(x))]"
  1086 apply (simp only: mem_eclose_def)
  1087 apply (intro FOL_reflections finite_ordinal_reflection eclose_n_reflection)
  1088 done
  1089 
  1090 
  1091 subsubsection\<open>The Predicate ``Is @{term "eclose(A)"}''\<close>
  1092 
  1093 (* is_eclose(M,A,Z) == \<forall>l[M]. l \<in> Z \<longleftrightarrow> mem_eclose(M,A,l) *)
  1094 definition
  1095   is_eclose_fm :: "[i,i]=>i" where
  1096     "is_eclose_fm(A,Z) ==
  1097        Forall(Iff(Member(0,succ(Z)), mem_eclose_fm(succ(A),0)))"
  1098 
  1099 lemma is_eclose_type [TC]:
  1100      "[| x \<in> nat; y \<in> nat |] ==> is_eclose_fm(x,y) \<in> formula"
  1101 by (simp add: is_eclose_fm_def)
  1102 
  1103 lemma sats_is_eclose_fm [simp]:
  1104    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
  1105     ==> sats(A, is_eclose_fm(x,y), env) \<longleftrightarrow> is_eclose(##A, nth(x,env), nth(y,env))"
  1106 by (simp add: is_eclose_fm_def is_eclose_def)
  1107 
  1108 lemma is_eclose_iff_sats:
  1109       "[| nth(i,env) = x; nth(j,env) = y;
  1110           i \<in> nat; j \<in> nat; env \<in> list(A)|]
  1111        ==> is_eclose(##A, x, y) \<longleftrightarrow> sats(A, is_eclose_fm(i,j), env)"
  1112 by simp
  1113 
  1114 theorem is_eclose_reflection:
  1115      "REFLECTS[\<lambda>x. is_eclose(L,f(x),g(x)),
  1116                \<lambda>i x. is_eclose(##Lset(i),f(x),g(x))]"
  1117 apply (simp only: is_eclose_def)
  1118 apply (intro FOL_reflections mem_eclose_reflection)
  1119 done
  1120 
  1121 
  1122 subsubsection\<open>The List Functor, Internalized\<close>
  1123 
  1124 definition
  1125   list_functor_fm :: "[i,i,i]=>i" where
  1126 (* "is_list_functor(M,A,X,Z) ==
  1127         \<exists>n1[M]. \<exists>AX[M].
  1128          number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *)
  1129     "list_functor_fm(A,X,Z) ==
  1130        Exists(Exists(
  1131         And(number1_fm(1),
  1132             And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))"
  1133 
  1134 lemma list_functor_type [TC]:
  1135      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_functor_fm(x,y,z) \<in> formula"
  1136 by (simp add: list_functor_fm_def)
  1137 
  1138 lemma sats_list_functor_fm [simp]:
  1139    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
  1140     ==> sats(A, list_functor_fm(x,y,z), env) \<longleftrightarrow>
  1141         is_list_functor(##A, nth(x,env), nth(y,env), nth(z,env))"
  1142 by (simp add: list_functor_fm_def is_list_functor_def)
  1143 
  1144 lemma list_functor_iff_sats:
  1145   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
  1146       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
  1147    ==> is_list_functor(##A, x, y, z) \<longleftrightarrow> sats(A, list_functor_fm(i,j,k), env)"
  1148 by simp
  1149 
  1150 theorem list_functor_reflection:
  1151      "REFLECTS[\<lambda>x. is_list_functor(L,f(x),g(x),h(x)),
  1152                \<lambda>i x. is_list_functor(##Lset(i),f(x),g(x),h(x))]"
  1153 apply (simp only: is_list_functor_def)
  1154 apply (intro FOL_reflections number1_reflection
  1155              cartprod_reflection sum_reflection)
  1156 done
  1157 
  1158 
  1159 subsubsection\<open>The Formula @{term is_list_N}, Internalized\<close>
  1160 
  1161 (* "is_list_N(M,A,n,Z) == 
  1162       \<exists>zero[M]. empty(M,zero) & 
  1163                 is_iterates(M, is_list_functor(M,A), zero, n, Z)" *)
  1164 
  1165 definition
  1166   list_N_fm :: "[i,i,i]=>i" where
  1167   "list_N_fm(A,n,Z) == 
  1168      Exists(
  1169        And(empty_fm(0),
  1170            is_iterates_fm(list_functor_fm(A#+9#+3,1,0), 0, n#+1, Z#+1)))"
  1171 
  1172 lemma list_N_fm_type [TC]:
  1173  "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_N_fm(x,y,z) \<in> formula"
  1174 by (simp add: list_N_fm_def)
  1175 
  1176 lemma sats_list_N_fm [simp]:
  1177    "[| x \<in> nat; y < length(env); z < length(env); env \<in> list(A)|]
  1178     ==> sats(A, list_N_fm(x,y,z), env) \<longleftrightarrow>
  1179         is_list_N(##A, nth(x,env), nth(y,env), nth(z,env))"
  1180 apply (frule_tac x=z in lt_length_in_nat, assumption)  
  1181 apply (frule_tac x=y in lt_length_in_nat, assumption)  
  1182 apply (simp add: list_N_fm_def is_list_N_def sats_is_iterates_fm) 
  1183 done
  1184 
  1185 lemma list_N_iff_sats:
  1186       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
  1187           i \<in> nat; j < length(env); k < length(env); env \<in> list(A)|]
  1188        ==> is_list_N(##A, x, y, z) \<longleftrightarrow> sats(A, list_N_fm(i,j,k), env)"
  1189 by (simp add: sats_list_N_fm)
  1190 
  1191 theorem list_N_reflection:
  1192      "REFLECTS[\<lambda>x. is_list_N(L, f(x), g(x), h(x)),  
  1193                \<lambda>i x. is_list_N(##Lset(i), f(x), g(x), h(x))]"
  1194 apply (simp only: is_list_N_def)
  1195 apply (intro FOL_reflections function_reflections 
  1196              is_iterates_reflection list_functor_reflection) 
  1197 done
  1198 
  1199 
  1200 
  1201 subsubsection\<open>The Predicate ``Is A List''\<close>
  1202 
  1203 (* mem_list(M,A,l) == 
  1204       \<exists>n[M]. \<exists>listn[M]. 
  1205        finite_ordinal(M,n) & is_list_N(M,A,n,listn) & l \<in> listn *)
  1206 definition
  1207   mem_list_fm :: "[i,i]=>i" where
  1208     "mem_list_fm(x,y) ==
  1209        Exists(Exists(
  1210          And(finite_ordinal_fm(1),
  1211            And(list_N_fm(x#+2,1,0), Member(y#+2,0)))))"
  1212 
  1213 lemma mem_list_type [TC]:
  1214      "[| x \<in> nat; y \<in> nat |] ==> mem_list_fm(x,y) \<in> formula"
  1215 by (simp add: mem_list_fm_def)
  1216 
  1217 lemma sats_mem_list_fm [simp]:
  1218    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
  1219     ==> sats(A, mem_list_fm(x,y), env) \<longleftrightarrow> mem_list(##A, nth(x,env), nth(y,env))"
  1220 by (simp add: mem_list_fm_def mem_list_def)
  1221 
  1222 lemma mem_list_iff_sats:
  1223       "[| nth(i,env) = x; nth(j,env) = y;
  1224           i \<in> nat; j \<in> nat; env \<in> list(A)|]
  1225        ==> mem_list(##A, x, y) \<longleftrightarrow> sats(A, mem_list_fm(i,j), env)"
  1226 by simp
  1227 
  1228 theorem mem_list_reflection:
  1229      "REFLECTS[\<lambda>x. mem_list(L,f(x),g(x)),
  1230                \<lambda>i x. mem_list(##Lset(i),f(x),g(x))]"
  1231 apply (simp only: mem_list_def)
  1232 apply (intro FOL_reflections finite_ordinal_reflection list_N_reflection)
  1233 done
  1234 
  1235 
  1236 subsubsection\<open>The Predicate ``Is @{term "list(A)"}''\<close>
  1237 
  1238 (* is_list(M,A,Z) == \<forall>l[M]. l \<in> Z \<longleftrightarrow> mem_list(M,A,l) *)
  1239 definition
  1240   is_list_fm :: "[i,i]=>i" where
  1241     "is_list_fm(A,Z) ==
  1242        Forall(Iff(Member(0,succ(Z)), mem_list_fm(succ(A),0)))"
  1243 
  1244 lemma is_list_type [TC]:
  1245      "[| x \<in> nat; y \<in> nat |] ==> is_list_fm(x,y) \<in> formula"
  1246 by (simp add: is_list_fm_def)
  1247 
  1248 lemma sats_is_list_fm [simp]:
  1249    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
  1250     ==> sats(A, is_list_fm(x,y), env) \<longleftrightarrow> is_list(##A, nth(x,env), nth(y,env))"
  1251 by (simp add: is_list_fm_def is_list_def)
  1252 
  1253 lemma is_list_iff_sats:
  1254       "[| nth(i,env) = x; nth(j,env) = y;
  1255           i \<in> nat; j \<in> nat; env \<in> list(A)|]
  1256        ==> is_list(##A, x, y) \<longleftrightarrow> sats(A, is_list_fm(i,j), env)"
  1257 by simp
  1258 
  1259 theorem is_list_reflection:
  1260      "REFLECTS[\<lambda>x. is_list(L,f(x),g(x)),
  1261                \<lambda>i x. is_list(##Lset(i),f(x),g(x))]"
  1262 apply (simp only: is_list_def)
  1263 apply (intro FOL_reflections mem_list_reflection)
  1264 done
  1265 
  1266 
  1267 subsubsection\<open>The Formula Functor, Internalized\<close>
  1268 
  1269 definition formula_functor_fm :: "[i,i]=>i" where
  1270 (*     "is_formula_functor(M,X,Z) ==
  1271         \<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M].
  1272            4           3               2       1       0
  1273           omega(M,nat') & cartprod(M,nat',nat',natnat) &
  1274           is_sum(M,natnat,natnat,natnatsum) &
  1275           cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) &
  1276           is_sum(M,natnatsum,X3,Z)" *)
  1277     "formula_functor_fm(X,Z) ==
  1278        Exists(Exists(Exists(Exists(Exists(
  1279         And(omega_fm(4),
  1280          And(cartprod_fm(4,4,3),
  1281           And(sum_fm(3,3,2),
  1282            And(cartprod_fm(X#+5,X#+5,1),
  1283             And(sum_fm(1,X#+5,0), sum_fm(2,0,Z#+5)))))))))))"
  1284 
  1285 lemma formula_functor_type [TC]:
  1286      "[| x \<in> nat; y \<in> nat |] ==> formula_functor_fm(x,y) \<in> formula"
  1287 by (simp add: formula_functor_fm_def)
  1288 
  1289 lemma sats_formula_functor_fm [simp]:
  1290    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
  1291     ==> sats(A, formula_functor_fm(x,y), env) \<longleftrightarrow>
  1292         is_formula_functor(##A, nth(x,env), nth(y,env))"
  1293 by (simp add: formula_functor_fm_def is_formula_functor_def)
  1294 
  1295 lemma formula_functor_iff_sats:
  1296   "[| nth(i,env) = x; nth(j,env) = y;
  1297       i \<in> nat; j \<in> nat; env \<in> list(A)|]
  1298    ==> is_formula_functor(##A, x, y) \<longleftrightarrow> sats(A, formula_functor_fm(i,j), env)"
  1299 by simp
  1300 
  1301 theorem formula_functor_reflection:
  1302      "REFLECTS[\<lambda>x. is_formula_functor(L,f(x),g(x)),
  1303                \<lambda>i x. is_formula_functor(##Lset(i),f(x),g(x))]"
  1304 apply (simp only: is_formula_functor_def)
  1305 apply (intro FOL_reflections omega_reflection
  1306              cartprod_reflection sum_reflection)
  1307 done
  1308 
  1309 
  1310 subsubsection\<open>The Formula @{term is_formula_N}, Internalized\<close>
  1311 
  1312 (*  "is_formula_N(M,n,Z) == 
  1313       \<exists>zero[M]. empty(M,zero) & 
  1314                 is_iterates(M, is_formula_functor(M), zero, n, Z)" *) 
  1315 definition
  1316   formula_N_fm :: "[i,i]=>i" where
  1317   "formula_N_fm(n,Z) == 
  1318      Exists(
  1319        And(empty_fm(0),
  1320            is_iterates_fm(formula_functor_fm(1,0), 0, n#+1, Z#+1)))"
  1321 
  1322 lemma formula_N_fm_type [TC]:
  1323  "[| x \<in> nat; y \<in> nat |] ==> formula_N_fm(x,y) \<in> formula"
  1324 by (simp add: formula_N_fm_def)
  1325 
  1326 lemma sats_formula_N_fm [simp]:
  1327    "[| x < length(env); y < length(env); env \<in> list(A)|]
  1328     ==> sats(A, formula_N_fm(x,y), env) \<longleftrightarrow>
  1329         is_formula_N(##A, nth(x,env), nth(y,env))"
  1330 apply (frule_tac x=y in lt_length_in_nat, assumption)  
  1331 apply (frule lt_length_in_nat, assumption)  
  1332 apply (simp add: formula_N_fm_def is_formula_N_def sats_is_iterates_fm) 
  1333 done
  1334 
  1335 lemma formula_N_iff_sats:
  1336       "[| nth(i,env) = x; nth(j,env) = y; 
  1337           i < length(env); j < length(env); env \<in> list(A)|]
  1338        ==> is_formula_N(##A, x, y) \<longleftrightarrow> sats(A, formula_N_fm(i,j), env)"
  1339 by (simp add: sats_formula_N_fm)
  1340 
  1341 theorem formula_N_reflection:
  1342      "REFLECTS[\<lambda>x. is_formula_N(L, f(x), g(x)),  
  1343                \<lambda>i x. is_formula_N(##Lset(i), f(x), g(x))]"
  1344 apply (simp only: is_formula_N_def)
  1345 apply (intro FOL_reflections function_reflections 
  1346              is_iterates_reflection formula_functor_reflection) 
  1347 done
  1348 
  1349 
  1350 
  1351 subsubsection\<open>The Predicate ``Is A Formula''\<close>
  1352 
  1353 (*  mem_formula(M,p) == 
  1354       \<exists>n[M]. \<exists>formn[M]. 
  1355        finite_ordinal(M,n) & is_formula_N(M,n,formn) & p \<in> formn *)
  1356 definition
  1357   mem_formula_fm :: "i=>i" where
  1358     "mem_formula_fm(x) ==
  1359        Exists(Exists(
  1360          And(finite_ordinal_fm(1),
  1361            And(formula_N_fm(1,0), Member(x#+2,0)))))"
  1362 
  1363 lemma mem_formula_type [TC]:
  1364      "x \<in> nat ==> mem_formula_fm(x) \<in> formula"
  1365 by (simp add: mem_formula_fm_def)
  1366 
  1367 lemma sats_mem_formula_fm [simp]:
  1368    "[| x \<in> nat; env \<in> list(A)|]
  1369     ==> sats(A, mem_formula_fm(x), env) \<longleftrightarrow> mem_formula(##A, nth(x,env))"
  1370 by (simp add: mem_formula_fm_def mem_formula_def)
  1371 
  1372 lemma mem_formula_iff_sats:
  1373       "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
  1374        ==> mem_formula(##A, x) \<longleftrightarrow> sats(A, mem_formula_fm(i), env)"
  1375 by simp
  1376 
  1377 theorem mem_formula_reflection:
  1378      "REFLECTS[\<lambda>x. mem_formula(L,f(x)),
  1379                \<lambda>i x. mem_formula(##Lset(i),f(x))]"
  1380 apply (simp only: mem_formula_def)
  1381 apply (intro FOL_reflections finite_ordinal_reflection formula_N_reflection)
  1382 done
  1383 
  1384 
  1385 
  1386 subsubsection\<open>The Predicate ``Is @{term "formula"}''\<close>
  1387 
  1388 (* is_formula(M,Z) == \<forall>p[M]. p \<in> Z \<longleftrightarrow> mem_formula(M,p) *)
  1389 definition
  1390   is_formula_fm :: "i=>i" where
  1391     "is_formula_fm(Z) == Forall(Iff(Member(0,succ(Z)), mem_formula_fm(0)))"
  1392 
  1393 lemma is_formula_type [TC]:
  1394      "x \<in> nat ==> is_formula_fm(x) \<in> formula"
  1395 by (simp add: is_formula_fm_def)
  1396 
  1397 lemma sats_is_formula_fm [simp]:
  1398    "[| x \<in> nat; env \<in> list(A)|]
  1399     ==> sats(A, is_formula_fm(x), env) \<longleftrightarrow> is_formula(##A, nth(x,env))"
  1400 by (simp add: is_formula_fm_def is_formula_def)
  1401 
  1402 lemma is_formula_iff_sats:
  1403       "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
  1404        ==> is_formula(##A, x) \<longleftrightarrow> sats(A, is_formula_fm(i), env)"
  1405 by simp
  1406 
  1407 theorem is_formula_reflection:
  1408      "REFLECTS[\<lambda>x. is_formula(L,f(x)),
  1409                \<lambda>i x. is_formula(##Lset(i),f(x))]"
  1410 apply (simp only: is_formula_def)
  1411 apply (intro FOL_reflections mem_formula_reflection)
  1412 done
  1413 
  1414 
  1415 subsubsection\<open>The Operator @{term is_transrec}\<close>
  1416 
  1417 text\<open>The three arguments of @{term p} are always 2, 1, 0.  It is buried
  1418    within eight quantifiers!
  1419    We call @{term p} with arguments a, f, z by equating them with 
  1420   the corresponding quantified variables with de Bruijn indices 2, 1, 0.\<close>
  1421 
  1422 (* is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o"
  1423    "is_transrec(M,MH,a,z) == 
  1424       \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M]. 
  1425        2       1         0
  1426        upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
  1427        is_wfrec(M,MH,mesa,a,z)" *)
  1428 definition
  1429   is_transrec_fm :: "[i, i, i]=>i" where
  1430  "is_transrec_fm(p,a,z) == 
  1431     Exists(Exists(Exists(
  1432       And(upair_fm(a#+3,a#+3,2),
  1433        And(is_eclose_fm(2,1),
  1434         And(Memrel_fm(1,0), is_wfrec_fm(p,0,a#+3,z#+3)))))))"
  1435 
  1436 
  1437 lemma is_transrec_type [TC]:
  1438      "[| p \<in> formula; x \<in> nat; z \<in> nat |] 
  1439       ==> is_transrec_fm(p,x,z) \<in> formula"
  1440 by (simp add: is_transrec_fm_def) 
  1441 
  1442 lemma sats_is_transrec_fm:
  1443   assumes MH_iff_sats: 
  1444       "!!a0 a1 a2 a3 a4 a5 a6 a7. 
  1445         [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A; a5\<in>A; a6\<in>A; a7\<in>A|] 
  1446         ==> MH(a2, a1, a0) \<longleftrightarrow> 
  1447             sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,
  1448                           Cons(a4,Cons(a5,Cons(a6,Cons(a7,env)))))))))"
  1449   shows 
  1450       "[|x < length(env); z < length(env); env \<in> list(A)|]
  1451        ==> sats(A, is_transrec_fm(p,x,z), env) \<longleftrightarrow> 
  1452            is_transrec(##A, MH, nth(x,env), nth(z,env))"
  1453 apply (frule_tac x=z in lt_length_in_nat, assumption)  
  1454 apply (frule_tac x=x in lt_length_in_nat, assumption)  
  1455 apply (simp add: is_transrec_fm_def sats_is_wfrec_fm is_transrec_def MH_iff_sats [THEN iff_sym]) 
  1456 done
  1457 
  1458 
  1459 lemma is_transrec_iff_sats:
  1460   assumes MH_iff_sats: 
  1461       "!!a0 a1 a2 a3 a4 a5 a6 a7. 
  1462         [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A; a5\<in>A; a6\<in>A; a7\<in>A|] 
  1463         ==> MH(a2, a1, a0) \<longleftrightarrow> 
  1464             sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,
  1465                           Cons(a4,Cons(a5,Cons(a6,Cons(a7,env)))))))))"
  1466   shows
  1467   "[|nth(i,env) = x; nth(k,env) = z; 
  1468       i < length(env); k < length(env); env \<in> list(A)|]
  1469    ==> is_transrec(##A, MH, x, z) \<longleftrightarrow> sats(A, is_transrec_fm(p,i,k), env)" 
  1470 by (simp add: sats_is_transrec_fm [OF MH_iff_sats])
  1471 
  1472 theorem is_transrec_reflection:
  1473   assumes MH_reflection:
  1474     "!!f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)), 
  1475                      \<lambda>i x. MH(##Lset(i), f'(x), f(x), g(x), h(x))]"
  1476   shows "REFLECTS[\<lambda>x. is_transrec(L, MH(L,x), f(x), h(x)), 
  1477                \<lambda>i x. is_transrec(##Lset(i), MH(##Lset(i),x), f(x), h(x))]"
  1478 apply (simp (no_asm_use) only: is_transrec_def)
  1479 apply (intro FOL_reflections function_reflections MH_reflection 
  1480              is_wfrec_reflection is_eclose_reflection)
  1481 done
  1482 
  1483 end